Nabil Kahouadji

Niky Kamran

Keti Tenenblat

SECOND-ORDER EQUATIONS AND LOCAL ISOMETRIC IMMERSIONS OF PSEUDO-SPHERICAL SURFACES

Keywords: evolution equations, nonlinear hyperbolic equations, pseudo-spherical surfaces, isometric immersions. MSC 2010: 35L60, 37K25, 47J35, 53B10, 53B25

. We answer the following question: Given a pseudo-spherical surface determined by a solution u of such an equation, do the coefficients of the second fundamental form of the local isometric immersion in R 3 depend on a jet of finite order of u? We show that, except for the sine-Gordon equation, where the coefficients depend on a jet of order zero, for all other differential equations, whenever such an immersion exists, the coefficients are universal functions of x and t, independent of u.

Introduction

The class of partial differential equations describing pseudo-spherical surfaces, which has been defined and studied in depth in a foundational paper by Chern and Tenenblat [START_REF] Chern | Pseudospherical surfaces and evolution equations[END_REF], contains a large subclass of equations enjoying remarkable integrability properties, such as the existence of infinite hierarchies of conservation laws, Bäcklund transformations and associated linear problems. Recall that a partial differential equation ∆(t, x, u, ∂u ∂x , ∂u ∂t , . . . , ∂ k u ∂t l ∂x k-l ) = 0,

is said to describe pseudo-spherical surfaces if there exist 1-forms

ω i = f i1 dx + f i2 dt, 1 ≤ i ≤ 3, (2)
where the coefficients f ij , 1 ≤ i ≤ 3, 1 ≤ j ≤ 2, are smooth functions of t, x, u and finitely many derivatives of u with respect to t and x, such that the structure equations

dω 1 = ω 3 ∧ ω 2 , dω 2 = ω 1 ∧ ω 3 , dω 3 = ω 1 ∧ ω 2 (3)
hold if, and only if, u is a solution of (1) for which ω 1 ∧ ω 2 = 0. In other words, every smooth solution of an equation (1) describing pseudo-spherical surfaces defines on its domain U ⊂ R 2 a Riemannian metric ds 2 = (ω 1 ) 2 + (ω 2 ) 2 , (4) of constant Gaussian curvature equal to -1, with ω 3 being the Levi-Civita connection 1-form of the metric [START_REF] Ding | On differential systems describing surfaces of constant curvature[END_REF].

One of the most important examples of a partial differential equation describing pseudospherical surfaces is the sine-Gordon equation

∂ 2 u ∂t∂x = sin u, (5)
for which a choice of 1-forms (2) satisfying the structure equations ( 3) is given by

ω 1 = cos u 2 (dx + dt), (6)
ω 2 = sin u 2 (dx -dt), (7)
ω 3 = u x 2 dx - u t 2 dt. (8)
It should be noted that this choice of 1-forms is by no means unique. In particular, we could also have used

ω 1 = 1 η sin u dt, (9)
ω 2 = η dx + 1 η cos u dt, (10)
ω 3 = u x dx, ( 11
)
where η is a continuous non-vanishing real parameter. This continuous parameter is closely related to the parameter appearing in the classical Bäcklund transformation for the sine-Gordon equation and accounts for the existence of infinitely many conservation laws for the sine-Gordon equation. More generally, partial differential equations (1) which describe pseudo-spherical surfaces and for which one of the components f ij (say f 21 ) can be chosen to be a continuous parameter will be said to describe η pseudo-spherical surfaces.

In [START_REF] Chern | Pseudospherical surfaces and evolution equations[END_REF], Chern and Tenenblat provided a complete classification of the evolution equations of the form u t = F (u, u x , ..., ∂u/∂x k ),

which describe pseudo-spherical surfaces under the assumption that f 21 = η, where η is a real parameter, providing an extensive class of non-linear partial differential equations, in two independent variables, describing pseudo-spherical surfaces. Rabelo in [START_REF] Rabelo | A characterization of differential equations of type uxt = F (u, ∂u/∂x, ..., ∂ k /∂x k ) which describe pseudospherical surfaces, An[END_REF], [START_REF] Rabelo | On equations which describe pseudospherical surfaces[END_REF] characterized equations of the form u xt = F (u, u x , ..., ∂u/∂x k ), with f 21 = η. The complete classification for equations of type u xt = F (u, u x ) and u t = u xxx + G(u, u x , u xx ) was given in [START_REF] Rabelo | On equations of the type uxt = F (u, ux) which describe pseudospherical surfaces[END_REF] and [START_REF] Rabelo | A classification of pseudospherical surfaces equations of the type ut = uxxx + G(u, ux, uxx) which describe pseudospherical surfaces[END_REF], respectively.

In general, the importance of the class of differential equations that describe pseudo-spherical surfaces is due to the fact that such a differential equation is always the integrability condition of a linear system of differential equations, which may be used in the inverse scattering method to solve the differential equation (see for example [START_REF] Beals | Bäcklund transformations and inverse scattering solutions for some pseudospherical surface equations[END_REF], where the method was applied to a subclass of equations obtained in [START_REF] Rabelo | On equations which describe pseudospherical surfaces[END_REF]). While the assumption of f 21 = η is natural in the context of the inverse scattering method, the problem of classifying the differential equations describing pseudo-spherical surfaces, without any other assumption, is important in its own right and was considered by Kamran and Tenenblat in [START_REF] Kamran | On differential equations describing pseudo-spherical surfaces[END_REF], where one can find a complete classification of evolution equations of the form [START_REF] Rabelo | On equations of the type uxt = F (u, ux) which describe pseudospherical surfaces[END_REF] which describe pseudo-spherical surfaces, as opposed to η pseudo-spherical surfaces. These results provide a systematic way of verifying if a given differential equation of this type describes pseudo-spherical surfaces. The results obtained in [START_REF] Kamran | On differential equations describing pseudo-spherical surfaces[END_REF] were extended by Reyes in [START_REF] Reyes | Pseudo-spherical surfaces and integrability of evolution equations[END_REF] to differential equations of the form u t = F (x, t, u, u x , ..., ∂u/∂x k ). The concept of a differential equation that describes pseudo-spherical surfaces was extended by Ding and Tenenblat in [START_REF] Ding | On differential systems describing surfaces of constant curvature[END_REF] to a system of differential equations that describes constant curvature surfaces (pseudo-spherical and also spherical), where classification results for such systems were obtained. More recently, in order to determine new classes of differential equations that describe pseudo-spherical surfaces, as a consequence of [START_REF] Kamran | On differential equations describing pseudo-spherical surfaces[END_REF], assuming that f 21 and f 31 are linear combinations of f 11 , Gomes [START_REF] Gomes Neto | Fifth-order evolution equations describing pseudospherical surfaces[END_REF] classified and obtained large new classes of such equations by considering fifth order equations of type [START_REF] Rabelo | On equations of the type uxt = F (u, ux) which describe pseudospherical surfaces[END_REF].

We should point out that the classification results mentioned above, contain not only general statements, but also examples of interesting new and well kown non linear differential equations. Other aspects of the theory of differential equations which describe pseudo-spherical surfaces and its applications thereof can be found in [START_REF] Cavalcante | Conservation laws for nonlinear evolution equations[END_REF], [START_REF] Górka | The modified Hunter-Saxton equation[END_REF], [START_REF] Jorge | Linear problems associated to evolution equations of type utt = F (u, ux, uxx, ut)[END_REF], [START_REF] Foursov | On formal integrability of evolution equations and local geometry of surfaces[END_REF], [START_REF] Reyes | Pseudo-spherical surfaces and integrability of evolution equations[END_REF]- [START_REF] Reyes | Correspondence theorems for hierarchies of equations of pseudo-spherical type[END_REF].

A classical theorem in the theory of surfaces states that any pseudo-spherical surface can be locally isometrically immersed into three-dimensional Euclidean space E 3 . This result can thus be applied to the metrics arising from the solutions u of any partial differential equation (1) describing pseudo-spherical surfaces, thereby associating to any solution u a local isometric immersion of a metric with constant Gaussian curvature equal to -1. This theorem is however largely an existence result, which does not give an explicit expression for the second fundamental form of the local isometric immersion. It is therefore a most remarkable property of the sine-Gordon equation that the second fundamental form of any such immersion can be expressed in closed form as a function of u and finitely many derivatives. Indeed, let us first recall that the components a, b, c of the second fundamental form of any local isometric immersion of a metric of constant curvature equal to -1 into E 3 are defined by the 1-forms ω 3 1 , ω 3 2 according to

ω 3 1 = aω 1 + bω 2 , ω 3 2 = bω 1 + cω 2 , ( 13
)
where these forms satisfy the structure equations

dω 3 1 = -ω 3 2 ∧ ω 3 , dω 3 2 = ω 3 1 ∧ ω 3 , (14)
and the Gauss equation ac -b 2 = -1. For the sine-Gordon equation, with the choice of 1-forms ω 1 , ω 2 and ω 3 given by ( 6), ( 7) and [START_REF] Kamran | On differential equations describing pseudo-spherical surfaces[END_REF], it is easily verified that the 1-forms ω 3 1 , ω 3 2 are given by

ω 3 1 = sin u 2 (dx + dt) = tan u 2 ω 1 , ω 3 2 = -cos u 2 (dx -dt) = -cot u 2 ω 2 .
In general, given a partial differential equation (1) describing pseudo-spherical surfaces, it is straightforward to derive a set of necessary and sufficient conditions, in terms of the coefficients f ij of the 1-forms (2), for a, b and c to be the components of the second fundamental form of a local isometric immersion corresponding to a solution of (1). We write

dω 3 1 = db(e 1 ) -da(e 2 ) ω 1 ∧ ω 2 -a ω 2 ∧ ω 3 + b ω 1 ∧ ω 3 , (15)
dω 3 2 = dc(e 1 ) -db(e 2 ) ω 1 ∧ ω 2 -b ω 2 ∧ ω 3 + c ω 1 ∧ ω 3 , (16)
where (e 1 , e 2 ) is the pair of vector fields dual to the coframe (ω 1 , ω 2 ), given by

f 11 f 21 f 12 f 22 e 1 = f 22 ∂ x -f 21 ∂ t , f 11 f 21 f 12 f 22 e 2 = -f 12 ∂ x + f 11 ∂ t .
Thus, using the notation D t and D x for the total derivative operators, we obtain

f 11 D t a + f 21 D t b -f 12 D x a -f 22 D x b -2b f 11 f 31 f 12 f 32 + (a -c) f 21 f 31 f 22 f 32 = 0, (17)
f 11 D t b + f 21 D t c -f 12 D x b -f 22 D x c + (a -c) f 11 f 31 f 12 f 32 + 2b f 21 f 31 f 22 f 32 = 0, (18)
where a, b and c, which are assumed to depend on t, x, u and finitely many derivatives of u with respect to t and x, satisfy the Gauss equation

ac -b 2 = -1. (19)
In view of the above discussion, it is is natural to ask the following question: Do there exist equations other than the sine-Gordon equation within the class of partial differential equations describing pseudo-spherical (or η pseudo-spherical) surfaces, for which the components a, b, c of the second fundamental form of the local isometric immersion depend on a jet of finite order of u, that is on x, t, u and finitely many derivatives of u?

If such equations were to exist, they would have an important geometric property in common with the sine-Gordon equation. In this paper, we give a complete answer to the above question in the case of second-order hyperbolic equations of the form

u xt = F (u, u x ), (20)
and evolution equations of the form

u t = F (u, u x , u xx ), (21)
which describe η pseudo-spherical surfaces as in [START_REF] Rabelo | On equations of the type uxt = F (u, ux) which describe pseudospherical surfaces[END_REF] and [START_REF] Chern | Pseudospherical surfaces and evolution equations[END_REF] .

We begin with the case of evolution equations ( 21), for which our main result is the following:

Theorem 1. Except for second-order evolution equations of the form

u t = f 12,ux f 11,u u xx + f 12,u f 11,u u x ∓ λf 11 -ηf 12 f 11,u , (22)
where f 11,u = 0 and f 12,ux = 0, there exists no second-order evolution equation describing η pseudo-spherical surfaces, given as in [START_REF] Chern | Pseudospherical surfaces and evolution equations[END_REF], with the property that the coefficients of the second fundamental forms of the local isometric immersions of the surfaces associated to the solutions u of the equation depend on a jet of finite order of u. Moreover, the coefficients of the second fundamental forms of the local isometric immersions of the surfaces determined by the solutions u of ( 22) are universal, i.e., they are universal functions of x and t, independent of u.

Theorem 1 suggests that there is no real analogue of the sine-Gordon equation within the class of second-order evolution equations describing η pseudo-spherical surfaces, from the perspective of the local isometric immersions of pseudo-spherical surfaces associated to their solutions. Indeed, even though the special class of evolution equations (22) has the property that the components of the second fundamental forms of the immersions associated to its solutions depend on jets of finite order of u, this dependence is given in terms of functions of x and t for all choices of solutions u (see Proposition 2).

The results for second-order hyperbolic equations (20) are similar, with the notable exception that they single out the sine-Gordon equation as the only equation, up to constants, for which the second fundamental form of the local isometric immersion is not universal. In order to state these results, we begin by recalling the classification theorem proved by Rabelo and Tenenblat [START_REF] Rabelo | On equations of the type uxt = F (u, ux) which describe pseudospherical surfaces[END_REF] for equations (20) describing pseudo-spherical surfaces:

Theorem 2 (Rabelo & Tenenblat [START_REF] Rabelo | On equations of the type uxt = F (u, ux) which describe pseudospherical surfaces[END_REF]). Let F be a differentiable function defined on an open connected subset U ⊂ R 2 . An equation

u xt = F (u, u x )
describes an η pseudo-spherical surface for η ∈ P ⊂ R, where P is a dense subset of R and F independent of η if, and only if, F satisfies one of the following: i) F is independent of u x and F ′′ (u) + αF (u) = 0, U = R 2 , P = R \ {0}, and α is a non-zero real constant. ii) F = νe δu β + γu 2

x , where U = {(u, z) ∈ R 2 ; β + γz 2 > 0}, P = R, δ, γ, β, ν are real constants, with δ, γ, ν nonzero, and β = 0 when γ = 1; or iii) F = λu + ζu x + τ , where U = R 2 , P = R \ {0}, and λ, ζ, τ are real constants.

The expressions of functions f ij of the 1-forms ω i for each equation of Theorem 2 are recalled in Section 4 (Lemmas 6-8). We are now ready to state our main result for the case of second-order hyperbolic equations (20). Theorem 3. Let F be an equation of the form u xt = F (u, u x ) that describes η pseudo-spherical surfaces as in [START_REF] Rabelo | On equations of the type uxt = F (u, ux) which describe pseudospherical surfaces[END_REF].

(1) If F is independent of u x and satisfies F ′′ (u) + αF (u) = 0, where α is a positive real constant, then there exists a local isometric immersion in R 3 of the pseudo-spherical surface determined by a solution u, for which the coefficients of the second fundamental form depend on a jet of finite order of u if, and only if, they depend on the jet of order zero.

(2) If F = λu + ζu x + τ , then there exists a local isometric immersion in R 3 of the pseudospherical surface determined by a solution u, for which the coefficients of the second fundamental form depend on a jet of finite order of u if, and only if, λ, ξ and τ do not vanish simultaneously, and the coefficients are independent of u, that is they are universal functions of x and t. (3) For the remaining equations, that is, if F is independent of u x and satisfies F ′′ (u) + αF (u) = 0, where α is a negative real constant, F = νe δu β + γu 2

x and F = 0, there is no local isometric immersion of the pseudo-spherical surface determined by a solution u, for which the coefficients of the second fundamental form depend on a jet of finite order of u.

The coefficients of the second fundamental form of the local isometric immersions stated in Theorem 3 are given explicitly in Section 4 (Propositions 3 and 5). Theorem 3 shows likewise that when viewed through the perspective of the local isometric immersions associated to its solutions, the sine-Gordon equation occupies a special position within the class of hyperbolic equations (20) as the unique equation, up to normalization constants, for which the coefficients of the second fundamental form of the local isometric immersion of the surface determined by a solution u, depends on a jet of finite order of u, without being universal, i.e. independent of u.

While Theorems 1 and 3 give a complete answer to the general question we have raised in this paper in the case of second-order evolution equations (21) and second-order hyperbolic equations (20), the question still remains open for all the other classes of equations describing pseudo-spherical surfaces. We believe that it should be possible to extend the proof of Theorem 1 to the case of k-th order evolution equations with k ≥ 3 in order to obtain a similar result to the effect that all the second-fundamental forms that depend only on jets of finite order of the solutions of evolution equation should be universal.

Our paper is organized as follows. In Section 2, we recall the results of Chern and Tenenblat [START_REF] Chern | Pseudospherical surfaces and evolution equations[END_REF] on the classification of evolution equations describing pseudo-spherical surfaces and use these to give an analogue of the normal forms of Theorem 2 for the case of second-order evolution equations (21). These normal forms are then used as the starting point in Section 3 of the proof of Theorem 1. Section 4 is devoted to the proof of Theorem 3. The proofs involve a careful analysis of the possible dependence on higher-order jets of u of the solutions of the system of differential constraints [START_REF] Reyes | Pseudo-potentials, nonlocal symmetries and integrability of some shallow water equations[END_REF] and [START_REF] Reyes | Correspondence theorems for hierarchies of equations of pseudo-spherical type[END_REF] that must be satisfied by components a, b, c of the second fundamental form, together with the algebraic constraint given by the Gauss equation (19).

The classification of second-order evolution equations describing η pseudo-spherical surfaces

In [START_REF] Chern | Pseudospherical surfaces and evolution equations[END_REF], Chern and Tenenblat obtained necessary and sufficient conditions in the form of differential equations on the functions f ij for the existence of an evolution equation of the form

∂u ∂t = F (u, . . . , ∂ k u ∂x k ), (23)
which describes η pseudo-spherical surfaces, i.e., with f 21 = η, where η is a nonzero parameter. They also performed a complete classification of the evolution equations of the form (23) which describe η pseudo-spherical surfaces. They obtained four classes of evolution equations (Theorems 2.2 to 2.5 in [START_REF] Chern | Pseudospherical surfaces and evolution equations[END_REF]). These four classes of equations are determined algebraically by f 11 , f 31 , f 22 and their derivatives, up to some differential constraints. In what follows, we consider only secondorder evolution equations of the form (23) and solve the differential constraints that f 11 , f 31 and f 22 must satisfy in order for (23) to describe η pseudo-spherical surfaces. We shall deal with two of the four classes (Theorems 2.2 and 2.4 in [START_REF] Chern | Pseudospherical surfaces and evolution equations[END_REF]) since the two remaining classes of evolution equations (Theorems 2.3 and 2.5 in [START_REF] Chern | Pseudospherical surfaces and evolution equations[END_REF]) lead to evolution equations of the first order, when k = 2.

It will be convenient to introduce the following notation for the spatial derivatives of u (used in [START_REF] Chern | Pseudospherical surfaces and evolution equations[END_REF] and also in [START_REF] Kamran | On differential equations describing pseudo-spherical surfaces[END_REF]),

z i = ∂ i u ∂x i , 0 i k, and to view (x, t, z 0 , z 1 , . . . , z k ) as local coordinates in an open subset U of a manifold.
Lemma 1 (Chern & Tenenblat [START_REF] Chern | Pseudospherical surfaces and evolution equations[END_REF]). Consider a second-order evolution equation of the form z 0,t = F (z 0 , z 1 , z 2 ) which describes an η pseudo-spherical surface with associated forms

ω i = f i1 dx + f i2 dt. If f ij are differentiable functions of z 0 , z 1 , z 2 , then f ij,z2 = 0, f 11,z1 = f 31,z1 = f 22,z1 = 0, (24)
f 2 11,z0 + f 2 31,z0 = 0. (25)
In order to state the results, we introduce the following notation

H = f 11 f 11,z0 -f 31 f 31,z0 , L = f 11 f 31,z0 -f 31 f 11,z0 , P = f 11,z0 f 31,z0z0 -f 31,z0 f 11,z0z0 , M = f 2 31,z0 -f 2 11,z0 . (26)
Lemma 2. Let f ij , 1 i 3, 1 j 2, be differentiable functions of z 0 , z 1 , z 2 such that (24) and (25) hold and f 21 = η a nonzero parameter. Suppose HL = 0. Then z 0,t = F (z 0 , z 1 , z 2 ) describes an η pseudo-spherical surface with associated 1-forms

ω i = f i1 dx + f i2 dt, if and only if F = ∓f 22,z0 η √ 1 -α 2 f 11,z0 z 2 ∓ f 22,z0z0 η √ 1 -α 2 f 11,z0 z 2 1 + (η 2 + f 2 11 -f 2 31 )f 22,z0 η (1 -α 2 )f 11 ∓ αη √ 1 -α 2 f 11,z0 + f 22 η z 1 , (27)
and

f 31 = αf 11 ± η 1 -α 2 , ( 28
)
f 12 = f 11 f 22 η ∓ f 22,z0 η √ 1 -α 2 z 1 , (29)
f 32 = (αf 11 ± η √ 1 -α 2 )f 22 η ∓ αf 22,z0 η √ 1 -α 2 z 1 , (30)
where f 22,z0 = 0, f 11,z0 = 0, and α 2 < 1.

Proof. If k = 2, Theorem 2.2 in [START_REF] Chern | Pseudospherical surfaces and evolution equations[END_REF] gives the general expression of second-order evolution equations z 0,t = F which describe η pseudo-spherical surfaces, namely

F = 1 L 1 i=0 z i+1 B zi + 1 HL -z 1 L η + f 2 31 -f 2 11 z 1 A 0 + B HL (z 1 M + ηL) + z 1 f 22 η , (31)
where

B = f 22,z0 z 1 , A 0 = 1 L (-z 1 P + ηM )B z1 + f 22,z1 H,
where the functions f 12 and f 32 are given by

f 12 = f 11 f 22 η + 1 H - f 11 A 0 η z 1 + f 31,z0 B , (32)
f 32 = f 31 f 22 η + 1 H - f 31 A 0 η z 1 + f 11,z0 B ,
and where (2.12) in [START_REF] Chern | Pseudospherical surfaces and evolution equations[END_REF] gives two differential equations that the functions f 11 , f 31 and f 22 must satisfy. When k = 2, these equations reduce to

L η f 22,z0 - L η z 1 A 0 H z0 + A 0 + M H B z0 + B H 2 (LP + M 2 ) = 0, (33)
L η f 22,z1 - L η z 1 A 0 H z1 + M H B z1 = 0. (34)
If L = 0, then the differential equation (34) leads to P f 22,z0 = 0. The vanishing of f 22,z0 contradicts the fact that F is a second order evolution equation. We conclude then that f 22,z0 = 0, P = 0.

(35)

Differentiating (33) with respect to z 1 leads to -L(M/HL) z0 + M 2 /L 2 = 0 and hence, the differential equation ( 33) leads to

M = - L 2 η 2 . ( 36
)
The vanishing of P implies that

f 31 = αf 11 + β, where α, β ∈ R. (37)
We have then

L = -βf 11,z0 , H = [(1 -α 2 )f 11 -αβ]f 11,z0 . (38)
The non-vanishing of L implies that β = 0 and f 11,z0 = 0. Substituting (37) in (36) and in the expression of M as in (26) leads to

β 2 = η 2 (1 -α 2 ). (39)
The non-vanishing of β and η, and the latter equation imply that α ∈ (-1, 1). Finally, substituting β = ±η √ 1 -α 2 , P = 0 and (37) in the expressions (31) and (32) leads to expressions ( 27), ( 28), (29), and (30).

If HL = 0, then there are three classes of evolution equations to consider, which are given in Theorems 2.3-2.5 in [START_REF] Chern | Pseudospherical surfaces and evolution equations[END_REF]. However, F is of second order only when H = L = 0, as in Theorem 2.4 in [START_REF] Chern | Pseudospherical surfaces and evolution equations[END_REF].

Lemma 3. Let f ij , 1 i 3, 1 j
2, be differentiable functions of z 0 , z 1 , z 2 such that (24) and (25) hold and f 21 = η a nonzero parameter. Suppose f 31 = ±f 11 = 0. Then z 0,t = F (z 0 , z 1 , z 2 ) describes an η pseudo-spherical surface with associated 1-forms

ω i = f i1 dx + f i2 dt, if
f 11 a t + ηb t -f 12 a x -f 22 b x -2b(f 11 f 32 -f 31 f 12 ) + (a -c)(ηf 32 -f 22 f 31 ) - ℓ i=0 (f 12 a zi + f 22 b zi )z i+1 + ℓ i=0 (f 11 a zi + ηb zi )z i,t = 0, (41)
and ( 18) becomes

f 11 b t + ηc t -f 12 b x -f 22 c x + (a -c)(f 11 f 32 -f 12 f 31 ) + 2b(ηf 32 -f 22 f 31 ) - ℓ i=0 (f 12 b zi + f 22 c zi )z i+1 + ℓ i=0 (f 11 b zi + ηc zi )z i,t = 0. ( 42
)
Since f 22,z0 = 0 and f 11,z0 = 0 for evolution equations (27), and f 11,z0 = 0 and f 12,z1 = 0 for evolution equations (40), differentiating (41) and (42) with respect to z ℓ+2 leads to f 11 a z ℓ +ηb z ℓ = f 11 b z ℓ + ηc z ℓ = 0, and hence

b z ℓ = - f 11 η a z ℓ , and c z ℓ = f 2 11 η 2 a z ℓ . (43)
Differentiating the Gauss equation ( 19) with respect to z ℓ leads to ca z ℓ + ac z ℓ -2bb z ℓ = 0, and substituting (43) in the latter leads to We have then

c + f 11 η 2 a + 2 f 11 η b a z ℓ = 0. ( 44
) If c + f 11 η 2 a + 2
D t b = - f 11 η D t a - a η f 11,z0 F, D t c = f 11 η 2 D t a + 2 η f 11 η a ∓ 1 f 11,z0 F, D x b = - f 11 η D x a - a η f 11,z0 z 1 , , D x c = f 11 η 2 D x a + 2 η f 11 η a ∓ 1 f 11,z0 z 1 ,
and hence

f 11 D t a + ηD t b = -af 11,z0 F, (45)
f 11 D t b + ηD t c = f 11 η a ∓ 2 f 11,z0 F, (46)
f 12 D x a + f 22 D x b = - ∆ 12 η D x a - af 22 η f 11,z0 z 1 , (47)
f 12 D x b + f 22 D x c = f 11 η ∆ 12 η D x a + ∆ 12 η 2 af 11,z0 z 1 + f 22 η f 11 η a ∓ 2 f 11,z0 z 1 . ( 48
)
where

∆ 12 = f 11 f 22 -ηf 12 .
Substituting the latter four equalities in ( 17) lead to

-af 11,z0 F + ∆ 12 η D x a + af 22 η f 11,z0 z 1 -2b(f 11 f 32 -f 31 f 12 ) + (a -c)(ηf 32 -f 31 f 22 ) = 0,
which is equivalent to

-af 11,z0 F + ∆ 12 η ℓ i=0 a zi z i+1 + af 22 η f 11,z0 z 1 -2b(f 11 f 32 -f 31 f 12 ) + (a -c)(ηf 32 -f 31 f 22 ) = 0. ( 49
)
Substituting the four equalities (45)-( 48) into ( 18) lead to

f 11 η a ∓ 2 f 11,z0 F - f 11 η ∆ 12 η D x a - ∆ 12 η 2 af 11,z0 z 1 - f 22 η f 11 η a ∓ 2 f 11,z0 z 1 + (a -c)(f 11 f 32 -f 31 f 12 ) + 2b(ηf 32 -f 31 f 22 ) = 0,
which is equivalent to

f 11 η a ∓ 2 f 11,z0 F - f 11 η ∆ 12 η ℓ i=0 a zi z i+1 - ∆ 12 η 2 af 11,z0 z 1 - f 22 η f 11 η a ∓ 2 f 11,z0 z 1 + (a -c)(f 11 f 32 -f 31 f 12 ) + 2b(ηf 32 -f 31 f 22 ) = 0. (50)
• If ℓ ≥ 2, then differentiating (49) with respect to z ℓ+1 leads to ∆ 12 a z ℓ = 0. Thus a z ℓ = 0 and also b z ℓ = c z ℓ = 0. • If ℓ = 1, then differentiating (49) and (50) with respect to z 2 lead to

-af 11,z0 F z2 + ∆ 12 η a z1 = 0, f 11 η a ∓ 2 f 11,z0 F z2 - f 11 η ∆ 12 η a z1 = 0.
The latter system leads to f 11,z0 F z2 = 0, which runs into a contradiction. • If ℓ = 0, then differentiating (49) and (50) with respect to z 2 lead to

-af 11,z0 F z2 = 0, f 11 η a ∓ 2 f 11,z0 F z2 = 0.
The latter system leads to f 11,z0 F z2 = 0, which runs into a contradiction.

Therefore, for all ℓ, (17), [START_REF] Reyes | Correspondence theorems for hierarchies of equations of pseudo-spherical type[END_REF] and the Gauss equation is an inconsistent system. Finally, if the functions a, b and c depend on a jet of finite order, then there are universal, i.e., they are functions of x and t only.

If c + f 11 η 2 a + 2 f 11 η b = 0, then a z ℓ = 0,
Proposition 1. For the second-order evolution equations which describe η pseudo-spherical surfaces as in Lemma 2, there is no local isometric immersion in R 3 of a pseudo-spherical surface determined by a solution u, for which the coefficients a, b, c of the second fundamental form depend on a jet of finite order of u.

Proof. Let a, b, and c be coefficients of the second fundamental form satisfying the Gauss equation ac -b 2 = -1. By Lemma 4, if a, b and c depend on a jet of finite order, then a, b and c depend only on x and t. From (28)-(30), we have

f 11 f 32 -f 12 f 31 = f 22,z0 z 1 , (51)
ηf 32 -f 22 f 31 = ∓ αf 22,z0 √ 1 -α 2 z 1 . ( 52
)
Taking into account the expressions (51), ( 52) and (29), equations ( 17) and ( 18) become

f 11 a t + ηb t - f 11 f 22 η ∓ f 22,z0 η √ 1 -α 2 z 1 a x -f 22 b x -2bf 22,z0 z 1 ∓ (a -c) αf 22,z0 √ 1 -α 2 z 1 = 0, (53)
f 11 b t + ηc t - f 11 f 22 η ∓ f 22,z0 η √ 1 -α 2 z 1 b x -f 22 c x + (a -c)f 22,z0 z 1 ∓ 2b αf 22,z0 √ 1 -α 2 z 1 = 0. (54)
Differentiating ( 53) and ( 54) with respect to z 1 and the fact that f 22,z0 = 0 lead to

a x b x = ±η √ 1 -α 2 αη αη ∓η √ 1 -α 2 2b a -c .
The determinant of the 2 × 2 matrix appearing in the above equation is non-zero, therefore, a x and b x can not vanish simultaneously. Otherwise, a -c = b = 0, and this contradicts the Gauss equation. ( 53) and (54) become then

ηf 11 a t + η 2 b t -f 11 f 22 a x -ηf 22 b x = 0, (55)
ηf 11 b t + η 2 c t -f 11 f 22 b x -ηf 22 c x = 0. (56)
Differentiating ( 55) and (56) with respect to z 0 , and dividing by ηf 11,z0 lead to

a t = (f 11 f 22 ) z0 ηf 11,z0 a x + f 22,z0 f 11,z0 b x , (57)
b t = (f 11 f 22 ) z0 ηf 11,z0 b x + f 22,z0 f 11,z0 c x . (58)
Observe that (f 11 f 22 ) z0 f 11,z0 and f 22,z0 f 11,z0 cannot both be constant. Otherwise, f 22,z0 = 0 which is a contradiction. Differentiating (57) and (58) with respect to z 0 leads to

(f 11 f 22 ) z0 ηf 11,z0 z0
a x + f 22,z0 f 11,z0 z0 b x = 0, (f 11 f 22 ) z0 ηf 11,z0 z0 b x + f 22,z0 f 11,z0 z0 c x = 0.
We conclude that

a x c x -b 2 x = 0. (59)
Subtracting (58) multiplied by a x from (57) multiplied by b x , it follows from (59) that

a x b t -a t b x = 0. (60)
From (55), we have f 11 (ηa t -f 22 a x ) + η 2 b t -ηf 22 b x = 0. Note that (ηa t -f 22 a x ) = 0. Otherwise, since f 22,z0 = 0, we have a x = a t = 0 and hence it follows from (59) that b x = 0, which runs into a contradiction. Therefore,

f 11 = η(ηb t -f 22 b x ) ηa t -f 22 a x . (61)
Differentiating (61) with respect to z 0 and taking into account (60) lead to f 11,z0 = 0, which is a contradiction.

Proposition 2. Let u t = F (u, u x , u xx ) be a second-order evolution equation which describes η pseudo-spherical surfaces, as in Lemma 3. There exists a local isometric immersion in R 3 of a pseudo-spherical surface, determined by a solution u, for which the coefficients of the second fundamental form (13) depend on a jet of finite order of u if, and only if, the coefficients are universal and are given by

a = le ±2(ηx+λt) -γ 2 e ±4(ηx+λt) -1, ( 62
) b = γe ±2(ηx+λt) , (63)
c = γ 2 e ±4(ηx+λt) -1 le ±2(ηx+λt) -γ 2 e ±4(ηx+λt) -1 , (64)
l, γ ∈ R, l > 0 and l 2 > 4γ 2 . The 1-forms are defined on a strip of R where

log l -l 2 -4γ 2 2γ 2 < ±(ηx + λt) < log l + l 2 -4γ 2 2γ 2 . ( 65
)
Moreover, the constants l and γ have to be chosen so that the strip intersects the domain of the solution of the evolution equation.

Proof. As for the previous proposition, if a, b and c depend on a jet of finite order, it follows from Lemma 4 that a, b and c depend only on x and t. We assume also that f 12,z1 = 0, otherwise, the evolution equation is not of second-order. Equations ( 17) and ( 18) become

f 11 a t + ηb t -f 12 a x -λb x ± (ηf 12 -λf 11 )(a -c) = 0, (66)
f 11 b t + ηc t -f 12 b x -λc x ± (ηf 12 -λf 11 )2b = 0. (67)
Differentiating (66) and (67) with respect to z 1 , and the fact that f 12,z1 = 0 lead to

a x ∓ η(a -c) = 0, (68)
b x ∓ 2ηb = 0. ( 69
)
Taking into account (68) and (69) and differentiating (66) and (67) with respect to z 0 leads to

a t ∓ λ(a -c) = 0, ( 70
) b t ∓ 2λb = 0, (71)
and hence, (66) and (67) become

ηb t -λb x = 0, ( 72
)
ηc t -λc x = 0. ( 73
)
Note that (69) and ( 71) imply (72), and (68) and (70) imply

ηa t -λa x = 0, (74)
and hence imply (73). From ( 69) and (71), we conclude that

b = γe ±2(ηx+λt) , γ ∈ R. (75)
Note that a = 0. Otherwise, if a = 0, then (68) implies that c = 0 and the Gauss equation leads to b = ±1 which contradicts (69). Therefore, from the Gauss equation we have c = (b 2 -1)a -1 . Then, in view of (75), equations (68) and (70) reduce to

aa x ∓ η(a 2 -γ 2 e ±4(ηx+λt) + 1) = 0, aa t ∓ λ(a 2 -γ 2 e ±4(ηx+λt) + 1) = 0.
The latter system leads then to

a = le ±2(ηx+λt) -γ 2 e ±4(ηx+λt) -1, l ∈ R,
which is defined wherever le ±2(ηx+λt) -γ 2 e ±4(ηx+λt) -1 > 0. Hence l > 0 and

l -l 2 -4γ 2 2γ 2 < e ±2(ηx+λt) < l + l 2 -4γ 2 2γ 2 ,
i.e., a is defined on the strip described by (65). Now, from either (68) or (70), we obtain

c = γ 2 e ±4(ηx+λt) -1 le ±2(ηx+λt) -γ 2 e ±4(ηx+λt) -1 .
A straightforward computation shows that the converse holds. Finally, we observe that given a solution of the evolution equation, in order to have an immersion, one has to choose the constants l and γ, such that the strip (65) intersects the domain of the solution in R 2 .

Proof of Theorem 3

We begin by introducing some notations. Given a differentiable function u(x, t), we denote its partial derivatives by

z i = ∂ i u ∂x i , w i = ∂ i u ∂t i , where z 0 = w 0 = u. (76)
We have therefore

z i,x = z i+1 , z i,t = ∂ i-1 u xt ∂x i-1 , w i,x = ∂ i-1 u xt ∂t i-1
, w i,t = w i+1 , and the total derivatives of a differentiable function ϕ = ϕ(x, t, z 0 , z 1 , w 1 , . . . , z ℓ , w ℓ ) are given by

D x ϕ = ϕ x + ℓ i=0 ϕ zi z i+1 + ℓ i=1 ϕ wi w i,x , (77)
D t ϕ = ϕ t + ℓ i=1 ϕ zi z i,t + ℓ i=0 ϕ wi w i+1 . ( 78
)
We also introduce the notation

∆ ij = f i1 f j2 -f j1 f i2 . (79)
Observe that

∆ 12 = 0, ∆ 2 13 + ∆ 2 23 = 0. (80)
In fact, ∆ 12 = 0 is equivalent to ω 1 ∧ ω 2 = 0. Moreover, ω 1 ∧ ω 3 = ∆ 13 dx ∧ dt and ω 2 ∧ ω 3 = ∆ 23 dx ∧ dt. If ∆ 13 = ∆ 23 = 0, then it follows from (3) that dω 1 = dω 2 = 0. Therefore, ω 3 (e 1 ) = ω 3 (e 2 ) = 0 and hence ω 3 = 0 that is in contradiction with

dω 3 = ω 1 ∧ ω 2 .
The classification theorem of Rabelo and Tenenblat (see Theorem 2) for hyperbolic equations describing η pseudo-spherical surfaces makes use of a number of lemmas. Its proof also provides the coefficients f ij of the 1-forms (2) for each equation of Theorem 2. We will need the lemmas and these coefficients for the proof of Theorem 3. We therefore recall them from [START_REF] Rabelo | On equations of the type uxt = F (u, ux) which describe pseudospherical surfaces[END_REF] without proof. However, the reader can easily check, in each case stated in Lemmas 6-8, that the structure equations (3) hold if, and only if, the corresponding hyperbolic equation holds.

Lemma 5. [START_REF] Rabelo | On equations of the type uxt = F (u, ux) which describe pseudospherical surfaces[END_REF] Let u xt = F (u, u x ) be a differential equation describing η pseudo-spherical surfaces, with associated one-forms

ω i = f i1 dx + f i2 dt, where f ij and F are real differentiable (C ∞ ) functions on a open connected set U ⊂ R 2 . Then f 11,u ≡ f 31,u ≡ 0, f 12,ux ≡ f 22,ux ≡ f 32,ux ≡ 0, f 2 11,ux + f 2 31,ux = 0, in U. Lemma 6. [12]
The coefficients f ij of the 1-forms (2) for the equation

u xt = F (u),
where

F ′′ (u) + αF (u) = 0, α ∈ R \ {0} (81)
are given by

  f 11 f 12 f 21 f 22 f 31 f 32   =   -α(Bz 1 -AQ) Aα(QF ′ -ηF )/(Q 2 α + η 2 ) η (ηF ′ + αQF )/(Q 2 α + η 2 ) -α(Az 1 -BQ) Bα(QF ′ -ηF )/(Q 2 α + η 2 )   , (82)
where

z 1 = u x , A, B, Q ∈ R are such that α = 1/(A 2 -B 2 ), A 2 -B 2 = 0 and Q 2 α + η 2 = 0 and η ∈ R \ {0}.
In particular, if B = 0 and hence A = 0, one has α = 1/A 2 > 0 and

  f 11 f 12 f 21 f 22 f 31 f 32   =   αAQ αA(QF ′ -ηF )/(Q 2 α + η 2 ) η (ηF ′ + QαF )/(Q 2 α + η 2 ) -αAz 1 0   . ( 83
)
Lemma 7. [START_REF] Rabelo | On equations of the type uxt = F (u, ux) which describe pseudospherical surfaces[END_REF] The coefficients f ij of the 1-forms (2) for the equation

u xt = νe δu β + γu 2 x , where δ, γ, ν ∈ R \ {0} and β = 0, when γ = 1, (84)
are given as follows:

a) If γ = 1, then   f 11 f 12 f 21 f 22 f 31 f 32   =   ηAδ -(Bz 1 ∓ A √ ∆)δ 2 /(γ -1) ±Aδνe δz0 η ±νe δz0 ηBδ -(Az 1 ∓ B √ ∆)δ 2 /(γ -1) ±Bδνe δz0   , (85)
where

z 0 = u, z 1 = u x , ∆ = β + γz 2 1 > 0, A 2 -B 2 = (γ -1)/δ 2 and η ∈ R \ {0}. b) If γ = 1,   f 11 f 12 f 21 f 22 f 31 f 32   =   1 2 ( 1 A + δ 2 A)z 1 + ηδA ±Aδνe δz0 η ±νe δz0 1 2 (-1 A + δ 2 A)z 1 ± ηδA Aδνe δz0   , (86)
where A, η ∈ R \ {0}.

Lemma 8. [12]

The coefficients f ij of the 1-forms (2) for the equation

u xt = λu + ξu x + τ, λ, ξ, τ ∈ R (87)
are given as follows:

a) If λ = ξ = τ = 0, then   f 11 f 12 f 21 f 22 f 31 f 32   =   z 1 0 η e z0 η e z0   , (88)
where

z 0 = u, z 1 = u x and η = 0. b) If λ = 0, then   f 11 f 12 f 21 f 22 f 31 f 32   =   ±ηT z 1 /λ T z 0 + τ T /λ η λ/η ∓ ξ ηT z 1 /λ ±T z 0 ± τ T /λ   ( 89
)
where

T, η ∈ R \ {0}. c) If λ = 0 and ξ 2 + τ 2 = 0, then   f 11 f 12 f 21 f 22 f 31 f 32   =   dz 1 /F (z 1 ) 1/η η 0 dz 1 /F (z 1 ) 1/η   ( 90
)
where η ∈ R \ {0}.

Having recalled these results from [START_REF] Rabelo | On equations of the type uxt = F (u, ux) which describe pseudospherical surfaces[END_REF], we are now ready to proceed with the proof of Theorem 3. The proof consists of a number of technical lemmas and propositions, in which we analyze the existence of solutions for the system of equations ( 17), ( 18) and ( 19) that depend on u and finitely many derivatives, for each of the classes of hyperbolic equations obtained by Rabelo and Tenenblat in Theorem 2.

With the notation introduced in (79), equations ( 17) and ( 18) are written as

f 11 D t a + ηD t b -f 12 D x a -f 22 D x b -2b∆ 13 + (a -c)∆ 23 = 0, (91)
f 11 D t b + ηD t c -f 12 D x b -f 22 D x c + (a -c)∆ 13 + 2b∆ 23 = 0. (92
∓2∆ 13 -c∆ 23 = 0, ( 94
)
ηD t c -f 22 D x c -c∆ 13 ± 2∆ 23 = 0. ( 95
)
It follows from ( 94) and (80) that ∆ 23 = 0 and c = ∓2∆ 13 /∆ 23 . Since ∆ 13 and ∆ 23 depend only on z 0 and z 1 , we conclude that c depends only on z 0 and z 1 and (95) reduces to

η(c z1 F + c z0 w 1 ) -f 22 (c z0 z 1 + c z1 z 2 ) -c∆ 13 ± 2∆ 23 = 0. ( 96
)
Taking the derivative of this equation with respect to z 2 and w 1 implies that f 22 c z1 = 0 and c z0 = 0. If f 22 = 0 then c is constant and (96) reduces to -c∆ 13 ± 2∆ 23 = 0 i.e., we have

-c ±2 ∓2 -c ∆ 13 ∆ 23 = 0 0 .
Since the determinant is nonzero, it implies that ∆ 13 = ∆ 23 = 0 which contradicts (80). If f 22 = 0 on an open set, then the functions f ij are given by (90) and hence ∆ 13 = 0 and ∆ 23 = 1.

Then (94) implies that c = 0 and (95) gives a contradiction. This concludes the proof of i).

ii) Observe that except for the functions f ij given by ( 83) with Q = 0, f 11 does not vanish on an open set. We will first show that if f 11 = 0 on an open set i.e, F satisfies (81) and f ij are given by ( 83) and Q = 0, then c = 0. In fact, for such f ij s we have 91) and (92) reduce to

∆ 13 = -A 2 α 2 F (u)z 1 /η, ∆ 23 = AαF ′ (u)z 1 /η, α = 1/A 2 > 0 and A = 0. Hence (
ηD t b -f 12 D x a -f 22 D x b -2b∆ 13 + (a -c)∆ 23 = 0, ηD t c -f 12 D x b -f 22 D x c + (a -c)∆ 13 + 2b∆ 23 = 0,
where f 12 = -αAF/η and f 22 = F ′ /η. Assume c = 0, then it follows from (19) that a = (b 2 -1)/c. Assume that a,b and c depend on a jet of order ℓ of u. For ℓ ≥ 1, taking derivatives of both equations with respect to w ℓ+1 implies that b w ℓ = c w ℓ = 0 and hence a w k = 0. Successive differentiation with respect to w k ,...w 1 imply that a, b and c do not depend on w ℓ ,...w 0 . Successive differentiation with respect to z ℓ+1 , ...z 2 imply that a, b and c do not depend on z ℓ ,...z 1 . Hence, a, b and c depend only on x and t. Therefore, the above system of equations reduce to

ηb t + αAF η a x - F ′ η b x + 2b α 2 A 2 F η z 1 + (a -c) αAF ′ η z 1 = 0, ηc t + αAF η b x - F ′ η c x + (a -c) α 2 A 2 F η z 1 + 2b αAF ′ η z 1 = 0.
Taking the derivative with respect to z 1 we get

2b a -c a -c 2b αAF F ′ = 0 0 . ( 97
)
Since αAF and F ′ are not zero we get a -c = ±2b and the derivative with respect to z 0 of any equation of (97) reduces to b(AF ′ ∓ F ) = 0 as a consequence of (81). If b = 0 then Gauss equation ( 19) reduces to a 2 = -1. If F = ±AF ′ then the derivative with respect to z 0 implies that αA 2 = -1. In both cases we get a contradiction. Therefore, c = 0.

Conversely, assume c = 0 on an open set, then (19) implies b = ±1 and (91) and (92) reduce to

f 11 D t a -f 12 D x a ∓ 2∆ 13 + a∆ 23 = 0, (98) a∆ 13 ± 2∆ 23 = 0. ( 99
)
It follows from ( 99) and (80) that ∆ 13 = 0 and a = ∓2∆ 23 /∆ 13 . Since ∆ 13 and ∆ 23 depend only on z 0 and z 1 , we conclude that a depends only on z 0 and z 1 and (98) reduces to

f 11 (a z1 F + a z0 w 1 ) -f 12 (a z0 z 1 + a z1 z 2 ) ∓ 2∆ 13 + a∆ 23 = 0. ( 100
)
Differentiation with respect to w 1 and z 2 implies

f 11 a z0 = f 12 a z1 = 0. ( 101
)
Since ∆ 12 = 0, we observe that f 11 and f 12 cannot vanish simultaneously.

If both f 11 = 0 and f 12 = 0 then from (101) we conclude that a is constant and (100) reduces to ∓2∆ 13 + a∆ 23 = 0. This equation with (99) implies that ∆ 13 = ∆ 23 = 0 which contradicts (80).

If f 12 = 0 on an open set, then f ij are given by (82) with A = 0, B = 0 or (85) with A = 0, B = 0 or (88). Since f 11 = 0, it follows from (101) that a z0 = 0 and (100) reduces to f 11 a z1 F ∓ 2∆ 13 + a∆ 23 = 0.

(102)

If f ij are given by ( 82) with A = 0, B = 0, then

∆ 13 = α(QF ′ -ηF Q 2 α + η 2 z 1 , ∆ 23 = -BαF, a z1 z 1 = -a.
Substituting into (102) and differentiating twice with respect to z 1 runs into a contradiction. If f ij are given by ( 85) with A = 0, B = 0, then

∆ 13 = ∓Bνδ 2 z 1 e δz0 , ∆ 23 = ±νδz 1 e δz0 , a = ± 2 Bδ .
Therefore, (102) reduces to ∓2∆ 13 + a∆ 23 = 0 which is in contradiction with (99). Finally if f ij are given by (88), then ∆ 23 = 0, hence it follows from (80) and (99) that a = 0 which is a contradiction. We conclude that if c = 0 on an open set, then f 11 = 0 i.e., f ij are given by ( 83) with

Q = 0. Therefore ∆ 13 = -A 2 α 2 F (u)z 1 /η, ∆ 23 = AαF ′ (u)z 1 /
η and hence (99) implies that a = ±2F ′ /(AαF ). Moreover, (100) is an identity since A 2 α = 1. This concludes the proof of Lemma 9.

Consider an equation u xt = F (u, u x ) describing η pseudo-spherical surfaces given by Lemmas 6-8. The existence of a local isometric immersion in R 3 of any pseudo-spherical surface, determined by a solution u, for which the coefficients a, b and c depend on x, t, z 0 , z 1 , w 1 , ...z ℓ , w ℓ , is equivalent to requiring that (91), ( 92) and ( 19) must be satisfied. Substituting the expressions of the total derivatives with respect to x and t given by ( 77) and (78), we rewrite (91) and ( 92) as

f 11 a t + ηb t + ℓ i=0 (f 11 a wi + ηb wi )w i+1 + ℓ i=1 (f 11 a zi + ηb zi ) ∂ i-1 F ∂x i-1 -(f 12 a x + f 22 b x ) - ℓ i=0 (f 12 a zi + f 22 b zi )z i+1 - ℓ i=1 (f 12 a wi + f 22 b wi ) ∂ i-1 F ∂t i-1 -2b∆ 13 + (a -c)∆ 23 = 0, (103)
and

f 11 b t + ηc t + ℓ i=0 (f 11 b wi + ηc wi )w i+1 + ℓ i=1 (f 11 b zi + ηc zi ) ∂ i-1 F ∂x i-1 -(f 12 b x + f 22 c x ) - ℓ i=0 (f 12 b zi + f 22 c zi )z i+1 - ℓ i=1 (f 12 b wi + f 22 c wi ) ∂ i-1 F ∂t i-1 + (a -c)∆ 13 + 2b∆ 23 = 0. (104)
Differentiating ( 103) and ( 104) with respect to w ℓ+1 leads to

f 11 a w ℓ + ηb w ℓ = 0 f 11 b w ℓ + ηc w ℓ = 0. (105)
Differentiation of the Gauss equation ( 19) with respect to w ℓ gives ca w ℓ + ac w ℓ -2bb w ℓ = 0.

Taking into account (105) in the latter, we obtain

c + f 11 η 2 a + 2 f 11 η b a w ℓ = 0. (106)
The following two lemmas will consider the cases in which the expression between brackets in (106) vanishes or not on an open set.

Lemma 10. Consider an equation u xt = F (u, u x ) describing η pseudo-spherical surfaces, with 1-forms ω i as in [START_REF] Cavalcante | Conservation laws for nonlinear evolution equations[END_REF] where the functions f ij are given by ( 82)-(90). Assume there is a local isometric immersion of a pseudo-spherical surface determined by a solution u(x, t), for which the coefficients a, b, c of the second fundamental form depend on a jet of finite order of u. If

c + f 11 η 2 a + 2 f 11 η b = 0 (107)
on a non empty open set, then i) For equation ( 81) with f ij as in (83) a, b and c are given by

a = ± 2η A(Q 2 α + η 2 ) ηF ′ αF + Q , b = ∓ 1 Q 2 α + η 2 2ηQ F ′ F + Q 2 α -η 2 , (108)
c = ± 2QAα Q 2 α + η 2 Q F ′ F -η ,
where α = 1/A 2 . In particular when Q = 0, a, b, c are given by (93). ii) For all equations, except those considered in i), equations ( 91), ( 92) and ( 19) form an inconsistent system.

Proof. If (107) holds then substituting c into the Gauss equation ( 19) leads to (f 11 a/η + b) 2 = 1, and hence

b = ±1 - f 11 η a and c = f 11 η 2 a ∓ 2 f 11 η . (109)
Therefore,

f 11 D t a + ηD t b = -af 11,z1 F, f 12 D x a + f 22 D x b = - ∆ 12 η D x a - af 22 f 11,z1 η z 2 , f 11 D t b + ηD t c = af 11 f 11,z1 η F ∓ 2f 11,z1 F, f 12 D x b + f 22 D x c = f 11 ∆ 12 η 2 D x a + ∆ 12 af 11,z1 η 2 z 2 + af 22 f 11 f 11,z1 η 2 z 2 ∓ 2 f 22 f 11,z1 η z 2 .
Equation (91) becomes

-af 11,z1 F + ∆ 12 η D x a+ af 22 f 11,z1 η z 2 ∓2∆ 13 +2 f 11 η a∆ 13 + 1- f 11 η 2 a∆ 23 ±2 f 11 η ∆ 23 = 0 (110)
and (92) becomes

af 11 f 11,z1 η F ∓ 2f 11,z1 F - f 11 ∆ 12 η 2 D x a - ∆ 12 af 11,z1 η 2 z 2 - af 22 f 11 f 11,z1 η 2 z 2 ± 2 f 22 f 11,z1 η z 2 + 1 - f 11 η 2 a∆ 13 ± 2 f 11 η ∆ 13 ± 2∆ 23 -2 f 11 η a∆ 23 = 0. (111)
If ℓ 2, then differentiating (110) with respect to z ℓ+1 leads to a z ℓ = 0. Successive differentiation with respect to z ℓ , ..., z 3 leads to 110) and (111)with respect to z 2 leads to

a z ℓ = a z ℓ-1 = • • • = a z2 = 0. If ℓ ≥ 1, then differentiating (
∆ 12 a z1 + af 22 f 11,z1 = 0, -f 11 ∆ 12 a z1 -∆ 12 af 11,z1 -af 22 f 11 f 11,z1 ± 2ηf 22 f 11,z1 = 0, which is equivalent to ∆ 12 a z1 + af 22 f 11,z1 = 0, (112)
(∆ 12 a ∓ 2ηf 22 )f 11,z1 = 0. (113)
i) For equation ( 81) with f ij given by (83) we have f 11,z1 = 0. Hence (113) is trivially satisfied and (112) implies that a z1 = 0. Moreover, (110) and (111) reduce to

∆ 12 η D x a + 2 f 11 η a ∓ 1 ∆ 13 + 1 - f 2 11 η 2 a ± 2 f 11 η ∆ 23 , (114)
-f 11 ∆ 12 η 2 D x a + 1 - f 2 11 η 2 a ± 2 f 11 η ∆ 13 -2 f 11 η a ∓ 1 ∆ 23 . (115)
Adding equation (114) multiplied by f 11 /η with (115) and cancelling a nonzero factor, we get

a∆ 13 - f 11 a η ∓ 2 ∆ 23 = 0. Since ∆ 13 -f 11 ∆ 23 /η = f 31 ∆ 12 /η, we conclude that a = ∓2∆ 23 η/(f 31 ∆ 12 ).
For the functions f ij as in (83) we have f 31 = -αAz 1 = 0 and

∆ 12 = αAF, ∆ 13 = α(QF ′ -ηF ) Q 2 α + η 2 z 1 , ∆ 23 = αA(ηF ′ + αQF ) Q 2 α + η 2 z 1 .
Therefore, we conclude that a is given by

a = ± 2η αQ 2 + η 2 ηA F ′ F + Q A .
A straightforward computation shows that substituting the expressions of a, D x a = a z0 z 1 , f 11 = αAQ and using the fact that αA 2 = 1 equation ( 114) is trivially satisfied. It follows from (109) that b and c are given as in (108). Observe that when Q = 0 then (108 reduces to (93).

ii) For all equations except those considered in i) we have f 11,z1 = 0. If ℓ = 0, then differentiating (110) and (111) with respect to z 2 leads to af 22 f 11,z1 = 0 and to ∆ 12 af 11,z1 + af 22 f 11 f 11,z1 ∓ 2ηf 22 f 11,z1 = 0. From Lemma 9 a = 0, hence f 22 f 11,z1 = 0 and ∆ 12 f 11,z1 = 0. This implies that f 11,z1 = 0 which is a contradiction. Therefore, ℓ ≥ 1.

If f 22 = 0, which is the case for equation (87) with f ij given by (90), then (112) and (113) leads to a = 0 which contradicts Lemma 9. Thus, (91), (92), and the Gauss equation form an inconsistent system.

If f 22 = 0, (which is the case for all equations except (87) with λ = 0, ξ 2 + τ 2 = 0) then dividing (113) by f 11,z1 leads to ∆ 12 a ∓ 2ηf 22 = 0, and differentiating the latter with respect to z 1 gives ∆ 12 a z1 + a∆ 12,z1 = 0, where from (79) we have ∆ 12,z1 = f 22 f 11,z1 . Therefore, (112) is a consequence of (113). From (113), we have

a = ±2η f 22 ∆ 12 , (116)
which means that a w1 = a x = a t = 0, i.e., a is a function of z 0 and z 1 only. Equations ( 110

Observe that we are considering f 22 = 0 and f 11,z1 = 0. A straightforward computation shows that (119) leads to a contradiction for equation (81) with f ij as in (82) with B = 0 and equations ( 84), (87) with f ij given as in ( 85) -( 90). This concludes the proof of Lemma 10.

Lemma 11. Consider an equation u xt = F (u, u x ) describing η pseudo-spherical surfaces, with 1-forms ω i as in [START_REF] Cavalcante | Conservation laws for nonlinear evolution equations[END_REF] where the functions f ij are given by ( 82)-(90). Assume there is a local isometric immersion of a pseudo-spherical surface, determined by a solution u(x, t), for which the coefficients a, b, c of the second fundamental form depend on a jet of finite order of u.

If c + f 11 η 2 a + 2 f 11 η b = 0, (120)
holds then a, b and c are functions of x and t, and thus universal.

Proof. If (120) holds then, it follows from Lemma 9 that c = 0 and f 11 = 0. Moreover, from (106) we get a w ℓ = 0 and hence (105) implies that b w ℓ = c w ℓ = 0. If ℓ = 0, then a, b, and c are functions of x and t, and thus universal. If ℓ 1, then consecutive differentiation of (103), ( 104) and ( 19) with respect to w ℓ , . . . w 1 lead to a wi = b wi = c wi = 0 for i = 0, . . . , ℓ. In particular, a, b and c do not depend on z 0 . Therefore, a, b, and c are functions of x, t, z 1 , . . . , z ℓ . Differentiating (103) and (104) with respect to z ℓ+1 leads to

f 12 a z ℓ + f 22 b z ℓ = 0 and f 12 b z ℓ + f 22 c z ℓ = 0. (121)
Differentiation of the Gauss equation ( 19) with respect to z ℓ gives

ca z ℓ + ac z ℓ -2bb z ℓ = 0. (122)
If f 22 = 0, which is the case for equation (87) with f ij as in (90), since f 12 = 0, (121) implies that a z ℓ = b z ℓ = 0, and (122) leads to ac z ℓ = 0. From Lemma [START_REF] Jorge | Linear problems associated to evolution equations of type utt = F (u, ux, uxx, ut)[END_REF] we have a = 0, hence c z ℓ = 0. Successive differentiation of (103), ( 104) and ( 19) with respect to z ℓ , . . . , z 2 leads to a zi = b zi = 0, and hence c zi = 0 for i = 1, . . . , ℓ. Therefore, a, b, and c are functions of x and t.

If f 22 = 0, then (121) leads to

b z ℓ = - f 12 f 22 a z ℓ and c z ℓ = f 2 12 f 2 22 a z ℓ . (123)
Substituting these expressions into (122) we get

c + f 12 f 22 2 a + 2 f 12 f 22 b a z ℓ = 0. (124) If c + f 12 f 22 2 a + 2 f 12 f 22 b = 0, (125)
Therefore,

f 11 D t a + ηD t b = ∆ 12 f 22 D t a -ηa f 12 f 22 z0 w 1 , f 12 D x a + f 22 D x b = -af 22 f 12 f 22 z0 z 1 , f 11 D t b + nD t c = 2ηf 12 f 22 a -f 11 a ∓ 2η f 12 f 22 z0 w 1 - f 12 f 2 22 ∆ 12 D t a, f 12 D x b + f 22 D x c = (af 12 ∓ 2f 22 ) f 12 f 22 z0 z 1 .
Therefore, equation (91) becomes

∆ 12 f 22 D t a -ηa f 12 f 22 z0 w 1 + af 22 f 12 f 22 z0 z 1 -2b∆ 13 + (a -c)∆ 23 = 0 and (92) becomes - f 12 f 22 ∆ 12 f 22 D t a+ 2ηf 12 f 22 a -f 11 a ∓ 2η w 1 -(af 12 ∓ 2f 22 )z 1 f 12 f 22 z0 +(a-c)∆ 13 +2b∆ 23 = 0.
Differentiating the first equation with respect to w 1 leads to ηa( f12 f22 ) z0 = 0. Since ηa = 0 we have ( f12 f22 ) z0 = 0 and the equations reduce to

∆ 12 f 22 D t a -2b∆ 13 + (a -c)∆ 23 = 0, (128)
- f 12 f 22 ∆ 12 f 22 D t a + (a -c)∆ 13 + 2b∆ 23 = 0, (129)
Adding (128) multiplied by f 12 /f 22 with (129) we get

a∆ 13 + ±2 - f 12 f 22 a ∆ 23 = 0, which reduces to f 32 f 22 ∆ 12 a ± 2∆ 23 = 0. (130)
Observe that we have f 22 = 0, f 12 = 0 and (f 12 /f 22 ) z0 = 0. Therefore, the only equation that satisfies these conditions is (84) with f ij as in Lemma 7.

If γ = 1, it follows from (86) that f 32 = 0 and (130) implies that a is constant hence, D t a = 0. Therefore, (128) and ( 129

Substituting the expressions of b and c as in (127) into (128) we get

∆ 12 f 22 D t a + 2 f 12 f 22 a ∓ 1 ∆ 13 + a - f 12 f 22 2 a ± 2 f 12 f 22 ∆ 23 = 0. (132)
Computing the total derivative of a with respect to t, using the expression of a as in (131), equation (132) leads to

F (∆ 23,z1 ∆ 12 -∆ 12,z1 ∆ 23 ) = -f 22 (∆ 2 13 + ∆ 2 23
), which in view of (84) and (85) reduces to (B 2 -A 2 γ)z 2 1 -A 2 β = 0, which is also a contradiction. Therefore, we conclude that the system (91), (92) and the Gauss equation is an inconsistent system. This concludes the proof of Lemma 11.

Lemma 12. Consider the equation u xt = F (u, u x ) which describes η pseudo-spherical surfaces where F is given by (81) and f ij as in (82). If the coefficients of the second fundamental form of the isometric immersion in R 3 of the pseudo-spherical surface, determined by a solution u, are universal, then the system of equations (91), (92) and the Gauss equation ( 19) is inconsistent.

Proof. If the coefficients of the second fundamental form of the isometric immersion of the η pseudo-spherical surfaces described by the differential equation are universal, then equations (91) and (92) reduce to:

f 11 a t + ηb t -f 12 a x -f 22 b x -2b∆ 13 + (a -c)∆ 23 = 0 f 11 b t + ηc t -f 12 b x -f 22 c x + (a -c)∆ 13 + 2b∆ 23 = 0,
where f ij are given by (82). Differentiating both equations with respect to z 1 leads to

-αBa t -2b α(QF ′ -ηF ) Q 2 α + η 2 + (a -c) αA(ηF ′ + αQF ) Q 2 α + η 2 = 0 (133) -αBb t + (a -c) α(QF ′ -ηF ) Q 2 α + η 2 + 2b αA(ηF ′ + αQF ) Q 2 α + η 2 = 0 (134)
Multiplying ( 133) and (134) by Q 2 α + η 2 /α, and differentiating with respect to z 0 , and taking into account that F ′′ = -αF , we obtain

2b αA(a -c) -(a -c) 2αAb αQF + ηF ′ QF ′ -ηF = 0 0 .
Since αQF + ηF ′ and QF ′ -ηF are not zero, we conclude that αA[4b 2 + (a -c) 2 ] = 0. If b = 0 and a = c then Gauss equation leads to a contradiction. If A = 0 then equations (133) and (134) reduce to

-αBa t -2b α(QF ′ -ηF ) Q 2 α + η 2 = 0, -αBb t + (a -c) α(QF ′ -ηF ) Q 2 α + η 2 = 0.
taking derivative with respect to z 0 of both equations, we conclude that b = a -c = 0 which is again a contradiction. Therefore, the system (91), (92) and the Gauss equation is inconsistent.

Proposition 3. Consider an equation

u xt = F (u), with F ′′ + αF = 0, α = 0,
describing η pseudo-spherical surfaces with f ij given by (82). There exists a local isometric immersion in R 3 of a pseudo-spherical surface, defined by a solution u, for which the coefficients of the second fundamental form depend on a jet of finite order of u, that is, a, b and c depend on x, t, u, w 1 . . . , ∂ ℓ u/∂x ℓ , w ℓ , where ℓ is finite if, and only if, α > 0 and f ij are given by (83), a, b, c depend on the jet of order zero of u and are given by (108).

Proof. Assume the local isometric immersion exists. If c+(f 11 /η) 2 a+2f 11 b/η = 0 on a non empty open set, then it follows from Lemma 10 that B = 0, i.e. α > 0 and f ij are given by (83). Moreover, a, b, c depend on the jet of order zero of u and are given by (108). If c + (f 11 /η) 2 a + 2f 11 b/η = 0, then Lemma 11 implies that a, b, c are universal. However, it follows from Lemma 12 that such an immersion does not exist.

Conversely, a straightforward computation shows that if f ij are given as in (83) and a, b, c as in (108), then the connection forms ω 3 1 and ω 3 2 given by ( 13) satisfy the structure equations ( 14) of an immersion in R 3 and the Gauss equation (19). x describing η pseudo-spherical surfaces, with f ij given by Lemma 7. There is no local isometric immersion in R 3 of a pseudospherical surface determined by a solution u of the equation, for which the coefficients of the second fundamental form depend on a jet of finite order of u. In both cases, i.e., γ = 1 or γ = 1, since ∆ 13,z1z0 ∆ 23,z1z0 = 0, these equations imply that b = 0 and a = c which is inconsistent with the Gauss equation.

Proposition 5. Consider an equation u xt = λu + ξu x + τ describing η-pseudospherical surfaces with f ij given by (88)-(90). There exists a local isometric immersion in R 3 of a pseudo-spherical surface, defined by a solution u, for which the coefficients of the second fundamental form a, b, c depend of a jet of finite order of u if, and only if, λ, ξ and τ do not vanish simultaneously and a, b, c are universal and given by:

i) When λ = 0, a = lL(x, t) -γ 2 L 2 (x, t) -1, b = γL(x, t), c = b 2 -1 a , (135)
where L(x, t) = e ±2[ηx+(λ/η∓ζ)t] l, γ ∈ R and l 2 > 4γ 2 and the 1-forms are defined on a strip of R where

log l -l 2 -4γ 2 2γ 2 < ±[ηx + (λ/η ∓ ζ)t] < log l + l 2 -4γ 2 2γ 2 . ( 136
)
ii) When λ = 0 and ξ 2 + τ 2 = 0,

a = le 2ηx -γ 2 e 4ηx -1, b = γe 2ηx , c = b 2 -1 a , (137)
l, γ ∈ R and l 2 > 4γ 2 and the 1-forms are defined on a strip of R 2 where

log l -l 2 -4γ 2 2γ 2 < ηx < log l + l 2 -4γ 2 2γ 2 . ( 138
)
Moreover, the constants l and γ have to be chosen so that the strip intersects the domain of the solution of the evolution equation. 137), that are defined on the strip given by (138).

The converse follows from a straightforward computation.

Finally, the proof of Theorem 3 follows from Propositions 3, 4 and 5.

f 11 η b = 0

 0 on an open set, then substituting the expression of c in the Gauss equation -ac + b 2 = 1 leads to (f 11 a/η + b) 2 = 1, so that b = ±1 -

  and hence b z ℓ = c z ℓ = 0, and successive differentiating leads to a zi = b zi = c zi = 0 for all i = 0, . . . , ℓ.

) Lemma 9 .

 9 Consider an equation u xt = F (u, u x ) describing η pseudo-spherical surfaces, with 1-forms ω i as in[START_REF] Cavalcante | Conservation laws for nonlinear evolution equations[END_REF] where the functions f ij are given by (82)-(90). Assume there is a local isometric immersion of any pseudo-spherical surface, determined by a solution u(x, t), for which the coefficients a, b, c of the forms ω 3 1 and ω 3 2 depend on a jet of finite order of u. Then i) a = 0 on any open set. ii) c = 0 on an open set U if, and only if, f 11 = 0 on U , i.e., F satisfies (81) and f ij are given by (83) with Q = 0. In this case, α = 1/A 2 > 0, If there is a local isometric immersion of the pseudo-spherical surface, then (91), (92) and (19) must be satisfied by a, b and c. i) Assume a = 0 on an open set, then it follows from (19) that b ± 1. Substituting into (91) and (92) leads to

then a z ℓ = 0

 0 and (123) implies that b z ℓ = c z ℓ = 0. Consecutive differentiations of (103) and (104) with respect to z ℓ , . . . , z 2 lead to a zi = b zi = c zi = 0 for i = 1, . . . , ℓ, and hence, a, b and c are functions of x and t only. If empty open set, then Lemma 9 and (120) imply that c = 0 and hence f 12 = 0. It follows from (126) and (19) that b = ±1 -

  It follows from (80) that b = 0 and a = c, which contradicts the Gauss equation.If γ = 1, the functions f ij are given by (85). If f 32 = 0 then B = 0 and (130) implies that ∆ 23 = 0. Then it follows from the expression of ∆ 23 that A = 0, which contradicts the fact that A 2 -B 2 = 0. If f 32 = 0 i.e., B = 0, then (130) implies that a = ∓2 ∆ 23 f 22 ∆ 12 f 32 .

Proposition 4 .

 4 Consider an equation of type u xt = νe δu β + γu 2

Proof.

  If the immersion exists, then Lemma 10 ii) implies that c + (f 11 /η) 2 a + 2f 11 b/η = 0, and it follows from Lemma 11 that a, b, c are universal. Therefore, equations (91) and (92) reduce tof 11 a t + ηb t -f 12 a x -f 22 b x -2b∆ 13 + (a -c)∆ 23 = 0, f 11 b t + ηc t -f 12 b x -f 22 c x + (a -c)∆ 13 + 2b∆ 23 = 0,where f ij are given by (85) if γ = 1 and (86) if γ = 1. Differentiating these equations with respect to z 1 and then with respect to z 0 leads to -2b a -c a -

Proof.

  If the coefficients of the second fundamental form of the local isometric immersion of η pseudo-spherical surfaces described by the equation of type iii) depend of a jet of finite order of u, then they are universal by Lemmas 10 and 11, and hence (91) and (92) becomesf 11 a t + ηb t -f 12 a x -f 22 b x -2b∆ 13 + (a -c)∆ 23 = 0,(139)f 11 b t + ηc t -f 12 b x -f 22 c x + (a -c)∆ 13 + 2b∆ 23 = 0.(140)If λ = ξ = τ = 0 and f ij are given by (88) then taking the derivative of both equations with respect to z 0 , and using the fact that ∆ 13 = e z0 z 1 and ∆ 23 = 0 we getb x + 2bz 1 = 0, c x -(a -c)z 1 = 0.Since a, b, c are universal we conclude that b = 0 and a = c which contradicts Gauss equation. Therefore the immersion does not exit. i) If λ = 0 and the functions f ij are as in (89) then ∆ 13 = 0. Differentiating (139) and (140) with respect to z 1 leads to (after dividing by f 11,z1 )a t = ±f 22 (a -c), (141) b t = ±2bf 22 .(142)Differentiating (139) and (140) with respect to z 0 leads to (after dividing by f 12,z0 )a x = ±η(a -c),(143)b x = ±2ηb. (144)and hence, (139) and (140) reduce toηb t -f 22 b x = 0, (145) ηc t -f 22 c x = 0. (146)The equations (141), (142), (143), (144), (145), and (146) are the same as (70), (71), (68), (69), (72), and (73) respectively, since f 22 is contant. Therefore, a is as in (62), b is as in (63), and c is as in (64) and are subject to (65), where λ is replaced by f 22 = λ/η ∓ ζ. Therefore, we obtain a, b, c given as in (135) defined on the strip (136).ii) If λ = 0, ξ 2 + τ 2 = 0 and the functions f ij are as in (90), then ∆ 13 = 0 and ∆ 23 = 1, hence (139) and (140) reduce tof 11 a t + ηb t -f 12 a x + (a -c) = 0,(147)f 11 b t + ηc t -f 12 b x + 2b = 0.(148)Differentiating with respect to z 1 leads to a t = b t = 0. Since from Lemma 9 we have a = 0, Gauss equation implies that c t = 0 and thus (147) and (148) become a x = η(a -c), b x = 2ηb, where c = (b 2 -1)/a. The arguments used in the proof of Proposition 2, with λ = 0 and ± replaced by +, imply that a, b, c are given by (

  and only if f 22 = λ, where λ is constant, f 32 = ±f 12 , and Lemma 4. Let u t = F (u, u x , u xx ) be a second-order evolution equation describing η pseudospherical surfaces as in Lemma 2 or in Lemma 3. If there exists a local isometric immersion of a surface determined by a solution u for which the coefficients of the second fundamental form[START_REF] Rabelo | A classification of pseudospherical surfaces equations of the type ut = uxxx + G(u, ux, uxx) which describe pseudospherical surfaces[END_REF] depend on a jet of finite order of u, i.e., a, b and c depend on x, t, u, . . . , ∂ ℓ u/∂x

	F =	f 12,z1 f 11,z0	z 2 +	f 12,z0 f 11,z0	z 1 ∓	λf 11 -ηf 12 f 11,z0	.	(40)
	Proof. Immediate when k = 2 in Theorem 2.4 in [3].
		3. Proof of Theorem 1

ℓ

, where ℓ is finite, then a, b and c are universal, i.e., a, b and c depend only on x and t.

Proof. Assume a, b and c depend on a jet of finite order, i.e., they depend on x, t, z 0 , . . . and z ℓ , where ℓ is fixed. Then (17) becomes

Research partially supported by a CRM-ISM post-doctoral Fellowship, by NSERC Grant RGPIN 105490-2011 and by the Ministério de Ciência e Tecnologia, Brazil, CNPq Proc. No. 303774/2009-6.


Maya Geagea

Ivaylo Genov

Zdravko Stoynov

Daria Vladikova

Anthony Chesnaud

Alain Thorel

Permeability of gases In the anode of An anode-supported SOFC

I. Introduction

In electrolyte-supported solid oxide fuel cell, the ohmic contribution to over-potential is large because of high resistivity of the electrolyte. These cells operate at high temperature (1000°C) where the mobility of mobile ions is enhanced. But high operating temperature induces high costs, requires specific sealing and sophisticated interconnects and may result in poor cell durability because of active diffusion of species. Therefore, anode-supported fuel cells with thin associated electrolyte layer were extensively investigated for intermediate temperature operation (600-800°C) in the last few years in order to improve the electrochemical performance and the cell durability for long-term operation. The performance of a SOFC is controlled by various polarizations or losses: ohmic losses (due to the electrical resistance of the components), activation losses (due to low kinetics of charge transfer of the electrochemical reactions; it mainly depends on the electrode-electrolyte interface) and concentration losses (when the transport of fuel and the transport of products of reaction are retarded). Even though thin electrolyte layer in anode-supported cell reduces drastically the electrolyte ohmic contribution, activation and concentration polarizations of the anode still penalize the cell performances.

Therefore, an optimization of the overall performance of the fuel cell requires optimizing the parameters which control the activation and concentration overpotentials. In this aim, several studies show the strong impact of microstructure parameters (1) on the electrode performance. According to Virkar et al. (2), Sakamoto et al (3) and Lee et al. (5), the microstructural characteristics of the anode in anodesupported SOFCs are considered having crucial effects on the electrochemical performance, and more specifically on the concentration polarization. This latter occurs when the demand for reactants exceeds the capacity of the porous cermet anode to supply them by gas diffusion mechanisms (4) and, in the case of a SOFC anode, when the rate at which water is produced exceeds the ability of the porosity to evacuate water. It is obviously related to the anode microstructure in terms of porosity (tortuosity, pore volume fraction, size and distribution). In consequence, an appropriate path for the gases reduces the diffusional (concentration) polarization loss. Since the various requirements for an efficient anode (high ionic and electronic conductivity, homogeneous distribution of all phases and uniform percolation, high gas permeability for transport of hydrogen and of reaction products, fine particle size for high electro-catalytic activity) could not be all investigated in one anode layer in order to achieve high power densities (6)(7)(8)(9), the investigation of our case study is only limited to the modification of the porosity characteristics by using graphite pore former particles and conventional materials (YSZ and Ni). Three potential diffusional mechanisms can occur once the gas is passing through porous electrodes. These mechanisms depend essentially on the nature of the gas and on the microstructure of the porous medium (anode). The three mechanisms are: Molecular, Viscous and Knudsen diffusion. According to W.He et al (10), in the case of porous media having the pore diameter much larger than the mean free path of gas molecules at a given temperature, the binary diffusion (also called the Darcy regime) takes the advantage over the Knudsen diffusion process. The effective diffusion coefficient is then defined based on Fick's Law as an approach to evaluate gas diffusion as Equation [1]: [1] where D ij is the binary diffusivity, ø the porosity (vol%) and τ the tortuosity of the porous electrode. However, if the diffusion is taking place in porous media having small pores, with low pressure, Knudsen diffusion will take place since the interactions involve more frequent collisions between gas molecules and walls of the porous media than with other gas molecules. Cases exist where both mechanisms Knudsen and molecular one are very important. Then the total diffusivity of gas species is written as shown in Equation [2]: [2] where D ij is the ordinary binary diffusion coefficient of two species (i in j) defined as following from the Chapman-Enskog theory Equation [3]: [3] M i and M j are molecular weight (g.mol -1 ), p is the total pressure (atm), R is the ideal gas constant (J.Kg -1 K -1 ), T is the temperature in K, σ ij and dimensionless ωD are Lennard-Jones parameters. D iK is the Knudsen diffusivity with d p as the mean pore size as given in Equation [4]: [4] Taking in consideration that in most porous media, especially in compact solid, paths for the diffusion of gas molecules within the pores are random and tortuous, the effective diffusion coefficient is therefore given by Eq [5]: [5] In this study, our effort to develop anode substrate for the practical application of the low or intermediate temperature anode-supported type SOFC is reported. Our studies focused on the optimization of the anode microstructure through permeability and mercury porosimetry tests for anode substrates prepared using cold pressing.

II. Experimental procedures

Yttria-stabilized zirconia (Tosoh, Japan) and nickel oxide (Novamet, USA) were used to prepare NiO-YSZ composite anodes. The average particle size of YSZ and NiO was 0.1 and 4 µm respectively. The pre-mixed powders were ball-milled for 3 hours. We added plate shaped graphite (0, 15, 20 and 25 vol % respectively) with 2 µm in diameter to provide the additional porosity of anode substrate to the appropriate level. The granules of NiO-YSZ mixture were compacted and uniaxially pressed into pellets with 1.5 mm in thickness and 25 mm in diameter, and then sintered at 1350 °C in air. The porosity of the NiO-YSZ composite was measured via Hg porosimetry. The microstructure of the various NiO-YSZ composite samples was investigated with a scanning electron microscope (SEM FEG FEI Nova NanoSEM 450) combined with Image Analysis. For this purpose, the investigated samples were impregnated in resin and polished. The numerical images were segmented to isolate pores and allow calculating the pore volume percentage using MATLAB software package. An average of 15 micrographs were taken for each sample to ensure the robustness of the analysis (homogeneous distribution of pores throughout the whole sample, at the edges as well as in the core). The maximum error for the analysis lies within 5%. Four different volume percentages of pore formers were investigated; for each percentage, two samples were studied before and after reduction. Table I summarizes the different samples under consideration. Samples with the annotation (G) represent unreduced samples and (B) is for reduced samples in H 2 . Reduction of samples was carried out in situ at 700°C in a 10% H 2 + 90% N 2 mixture for 2 hours prior to measurements. The fuel flow rate was maintained at 140 mL/min.

III. Results and discussion

III.1 Microstructural analysis

In an attempt to promote a high porosity, and to favour a percolating distribution of Ni within the anode, a hierarchical microstructure was looked for in the final material by purposely interrupting the grinding step prior to sintering. Therefore, mixing NiO particles with YSZ granulates will lead in the final anode to the repartition of a network of pores and Ni surrounding a skeleton of significantly large YSZ grains. This hierarchical microstructure appears positive in terms of conductivity and limited concentration overpotentials, but must be thoroughly tailored since large final granulates do not play in favour of an increase of TPB. SEM micrographs of the four samples A1, A2, A4, and A6 are given in Figure 1(a)-(b). As expected, all samples present significant granulates. Interestingly, for the same grinding time granulates are increasingly eroded as a function of increasing amount of pore formers (graphite). As anticipated pores and NiO cover the YSZ skeleton, which leads to an easier percolation for electronic conductivity and gas transport. It is seen that NiO-YSZ sample with 15% volume graphite has a structure close to that of A1B sample without graphite, but with smaller granulates. On the other hand, the compacted pellet with 20%vol graphite shows similar microstructure to the one with 25%vol with a serious reduction of granulates size and a drastic increase of porosity. Results of Image Analysis of the microstructure of all the investigated samples (Figure 2) are summarized in Table II. Though a general trend toward an increase of porosity as a function of increasing amount of pore former is found, a significant discrepancy is observed between the measurements of porosity by Hg porosimetry and by Image Analysis, especially for large volume fractions. Though Hg porosimetry is known to be less efficient to assess very small pores, small pores do not account for much in the total volume fraction. Though Image Analysis has been carried out thoroughly on back scattered SEM images on sample impregnated with resin, then polished, it seems more probable that the underestimation of the porosity for large volume fraction of pores is due to a segmentation artifact at interfaces between solid phases and resin. On the other hand, about the same value is recorded by Hg intrusion porosimetry for samples A2 and A4, though A4 has 5% more of pore former. And finally it is found that the porosity of sample A4 after reduction is smaller than before, which is also counter-intuitive and requires further investigation

TABLE I. Volume fraction of pores in different samples

Samples A1(G) A1(B) A2(G) A2(B) A4(G) A4(B) A6(G) A6(B)

Graphite

III.2 Conductivity measurements

Conductivity measurements were evaluated on reduced samples by DC resistivity. Measurements were carried out at room temperature applying advanced four points method. No conductivity was noticed for samples with 0 and 15% pore former while electronic conductivity was recorded for higher amounts. This clearly indicates that the percolation of Ni is closely related to the distribution of pore formers.

III.3 Permeability analysis

Permeability analysis was performed on a new testing system especially designed for this purpose. These studies consisted on measuring the permeability resistance as gas penetrates through porous media. It is defined as the ratio of the pressure P (mm H 2 O) over the gas flow q flow [mL/min]: [8] Equation [8] is very similar to the Darcy low; however, it is more convenient and illustrative for gases permeability studies in porous ceramic media. The permeability measurements were carried out at room temperature with different gases: air, O 2 , N 2 , Ar, dry and humidified H 2 . The gas flows varied from 0 to 140 mL/min. The humidity (3% H 2 O) was ensured by passing the gas flow through watered vessel at room temperature. Two configurations of the gas flow set-up, (i) lateral, and (ii) transversal, were investigated (Figure 4). Measurements in both configurations showed that the lateral configuration could be neglected in the mixed set-up. Moreover, all the investigations were performed using transversal configuration, which combines both directions of gas flow. As different gases were supplied, the flow of species through the anode layer was provided by the pore structure of the anode. As anticipated and as shown in Figure 5(a), the gas permeability increases with the increase of volume fraction of pore former for non-reduced ones. The correlation with Hg porosimetry and Image Analysis (IA) on SEM images (see Table II) is especially valid for the highest amounts of pore formers, whereas the difference for lower volume fraction (samples A1G, A2G and A4G) is not significant. Although reduced samples showed higher permeability due to the additional porosity after reduction, (as expected and certified by porosimetry mercury and Image Analysis results listed in Table II), sample A2 showed an increase of the permeability resistance after reduction (See Figure 4(b)). This increase can be explained by a reduction problem. Further investigations on samples with the same pore former amount are actually in course to explain more this behavior. Measurements at room temperature and constant conditions (set-up configuration), for different gases having various molecular weight, registered an increase of the permeability with the decrease of the molecular weight for all the samples before, as well as after reduction (Figure 5(a),5(b)). Argon molecule shows higher permeability even though its molecular weight is the highest among investigated gases. According to James C.Maxwell, since gas molecules have different speeds and different degrees of freedom (3 for monoatomic, 5 for diatomic) to each degree of freedom of a molecule is associated a certain kinetic energy. Assuming as a first approximation that the thermal energy will be fully used to ensure translations in the three directions of space (three degrees of freedom), E kin is written: (6) Diatomic gases such as H 2 , O 2 , N 2 and Air have 5 degrees of freedom, and can therefore store more thermal energy (5/2.k B T); these molecules have the potential to exhibit a higher translational kinetic energy that could result in a lower permeability than monoatomic gases, as seen in Figure 5 where a kink in permeability is observed for the only monoatomic gas studied in this work. This experimental fact needs further studies and explanation. The observed graphs showed a kink in the vicinity of molecular weights to 28-30 (nitrogen/air). For higher molecular weight (as that of Argon) the resistivity increases faster. This experimental fact needs further studies and explanation.

IV. Conclusion:

In this study, gas permeation behavior through various pore former volume fractions in anode supports for Intermediate temperature application was investigated. The study was achieved using different gases. Investigations were performed using different analysis techniques such as permeability tests, image analysis and porosimetry mercury intrusion. The results showed that the gas permeability improved by increasing the porosity thus decreasing the size of powder particles. Therefore, the effect of molecular diffusion will become low after a certain volume fraction of pore former and the effect of Knudsen diffusion will be significant. However, gas permeability also depends on molecular weight. The results showed that at a certain pore former volume fraction, the diffusion phenomenon is in a transition region between Knudsen and Molecular diffusion.

Figure 1

 1 Figure 1(a): Backscattered electrons SEM micrographs

Figure 1

 1 Figure 1(b): Backscattered electrons SEM micrographs of samples sintered with 20%, 25% graphite

Figure 3 .

 3 Figure 3.Gas flow configurations of the experimental set-up: (i) lateral; (ii) transversal

Figure 4 :Fig. 5 .

 45 Figure 4: Permeability Resistance as a function of porosity

Table II .

 II Microstructural analysis of samples

	Samples/ pore former	A1(G) 0%	A1(B) 0%	A2(G) 15%	A2(B) 15%	A4(G) 20%	A4(B) 20%	A6(G) 25%	A6(B) 25%
	Poro Hg	15	24	17	20	15	48	40	55
	Poro (I.A)	-	24	-	23	32,5	30,6	33,1	33,1
	Mean
	Pore ø
	Volume	1,5	2,8	2,5	2,4	2,9	3,6	4	4,4
	(µm)

Figure 2. Comparision of distribution of pores after reduction through Image Analysis 50µm A4B 50µm A6B


Ghislain Atemezing

Hacene Cherfi

Anh Huynh

Martin Coste

Towards A Smarter Online Debates Analysis: A French Public Portal Use-Case

Keywords: speeches, debates, semantic annotations, interoperability, TopicPages, ITM, CAM-Manager

This paper describes a semantic annotation workflow and visualization of politician speeches from the French Government Portal. The pipeline consists of an ontology modeling, a pre-processing and customized annotations of 7200 random speeches of public actors during the last 30 years using Content Augmentation Manager (CA-Manager) and a visualization component based on the central notion of a Topic. The implementation of the proof-of-concept shows that adding semantic annotations to the current publication of official speeches in RDF leads to efficient search of debates and better interconnection with external datasets.

Introduction

Information is flowing on the Web in many formats, such as CSV, PDF or HTML. For human readable formats such as HTML, the basic challenge is to add href links between pages as it was at the early days of the Web. In the recent years, the idea of publishing data and linking them has gone mainstream since the first draft by Sir. Tim Berners-Lee [START_REF] Berners-Lee | Linked data: Design issues[END_REF]. Almost twenty years after, publishers and governments bodies are now witnessing the benefits of using semantic technologies to publish and better discover their content on the Web. Although there exist some guidelines to transform legacy government data into linked data [START_REF] Hyland | Best Practices for Publishing Linked Data[END_REF], there are some use-cases where a customized pipeline based on specific types of data provide efficient answer. It is the case for the French portal vie-publique service which contains more than 100,000 speeches in ATOM format, and is seeking to offer a novel approach to discover and explore its content based on semantic technologies.

We propose in this paper a semantic annotation workflow and a visualization based on the concept of a Topic. A TopicPage is the central element of the visualization obtained after annotating and enriching a pivot page with the concepts extracted by using the ontology for debates. The paper is described as follow: we first introduce the background and motivation (Section 2), and further detailed the components of the semantic annotation workflow (Section 3) and the ontology modeled for the debates online (Section 4). We then exemplify the implementation in Section 5, followed by the steps for the automatic generation of the TopicPage (Section 6). A brief conclusion remark is provided at the end of the paper (Section 7).

Motivation and Background

The French portal vie-publique.fr service provides citizens with resources and useful data in order to understand the main topics which are subject to debate of the general public in France. The website http://www.vie-publique.fr also contains a collection of more than 100,000 speeches from public actors, such as the President, government, members of congress or mayors during the last thirty years. The main formats used by the portal is the atom syndication format (ATOM), which is an XML format, and which is in turn converted into HTML. Our aim is to extract relevant entities described in an ontology for speeches, and be able to extract persons, organizations, location and related information that can be useful to retrieve in a speech.

Semantic Annotation Workflow

The worflow used is based on a central component: the Content Augmentation Manager (CA-Manager). The content augmentation manager (CA-Manager) is in charge of processing XML and HTML documents. This module extracts the concepts and entities detected using text mining techniques with the resources coming from the ITM module. In the scenario presented in this paper, we use the GATE tool [START_REF] Cunningham | GATE: an architecture for development of Robust HLT applications[END_REF] for the entity extraction. In [START_REF] Cherfi | Ca-manager: a middleware for mutual enrichment between information extraction systems and knowledge repositories[END_REF], Hacene et al., CA-Manager uses an ontology-based annotation schema to transform heterogeneous content (text, image, video, etc.) into semantically-driven and organized contents. Figure 1 represents the five components that support the building and customized management pipelines for semantic contents annotations. These components are the following:

1. Extraction: identify and tag domain-oriented knowledge (terms, named entities, relations) from content, performed by exiting Information Extraction (IE) tools (such as GATE, Luxid, GeolSemantics, EIDON, etc.); 2. Consolidation: reconcile extracted knowledge with the domain ontology and the content of the knowledge repository (instances and property values); 3. Storage: export and store the reconciled knowledge; 4. Validation: let the human user validate the suggested annotations and knowledge; 5. Enrichment: export new validated terms and entities into the IE's linguistic resources (gazetteers, grammars, named entity lists).

The rest of the components combined with CA-Manager to obtain a customized TopicPage visualization (Section 6) of any content are the following: 4 Debatescore: A vocabulary for Online Debate

The debatescore ontology3 is used during the annotation process. It is an extension of the Dublin Core [START_REF] Initiative | Dublin core metadata element set, version 1.1: Reference description[END_REF] vocabulary with a unique class Debate4 , aiming at giving to citizens and others to contribute or participate in taking public decision or political one. A debate should have the following requirements:

-An issued date captured with the property dcterms:issued with an xsd:date as range. -A title captured with the property dcterms:title.

-A debate belongs to a category, which are five (05): mediation, concertation, public debate, consultation and public pool. Those values are modeled as SKOS [START_REF] Miles | Skos simple knowledge organization system reference[END_REF] concepts. -A debate is target to a specific audience with a status. Some controlled status are used to represent the status of a debate.

The ontology is loaded into ITM for annotating the speeches accordingly. Each speech is modeled in RDF as an individual of the Debate class.

Implementation

We extracted 7,100 XML documents that are first encoded in UTF-8. During the annotation, 11% of the documents were rejected (i.e., 789 documents) because they were malformed or were non-UTF-correctly encoded. For these documents, we need further investigation. However, we still have enough material for our experiment. The Figure 3 shows how one can manually add documents in our knowledge base in order to semantically annotate them against the terminology set shown on the left hand side of the page (Public information, organizations, persons, etc.). Each document annotation in CA-Manager leads to the creation of its corresponding object in the knowledge base management system (ITM). The most important part of each annotation shown in Figure 5 is the set of list of subjects identified from the terminologies (e.g., Protocole, MINISTRE ) or suggested by the CAM system (e.g., France which exists in our base as an organization but not as a location) as shown in Figure 5.

Automatic TopicPage Generation

The notion of topic here has its historical origin in the the Topic Maps standard [START_REF] Biezunski | Topic maps at a glance[END_REF], meaning a unique subject, concept or business object. A TopicPage is the visualization component of a "Topic" with relevant links to resources that are internal or external to the Enterprise knowledge base.

Four steps are needed to generate a TopicPage:

1. Selection of the pivot page. In this step, the user identifies which is the HTML or XML page containing the central topic, or the pivot page. 2. Convert into RDF all the corpus annotated and Pivot Page. The database obtained can be loaded in a SPARQL endpoint. 3. Creation of a new dataset based on the concepts from the Pivot page by using SPARQL filtering and CONSTRUCT queries. 4. Generate visualization, by using a templating framework (such as RDF2XML) to automatically generate relevant views of the TopicPage, and adding links to additional information.

The result of the visualization is available at http://labs.mondeca.com/poc/ vp/topicPageNosRegions.html corresponding to the central topic "Our region" located at http://www.vie-publique.fr/forums/nos-regions-demain-parlons-en. html.

Note that for the TopicPage, the rendering of the speeches view is better interpreted using Chrome browser. We will further investigate the compatibility with other browsers.

Conclusion and Future Work

We have presented in this paper a customized pipeline for annotating and enriching a data portal on public speeches and debates. We have first presented the motivation of our approach, followed by the description of the main components of the workflow, to extracting and generating knowledge based on the debatescore vocabulary. We have further implemented our approach on 7200 random speeches of public actors during the last 30 years in France. By doing so, we claim that such approach leads to a new way of publishing, discovering and navigating contents based on a pivot page or TopicPage, which aggregates personalized information surrounding a topic. Our next challenge is to index the RDF dataset to build a search index on-top of the RDF dataset. Another direction is to identify other similar datasets on debates to link to, in order to have fully 5-star Linked Data.

Fig. 1 .

 1 Fig. 1. CA-Manager's main components

Fig. 2 .

 2 Fig. 2. Main components of the workflow for storing vocabularies, detecting and reconciling concepts in text, export and topicPage visualization.

Fig. 3 .

 3 Fig. 3. An example of the annotation process for 5 speeches extracted from our corpus.

Fig. 4 .

 4 Fig. 4. An example of the annotations stored in ITM which have been extracted from a document Speech annotated in CA-Manager.

Figure 6

 6 Figure 6 depicts for our use-case, the concepts retrieved associated respectively to EuroVoc multilingual thesaurus 5 reference data, specific concepts annotated in the portal website and the locations.

Fig. 5 .

 5 Fig. 5. An example of entities and concepts retrieved of a document in CA-Manager.

Seclect&Pivot&Page&Fig. 6 .

 6 Fig. 6. Steps for generating customized TopicPage visualization.

Fig. 7 .

 7 Fig. 7. Four categories detected in a politician speech. From left to right, persons, subjects, organizations and locations.

http://www.w3.org/TR/skos-reference/

https://github.com/MondecaLabs/rdf2xml

https://raw.githubusercontent.com/MondecaLabs/vocabs/master/ debatescore.ttl

goo.gl/d712R4

http://eurovoc.europa.eu/

Acknowledgments. The authors want to thank Olivier Garry and team at DILA for the interactions to provide us with input and comments.