J L Bona

S Vento

F B Weissler

Singularity formation and blowup of complex-valued solutions of the modified KdV equation

The dynamics of the poles of the two-soliton solutions of the modified Korteweg-de Vries equation

are determined. A consequence of this study is the existence of classes of smooth, complex-valued solutions of this equation, defined for -∞ < x < ∞, exponentially decreasing to zero as |x| → ∞, that blow up in finite time.

Introduction

Studied here is the modified Korteweg-de Vries equation

u t + 6u 2 u x + u xxx = 0, (1.1)
which has been derived as a rudimentary model for wave propagation in a number of different physical contexts. The present paper is a sequel to the recent work [START_REF] Bona | Pole dynamics of interacting solitons and blowup of complex-valued solutions of KdV[END_REF] wherein the dynamics of the complex singularities of the two-soliton solution of the Korteweg de Vries equation,

u t + 6uu x + u xxx = 0, (1.2) 1
were examined in detail.

The study of the pole dynamics of solutions of the Korteweg-de Vries equation and its near relatives began with some remarks of Kruskal [START_REF]The Korteweg-de Vries equation and related evolution equations[END_REF] in the early 1970's. More comprehensive work was carried out later, see [START_REF] Thickstun | A system of particles equivalent to solitons[END_REF], [START_REF] Bowtell | A particle representation of the Kortweg-de Vries soliton[END_REF] and [START_REF] Bryan | On the dynamics of soliton interactions for the Korteweg-de Vries equation[END_REF]. One goal in our preceding paper [START_REF] Bona | Pole dynamics of interacting solitons and blowup of complex-valued solutions of KdV[END_REF] was to understand in more detail the propagation of solitons in a neighborhood of the interaction time. Another motivation was an idea to be explained presently concerning singularity formation in nonlinear, dispersive wave equations. More particularly, we are interested in both the generalized Korteweg-de Vries equation

u t + u xxx + u p u x = 0 (1.3)
and coupled systems of Korteweg-de Vries type, viz.

u t + u xxx + P (u, v) x = 0, v t + v xxx + Q(u, v) x = 0, (1.4)
where P and Q are, say, homogeneous polynomials.

Concerning singularity formation, it is an open question whether or not smooth, rapidly decaying, real-valued solutions of (1.3) develop singularities in finite time in the supercritical case p ≥ 5. In the critical case p = 4, blowup in finite time has been established by Martel and Merle [START_REF] Merle | Existence of blow-up solutions in the energy space for the critical generalized KdV equation[END_REF], [START_REF] Martel | Stability of blow-up profile and lower bounds for blow-up rate for the critical generalized KdV equation[END_REF] whilst for subcritical values p = 1, 2, 3, there is no singularity formation for data that lies at least in the Sobolev space H 1 (R). (However, solutions corresponding to infinitely smooth initial values lying only in L 2 (R) can develop singularities; see [START_REF] Bona | Dispersive blowup of generalized Kortewegde Vries equations[END_REF].) Numerical simulations reported in [START_REF] Bona | Conservative, high-order numerical schemes for the generalized Korteweg-de Vries equation[END_REF] of solutions of (1.3) initiated with an amplitude-modified solitary wave indicate blowup in finite time. Such initial data has an analytic extension to a strip symmetric about the real axis in the complex plane. The results of Bona, Grujic and Kalisch [START_REF] Grujiç | Local well-posedness of the generalized Korteweg-de Vries equation in spaces of analytic functions[END_REF], [START_REF] Bona | Spatial analyticity properties of nonlinear waves[END_REF] and [START_REF] Bona | Algebraic lower bounds for the uniform radius of spatrial analyticity for the generalized Korteweg-de Vries equation[END_REF] indicate that blowup at time t has to be accompanied by the width of the strip of analyticity shrinking to zero at the same time. This points to the prospect of a pair of complex conjugate singular points coliding at a spatial point on the real axis, thereby producing non-smooth behavior of the real-valued solution. It was shown in [START_REF] Bona | Pole dynamics of interacting solitons and blowup of complex-valued solutions of KdV[END_REF] that, in certain cases, curves of singularities do merge together. This happens at the moment of interaction of the two solitons when the amplitudes are related in a particular way. This result provides some indication that the blowup seen in the numerical simulations might occur because of the coalescence of curves of complex singularities. Such ruminations seem to justify interest in the pole dynamics in the context of (1.3), even in the case where the initial data is real-valued.

If one considers instead complex-valued solutions, it was shown in [START_REF] Bona | Blow-up of spatially periodic complexvalued solutions of nonlinear dispersive equations[END_REF] (and see also [START_REF] Bona | Finite time blowup of spatially periodic, complex-vlaued solutions of the Kortweg-de Vries equation[END_REF]) that, in the case of spatially periodic boundary conditions, equation (1.3) has solutions which blow up in finite time for all integers p ≥ 1. Explicit examples of smooth, complex-valued solutions defined for x, t ∈ R of the Korteweg-de Vries equation (1.2) that blow up in finite time have been given in [START_REF] Birnir | An example of blow-up for the complex KdV-equation and existence beyond the blow-up[END_REF][START_REF] Bona | Pole dynamics of interacting solitons and blowup of complex-valued solutions of KdV[END_REF][START_REF] Li | Simple explicit formulae for finite time blowup solutions to the complex KdV equation[END_REF][START_REF] Wu | The complex KdV equation with or without dissipation[END_REF]. One outcome of the present paper is an explicit example of a blowing-up soution of the modified Korteweg-de Vries equation (1.1).

For the system (1.4) where P and Q are homogeneous quadratic polynomials, conditions on the coefficients are known that imply global wellposedness for real-valued initial data (u 0 , v 0 ) (see [START_REF] Bona | Global well posedness for a system of KdV-type equations with coupled quadratic nonlinearities[END_REF]). And, the pole dynamics investigated in [START_REF] Bona | Pole dynamics of interacting solitons and blowup of complex-valued solutions of KdV[END_REF] for the Korteweg-de Vries equation (1.2) itself, (KdV-equation henceforth) reveals that the choice

P (u, v) = u 2 -v 2 and Q(u, v) = 2uv
(1.5) leads to a system (1.4) possessing solutions that develop singularities in finite time. (This is the system that obtains if complex-valued solutions of the KdV-equation are broken up into real and imaginary parts.) The latter result is obtained by a careful study of the pole dynamics of the explicit two-soliton solution U = U(z, t) = U(x + iy, t) of the KdV-equation in the complex z-plane. It transpires that as a function of time, most of the singularities of this exact solution, which are all poles, move vertically in the y-direction in the complex plane as well as propagating horizontally in the x-direction. As a consequence, by choosing y 0 appropriately, one can arrange that the function u(x, t) = U(x + iy 0 , t) is a complex-valued solution of the KdV-equation that, at t = 0, is infinitely smooth and decays to zero exponentially rapidly as x → ±∞, but which blows up for a positive value t > 0. If we write u(x, t) = v(x, t) + iw(x, t), then the pair (v, w) is a solution of (1.4) with the choice (1.5) that starts at (v 0 , w 0 ) = (v(•, 0), w(•, 0)) smooth and rapidly decaying, but which forms a singularity in finite time. It is worth noting that by an appropriate choice of the particular two-soliton solution, the initial data (v 0 , w 0 ) can be taken to be arbitrarily small in any of the usual function spaces used in the analysis of such equations.

The motivation for the current paper was to see if the phenomenon just described, i.e. curves of singularities merging together, can occur for the modified KdV-equation, and if this behavior might provide additional insight into the possible ways a singularity is produced in nonlinear, dispersve wave equaions. Indeed, it turns out that the same phenomenon does occur. In certain cases, curves of singularities do converge together at one point. On the other hand, the behavior of these curves is not so different from what occurs in the context of the KdV-equation.

The two-soliton solutions

Preliminary analysis of the two-soliton solutions of the mKdV-equation

u t + u xxx + 6u 2 u x = 0, x ∈ C, t ∈ R, (2.1)
are set forth here in preparation for the investigation of their pole dynamics. We begin with a standard transformation enabling one to express solutions of (2.1) in terms of solutions of another equation. Let u = v x where u is a solution of (2.1). Then v satisfies the equation

d dx v t + v xxx + 2v 3 x = 0.
Assume v and its derivatives vanish at infinity and search for solutions of the latter equation of the form v = 2 arctan(g). A calculation shows that v satisfies

v t + v xxx + 2v 3 x = 0 (2.2)
if and only if

(1 + g 2 )(g t + g xxx ) + 6g x (g 2 x -gg xx ) = 0. (2.3)
This yields a solution to (2.1) having the form

u(x, t) = 2 arctan g(x, t) x = 2g x (x, t) 1 + g(x, t) 2 .
(2.4)

It is important to note that equation (2.1) and (2.3) are both invariant under change of sign. That is, u is a solution of (2.1) if and only if -u is a solution and, likewise, g is a solution of (2.3) if and only if -g is a solution. Also, replacing g by 1/g in (2.4) has the effect of multiplying the solution u by -1.

More precisely, if u is given by (2.4), then

2 arctan 1 g(x, t) x = -2 g x (x, t) 1 + g(x, t) 2 = -u(x, t).
(2.5)

The well-known soliton solution of (2.1) has a hyperbolic secant profile. In detail, for any amplitude value k > 0, it is straightforward to check that

g(x, t) = exp -k(x -x 0 ) + k 3 t = exp -k(x -x 0 -k 2 t) (2.6)
is a solution of (2.3). The corresponding solution u of (2.1) is the soliton solution with speed k 2 and is given explicitly as

u(x, t) = 2 arctan g(x, t) x = -2k e -k(x-x 0 )+k 3 t 1 + e -2k(x-x 0 )+2k 3 t (2.7) = -k sech -k(x -x 0 ) + k 3 t . (2.8)
Note that the choice of sign in the exponential function g in (2.6) produces the negative soliton. Replacing g by either -g or 1/g will produce the positive soliton.

The two-soliton solutions are a little more complicated. The formulation presented here is based on that appearing in [START_REF] Bi | Novel solutions of MKdV-equation with the modified Bäcklund transformation[END_REF]. As just noted, there are both positive and negative soliton solutions of (2.1). Consequently, there are two types of two-soliton solutions, namely interacting solitons of the same or opposite signs.

Suppose first that 0 < k 1 < k 2 . Define the functions f j by

f j (x, t) = exp(-k j x + k 3 j t), j = 1, 2. (2.9)
Of course, this definition omits two arbitary spatial translations; these will be added later. Define two auxilliary functions, g + and g -by

g + (x, t) = -γ f 1 (x, t) + f 2 (x, t) 1 -f 1 (x, t)f 2 (x, t) , (2.10)
g -(x, t) = γ f 1 (x, t) -f 2 (x, t) 1 + f 1 (x, t)f 2 (x, t) , (2.11)
where

γ = k 2 + k 1 k 2 -k 1 > 1.
(2.12)

Proposition 2.1. The functions g + and g -defined in (2.10)-(2.11) are solutions to (2.3).

Proof. It suffices to provide the proof for g -. Indeed, if one replaces f 1 by -f 1 , then g -is transformed into g + , and all the calculations below remain valid in this case. Temporarily, set g = g -. Notice that f jx = -k j f j and f jt = k 3 j f j for j = 1, 2. Thus, the quantities g t , g x , g xx and g xxx may be expressed in terms of f j and k j . First, differentiate with respect to time and come to the expression

g t = γ k 3 1 f 1 -k 3 2 f 2 + k 3 1 f 1 f 2 2 -k 3 2 f 2 1 f 2 (1 + f 1 f 2 ) 2 .
Similarly, the derivative with respect to x is

g x = γ k 2 f 2 -k 1 f 1 -k 1 f 1 f 2 2 + k 2 f 2 1 f 2 (1 + f 1 f 2 ) 2 .
Differentiating the latter expression leads to

g xx = γ (1 + f 1 f 2 ) 3 k 2 1 f 1 -k 2 2 f 2 +(k 2 1 +4k 1 k 2 +k 2 2 )(f 1 f 2 2 -f 2 1 f 2 )-k 2 1 f 2 1 f 3 2 +k 2 2 f 3 1 f 2 2 .
Differentiating once more and simplifying gives

g xxx = γ (1 + f 1 f 2 ) 4 k 3 2 f 2 -k 3 1 f 1 -(k 3 1 + 4k 3 2 + 6k 2 1 k 2 + 12k 1 k 2 2 )(f 1 f 2 2 + f 3 1 f 2 2 ) + (4k 3 1 + k 3 2 + 6k 1 k 2 2 + 12k 2 1 k 2 )(f 2 1 f 2 + f 2 1 f 3 2 ) -k 3 1 f 3 1 f 4 2 + k 3 2 f 4 1 f 3 2 ) .
It follows that

g t + g xxx = 6γ(k 1 + k 2 ) 2 (1 + f 1 f 2 ) 4 k 1 f 2 1 f 2 -k 2 f 1 f 2 2 + k 1 f 2 1 f 3 2 -k 2 f 3 1 f 2 2
on the one hand, and

g 2 x -gg xx = γ 2 (1 + f 1 f 2 ) 4 (k 1 -k 2 ) 2 (f 1 f 2 + f 3 1 f 3 2 )- 8k 1 k 2 f 2 1 f 2 2 + (k 1 + k 2 ) 2 (f 3 1 f 2 + f 1 f 3 2 )
on the other. It is now straightforward to check that g satisfies equation (2.3).

The above proposition implies that

u ± (x, t) = 2 arctan g ± (x, t) x = 2g ± x (x, t) 1 + (g ± ) 2 (x, t)
are solutions to (2.1). A further calculations reveals that

u + = 2γ G + F + (2.13)
and

u -= 2γ G - F - (2.14)
where the new combinations

G + = k 1 f 1 (1 + f 2 2 ) + k 2 f 2 (1 + f 2 1 ), (2.15) G -= -k 1 f 1 (1 + f 2 2 ) + k 2 f 2 (1 + f 2 1 ), (2.16
)

F + = (1 -f 1 f 2 ) 2 + γ 2 (f 1 + f 2 ) 2 ,
(2.17)

F -= (1 + f 1 f 2 ) 2 + γ 2 (f 1 -f 2 ) 2 , (2.18)
have been introduced. Note that the functions with a superscript "+" are obtained from the functions with a superscript "-" simpy by replacing f 1 by -f 1 . If every occurence of k 1 is replaced by -k 1 in formula (2.14) for u -, then f 1 is replaced by 1/f 1 and γ is replaced by 1/γ. Simplifying the resulting expression gives exactly the formula (2.13) for u + . In other words, the formulas for u + and u -can be obtained from each other by replacing every occurence of k 1 by -k 1 .

The function u + is a two-soliton solution of (2.1) having two positive interacting solitons whist u -is a two-soliton solution with two interacting solitons of opposite sign, the faster one being the positive one. One can see this graphically using MAPLE or Mathematica, for example. Analytically, there is an explicit relationship between the fomulas for u ± given by (2.13) and (2.14) and the single soliton solutions of speeds k 2 1 and k 2 2 . As in (2.7), let u j be the centered, positive, soliton solution

u j (x, t) = k j sech(-k j x + k 3 j t) = 2k j f j (x, t) 1 + f j (x, t) 2 . ( 2
u + = γ u 1 + u 2 D + (2.20)
and

u -= γ -u 1 + u 2 D - (2.21)
where

D + = (1 -f 1 f 2 ) 2 + γ 2 (f 1 + f 2 ) 2 (1 + f 2 1 )(1 + f 2 2 ) , D -= (1 + f 1 f 2 ) 2 + γ 2 (f 1 -f 2 ) 2 (1 + f 2 1 )(1 + f 2 2 )
.

Formulas (2.20) and (2.21) show in particular that u + (x, t) > 0 for all x ∈ R and that u -(x, t) > 0 precisely for those x ∈ R and t ∈ R for which u 2 (x, t) > u 1 (x, t).

In Section 4, the asymptotic behavior of the singularities of u ± are examined for large positive and negative time. We will see that they separate into two groups, corresponding to the two single solitons. More remarks on the shape of the two-soliton solutions during their interaction are to be found in Section 8.

Before ending this section, we return to the issue of spatial shifts. For 0

< k 1 < k 2 and x 1 , x 2 ∈ R, let f1 (x, t) = exp(-k 1 (x -x 1 ) + k 3 1 t), f2 (x, t) = exp(-k 2 (x -x 2 ) + k 3 2 t), and
g + (x, t) = - f1 (x, t) + f2 (x, t) 1 -γ -2 f1 (x, t) f2 (x, t) , g -(x, t) = f1 (x, t) -f2 (x, t) 1 + γ -2 f1 (x, t) f2 (x, t) .
Define the interaction time t 0 and the interaction center x 0 of g ± (x, t) to be

t 0 = - x 2 -x 1 k 2 2 -k 2 1 - 1 (k 2 + k 1 )k 1 k 2 log γ and (2.22) x 0 = k 2 2 x 1 -k 2 1 x 2 k 2 2 -k 2 1 - k 2 1 + k 1 k 2 + k 2 2 (k 2 + k 1 )k 1 k 2 log γ, (2.23)
respectively.

Proposition 2.2. Let t 0 and x 0 be the interaction time and interaction center for g + (x, t) and g -(x, t). Then, for any t ∈ R and x ∈ C, we have

g ± (x, t) = g ± (x -x 0 , t -t 0 ), (2.24)
where g ± are defined in (2.10)- (2.11). Moreover, the functions u ± (•, t 0 ) = 2(arctan g ± (•, t 0 )) x are symmetric about the point x 0 on both R and C.

Proof. It suffices to find (x 0 , t 0 ) ∈ R 2 such that fj (x, t) = γf j (x -x 0 , t -t 0 ) (j = 1, 2) for all t ∈ R, x ∈ C. Equivalently, (2.24) will be satisfied if

γe k 1 x 0 -k 3 1 t 0 = e k 1 x 1 , γe k 2 x 0 -k 3 2 t 0 = e k 2 x 2 .
Since (x 0 , t 0 ) given by (2.22)-(2.23) is the solution for this system, the first assertion is proved. To see the symmetry of the functions u ± (•, t 0 ) about x 0 , it is only necessary to deduce from (2.24) that u ± (•, 0) = 2(arctan(g ± (•, 0)) x is an even function. This is easily verified since g ± (-x, 0) = -g ± (x, 0) for all x ∈ C.

Since (mKdV) is invariant under time-and space-translation, it is concluded from Proposition 2.2 that the functions u ± (x, t) = 2(arctan( g ± (x, t)) x are also solutions to (2.1). The point is that, for a general two-soliton solution, the time and place of the interaction are given by t 0 and x 0 in (2.22) and (2.23), respectively. In particular, the solutions u ± are already normalized so that the interaction time is t = 0, and the center of the interaction is at x = 0. It is interesting to note that the values of t 0 and x 0 in the above proposition are precisely the same as for the two-soliton solution of the Korteweg deVries equation, as given in Theorem 1 of [START_REF] Bona | Pole dynamics of interacting solitons and blowup of complex-valued solutions of KdV[END_REF].

Singularities of the two-soliton solutions in the complex plane

As mentioned in the introduction, the two-soliton solutions u(x, t) of (2.1) are viewed as meromorphic functions in the complex variable x. We aim to determine how the dynamics of the singularities in C reflect the behavior of u(x, t) when x is restricted to the real axis.

Consider first the (one)-soliton solution, given by (2.7). It is immediate that the singularities of these solutions are precisely

x = x 0 + k 2 t + mπi 2k (3.1)
where m runs through the odd integers. Moreover, these singularities of u are all simple poles. Thus, the speed of the soliton is exactly the speed of its poles in the complex plane, while the position of the maximum point of the soliton at time t is the real part of the position of these poles. The imaginary part of the singularity remains constant in time of course.

The poles of the two-solitons solutions (2.13) and (2.14), correspond to the zeros of the functions F ± and G ± defined in (2.15), (2.16), (2.17) and (2.18). It will turn out that the singularities of u ± are all simple poles, just as for the one-solitons. In all but a specfic class of exceptional cases, these poles correspond to simple zeros of F ± .

Recall that k 1 and k 2 are called commensurable if there exist positive integers p 1 and p 2 such that

k 2 k 1 = p 2 p 1 . (3.2)
Without loss of generality, we may take it that gcd(p 1 , p 2 ) = 1. In what follows, we will always assume that the integers p 1 and p 2 appearing in (3.2) are without common prime factors. In this case, F ± (x, t) and G ± (x, t) (and thus u ± (x, t)) are periodic in x with minimal imaginary period 2πλi where

λ = p 1 k 1 = p 2 k 2 . (3.3)
Lemma 3.1. For any t ∈ R, the zeros of F ± (•, t) are simple, except for the following special case. If k 1 and k 2 are commensurable, and if p 1 , p 2 ∈ N and λ > 0 are given by (3.2) and (3.3), with p 1 ± p 2 ∈ 4N, then there is a fourth order zero of F ± (•, 0) at x = ( 1 2 + q)λπi for all q ∈ Z. Proof. From (2.18), we infer

F - x = 2 (1 + f 1 f 2 )(f 1x f 2 + f 1 f 2x ) + γ 2 (f 1 -f 2 )(f 1x -f 2x ) . Noticing that f jx = -k j f j , (j = 1, 2) and setting X = f 1 and Y = f 2 , it follows that if x is a zero of F -(•, t) of order greater than or equal to 2, then (1 + XY ) 2 + γ 2 (X -Y ) 2 = 0 and -(k 1 + k 2 )(1 + XY )XY + γ 2 (X -Y )(k 2 Y -k 1 X) = 0. (3.4)
Let (X, Y ) be a solution of this system. Then, from the first equation we deduce that

1 + XY = iεγ(X -Y ), (3.5)
where ǫ is either 1 or -1. Inserting this into the second equation, it is found

that iε(k 2 -k 1 )XY = k 2 Y -k 1 X, provided X = Y .
Extract the product XY and inject it in (3.5) to obtain the linear relation

Y = k 2 k 1 X + iε k 2 -k 1 k 1 .
Then formla (3.5) implies that X must satisfy

1 + X k 2 k 1 X + iε k 2 -k 1 k 1 = iεγ 1 - k 2 k 1 X -iε k 2 -k 1 k 1 ,
which simplifies to

X 2 + 2iεX -1 = 0.
It follows that X = Y = ±i. Thus, it must be that e -k 1 x+k 3 1 t = e -k 2 x+k 3 2 t = ±i, from which we deduce that -k 1 x + k 3 1 t and -k 2 x + k 3 2 t are both purely imaginary. Therefore, t = 0 and x is purely imaginary. Additionally, we have

k 1 x = ( 1 2 + q 1 )πi, k 2 x = ( 1 2 + q 2 )
πi, for some q 1 , q 2 ∈ Z.

Thus k 1 and k 2 are commensurable and in fact,

k 2 /k 1 = r 2 /r 1 with r j = 1 + 2q j . Noticing that q 2 -q 1 ∈ 2Z (because k 1 x -k 2 x ∈ 2iπZ),
x = ± λ 2 πi. It is straightforward to ascertain that for such values of x, F - xx (x, 0) = F - xxx (x, 0) = 0 and F - xxxx (x, 0) = 0.
The same arguments hold for F + . Lemma 3.2. For any t ∈ R, the zeros of F -(•, t) and G -(•, t) are distinct, except the following special case. If k 1 and k 2 are commensurable, and if p 1 , p 2 ∈ N and λ > 0 are given by (3.2) and (3.3), with p 1 ± p 2 ∈ 4N, then there is a third order zero of F -(•, 0) at x = ( 1 2 + q)λπi for all q ∈ Z. The same statement is true with F -and G -replaced by F + and G + .

Proof. In the notation of the proof of Lemma 3.1, the result follows if the system (1

+ XY ) 2 + γ 2 (X -Y ) 2 = 0, -k 1 X(1 + Y 2 ) + k 2 Y (1 + X 2 ) = 0, (3.6)
admits as its only solution X = Y = ±i. As before, if (X, Y ) is a solution, there exists ε ∈ {-1, 1} for which

1 + XY = iεγ(X -Y ).
Extracting from this the variable Y and computing 1 + Y 2 leads to

Y = iεγX -1 X + iεγ
and so

1 + Y 2 = (1 -γ 2 ) 1 + X 2 (X + iεγ) 2 .
Now insert this into the second equation of the system (3.6) and, assuming by contradiction that 1 + X 2 is not zero, simplify the outcome. It follows that

k 1 k 2 (1 -γ 2 )X = (iεγX -1)(X + iεγ).
After further simplifications, this becomes

X 2 + 2iεX -1 = 0,
and the claim follows.

Definition 3.1. The values of k 1 and k 2 for which F ± (•, 0) has multiple zeros are collectively referred to as the exceptional case. This occurs when k 1 and k 2 are commensurable, p 1 and p 2 are odd, and

p 2 -p 1 ∈ 4N
when considering F -, and

p 2 + p 1 ∈ 4N when considering F + .
Lemmas 3.1 and 3.2 show that all singularities of u ± (•, t) are simple poles. In all but the exceptional case, they correspond to simple zeros of F ± . In the exceptional case, they correspond to a third order zero of g ± coinciding with a fourth order zero of F ± . It follows from Rouché's Theorem that in the exceptional case, four simple poles converge, as t → 0, to the simple pole at ( 1 2 + q)λπi for all q ∈ Z. Moreover, by the residue theorem, the sum of the residues of u ± (•, t) at these four poles converge, as t → 0, to the residue of the simple pole at ( 1 2 + q)λπi for all q ∈ Z. Proposition 3.1. In all but the exceptional case, the poles of u ± (•, t), i.e. the zeros of F ± (•, t), are described by analytic curves x : R → C. In the exceptional case, these curves are defined and analytic separately for t < 0 and t > 0.

Proof. Consider the case of u + and F + . Since all the zeros of F + (•, t) are simple, the implicit-function theorem shows that, for a fixed time t 0 , a zero of F + (•, t 0 ) can be locally and uniquely continued as an analytic curve x(t) such that F + (x(t), t) = 0. Such a curve x(t) can be continued as long as it remains in a bounded region of C. Thus, we need to show that |x(t)| must stay bounded as long as t remains in a bounded interval of R. First, it follows from Proposition 6.3 in Section 6 below that the imaginary part of x(t) must remain bounded. Furthermore, if Re x(t) → ∞ in finite time, then the equation F + (x(t), t) = 0 implies 1 = 0. If Re x(t) → -∞ in finite time, then the equation

F + (x(t),t)
1-f 1 (x(t),t) 2 f 2 (x(t),t) 2 = 0 also implies that 1 = 0. A similar argument works for u -and F -. This section is closed with a specific example of an element in the exceptional case, namely k 1 = 1, k 2 = 5 (calculations done with MAPLE). In this case, λ = 1 and F -and G -may be rewritten as G -(x, t) = -e 251t y 11 + 5e 127t y 7 + 5e 125t y 5 -e t y and where y = e -x . Note that at time t = 0, F -and G -are symmetric polynomials in y, and that i and -i are third order zeros of G -, and fourth order zeros of F -. All the other zeros are simple. We also have the decomposition

u -(x, 0) = 3 G -(x, 0) F -(x, 0) = -2 y -i - 2 y + i + y -4i 2y 2 -iy -2 + y + 4i 2y 2 + iy -2 .
4 Large-time asymptotic behavior of the singularities

In this section we show that the poles of u ± , that is the zeros of F ± , separate out into two groups, as t → ±∞, behaving asymptotically as poles of single solitons, of speeds k 2 1 and k 2 2 , respectively. This, of course, reflects the fact that u ± are in fact "two-soliton" solutions of (2.1).

The situation is nearly identical to that obtaining for the two-soliton solution for the Korteweg-de Vries equation (compare the following theorem with Theorem 2 in [START_REF] Bona | Pole dynamics of interacting solitons and blowup of complex-valued solutions of KdV[END_REF]). In particular, the backward shift of the slower wave, well-known in the case of the KdV-solitons, is also present for the modified KdV-equation.

Theorem 4.1. The asymptotic behaviors as t → ±∞ of the curves x(t) of zeroes of F ± (x, t) whose existence was determined in Proposition 3.1 are completely described as follows.

1. For every odd integer m ∈ Z, there exists a unique curve

x m,s -(t) of zeros of F ± (•, t) such that x m,s -(t) = k 2 1 t + 1 k 1 log γ + mπi 2k 1 + o(1), as t → -∞.
2. For every odd integer m ∈ Z, there exists a unique curve x m,s + (t) of zeros of F ± (•, t) such that

x m,s + (t) = k 2 1 t - 1 k 1 log γ - mπi 2k 1 + o(1),
as t → ∞.

3. For every odd integer n ∈ Z, there exists a unique curve

x n,f -(t) of zeros of F ± (•, t) such that x n,f -(t) = k 2 2 t - 1 k 2 log γ + nπi 2k 2 + o(1), as t → -∞.
4. For every odd integer n ∈ Z, there exists a unique curve x n,f + (t) of zeros of F ± (•, t) such that

x n,f + (t) = k 2 2 t + 1 k 2 log γ - nπi 2k 2 + o(1),
as t → ∞.

To prove Theorem 4.1, it is necessary to study the zeros of F ± (•, t) with respect to frames of reference which move at the speed of each constituant soliton. As in Section 4 of [START_REF] Bona | Pole dynamics of interacting solitons and blowup of complex-valued solutions of KdV[END_REF], we set

z = x -k 2 1 t, (4.1) w = x -k 2 2 t, (4.2) r = exp(k 2 (k 2 2 -k 2 1 )t), (4.3) s = exp(k 1 (k 2 2 -k 2 1 )t). (4.4) If H ± (z, r) = (1 ∓ re -(k 1 +k 2 )z ) 2 + γ 2 (e -k 1 z ± re -k 2 z ) 2 (4.5)
and

I ± (w, s) = (s ∓ e -(k 1 +k 2 )w ) 2 + γ 2 (e -k 1 w ± se -k 2 w ) 2 , (4.6)
then

F ± (x, t) = H ± (z, r) = s -2 I ± (w, s).
Solutions z(r) of H ± (•, r) which remain bounded in C as r → 0 and as r → ∞ correspond to zeros x(t) of F ± (•, t) asymptotically traveling at speed k 2 1 as t → ±∞. Likewise, solutions w(s) of I ± (•, s) which remain bounded in C as s → 0 and as s → ∞ correspond to zeros x(t) of F ± (•, t) asymptotically traveling at speed k 2 2 as t → ±∞. The following is true of the curves z(r). Proposition 4.1. For every odd integer m ∈ Z, there exists a smooth curve z ± m (r) defined in some interval of r ≥ 0, such that H ± (z ± m (r), r) = 0 and

z ± m (r) = 1 k 1 log γ + mπi 2k 1 ∓ (-1) m-1 2 4k 2 k 2 2 -k 2 1 γ -k 2 /k 1 e - k 2 2k 1 mπi r + o(r) (4.7)
as r → 0 + . For every odd integer n ∈ Z, there exists a smooth curve w ± n (s) defined in some interval of s ≥ 0, such that I ± (w ± n (s), s) = 0 and

w ± n (s) = - 1 k 2 log γ + nπi 2k 2 ∓ (-1) n-1 2 4k 1 k 2 2 -k 2 1 γ -k 1 /k 2 e k 1 2k 2 nπi s + o(s) (4.8)
as s → 0 + .

Proof. The relation H ± (z, 0) = 0 is satisfied if and only if there exists an odd integer m ∈ Z such that

z = 1 k 1 log γ + mπi 2k 1 ,
and similarly, I ± (w, 0) = 0 is equivalent to

w = - 1 k 2 log γ + nπi 2k 2
for some odd integer n ∈ Z. Applying the implicit function theorem, there exist smooth curves z ± m (r) and w ± n (s) defined in a neighborhood of z 0 = 1/k 1 log γ + mπi/2k 1 and w 0 = -1/k 2 log γ + nπi/2k 2 , respectively, such that

H ± (z ± m (r), r) = 0, z ± m (0) = z 0 and I(w ± n (s), s) = 0, w ± n (0) = w 0 . It remains to calculate (z ± m ) ′ (r) and (w ± n ) ′ (r). Differentiating the equation H ± (z ± m (r), r) = 0 with respect to r yields -(k 1 +k 2 )(1∓re -(k 1 +k 2 )z )re -(k 1 +k 2 )z +γ 2 (e -k 1 z ±re -k 2 z )(k 2 re -k 2 z ±k 1 e -k 1 z ) z ′ = -(1 ∓ re -(k 1 +k 2 )z )e -(k 1 +k 2 )z + γ 2 (e -k 1 z ± re -k 2 z )e -k 2 z
where z = z ± m (r). Taking r = 0 gives

±k 1 γ 2 e -2k 1 zm(0) (z ± m ) ′ (0) = (γ 2 -1)e -(k 1 +k 2 )z ± m (0) ,
and using that

e -(k 1 +k 2 )z ± m (0) = 1 γ (-1) m-1 2 γ -k 2 /k 1 e - k 2 2k 1 mπi leads to (z ± m ) ′ (0) = ∓(-1) m-1 2 4k 2 k 2 2 -k 2 1 γ -k 2 /k 1 e - k 2 2k 1
mπi .

The derivative (w ± n ) ′ (0) is similarly calculated. Differentiating the equation I ± (w ± n (s), s) = 0 with respect to s leads to

[±(k 1 +k 2 )(s∓e -(k 1 +k 2 )w )e -(k 1 +k 2 )w +γ 2 (e -k 1 w ±se -k 2 w )(∓k 2 se -k 2 w -k 1 e -k 1 s )]w ′ = -(s ∓ e -(k 1 +k 2 )w ) ∓ γ 2 (e -k 1 w ± se -k 2 w )e -k 2 w .
At s = 0, there obtains

[-(k 1 + k 2 )e -2(k 1 +k 2 )w ± n (0) -k 1 γ 2 e -2k 1 w ± n (0) ](w ± n ) ′ (0) = ∓(γ 2 -1)e -(k 1 +k 2 )w ± n (0)
and since

e -(k 1 +k 2 )w ± n (0) = - 1 γ (-1) n-1 2 γ -k 1 /k 2 e - k 1 2k 2 nπi , we conclude (w ± n ) ′ (s) = ∓(-1) n-1 2 4k 1 k 2 2 -k 2 1 γ -k 1 /k 2 e k 1 2k 2 nπi .
A proof of Theorem 4.1 is now readily available. The last proposition immediately gives all the curves of zeros of F ± (•, t) claimed in Theorem 4.1. The only remaining issue is to prove that there are no additional zeros of F ± (•, t). If k 1 and k 2 are commensurable, this is straightforward. For each t ∈ R, F ± (•, t) is a polynomial in e -xλ of degree 2(p 1 + p 2 ), and thus must have precisely 2(p 1 + p 2 ) zeros in the complex plane. The zeros of F ± (•, t) described in Theorem 4.1 account for all of them, for large |t|, and so no other zeros can exist.

If k 1 and k 2 are not commensurable, the desired result can be obtained by approximating k 1 and k 2 by sequences {k ν 1 } ∞ ν=1 and {k ν 2 } ∞ ν=1 , which are commensurable. [START_REF] Bona | Spatial analyticity properties of nonlinear waves[END_REF] The nature of the singularity in the exceptional case

In this section a more detailed analysis is undertaken of the singularity of u ± in the exceptional case (see Definition 3.1). As described just after this definition, the singularity at ( 1 2 + q)λπi, for any q ∈ Z, corresponds to a fourth order zero of F ± (•, 0) and is approached by four simple zeros of F ± as t → 0. The goal is to understand the behavior of a smooth curve of zeros of F (•, t) in a neighborhood of such a fourth order zero.

We claim it suffices to analyse the singularity at λ 2 πi, i.e. the case q = 0. Indeed, since F ± (•, t) is 2πλi periodic, it is enough to consider q = -1, 0. Furthermore, F ± (x, t) = F ± (x, t), and so F ± (x(t), t) = 0 if and only if F ± (x(t), t) = 0. Hence, only the case q = 0 need be examined.

Remark that

F ± (x, t) = e -2(k 1 +k 2 )x+2(k 3 1 +k 3 2 )t F ± (-x, -t).
From this, it is deduced that F ± (x(t), t) = 0 if and only if F ± (-x(t), -t) = 0. In other words, if x(t) is a curve of zeros approaching λ 2 πi as t ր 0, then -x(t) is a curve of zeros approaching λ 2 πi as t ց 0. Theorem 5.1. Suppose we are in the exceptional case wherein k 1 and k 2 are commensurable and the odd integers p 1 , p 2 ∈ N and λ > 0 are as in (3.2) and (3.3). In the case of F + , it is assumed that p 1 + p 2 ∈ 4N and in the case of F -, it is presumed that p 1 -p 2 ∈ 4N. Let x(t) be a smooth curve, defined for t close to 0, such that F ± (x(t), t) = 0 and x(t) = λ 2 πi but x(t) → λ 2 πi as t → 0. Then, either

lim t→0 (x(t) -λ 2 πi) 3 t = -12, (5.1)
or lim t→0 x(t) -λ 2 πi t = k 2 1 + k 2 2 .
(5.2)

Remark 5.1. The similarity of the result in Theorem 5.1 with the "exceptional case" for the two-soliton solution of sthe KdV-equation is quite striking. The behavior of the curves described in 5.1 is exactly the same as given by Proposition 4.9 in [START_REF] Bona | Pole dynamics of interacting solitons and blowup of complex-valued solutions of KdV[END_REF]. However, the exceptional case for the KdV-equation corresponds to p 1 being an odd integer, and p 2 being an even integer. For the mKdV-equation, the "exceptional pole" is located at λ 2 πi, rather than at λπi as it is for the KdV-equation.

Also, how does one explain the behavior described by (5.2)? In the case of the two-soliton solution of the KdV-equation, if p 1 is even and p 2 is odd, there is a horizontally moving pole approaching the singularity at πλi exactly as described by (5.2). This calculation was not carried out in [START_REF] Bona | Pole dynamics of interacting solitons and blowup of complex-valued solutions of KdV[END_REF], but can be obtained by a simple modification of the proof of Proposition 4.9 in [START_REF] Bona | Pole dynamics of interacting solitons and blowup of complex-valued solutions of KdV[END_REF]. Thus, it appears that the exceptional case for the two-soliton solution of the mKdV-equation (2.1) includes the behavior of curves of singularities from two different cases of the two-soliton solution of the KdV-equation, namely the "exceptional case", where p 1 is odd and p 2 is even, as well as the case where p 1 is even and p 2 is odd. It is precisely in these two cases that there is a pole located at the same place πλi (at time 0).

Proof. The proof is provided for F -. Similar arguments apply for F + . Let z(r) = x(t) -k 2 1 t where r is defined in (4.3). It follows that z(r) is a smooth curve such that H -(z(r), r) = 0, where H is as in (4.5). Notice that z(r)

→ λ 2 πi as r → 1. Now, H -(z, r) = 0 ⇔ 1 + re -(k 1 +k 2 )z = ±iγ(e -k 1 z -re -k 2 z ). (5.3)
If the minus sign is chosen on the right-hand side of (5.3), we come to

r = - iγe -k 1 z + 1 e -(k 1 +k 2 )z -iγe -k 2 z .
Differentiating this relation with respect to r, it follows that

1 = 1 (e -(k 1 +k 2 )z -iγe -k 2 z ) 2 iγk 1 e -k 1 z e -(k 1 +k 2 )z -iγe -k 2 z + iγe -k 1 z + 1)(-(k 1 + k 2 )e -(k 1 +k 2 )z + iγk 2 e -k 2 z ) z ′ (r).
This may be rewritten as

1 = iγk 2 e -k 2 z 1 + ie -k 1 z 2 e -(k 1 +k 2 )z -iγe -k 2 z 2 z ′ ,
or what is the same,

z ′ = - ie k 2 z e -(k 1 +k 2 )z -iγe -k 2 z 2 k 2 γ 1 + ie -k 1 z 2 . ( 5.4)
Since p 1 and p 2 are both odd with p 2 -p 1 ∈ 4N and z(r) → λ 2 πi as r → 1, it must be that e -k 1 z → e 1 πi/2 and e -k 2 z → e -p 2 πi/2 as r → 1. If p 1 and p 2 are both in 4N + 1, then e -k 1 z and e -k 1 z both converge to -i as r → 1, while if p 1 and p 2 are both in 4N + 3, then e -k 1 z and e -k 1 z both converge to i as r → 1. We suppose first that p 1 and p 2 are both in 4N + 1. In this case, it follows from (5.4) that

z ′ (r) → (γ + 1) 2 4k 2 γ = k 2 k 2 2 -k 2 1
(5.5) as r → 1, and thus

lim r→1 z(r) -λ 2 πi r -1 = k 2 k 2 2 -k 2 1 .
Turning back to x(t), and using the fact that (r -

1)/t → k 2 (k 2 2 -k 2 1 ) as t → 0, it follows that lim t→0 x(t) -k 2 1 t -λ 2 πi t = k 2 2 from which it is concluded that lim t→0 x(t) -λ 2 πi t = k 2 1 + k 2 2 .
If instead the plus sign is chosen on the right-hand side of (5.3), then it is immediately inferred that

r = iγe -k 1 z -1 e -(k 1 +k 2 )z + iγe -k 2 z .
Differentiating this equation with respect to r, we get

1 = 1 (e -(k 1 +k 2 )z + iγe -k 2 z ) 2 -iγk 1 e -k 1 z e -(k 1 +k 2 )z + iγe -k 2 z -iγe -k 1 z -1 -(k 1 + k 2 )e -(k 1 +k 2 )z -iγk 2 e -k 2 z z ′ ,
which may be simplified to

1 = - iγk 2 e -k 2 z (1 -ie -k 1 z ) 2 (e -(k 1 +k 2 )z + iγe -k 2 z ) 2 z ′ .
From this, it is inferred that

(1 -ie -k 1 z ) 2 z ′ = ie k 2 z k 2 γ (e -(k 1 +k 2 )z + iγe -k 2 z ) 2 .
(5.6)

Since p 1 and p 2 are both in 4N + 1, it is concluded that

(1 -ie -k 1 z ) 2 z ′ → - (γ -1) 2 k 2 γ = -4k 2 1 k 2 (k 2 2 -k 2 1 )
(5.7) as r → 1. A consequence of this is that

d dr (1 -ie -k 1 z ) 3 = 3(1 -ie -k 1 z ) 2 (ik 1 e -k 1 z )z ′ → -12k 3 1 k 2 (k 2 2 -k 2 1 )
.

L'Hopital's rule comes to the rescue and it is found that

lim r→1 (1 -ie -k 1 z ) 3 r -1 = -12k 3 1 k 2 (k 2 2 -k 2 1 )
.

Since lim

z→ λ 2 πi 1 -ie -k 1 z z -λ 2 πi = -i lim z→ λ 2 πi e -k 1 z -e -k 1 λ 2 πi z -λ 2 πi = -ik 1 e -k 1 λ 2 πi = k 1 , we must have lim r→1 (z(r) -λ 2 πi) 3 r -1 = -12 k 2 (k 2 2 -k 2 1 )
.

Reverting to the original variable x(t) and using again that (r -

1)/t → k 2 (k 2 2 -k 2 1 ) as t → 0, it follows that lim t→0 (x(t) -k 2 1 t -λ 2 πi) 3 t = -12
and thus lim t→0

(x(t) -λ 2 πi) 3 t = -12.
It remains to treat the situation where p 1 , p 2 ∈ 4N + 3, in which case e -k j z(r) → i as r → 1. In fact, this case is "dual" to the one just treated and the same calculations lead to the result. To see this, assume first that the minus sign obtains on the right-hand side of (5.3). Then (5.4) implies that

(1 + ie -k 1 z ) 2 z ′ = - ie k 2 z k 2 γ (e -(k 1 +k 2 )z -iγe -k 2 z ) 2 → - (γ -1) 2 k 2 γ = -4k 2 1 k 2 (k 2 2 -k 2 1
) as r → 1. On the other hand, it is straightforward that lim

z→ λ 2 πi 1 + ie -k 1 z z -λ 2 πi = k 1 .
Following the same line of development as pursued for the positive sign in the previous situation where both p 1 and p 2 lie in 4N + 1 leads to

lim t→0 (x(t) -λ 2 πi) 3 t = -12.
Now assume we have the positive sign on the right-hand side of (5.3). From (5.6) we deduce

z ′ = ie k 2 z k 2 γ (e -(k 1 +k 2 )z + iγe -k 2 z ) 2 (1 -ie -k 1 z ) 2 → (γ + 1) 2 4k 2 γ = k 2 k 2 2 -k 2 1 21 as r → 1. Then, it is clear that lim t→0 x(t) -λ 2 πi t = k 2 1 + k 2 2 .

Vertical movement of poles

In this section we study the vertical motion of the singularities of the twosoliton solutions u ± of (2.1). As seen in Section 2, this comes down to studying the zeroes of F ± given by (2.17)- (2.18). Recall the solution u + represents two interacting solitons of the same sign, while u -represents two interacting solitons of opposite sign. Observe that

F + (x, t) = F + 1 (x, t)F + 2 (x, t) (6.1) F -(x, t) = F - 1 (x, t)F - 2 (x, t) (6.2)
where

F + 1 = 1 + iγf 1 + iγf 2 -f 1 f 2 (6.3) F + 2 = 1 -iγf 1 -iγf 2 -f 1 f 2 (6.4) F - 1 = 1 + iγf 1 -iγf 2 + f 1 f 2 (6.5) F - 2 = 1 -iγf 1 + iγf 2 + f 1 f 2 (6.6)
and f 1 and f 2 are as in (2.9). Note the similarity in form between the above functions and the function F defined by formula (2.13) in [START_REF] Bona | Pole dynamics of interacting solitons and blowup of complex-valued solutions of KdV[END_REF]. Since F + 1 (x, t) = 0 if and only if F + 2 (x, t) = 0, to study the zeros of F + , it suffices to study the zeros of F + 1 . Similarly, since F - 1 (x, t) = 0 if and only if F - 2 (x, t) = 0, to study the zeros of F -, it suffices to study the zeros of F - 1 . As before, it is sometimes necessary to distinguish between the cases where k 1 and k 2 are commensurable and the cases where they are not. In the former case, denote by p 1 and p 2 the relatively prime positive integers such that (3.2) holds, and let λ be defined as in (3.3). Hence, if k 1 and k 2 are commensurable, the functions f 1 and f 2 are 2πλi periodic in x. Thus, in the commensurable case, all four of the functions F ± 1 , F ± 2 , must also be 2πλi periodic in x. Proposition 6.1. The zeroes of F + and F -in the complex plane lie off the real axis. Moreover, in the commensurable case, if either F + (x, t) = 0 or F -(x, t) = 0, then Im x = 2mπλ for all m ∈ Z.

Proof. Suppose x ∈ R and F + (x, t) = 0. It follows that F + 1 (x, t) = F + 2 (x, t) = 0. Taking real and imaginary parts produces the coupled system

1 = f 1 f 2 , f 1 = -f 2 , from which it follows that f 1 (x, t) 2 = -1, which is impossible. If x ∈ R and F -(x, t) = 0, it similarly follows that 1 = -f 1 f 2 , f 1 = f 2 ,
from which one observes that f 1 (x, t) 2 = -1, which is again impossible.

The last statement follows from the 2πλi periodicity which holds in the commensurable case.

If k 1 and k 2 are commensurable, further information about the location of the zeros of F ± can be obtained. Proposition 6.2. Suppose that k 1 and k 2 are commensurable. If either F + (x, t) = 0 or F -(x, t) = 0, then Im x = (2m + 1)πλ for any m ∈ Z.

Proof. Because of 2πλi-periodicity, it suffices to consider m = 0. Suppose Im x = πλ and F + 1 (x, t) = 0. Since x -x = 2πλi, it follows from 2πλiperiodicity that F + 1 (x, t) = 0. But this says precisely that F + 2 (x, t) = 0. Adding and subtracting the two equations, F + 1 (x, t) = 0 and

F + 2 (x, t) = 0 gives 1 = f 1 f 2 , f 1 = -f 2 ,
from which it is concluded that f 1 (x, t) 2 = -1, i.e. f 1 (x, t) = ±i and f 2 (x, t) = ∓i. The latter system together with the definition (2.9) of the f j 's implies

Re x = k 2 1 t, Re x = k 2 2 t,
and so t = Re x = 0. Thus x = πλi. However,

f 1 (πλi, 0) = exp(-k 1 πλi) = exp(-p 1 πi) = ±1,
since p 1 is an integer. This contradiction shows that F + 1 (x, t) can not be zero if Im x = πλ. A similar argument applies to the other functions

F + 2 , F - 1 and F - 2 .
To show that most of the poles of the two-soliton solution of (2.1) move vertically, we need to reproduce calculations which are similar to those found in Section 3 of [START_REF] Bona | Pole dynamics of interacting solitons and blowup of complex-valued solutions of KdV[END_REF]. Unfortunately, it seems that the calculations in [START_REF] Bona | Pole dynamics of interacting solitons and blowup of complex-valued solutions of KdV[END_REF] do not directly apply to the present situation in all cases. In an effort at economy, we only prove here that, with certain very specific exceptions, the poles in the two-soliton solution always feature vertical movement. Consequently, we do not need to reproduce the entirety of Section 3 of [START_REF] Bona | Pole dynamics of interacting solitons and blowup of complex-valued solutions of KdV[END_REF]. Nonetheless, we cannot avoid certain, somewhat tedious calculations, closely modeled on those in [START_REF] Bona | Pole dynamics of interacting solitons and blowup of complex-valued solutions of KdV[END_REF].

The following notation α = -Im x,

A 1 = e -k 1 Re x+k 3 1 t , A 2 = e -k 2 Re x+k 3 2 t ,
is taken from [START_REF] Bona | Pole dynamics of interacting solitons and blowup of complex-valued solutions of KdV[END_REF]. The function F + 1 may be rewritten in this notation, viz.

F + 1 (x, t) = 1 + iγA 1 e ik 1 α + iγA 2 e ik 2 α -A 1 A 2 e i(k 1 +k 2 )α . (6.7)
Attention is first focussed on the solution u + . To investigate possible vertical movement of poles of u + , we examine more closely the zeros of F + 1 (x, t). Proposition 6.3. Suppose F + 1 (x, t) = 0. Then, cos k 1 α = 0 if and only if cos k 2 α = 0. Moreover, the relation

A 2 - 1 A 2 cos k 1 α + A 1 - 1 A 1 cos k 2 α = 0 (6.8)
always holds. In case cos k 1 α = 0 and cos k 2 α = 0, then A 1 = 1 if and only if A 2 = 1, and this can only happen if t = 0 and Re x = 0.

Proof. First, multiply the equation F + 1 (x, t) = 0 by 1 -iγA 1 e -ik 1 α , express this using the representation (6.7) and take the imaginary part of the result. This leads to the formula

A 1 + 1 A 1 cos k 2 α = 1 γ sin(k 2 + k 1 )α -γ sin(k 2 -k 1 )α. ( 6.9)
In the same way, multiplying by 1 -iγA 2 e -ik 2 α and subsequently extracting the imaginary part yields

A 2 + 1 A 2 cos k 1 α = 1 γ sin(k 2 + k 1 )α + γ sin(k 2 -k 1 )α. (6.10)
If cos k 2 α = 0, so that in particular sin k 2 α = ±1, it follows from (6.9), using the formulas for the sine of the sum and difference, that cos k 1 α = 0. In the same way, using (6.10), if cos k 1 α = 0, then cos k 2 α = 0. This proves the first assertion in the proposition.

For future reference, notice that it follows by subtracting (6.9) from (6.10) that

γ sin(k 2 -k 1 )α = 1 2 A 2 + 1 A 2 cos k 1 α - 1 2 A 1 + 1 A 1 cos k 2 α. (6.11)
Taking the real and imaginary parts of (6.7) and setting them equal to zero yields

1 -γA 1 sin k 1 α -γA 2 sin k 2 α -A 1 A 2 cos(k 1 + k 2 )α = 0, (6.12) γA 1 cos k 1 α + γA 2 cos k 2 α -A 1 A 2 sin(k 1 + k 2 )α = 0. ( 6.13)
Multiply the first equation above by sin(k 1 + k 2 )α, the second equation by cos(k 1 + k 2 )α and form the difference of the outcomes. The formula

sin(k 1 + k 2 )α = γA 2 cos k 1 α + γA 1 cos k 2 α (6.14)
emerges from these machinations. But, (6.13) implies

sin(k 1 + k 2 )α = γ A 2 cos k 1 α + γ A 1 cos k 2 α. (6.15)
The last two equations taken together imply (6.8) This proves the second assertion of the proposition. Finally, it is clear that A 1 = A 2 = 1 if and only if t = 0 and Re x = 0. Proposition 6.4. Suppose that cos k 1 α = cos k 2 α = 0. It follows that k 1 and k 2 are commensurable, that p 1 and p 2 are both odd and that α is an odd multiple of πλ/2.

Proof. If cos k 1 α = cos k 2 α = 0, then there exist integers m and n such that

k 1 α = (2m + 1) π 2 , k 2 α = (2n + 1) π 2 .
Thus it is clear that k 1 and k 2 are commensurable and

p 1 p 2 = k 1 k 2 = 2m + 1 2n + 1 , whence p 1 (2n + 1) = p 2 (2m + 1). (6.16)
It follows that p 2 -p 1 is even, and since they are also relatively prime, they both must be odd. Next, equation (6.16) implies that p 1 = (2m+1)/c, where

c = gcd(2m + 1, 2n + 1).
Consequently,

α = 2m + 1 k 1 π 2 = 2m + 1 p 1 πλ 2 = c πλ 2 ,
which concludes the proof since c is necessarily odd. Proposition 6.5. Let x(t) be a smooth curve such that F + 1 (x(t), t) = 0 and ∂ x F + 1 (x(t), t) = 0. It follows that Im x ′ (t) has the same sign as

A 1 - 1 A 1 cos k 2 α = -A 2 - 1 A 2 cos k 1 α.
Proof. Let x(t) be a smooth curve such that F + 1 (x(t), t) = 0 and ∂ x F + 1 (x(t), t) = 0. Implicit differentiation provides the relation

x ′ (t) = - ∂ t F + 1 (x(t), t) ∂ x F + 1 (x(t), t) = k 3 1 iγA 1 e ik 1 α + k 3 2 iγA 2 e ik 2 α -(k 3 1 + k 3 2 )A 1 A 2 e i(k 1 +k 2 )α k 1 iγA 1 e ik 1 α + k 2 iγA 2 e ik 2 α -(k 1 + k 2 )A 1 A 2 e i(k 1 +k 2 )α = k 3 1 γA 1 e -ik 2 α + k 3 2 γA 2 e -ik 1 α + i(k 3 1 + k 3 2 )A 1 A 2 k 1 γA 1 e -ik 2 α + k 2 γA 2 e -ik 1 α + i(k 1 + k 2 )A 1 A 2 = 1 k 1 γA 1 e -ik 2 α + k 2 γA 2 e -ik 1 α + i(k 1 + k 2 )A 1 A 2 2 k 3 1 γA 1 e -ik 2 α + k 3 2 γA 2 e -ik 1 α +i(k 3 1 + k 3 2 )A 1 A 2 k 1 γA 1 e ik 2 α + k 2 γA 2 e ik 1 α -i(k 1 + k 2 )A 1 A 2 .
The imaginary part of the numerator in this last expression is equal to

k 1 k 2 (k 2 2 -k 2 1 )γ 2 A 1 A 2 sin(k 2 -k 1 )α+k 1 k 2 (k 1 +k 2 ) 2 A 1 A 2 (A 1 cos k 2 α-A 2 cos k 1 α),
which is a positive mulitple of, and so has the same sign as,

γ sin(k 2 -k 1 )α + A 1 cos k 2 α -A 2 cos k 1 α. (6.17)
Finally, formulas (6.11) and (6.8) reveal that the expression in (6.17) is equal to

A 1 cos k 2 α - 1 A 1 cos k 2 α.
Corollary 6.1. Suppose either that k 1 and k 2 are not commensurable, or that they are commensurable with p 1 and p 2 having opposite parity. Let x(t) be a smooth curve such that F + 1 (x(t), t) = 0. If either t = 0 or if Re x = 0, then Im x ′ (t) = 0.

Proof. We know from Lemma 3.1 that the roots of F + (•, t) are all simple unless k 1 and k 2 are commensurable and p 1 and p 2 are both odd. It follows that the same is true for F + 1 (•, t). Thus, it must be the case that ∂ x F + 1 (x(t), t) = 0. The last three propositions can now be brought to bear to establish the claim.

To recapitulate, it has been shown, except in the commensurable case with p 1 and p 2 both odd, that if either t = 0 or if Re x = 0, then Im x ′ (t) = 0, so the poles of the solution u + are always moving vertically. The same is true in the case that p 1 and p 2 are both odd, so long as the imaginary part of the pole is not an odd multiple of πλ/2. The same conclusions are true about the poles of u -. The calculations leading to this conclusion, while not the same as those for u + , are analogous enough that we pass over the details. Interestingly, in the commensurable case when p 1 and p 2 have opposite parity, it turns out that the poles of u -and those of u + bear a very simple relationship to one another. Proposition 6.6. Suppose that p 1 and p 2 have opposite parity (one odd, one even). It follows that there exists θ ∈ R such that

F + (x -iθ, t) = F -(x, t) (6.18)
for all x ∈ C and t ∈ R. In other words, the poles of u -are precisely given by a vertical translation of the poles of u + .

Proof. Suppose θ ∈ R is such that

f 1 (x -iθ, t) = f 1 (x, t), (6.19) f 2 (x -iθ, t) = -f 2 (x, t), (6.20)
for all x ∈ C and t ∈ R. It would then follow that (6.18) holds for all x ∈ C and t ∈ R. The same would be true if we had instead

f 1 (x -iθ, t) = -f 1 (x, t), (6.21) f 2 (x -iθ, t) = f 2 (x, t). (6.22)
For (6. [START_REF] Wu | The complex KdV equation with or without dissipation[END_REF]) and ( 6.20) to be valid, it is necessary and sufficient that exp(ik 1 θ) = exp(ip 1 θ/λ) = 1, (6.23) exp(ik 2 θ) = exp(ip 2 θ/λ) = -1.

(6.24)

For these latter conditions to hold, it is necessary for there to be two integers m and n such that p 1 θ/λ = 2mπ, (6.25) p 2 θ/λ = (2n + 1)π, (6.26) or, what is the same,

θ λπ = 2m p 1 = 2n + 1 p 2 . ( 6

.27)

If p 1 is even and p 2 is odd, it is clear that one may choose appropriate values of θ, m and n so that the last equation holds In the opposite case, if p 1 is odd and p 2 is even, a similar argument shows that there exists θ, mand n such that (6.21) and (6.22) hold.

It remains to consider the case where k 1 and k 2 are commensurable, with p 1 and p 2 odd. To establish the existence of poles with non-trivial vertical movement, it suffices by Proposition 6.4 to establish the existence of poles whose imaginary parts are not an odd multiple of πλ/2 (with either t = 0 or with non-zero real part).

As shown in [START_REF] Bona | Pole dynamics of interacting solitons and blowup of complex-valued solutions of KdV[END_REF], for the two-soliton solution of the KdV-equation in the commensurable case, there are always poles with vertical movement, and at least one pole moving precisely horizontally. The proof of this fact requires the full force of the delicate and technical analysis in [START_REF] Bona | Pole dynamics of interacting solitons and blowup of complex-valued solutions of KdV[END_REF]. We would like to avoid such technical calculations in this paper to the extent possible.

Thus, for the modified KdV-equation (2.1), we should not expect that all the poles will be moving vertically in the remaining cases. The fact that in the case of opposite parity, all the poles move vertically (except if t = 0 and Re x = 0) is already an interesting difference in behavior between the two equations.

It turns out that in the commensurable case, with p 1 and p 2 both odd, the movement of the poles of u + can in fact be reduced to the movement of the poles of the 2-soliton solution of the KdV-equation if p 2 -p 1 ∈ 4N. The same is true for u -if p 2 + p 1 ∈ 4N. Here are the precise statements. Proposition 6.7. Suppose that p 1 and p 2 are both odd. If p 2 -p 1 ∈ 4N then there exist θ 1 , θ 2 ∈ R such that

F + 1 (x -iθ 1 , t) = 1 + γf 1 (x, t) + γf 2 (x, t) + f 1 (x, t)f 2 (x, t) (6.28)
and

F + 2 (x -iθ 2 , t) = 1 + γf 1 (x, t) + γf 2 (x, t) + f 1 (x, t)f 2 (x, t) (6.29)
for all x ∈ C and t ∈ R.

Proof. Suppose θ 1 ∈ R is such that f 1 (x -iθ 1 , t) = -if 1 (x, t), (6.30) f 2 (x -iθ 1 , t) = -if 2 (x, t), (6.31)
for all x ∈ C and t ∈ R. It would follow that (6.28) holds for all x ∈ C and t ∈ R. In addition, if we have

f 1 (x -iθ 2 , t) = if 1 (x, t), (6.32) f 2 (x -iθ 2 , t) = if 2 (x, t), (6.33)
then (6.29) would be true.

For the system (6.30)-( 6.31) to be valid, it is necessary and sufficient that exp(ik 1 θ) = exp(ip 1 θ/λ) = -i, (6.34) exp(ik 2 θ) = exp(ip 2 θ/λ) = -i. (6.35) For this, we need to find two integers m and n such that The proof for F + 2 is similar, but with 4m + 1 replacing 4m -1 and 4n + 1 replacing 4n -1. Proposition 6.8. Suppose that p 1 and p 2 are both odd. If p 2 + p 1 ∈ 4N then there exist θ 1 , θ 2 ∈ R such that

p 1 θ λ = (4m -1) π 2 , ( 6
F - 1 (x -iθ 1 , t) = 1 + γf 1 (x, t) + γf 2 (x, t) + f 1 (x, t)f 2 (x, t) (6.40) and
F - 2 (x -iθ 2 , t) = 1 + γf 1 (x, t) + γf 2 (x, t) + f 1 (x, t)f 2 (x, t) (6.41)
for all x ∈ C and t ∈ R.

The expression on the right side of the four formulas (6.28), (6.29), (6.40), and (6.41), i.e.

1 + γf 1 (x, t) + γf 2 (x, t) + f 1 (x, t)f 2 (x, t), (6.42)
is exactly the function F in formula (2.13) of [START_REF] Bona | Pole dynamics of interacting solitons and blowup of complex-valued solutions of KdV[END_REF] whose zeros correspond to the poles of the 2-soliton solution of the KdV-equation. Thus, we may use the results of [START_REF] Bona | Pole dynamics of interacting solitons and blowup of complex-valued solutions of KdV[END_REF] to describe the behavior of the poles of u ± in the cases under consideration.

More precisely, we may now affirm that in the case where p 1 and p 2 are both odd and p 2 -p 1 ∈ 4N, the solution u + has two poles moving horizontally on the line Im x = πλ/2 and also on the line Im x = -πλ/2. There are 2(p 1 + p 2 -2) other poles with imaginary part between ±πλ, and they will move vertically for all t = 0. The same will be true for u -in the case p 1 and p 2 are both odd if p 2 + p 1 ∈ 4N. These configurations will repeat with 2πλi periodicity, so that horizontally moving poles are found with imaginary part equal to every odd multiple of πλ/2.

The last case is one where four poles meet at t = 0, as described in the previous section, at poles on the imaginary axis whose imaginary part is an odd multiple of πλ/2. At each of these points, the analysis shows that (at least) two of these poles have vertical movement. We refrain from going into the detailed considerations needed to establish that all the poles move vertically for all t = 0, excepting the two poles approaching the singular points horizontally. In any event, we have shown in this case the existence of poles with vertical movement.

Finite time blowup of solutions

The results of the previous section immediately imply that there exist complexvalued solutions to (2.1) on R which blow up in finite time. Indeed, fix α ∈ R, and let u be given by u(x, t) = u ± (x -iα, t),

where we may use either of the two soliton solutions u + or u -. As long as the set {(x -iα, t 0 ) : x ∈ R} does not contain a pole of u ± , u is a smooth, complex-valued solution of (2.1) for t in a neighborhood of t 0 . Since at any given time, the imaginary parts of the collection of all poles of u + or u -form a discrete set, and since both u + and u -have poles whose imaginary parts move continuously in time, there exist α and t 0 such that (x -iα, t 0 ) is a pole of u + , say, but (x -iα, t) is not a pole of u + if t is sufficiently close to, but not equal to, t 0 . It follows that the solution u defined in (7.1) with this choice of α is a regular complex-valued solution of (2.1) on R, decaying exponentially to zero as x → ±∞ for t close to t 0 , but which is singular at t = t 0 . In other words, the solution blows up in finite time.

It is interesting to note that this result of singularity formation for complexvalued solutions of the mKdV-equation can be interpreted as a blow-up result for real-valued solutions of a system of dispersive equations. Let u be a solution of (2.1) and let r = Re u and s = Im u. It follows that r and s satisfy the real-valued system r t + r xxx + 6(r 2 -s 2 )r x -2rss x = 0, (7.2) s t + s xxx + 2rsr x + 6(r 2 -s 2 )s x = 0.

(7.3) Thus, we have shown that this coupled, dispersive system admits real-valued solutions (exponentially decaying in space) which blow up in finite time.

8 Some formal calculations (All the computations in this section have been carried out using MAPLE.) As noted at the end of Section 2, the interaction time for the two-soliton solutions u ± , given in (2.13) and (2.14), is t = 0, the center of the interaction is x = 0 and at t = 0, the solution is even in x. The explicit formulas for u ± allow us to observe and calculate certain aspects of these solutions at the moment of interaction. In particular, it is interesting to know whether there is a single maxima during the interaction or not, and it is likewise interesting to know the speed of the two solitons at the moment of interaction.

In the case of u + , one observes that that the solution has one centered maximum if the ratio k 2 /k 1 is large enough (bigger than around 2.6), and two symmetrically located maxima for smaller values of k 2 /k 1 . Partial confirmation of this can be obtained by computing u + xx (0, 0). Since, by symmetry, u +

x (0, 0) = 0, the sign of the second derivative will tell us if it is a local maximum or a local minimum. A MAPLE-implemented calculation shows that

u + xx (0, 0) = -(k 2 -k 1 )(k 2 1 -3k 1 k 2 + k 2 2
). Therefore, if 1 < k 2 /k 1 < (3 + √ 5)/2, then u + xx (0, 0) > 0, which means that u + (•, 0) has a local minimum at x = 0. Thus, there are (at least) two maxima at the moment of interaction. If k 2 /k 1 > (3 + √ 5)/2, then u + (•, 0) has a local maximum at x = 0, which is consistent with there being a single maximum at the moment of interaction.

Similarly, we calculate that

u - xx (0, 0) = -(k 2 + k 1 )(k 2 1 + 3k 1 k 2 + k 2 2 ),
which means u -(•, 0) has a local maximum at x = 0 for all values 0 < k 1 < k 2 , which is consistent with the graphical observations of the solution itself.

If we are interested in the "speed" of the two-soliton solution at the moment of interaction, one idea is to calculate the speed of the maximum or the local minimum of the solution as (x, t) approaches (0, 0). One might argue that the movement of the maximum is some kind of speed. It must be acknowledged that the interpretation is less clear when one is tracking a local minimum. To calculate this speed, suppose y(t) = y ± (t) is a real-valued curve such that u ±

x (y(t), t) = 0, i.e. y ± (t) is always at an extemal point of the solution u = u ± . Suppose also that y ± (0) = 0, which is to say the curve is at the interaction center at the interaction time t = 0. Differentiating with respect to t, we see that u xx (y(t), t)y ′ (t) + u xt (y(t), t) = 0, or y ′ (t) = -u xt (y(t), t) u xx (y(t), t) .

This gives, in turn, y ′ (0) = -u xt (0, 0) u xx (0, 0) , (

which might be thought of as representing the speed of the two-soliton solution at the moment of interaction of the two solitons. The value of y ′ (0) can be calculated explicitly from (2.13) and (2.14). The results are as follows, for both u + and u -,

y ′ + (0) = k 4 1 -3k 3 1 k 2 + 3k 2 1 k 2 2 -3k 1 k 3 2 + k 4 2 k 2 1 -3k 1 k 2 + k 2 2 , y ′ -(0) = k 4 1 + 3k 3 1 k 2 + 3k 2 1 k 2 2 + 3k 1 k 3 2 + k 4 2 k 2 1 + 3k 1 k 2 + k 2 2 .
For u -, where there is always a maximum at x = 0 at the moment of interaction, the maximum is moving with a posistive speed. What that speed represents is not entirely clear. In the case of u + , where the midpoint is a local maximum only if k 2 /k 1 > (3 + √ 5)/2, we see that for these values of 0 < k 1 < k 2 , we have indeed y ′ + (0) > 0. On the other hand, y ′ + (0) < 0 for at least some values of 0 < k 1 < k 2 with k 2 /k 1 < (3 + √ 5)/2. (The lower bound on k 2 /k 1 for which this speed is negative is around 2.15.) This negative speed represents the speed of the local minimum, between the two maxima. It is curious that in some cases this minimum is moving backwards.

It is also interesting to do this for the two-soliton solution of the KdVequation, a calculation which was not carried out in [START_REF] Bona | Pole dynamics of interacting solitons and blowup of complex-valued solutions of KdV[END_REF]. In this case, we obtain

y ′ (0) = k 4 1 + 2k 2 1 k 2 2 -k 4 2 3k 2 1 -k 2 2 .
As is known, the two-soliton solution of the KdV-equation has one maximum at the interaction time if k 2 /k 1 > √ 3 and two maxima, symmetrically located about the interaction center, if 1 < k 2 /k 1 < √ 3. Here we see that y ′ (0) < 0 for 1 + √ 2 < k 2 /k 1 < √ 3. In these case, of course, it is the minimum which is moving backward.

it transpires that r 2 -

 2 r 1 = 4k for some k ∈ Z. It follows that d = gcd(r 1 , r 2 ) divides 4k and therefore d divides k since d is necessarily an odd integer. Define p 1 and p 2 by k 2 /k 1 = p 2 /p 1 with gcd(p 1 , p 2 ) = 1. With this definition, p 2 -p 1 = (r 2 -r 1 )/d = 4k/d ∈ 4N. There are precisely two such complex numbers in the fundamental strip S = {x ∈ C, -λπ < Im x < λπ}, and they are

F 4 e 250t y 10 - 5 2 e 126t y 6 + 9 4 e

 44 -(x, t) = e 252t y 12 + 9 2t y 2 + 1,


Yu-Jun Cui

Anh-Minh Tang

Li-Xin Qian

Wei-Min Ye

Bao Chen

Behaviour of GMZ bentonite 1 THERMAL-MECHANICAL BEHAVIOR OF COMPACTED GMZ BENTONITE

Keywords: high-level radioactive waste disposals, compacted GMZ bentonite, suction, temperature, thermo-mechanical behavior, comparison (IGC: D05/D08)

The THM behavior of compacted GMZ bentonite has been investigated using a suction-temperature controlled isotropic cell. The results obtained were compared with the existing results on other reference bentonites (MX80, FEBEX, FoCa, and Kunigel-V1). It has been observed that the coefficient of thermal expansion of the compacted GMZ bentonite is 2

x 10 -4 °C-1 , similar to the values of compacted MX80 and FEBEX bentonites. The heating tests of the GMZ bentonite also show that the suction is an important parameter that governs the thermal volumetric behavior of unsaturated soils. Unlike temperature, suction has a significant effect on the compressibility parameters. Examination of the mineralogy of various bentonites showed that a good correlation can be generally established between the montmorillonite content and the cations exchange capacity (CEC) or the specific surface area (S). Nevertheless, both the basic geotechnical properties and the swelling potential seem to depend not only on the montmorillonite content but also on other factors such as the nature of base exchangeable cations. The quartz content of the GMZ bentonite is relatively high (11.7%). This could explain its relatively large values of thermal conductivity.

Introduction

Considering the key role played by the engineered barrier systems (EBS) in deep radioactive waste disposals, notably related to i) isolation functions, ii) containment functions, and iii) retrievability, compacted bentonites are often considered the best candidates for engineered barrier materials. As an assessment of the EBS performance under complex thermo-hydro-mechanical (THM) loadings is crucial in demonstrating the long-term safety of disposal systems, large investigations are generally needed for a selected bentonite. The experiments in these investigations must cover various scales, from small-scale elementary laboratory tests through to mid-scale to large-scale tests in underground research facilities.

Various reference bentonites have been studied in various countries for this purpose:

Kunigel-V1 is used as the reference material for all the clay-based components in the repository concepts in Japan [START_REF] Sugita | The Japanese approach to developing clay-based repository concepts -An example of design studies for the assessment of sealing strategies[END_REF]; MX80 was used in the Swedish prototype repository project [START_REF] Johannesson | Prototype repository: A full scale experiment at Äspö HRL[END_REF]; the FoCa clay from the Paris Basin and the Serrata clay from Spain (which is equally known as the FEBEX bentonite [START_REF] Lloret | Advances on the knowledge of the thermo-hydro-mechanical behaviour of heavily compacted "FEBEX" bentonite[END_REF] were considered in the RESEAL project performed in Belgium [START_REF] Van Geet | Efficiency of a borehole seal by means of pre-compacted bentonite blocks[END_REF].

The preliminary long-term plan for the implementation of China's high-level radioactive waste repository [START_REF] Wang | Deep geological disposal of high-level radioactive wastes in China[END_REF] suggests that a high-level radioactive waste repository will be built in the middle of the 21 st Century. In the Chinese concept of geological disposal, bentonite has been selected as the buffer/backfill material. The Gaomiaozi (GMZ) Na-bentonite taken from a large-scale deposit located in the North Chinese Inner Mongolia Autonomous Region (300 km northwest of Beijing) has been chosen for this purpose. After [START_REF] Wang | Deep geological disposal of high-level radioactive wastes in China[END_REF], preliminary research conducted on the swelling, mechanical, hydraulic, and thermal properties have shown that the GMZ bentonite is a good buffer/backfill material. Indeed, as reported by [START_REF] Wang | Deep geological disposal of high-level radioactive wastes in China[END_REF], it has relatively high thermal conductivity (K = 1.51 W/mK at a dry density of 1.6 Mg/m 3 and a water content of 26.7%), quite low water permeability (at saturated state, k = 1.94 x 10 -13 m/s at a dry density of 1.6 Mg/m 3 and a temperature of 25 °C), a relatively high unconfined compression strength (1.74 MPa at a dry density of 1.6 Mg/m 3 and a water content of 23.6%), and quite a high swelling pressure (3.17 MPa at a dry density of 1.6 Mg/m 3 ). [START_REF] Chen | Soil-water characteristic curves of Gaomiaozi bentonite[END_REF] completed the experimental investigation by determining the water retention curves of the GMZ bentonite and showed its high retention capacity which is necessary for ensuring the containment function of the EBS.

Though sufficient preliminary experimental data have been obtained, allowing the candidature of the GMZ bentonite to be retained, there is still a great need of experimental data needed in terms of THM coupling behavior in order to confirm this candidature. In this regard, there have been some results allowing a relevant analysis. In one study, Ye et al. (2009) determined the unsaturated hydraulic conductivity of the GMZ bentonite under constant volume conditions and they observed a strong coupling between water flux and mechanical confining: the hydraulic conductivity determined was found first to decrease and then increase with decreasing suction. The decrease can be attributed to the large pore clogging due to soft gel creation by the exfoliation process. This coupling phenomenon was also observed on a mixture of the Kunigel-V1/Hostun sand [START_REF] Cui | Determining the unsaturated hydraulic conductivity of a compacted bentonite-sand mixture under confined and free-swell conditions[END_REF]). In the present work, a complementary study was performed to further investigate the coupled thermo-mechanical behavior of the GMZ bentonite. The suction-temperature controlled isotropic cell developed by [START_REF] Tang | A new isotropic cell for studying the thermo-mechanical behavior of unsaturated expansive soil[END_REF] was used for this purpose. The results obtained cover the swelling upon wetting, the volumetric changes upon heating, and the compressibility at controlled suction and temperature. Furthermore, the properties found for GMZ bentonite in this investigation were compared with those of other reference bentonites available in the literature.

Materials and methods

Materials

Table 1 presents the mineralogical properties of the GMZ bentonite, as well as other reference bentonites: the Kunigel-V1, FoCa, MX80 and FEBEX. These bentonites contain mainly montmorillonite, which is an essential mineral to ensure sealing properties. Beside the montmorillonite, there is quartz, which is also an important mineral for its particular influence on thermal conductivity (Tang et al., 2008b). Note that a high thermal conductivity of the engineered barrier allows the heat to dissipate quickly from the canisters, consequently reducing the maximum temperature in the EBS. This is one of the reasons why sand/bentonite mixtures are equally studied as buffer material in the repository concepts of Canada [START_REF] Martino | The tunnel sealing experiment: An international study of full-scale seals[END_REF] and Japan [START_REF] Sugita | The Japanese approach to developing clay-based repository concepts -An example of design studies for the assessment of sealing strategies[END_REF].

The physical properties of GMZ bentonite are presented in Table 2. Compared to other bentonites, GMZ bentonite has high montmorillonite content, which gives it a high Cations Exchange Capacity (CEC = 77.30 meq/100g), a large plasticity index (I p = 275), and a large specific surface area (S = 570 m 2 /g). Note also that the main base cations are Na and Ca.

Methods

As mentioned previously, the present work aims at investigating the thermo-mechanical behavior of the GMZ bentonite. For the samples preparation, soil powder with an initial water content of 12.2% was firstly compacted in an isotropic cell under a static pressure of 30 MPa.

The dry density after compaction was about 1.70 Mg/m 3 . The compacted sample was then cut and machined to obtain smaller samples of size 80 mm diameter and 10 mm high. All the samples were then put in a sealed box with a relative humidity controlled by a saturated solution of K 2 CO 3 for about one week (see technical details in [START_REF] Delage | The relationship between suction and swelling properties in a heavily compacted unsaturated clay[END_REF]. That permitted an initial suction of 110 MPa to be imposed on the samples. The soil volume changes during this suction initialization have been found to be negligible. That is in agreement with the water retention curve obtained by [START_REF] Chen | Soil-water characteristic curves of Gaomiaozi bentonite[END_REF], showing a suction of about 110 MPa at a water content of about 12% for a sample compacted to a dry density of 1.70 Mg/m 3 .

The isotropic cell developed by [START_REF] Tang | A new isotropic cell for studying the thermo-mechanical behavior of unsaturated expansive soil[END_REF], which enables the study of mechanical properties of compacted expansive soils using simultaneous control of suction and temperature, was used in the present work. The basic scheme of the cell is presented in Fig. 1.

The soil specimen (80 mm in diameter and 10 mm high) was sandwiched between two dry porous stones, both of which were embedded in metallic plates. Small holes (2 mm diameter) were drilled in the lower plate, allowing the moisture exchange between the soil specimen and chamber below the lower plate. A glass cup containing an over-saturated saline solution was placed in the chamber to control the soil suction. A neoprene membrane (1.2 mm thick) covered the soil specimen and the two metallic plates, avoiding any exchange between the confining water in the cell and the soil pore water. A thermocouple installed inside the cell was used to monitor the cell temperature, which was considered equal to the temperature of the soil specimen during the test. The cell was immersed in a temperature-controlled bath within a temperature fluctuation of ± 0.1 °C. A volume/pressure controller was used to apply the confining pressure in the cell. This volume/pressure controller was also used to monitor the volume change of the soil specimen through the volume change of the confining water in the cell. During the test, volume and pressure readings of the volume/pressure controller as well as the cell temperature were recorded by a computer.

Following the tests carried out by Tang et al. (2008a) on compacted MX80 bentonite, thermal loadings were applied over a short duration of less than 24 h, and the soil suction was assumed to remain unchanged within this short time. The soil suction was then equal to the total suction imposed by the saturated salt solution. [START_REF] Tang | Controlling suction by the vapour equilibrium technique at different temperatures and its application in determining the water retention properties of MX80 clay[END_REF] measured the suctions generated by saturated salt solutions at different temperatures, and these results were applied in this investigation to determine the imposed suction at a given temperature. The saturated salt solution, which filled the glass cup of the cell, was chosen according to the desired values of suction and temperature during the test. Three suctions were considered: 9 MPa (KNO 3 at 25 °C and K 2 SO 4 at 60 °C); 39 MPa (NaCl at 25 °C and 60 °C); and 110MPa (K 2 CO 3 at 25 °C, MgNO 3 at 60 °C).

For tests at suctions lower than the initial suction of 110 MPa, the total suction was initially imposed outside the cell by putting the samples in a sealed box containing saturated KNO 3 solution (9 MPa suction) or NaCl solution (39 MPa suction). The soil sample volume change due to this suction decrease was monitored using a precision caliper. The mass of the soil sample was determined every 3 days until it reached a stable value. The sample was installed inside the isotropic cell only after this suction equilibrium process. Prior to the test, the dimensions of the sample were adjusted to fit the required size (80 mm in diameter and 10 mm high).

A total of six tests were performed (T1 to T6) and their stress paths are presented in Table 3 and Fig. 2, in a space of total suction (s), pressure (p) and temperature (T). During a mechanical compression test the cell pressure was increased in steps from 0.1 to 0.2, 0.5, 1, 2, 5, 10, 20 and 50 MPa. The volume change of water in the volume/pressure controller was recorded for each step. Calibration tests at various temperatures and pressures were performed with a metallic specimen that had the same dimensions as the soil specimen. An example of the determination of soil volume change for a loading step from 1 to 2 MPa (test T1) is shown in Fig. 3. The deformation of the metallic specimen was assumed to be negligible within this range of pressure (lower than 50 MPa) and the water volume change recorded during the calibration corresponds mainly to the deformation of the cell and the tubing. The volume change of the soil sample was then deduced from the difference between the calibration curve and the curve from test, as depicted in Fig. 3. This volume change was used to calculate the volumetric strain of the soil sample upon mechanical loading. Fig. 4 shows the determination of the soil volume change during heating from 25 °C to 60 °C under a constant pressure of 0.1 MPa (test T4). Upon heating, the water in the cell expanded and moved into the volume/pressure controller. According to [START_REF] Romero | Thermo-hydro-mechanical behaviour of heavily overconsolidated clays[END_REF], the thermal expansion coefficient of compacted expansive soil is approximately 10 -4 °C-1 , much higher than the metal specimen (10 -6 °C-1 ) used for the calibration. Thus, it can be reasonably assumed that the thermal expansion of the metal was negligible in comparison with that of the soil. The soil volume change during heating can be then determined from the difference between two curves obtained during the calibration and the test (Fig. 4a). In the case of test T4 while heating from 25 to 60 °C, the soil volume increased by 310 mm 3 (Fig. 4b), which corresponds to a volumetric strain ε v of -0.622%.

Experimental results

Soil volume change during the initial humidification

Fig. 5 shows a typical result of volumetric strain of the soil specimens due to wetting from the initial suction (110 MPa) to 39 and 9 MPa, as well as axial and radial strains. It can be seen that the radial strain is almost equal to the axial strain, showing isotropic behavior in the soil specimen. This observation is consistent with the procedure of compaction under isotropic pressure adopted for the specimen preparation. A similar observation was also made by Tang et al. (2008a). Furthermore, it can be noticed that the total volume of the soil specimen is increased by 32.9% when the suction is decreased to 9 MPa from 110 MPa.

Volume change under thermal loading

As shown in Fig. 2, the initial state was defined by zero pressure, 25 °C temperature and 110 MPa suction. For test T4, the soil specimen in its initial state was confined to a pressure p = 0.1 MPa, and heated to a temperature T = 60 °C under p = 0.1 MPa. For test T5, a decrease of suction from 110 to 39 MPa at T = 25 °C was first undertaken during the initial wetting before the application of the confining pressure p = 0.1 MPa; heating to T = 60 °C then followed. For test T6, the soil specimen was wetted to a suction of 9 MPa and then loaded to p = 0.1 MPa and heated to T = 60 °C. Fig. 6 shows the results for the thermal volume change under p = 0.1 MPa. The results from tests T4 (110 MPa) and T5 (39 MPa) show that heating induces expansion and the coefficients of thermal expansion is α = 2×10 -4 °C-1 . On the contrary, the result from test T6 ( 9MPa) shows a volume decrease upon heating.

Volume change under mechanical loading

As shown in Fig. 2, the mechanical loadings in tests T1-T5 were performed at constant suction and temperature. Two temperatures (25 and 60 °C) and three suctions (110, 39 and 9 MPa) were applied. The results (the void ratio e against the logarithm of pressure logp) for all these tests are presented in Fig. 7. The compressibility parameters, such as the yield pressure p 0 , defined as the pressure at the intersection between the two quasi-linear segments, the elastic compressibility parameter, κ, which is related to the slope of small changes in the void ratio, and the plastic compressibility parameter, λ(s) , which is related to the slope of large changes in void ratio, can be determined from these curves. It is to be noted that the notations defined by [START_REF] Alonso | A constitutive model for partially saturated soils[END_REF] are adopted here for the compressibility parameters.

From Fig. 7a it can be observed that temperature increase from 25 °C to 60 °C raises the void ratio from 0.508 to 0.518 at a pressure of 0.1 MPa and constant suction of 110 MPa. Furthermore, there is no increase in the void ratio due to the temperature change after a pressure of 11 MPa. This indicates thermal expansion due to heating at low pressure which is consistent with the results shown in Fig. 6. The effect of heating on volume change is much smaller than the volume change under subsequent mechanical loading. This phenomenon can be observed again at constant suction of 39 MPa, as depicted in Fig. 7b. Heating from 25 °C to 60 °C slightly increased the void ratio e from 0.754 to 0.768. Again, the volume change due to heating is small compared to that due to subsequent mechanical loading. When comparing the compression curves of T2 (25 °C) and T5 (60 °C), it is observed that the difference is not significant. Moreover, the shape of the compression curve is similar to that obtained for T1 and T4. At 9 MPa suction, only one loading test was performed (T3); a technical problem occurred during test T6. The compression curve for test T3 is presented in Fig. 7c.

Though the temperature effects on λ(s) and κ are found to be negligible, this is not the case for the suction effect. Indeed, Fig. 8 shows that λ(s) and κ increase significantly when the suction is decreased. When the suction is decreased from 110 to 9 MPa, κ increases from 0.023 to 0.036 and λ(s) increases from 0.120 to 0.165. Fig. 9 shows the yield pressures p 0 determined from Fig. 7

Discussions

In this section the thermo-mechanical behavior of the GMZ bentonite identified will be discussed by comparing it with that of other reference bentonites (listed in Table 1) which have been studied worldwide.

Mineralogical properties

Table 1 shows that the montmorillonite content of the GMZ bentonite is lower than that of the MX80 bentonite and the FEBEX bentonite but it is higher than that of the Kunigel-V1 bentonite. The FoCa bentonite contains 80% of interstratified smectite/kaolinite. The quartz content of the Kunigel-V1 and the GMZ bentonites is relatively high: 29-38% and 11.7% respectively (smaller than 6% for the others). Good correlations can be established between the montmorillonite content and the CEC values (Fig. 10a). Indeed, the higher the montmorillonite content, the larger the CEC; the FEBEX bentonite with the highest montmorillonite content has the largest values of CEC. The FoCa clay with an 80% interstratified smectite/kaolinite content is not considered in these correlations. As far as the other properties are concerned (Table 2), it seems that the specific surface area (S), liquid limit (w L ) and the plasticity index (Ip) are not directly related to the montmorillonite content (Fig. 10b,c,d). For instance, the FEBEX bentonite with the highest montmorillonite content has the smallest values of w L and Ip while the MX80 bentonite (79% montmorillonite) has the highest values of w L and Ip. This shows that the montmorillonite content is not the only factor which influences the basic geotechnical properties; other factors, such as the nature of the base cations exchange (see Table 2), must play an important role also.

Thermal conductivity Tang et al. (2008b) studied the thermal conductivity of the compacted MX80, FEBEX and Kunigel bentonites. Tests on the compacted specimens having various dry densities (ranging from 1.45 to 1.85 Mg/m 3 ) and water contents show that the thermal conductivity of bentonites is strongly dependent on these two parameters: at the same water content, the higher the density the higher the thermal conductivity, and at the same dry density, the higher the water content, the higher the thermal conductivity. After testing various models for predicting the thermal conductivity, Tang et al. (2008b) found that a unique relationship can be established between the thermal conductivity K and the air-pore volume fraction V a /V for all these soils, as follows:

sat K K + = /V) (V a α (1)
where α is the slope of K -V a /V plot; K sat is the thermal conductivity at saturated state which corresponds to the intersection of K -V a /V plot with the K axis. Fig. 11 shows the K -V a /V plot for the compacted GMZ bentonite that was drawn using data reported by [START_REF] Wang | Deep geological disposal of high-level radioactive wastes in China[END_REF]. It can be seen that there is also a good linear relationship for this bentonite. From Fig. 11 W/mK). Even though relatively larger values of the thermal conductivity at saturated state (K sat ) for the GMZ and Kunigel-V1 bentonites can be related to their higher quartz contents (see Table 1), it appears that the difference is too small to allow establishing a relevant correlation. Further study is needed to clarify this phenomenon by working on the different bentonites but using the same test protocol and measurement technique.

Swelling behavior

Fig. 5 shows that for the GMZ bentonite compacted at 1.70 Mg/m 3 of dry density (initial suction of 110 MPa), wetting to suction at 9 MPa induced a swelling volumetric strain of 32.9%. The volumetric swelling strain of GMZ is plotted together with that of other soils in Fig. 12. A higher value of 50% was obtained by Tang et al. (2008a) when wetting from a suction of 110 MPa to 9 MPa the MX80 bentonite compacted at 1.78 Mg/m 3 of dry density. [START_REF] Delage | The relationship between suction and swelling properties in a heavily compacted unsaturated clay[END_REF] showed that wetting the compacted FoCa clay (1.85 Mg/m 3 of dry density, initial suction of 110 MPa) to a suction of 9 MPa induced a volumetric strain of 17%. After [START_REF] Enresa | Full-scale engineered barriers experiment for a deep geological repository for high-level waste in crystalline host rock[END_REF], wetting the FEBEX bentonite (initially compacted at 1.67 Mg/m 3 of dry density) from a suction of 110 MPa to a suction of 9 MPa induced a volumetric strain of 19%. [START_REF] Kanno | Moisture adsorption and volume change of partially saturated bentonite buffer materials[END_REF] studied the volume change behavior of the compacted Kunigel V1 (initially compacted at 1.82 Mg/m 3 of dry density) and obtained a volumetric strain of 13% when wetting from 110 to 9 MPa suction. The comparison shows that the swelling potential of the GMZ and the MX80 is higher than that of FoCa, Kunigel-V1 and FEBEX.

This is due to their montmorillonite contents (a higher montmorillonite content gives rise to a higher swelling potential, see [START_REF] Marcial | On the high stress compression of bentonites[END_REF] and the nature of exchangeable basic cations (Na-based bentonites show higher swelling potential than Ca-based bentonites, see [START_REF] Marcial | On the high stress compression of bentonites[END_REF].

Thermal-mechanical volume changes

The coefficient of thermal expansion of the compacted GMZ bentonite deduced from Fig. 6 is 2×10 -4 °C-1 (mean value determined from the thermal expansion curve of tests T4 and T5).

Similar values were obtained for the compacted MX80 bentonite (Tang et al., 2008a) and the compacted FEBEX bentonite [START_REF] Romero | Thermo-hydro-mechanical behaviour of heavily overconsolidated clays[END_REF]. As far as the nature of the volume changes upon heating is concerned, Tang et al. (2008a) observed that heating the compacted MX80 bentonite under 0.1 MPa pressure gave rise to thermal expansion at high suctions (39 and 110 MPa) and to thermal contraction at a lower suction (9 MPa). The same phenomenon was observed in the present work on the GMZ bentonite. This behavior was explained using an elasto-plastic constitutive model proposer by [START_REF] Tang | Modelling the thermo-mechanical behaviour of compacted expansive clays[END_REF]. The thermal expansion corresponds to an elastic behavior when the heating path remains in the elastic zone while the thermal contraction corresponds to a plastic behavior when the heating path under constant stress and suction touch the yield surface.

The compression curves obtained at controlled suction and temperature (Fig. 7) show an elasto-plastic behavior with a clear change in slope. That allowed the determination of the The effects of suction and temperature on κ and λ(s) of GMZ bentonite are similar to that of MX80: κ and λ(s) increase with the decrease of suction but are independent of the temperature changes (between 25 and 60 °C). [START_REF] Lloret | FEBEX II Project. Final report on thermo-hydro-mechanical laboratory tests[END_REF], however, observed that both the increase of temperature and the decrease of suction induced an increase of λ(s) in FEBEX bentonite. Comparing the values of λ(s), GMZ bentonite shows the highest values: λ(s) = 0.12 -0.16 at s = 9 -110 MPa for GMZ bentonite; λ(s) = 0.08 -0.12 at s = 9 -110 MPa for MX80 bentonite; λ(s) = 0.065 -0.080 at s = 8 -14 MPa for FEBEX bentonite (from suction-controlled oedometer tests by [START_REF] Lloret | Mechanical behaviour of heavily compacted bentonite under high suction changes[END_REF].

The effect of temperature on the yield stress p 0 of GMZ bentonite and MX80 bentonite (see Tang et al. 2008a) is similar; only a slight decrease of p 0 was observed for high suction values (s = 110 and 39 MPa). In contrast, for FEBEX bentonite, at s = 120 MPa, [START_REF] Lloret | FEBEX II Project. Final report on thermo-hydro-mechanical laboratory tests[END_REF] showed that p 0 decreased drastically from 20 MPa (at 20°C) to 3 MPa (at 50 °C).

In order to compare the effect of suction decrease on the yield stress, the results of three bentonites (GMZ, MX80 and FEBEX) are plotted together in Fig. 13. At high suctions (s = 110 -127 MPa), the yield stresses of the three soils are close to 20 MPa. Upon wetting, the yield stress decreases in similar pattern for GMZ bentonite and FEBEX bentonite but at a lower rate than MX80 bentonite: at s = 9 MPa, p 0 of GMZ and FEBEX bentonites decreases to 2 MPa while that of MX80 is much lower (0.38 MPa).

Conclusions

The thermo-mechanical behavior of GMZ bentonite, which has been selected as a possible material for the engineered barriers in the high-level radioactive waste disposals in China, was experimentally investigated. The results obtained were compared with other bentonites which have been widely investigated as reference materials in the projects carried out in other countries (MX80, FEBEX, FoCa, and Kunigel-V1). The following conclusions can be drawn from the present study:

i)

The montmorillonite content of GMZ bentonite is lower than that of MX80 bentonite and FEBEX bentonite but it is higher than that of Kunigel-V1 bentonite. That explains the large values of its cations exchange capacity (CEC) and specific surface area S. In general, good correlations can be established between the montmorillonite content and the CEC values or the specific surface area S (Table 2). Nevertheless, no direct correlation can be made between the montmorillonite content and the basic geotechnical properties; other factors, such as the nature of base exchangeable cations, also appear to have a significant influence.

ii) The same observation has been made in terms of swelling potential. In general, the higher the montmorillonite content, the higher the swelling potential. However, a

Ca-based bentonite generally shows lower swelling potential than a Na-based bentonite.

For GMZ bentonite it was observed that wetting (suction decreased from 110 to 9 MPa) induced a swelling volumetric strain of 30%. That is lower than MX80 (50%) and higher than that of other bentonites (less than 20%).

iii) The quartz content of GMZ bentonite is relatively high (11.7%) just behind Kunigel V1 bentonite (29-38%). This could explain their relatively large values of thermal conductivity. Note that compared to the values for other bentonites, the difference is too small to allow for a relevant correlation to be drawn between their thermal conductivity and the quartz content.

iv) The coefficient of thermal expansion of the compacted GMZ bentonite is 2.10 -4 °C-1 , that is similar to the values obtained for the compacted MX80 bentonite (Tang et al., 2008a) and the compacted FEBEX bentonite [START_REF] Romero | Thermo-hydro-mechanical behaviour of heavily overconsolidated clays[END_REF]. As the compacted MX 80 bentonite, the GMZ bentonite also showed a thermal expansion upon heating at high suctions (39 and 110 MPa) and a thermal contraction at lower suction (9 MPa).

v) The effect of suction and temperature on κ and λ(s) of GMZ bentonite is similar to MX80: κ and λ(s) increase with decreasing suction but are independent of the temperature changes. Compared to other bentonites, GMZ bentonite has the highest values of λ(s): 0.12 -0.16 at s = 9 -110 MPa.

vi) The effect of temperature on the yield stress p 0 of the GMZ bentonite has been found to be insignificant. A similar observation was made for MX80 bentonite by Tang et al. (2008a). In contrast, a significant suction effect was identified. As is also the case for MX80 bentonite and FEBEX bentonite, the yield stress p 0 of the GMZ bentonite decreased significantly, at a rate similar to that of FEBEX bentonite but lower than that of MX80 bentonite. Table 1. Mineral composition of some bentonites Table 2. Physical properties of some bentonites Table 3. Stress paths of thermo-mechanical tests

List of Tables

List of Figures

  at various suctions and temperatures. It can be seen that p 0 decreases from 21 MPa at the initial suction s = 110 MPa to 10 MPa at s = 39 MPa and 2 MPa at s = 9 MPa for the tests at 25 °C. As far as the temperature effect is concerned, at s = 110 MPa, heating from T= 25 °C to 60 °C decreases p 0 from 21 to 16 MPa. At s = 39 MPa heating decreases p 0 from 10 to 7.3 MPa.

  compressibility parameters: κ, λ(s) and p 0 . Similar observations were made byTang et al. (2008a) on the MX80 bentonite and by[START_REF] Lloret | FEBEX II Project. Final report on thermo-hydro-mechanical laboratory tests[END_REF] on the FEBEX bentonite. The plastic behavior in compacted soil is usually associated with the collapse of macro-pores that can be evidenced by MIP (Mercury Intrusion Porosimetry) test as in the works of[START_REF] Lloret | Mechanical behaviour of heavily compacted bentonite under high suction changes[END_REF],Tang &[START_REF] Tang | Modelling the thermo-mechanical behaviour of compacted expansive clays[END_REF][START_REF] Tang | A study of the hydro-mechanical behaviour of compacted crushed argillite[END_REF]. However,[START_REF] Cui | A model for the volume change behaviour of heavily compacted swelling clays[END_REF] performed suction-controlled isotropic compression tests on the highly compacted FoCa clay (compacted initially at 1.85 Mg/m 3 of dry density) and observed a reversible volume change behavior in a range of pressures up to 60 MPa. The authors explained this by the absence of collapsible macro-pores in this material. That mean the volume change behavior of compacted clays can be strongly dependent on the dry density; the higher the dry density (or the lower the volume of macro-pores) the higher the yield stress. Note that the initial dry densities of bentonite in other studies are lower than in[START_REF] Cui | A thermo-mechanical model for saturated clays[END_REF]: 1.70 Mg/m 3 for FEBEX bentonite in[START_REF] Lloret | Mechanical behaviour of heavily compacted bentonite under high suction changes[END_REF]; 1.78 Mg/m 3 for MX80 bentonite in[START_REF] Tang | Modelling the thermo-mechanical behaviour of compacted expansive clays[END_REF]. In the study of[START_REF] Tang | A study of the hydro-mechanical behaviour of compacted crushed argillite[END_REF], even at a high dry density of 2.00 Mg/m 3 , the volume of macro-pores measured by MIP tests is still high. It shows that isotropically compacting GMZ bentonite at 30 MPa pressure (to a dry density of 1.70 Mg/m 3 ) can not avoid macro-pores. Further study is necessary to clarify this point.

Fig. 1 .

 1 Fig. 1. Basic scheme of the suction-temperature controlled isotropic cell (after Tang et al., 2007) Fig. 2. Stress paths of tests T1-T6 Fig. 3. Determination of soil volume change during loading step from 1 to 2 MPa Fig. 4. Determination of soil volume change during heating under confining pressure of 0.1 MPa Fig. 5. Axial, radial and volumetric strain during wetting Fig. 6. Volumetric strain during thermal loading under constant pressure at 0.1 MPa: tests T4-T6 Fig. 7. Results of mechanical loading at constant suction and temperature: tests T1-T6 Fig. 8. Compressibility parameters κ and λ(s) versus suction for tests at 25°C Fig. 9. Yield pressure p 0 versus temperature Fig. 10. Correlation between the montmorillonite content with (a) Cations Exchange Capacity, CEC; (b) specifice surface area, S; (c) Liquid limit, w L ; and (d) Plasticity index, I P . Fig. 11. Variations of thermal conductivity with air-volume fraction for GMZ bentonite (after Wen, 2006) Fig. 12. Swelling of various compacted bentonite from a initial suction of 110 MPa Fig. 13. Relationship between yield stress and the applied suction for various bentonites

Fig. 13 .

 13 Fig. 1. Basic scheme of the suction-temperature controlled isotropic cell (after Tang et al., 2007)

  the parameters of Eq. [1] deduced are as follows:

	α	=	-	2	.	81	W/mK and	K	sat	=	1	.	57	W/mK
					These values are slightly larger than those for other bentonites: FEBEX (	α	=	-	2	.	29	W/mK
	and			K		sat	=	1	.	30	W/mK), MX80 ( α = -1.79 W/mK and sat K = 1.1 W/mK), Kunigel V1
	(	α	=	-	2	.	36	W/mK and	=	1	.	39

sat K

  Mechanics and Engineering, 25 (4): 649-658. Wen, Z. J. 2006. Physical property of China's buffer material for high-level radioactive waste repositories. Chinese Journal of Rock Mechanics and Engineering, 25 (4): 794-800. Ye, W.M., Cui, Y.J., Qian, L.X. and Chen, B. 2009. An experimental study of the water transfer through compacted GMZ bentonite. Engineering Geology 108, 169 -176.

Table 1 . Mineral composition of some bentonites

 1

	Mineral	Kunigel V1 a FoCa b	MX80 c FEBEX d GMZ e
	Montmorillonite (%)	46-49	80% (interstratified smectite/kaolinite)	79	92±3	75.4
	Plagioclase (%)	-	-	9.2	2±1	-
	Pyrite (%)	0.5-0.7	-	0.-	0.02±0.01 -
	Calcite (%)	2.1-2.6	1.4	0.8	Traces	0.5
	Dolomite (%)	2.0-2.8	-	-	0.60±0.13 -
	Gypsum (%)	-	0.4	-	0.14±0.01 -
	Halite (%)	-	-	-	0.13±0.02 -
	Analcite (%)	3.0-3.5	-	-	-	-
	Mica (%)	-	-	<1	-	-
	Feldspar (%)	2.7-5.5	-	2.0	Traces	4.3
	Cristobalite (%)	-	-	-	2±1	7.3
	Kaolinite (%)	-	4	-	-	0.8
	Quartz (%)	29-38	6	2.8	2±1	11.7
	Field organic	0.31-0.34	-	0.1	0.35±0.05 -
	a JNC (2000); b Guillot et al. (2002); c Montes-H. et al.(2003); d ENRESA (2000); e Wen
	(2006).

Table 2 . Physical properties of some bentonites

 2 FoCa b MX80 c FEBEX d GMZ e

	Parameter Kunigel V1 a particle < 2µm (%) 64.5	-	60	68	60
	CEC(meq/100g) 73.2	54	82.3	102 j	77.30
	Base cations exchange	Na-Ca	Ca	Na	Ca-Mg	Na-Ca
	w L (%)	474	112	519	102	313
	w P (%)	27	50	35	53	38
	I P	447	62	484	49	275
	ρ s (Mg/m 3 )	2.79	2.67	2.76	2.70	2.66
	S (m 2 /g)	687	300	522	725	570
	a Komine (2004); b Marcial et al. (2002); c Tang and Cui (2005); d ENRESA (2000); e Wen
	(2006).

Table 3 . Stress paths of thermo-mechanical tests

 3

Acknowledgements

The authors are grateful to the National Natural Science Foundation of China (No 40728003 and No 41030748), Kwang-Hua Fund for College of Civil Engineering at Tongji University, and Program for Changjiang Scholars and Innovative Research Team in University (PCSIRT, IRT1029).