Pierre Dolbeault
email: pierre.dolbeault@upmc.fr

Pierre Dolbeault On

On quaternionic functions

Keywords: 2010 Mathematics Subject Classification. Primary 53C26; Secondary 30D30, 32A27 Cauchy-Fueter, hyperholomorphy, residue

Several sets of quaternionic functions are described and studied. Residue current of the right inverse of a quaternionic function is introduced in particular cases.

Introduction

We will work with the definition of quaternions using pairs of complex numbers and with a modified Cauchy-Fueter operator that have been introduced in [CLSSS 07]. We will only use right multiplication; the (right) inverse of a nonzero quaternion is defined. We will consider (for simplicity) C ∞ IHI-valued quaternionic functions defined on an open set U of IHI containing 0. If such a function does not vanish over U, it has an (algebraic) inverse which is defined almost everywhere on U. Examples are given (section 2).

The origin of this research is a tentative of extension to right inverse of a quaternionic function of the notion of residue current of a meromorphic differential 1-form of one complex variable, which will be developed in section 5. In one complex variable, if the given function is holomorphic, with isolated zeros of finite multiplicity, its inverse is meromorphic, then holomorphic outside the set of poles; so it is natural to search when this property extends to hyperholomorphic functions.

In section 3, we characterize the quaternionic functions which are hyperholomorphic and whose inverses are hyperholomorphic almost everywhere, on U, as the solutions of a system of two non linear PDE. We only find non trivial examples of a solution, showing that the considered space of functions is significant: we will call theses functions hypermeromorphic. This also defines a space of germs of functions at 0.

In section 4, we try to describe a subspace H U of hyperholomorphic and hypermeromorphic functions defined almost everywhere on U, having "good properties for addition and multiplication"; we obtain again systems of non linear PDE, and we give first results on, mainly unknown, spaces of functions.

In section 5, we first recall Cauchy principal value and residue current in C, locally at 0. Afterwards, we define and study, locally, Cauchy principal value and residue current for the inverse of a quaternionic function, in very particular cases, and in relation with the classical theory in two complex variables.

This paper is a first announcement of a more complete one in progress.

Quaternions. IHI-valued functions. [CLSSS 07]

2.1. Quaternions. If q ∈ IHI, then q = z 1 + z 2 j where z 1 , z 2 ∈ C, hence IHI ∼ = C 2 ∼ = IR 4 as complex or real vector space. We have: z 1 j = jz 1 (by computation in real coordinates); by definition, the modulus of

q is || q ||= (|z 1 | 2 + |z 2 | 2 ) 1 2 . The conjugate of q is q = z 1 -z 2 j. Let * denote the (right) multiplication in IHI: q * q = (z 1 + z 2 j) * (z 1 -z 2 j) = |z 1 | 2 -z 1 z 2 j + z 2 jz 1 -z 2 jz 2 j = |z 1 | 2 + |z 2 | 2 , then: the (right) inverse of q = z 1 + z 2 j is: (|z 1 | 2 + |z 2 | 2 ) -1 q = (|z 1 | 2 + |z 2 | 2 ) -1 (z 1 -z 2 j). Moreover: (|z 1 | 2 + |z 2 | 2 ) -1 (z 1 -z 2 j) * (z 1 + z 2 j) = 1, so the right inverse of q -1 is q. 2.2. Quaternionic functions. Let U be an open set of IHI ∼ = C 2 and f ∈ C ∞ (U, IHI), then f = f 1 + f 2 j, where f 1 , f 2 ∈ C ∞ (U, C).
The complex valued functions f 1 , f 2 will be called the components of f . Remark that IHI is a real vector space in which real analysis is valid, in particular differential forms, distributions and currents are defined in IHI.

Remark that

∂f 1 ∂z 1 j = j ∂f 1 ∂z 1 and analogous relations.

Modified Cauchy

-Fueter operator D. Hyperholomorphic func- tions. ([CLSSS 07], [F 39]). For f ∈ C ∞ (U, IHI), with f = f 1 +f 2 j, where f 1 , f 2 ∈ C ∞ (U, C), Df (q) = 1 2 ∂ ∂z 1 + j ∂ ∂z 2 f (q) = 1 2 ∂f 1 ∂z 1 - ∂f 2 ∂z 2 (q) + j 1 2 ∂f 1 ∂z 2 + ∂f 2 ∂z 1 (q). A function f ∈ C ∞ (U, IHI) s said to be hyperholomorphic if Df = 0.
Characterization of the hyperholomorphic function f on U:

∂f 1 ∂z 1 - ∂f 2 ∂z 2 = 0; ∂f 1 ∂z 2 + ∂f 2 ∂z 1 = 0, on U.
The conditions:f 1 is holomorphic and: f 2 is holomorphic are equivalent. So holomorphic functions will be identified with hyperholomorphic functions f such that f 2 = 0.

Let

f ′ = f ′ 1 + f ′ 2 j, f " = f " 1 + f " 2 j be two hyperholomorphic functions. For every α ∈ IHI, D(f ′ α) = 0, D(f ′ + f ") = Df ′ + Df " = 0.
Proposition 2.1. The set H of almost everywhere defined hyperholomorphic functions is an IHI-right vector space.

Proposition 2.2. Let f ′ , f " be two hyperholomorphic functions. Then, their product f ′ * f " satisfies:

D(f ′ * f ") = Df ′ * jf " + f ′ ( ∂ ∂z 1 ) + f ′ j ∂ ∂z 2 f " Proof. f ′ = f ′ 1 + f ′ 2 j, f " = f " 1 + f " 2 j be two hyperholomorphic functions. We have: f ′ * f " = (f ′ 1 +f ′ 2 j)(f " 1 +f " 2 j) = f ′ 1 f " 1 -f ′ 2 f " 2 +(f ′ 1 f " 2 +f ′ 2 f " 1 )j Compute 1 2 ∂ ∂z 1 + j ∂ ∂z 2 f ′ 1 f " 1 -f ′ 2 f " 2 + (f ′ 1 f " 2 + f ′ 2 f " 1 )j
By derivation of the first factors of the sum f ′ * f ", we get the first term:

1 2 ∂f ′ 1 ∂z 1 + j ∂f ′ 1 ∂z 2 (f " 1 + f " 2 j) + 1 2 ∂f ′ 2 ∂z 1 + j ∂f ′ 2 ∂z 2 jj(f " 2 -f " 1 j) = 1 2 ∂f ′ 1 ∂z 1 + j ∂f ′ 1 ∂z 2 (f " 1 + f " 2 j) + 1 2 ∂f ′ 2 j ∂z 1 + j ∂f ′ 2 j ∂z 2 j(f " 2 j + f " 1 ) = Df ′ * jf " By derivation in 1 2 ∂ ∂z 1 + j ∂ ∂z 2 f ′ 1 f " 1 + f ′ 2 jf " 2 j + (f ′ 1 f " 2 j + f ′ 2 jf " 1 )
of the second factors of the sum f ′ * f ", we get the second term (up to factor 1 2 ):

f ′ 1 ∂f " 1 ∂z 1 +f ′ 1 j ∂f " 1 ∂z 2 +f ′ 1 ∂f " 2 ∂z 1 j+f ′ 1 j ∂f " 2 ∂z 2 j+f ′ 2 j ∂f " 2 ∂z 1 j+f ′ 2 j ∂f " 2 ∂z 2 +f ′ 2 j ∂f " 1 ∂z 1 +f ′ 2 jj ∂f " 1 ∂z 2 = (f ′ 1 +f ′ 2 j)( ∂ ∂z 1 )(f " 1 +f " 2 j)+(f ′ 1 +f ′ 2 j)j ∂ ∂z 2 (f " 1 +f " 2 j) = (f ′ 1 +f ′ 2 j)( ∂ ∂z 1 )+(f ′ 1 +f ′ 2 j)j ∂ ∂z 2 (f " 1 +f " 2 j) = f ′ ( ∂ ∂z 1 ) + f ′ j ∂ ∂z 2 f "
If the components of f ′ and f " are real, the second term is:

1 2 (f ′ 1 + f ′ 2 j)( ∂ ∂z 1 + j ∂ ∂z 2 )(f " 1 + f " 2 j) = f ′ * Df " i.e.
Corollary 2.3. The set H IR of almost everywhere defined hyperholomorphic functions whose components are real is an IR-right algebra.

2.4. Null set and inverse of a quaternionic function. We call inverse of a function f : q → f (q), the function

f -1 : q → f (q) -1 . Let f = f 1 + f 2 j be a quaternionic function on U. The null set Z(f ) satisfies: f 1 = 0; f 2 = 0, then Z(f ) is of measure 0 in U. Ex.: f 1 = z 1 ; f 2 = z 2 , then Z(f ) = {0}. Note that if f is holomorphic, then, f 2 ≡ 0 and Z(f ) is a complex hypersurface in C 2 .
Inversion and hyperholomorphy. The inverse of the quaternionic function f is the peculiar quaternionic function defined almost everywhere on U:

1 f = (|f 1 | 2 + |f 2 | 2 ) -1 (f 1 -f 2 j) = |f | -1 f
where f is the (quaternionic) conjugate of f .

Assume f to be hyperholomorphic and

Z(f ) = {0}, then 1 f is not neces- sarily hyperholomorphic outside {0}. Ex.: f = z 1 + z 2 j, then 1 f = (z 1 z 1 + z 2 z 2 ) -1 (z 1 -z 2 j); D( 1 f ) = 0,
Example of a function hyperholomorphic outside 0.

H(q) = (z 1 z 1 + z 2 z 2 ) -2 (z 1 -z 2 j) is hyperholomorphic since: DH(q) = 1 2 (z 1 z 1 + z 2 z 2 ) -3 (-2z 1 z 1 + z 1 z 1 + z 2 z 2 -2z 2 z 2 + (z 1 z 1 + z 2 z 2 )) = 0. But F = z 1 +z 2 j is not hyperholomorphic: the conjugate of F is F = z 1 -z 2 j; (z 1 + z 2 j) * (z 1 -z 2 j) = z 1 z 1 + z 2 z 2 . So F -1 = (z 1 -z 2 j)(z 1 z 1 + z 2 z 2 ) -1
, and

H(q) = F -1 (z 1 z 1 + z 2 z 2 ) -1 = F -1 |F | .
(H is the Cauchy kernel for the modified Cauchy-Fueter operator D). Inverse of a holomorphic function.

Let f = f 1 + 0j be a hyperholomorphic function. Then f -1 = f -1 1 + 0j and f -1 is hyperholomorphic outside of the complex hypersurface Z(f ). Remark that Z(f ) is a subvariety of complex dimension 1, then of measure zero, in U.

We will consider almost everywhere defined hyperholomorphic functions on U. Ex.: holomorphic, meromorphic functions.

Hyperholomorphic functions whose inverses are

hyperholomorphic almost everywhere.

Proposition 3.1. The following conditions are equivalent (i) the function f = f 1 + f 2 j and its right inverse are hyperholomorphic, when they are defined;

(ii) we have the equations:

(3) (f 1 -f 1 ) ∂f 1 ∂z 1 -f 2 ∂f 2 ∂z 1 -f 2 ∂f 1 ∂z 2 = 0, (4) f 2 ∂f 1 ∂z 1 + ∂f 2 ∂z 1 (f 1 -f 1 ) -f 2 ∂f 2 ∂z 2 = 0. Proof. Let f = f 1 + f 2 j be a hyperholomorphic function and g = g 1 + g 2 j = |f | -1 (f 1 -f 2 j) its inverse; so g 1 = |f | -1 f 1 ; g 2 = -|f | -1 f 2 , where |f | = (f 1 f 1 + f 2 f 2 ). Dg(q) = 1 2 ∂ ∂z 1 + j ∂ ∂z 2 g(q) = 1 2 ∂g 1 ∂z 1 - ∂g 2 ∂z 2 (q) + j 1 2 ∂g 1 ∂z 2 + ∂g 2 ∂z 1 (q) ∂g 1 ∂z 1 = |f | -1 ∂f 1 ∂z 1 -|f | -2 f 1 ∂f 1 ∂z 1 f 1 + f 1 ∂f 1 ∂z 1 + ∂f 2 ∂z 1 f 2 + f 2 ∂f 2 ∂z 1 , etc ( * * ) 2|f | 2 Dg = (f 1 f 1 + f 2 f 2 )( ∂f 1 ∂z 1 + ∂f 2 ∂z 2 ) -f 1 f 1 ∂f 1 ∂z 1 -f 1 f 1 ∂f 1 ∂z 1 -f 1 f 2 ∂f 2 ∂z 1 -f 1 f 2 ∂f 2 ∂z 1 -f 1 f 2 ∂f 1 ∂z 2 -f 1 f 2 ∂f 1 ∂z 2 . -f 2 f 2 ∂f 2 ∂z 2 -f 2 f 2 ∂f 2 ∂z 2 +j (f 1 f 1 + f 2 f 2 )( ∂f 1 ∂z 2 - ∂f 2 ∂z 1 ) -f 1 f 1 ∂f 1 ∂z 2 -f 1 f 1 ∂f 1 ∂z 2 -f 1 f 2 ∂f 2 ∂z 2 -f 1 f 2 ∂f 2 ∂z 2 +f 1 f 2 ∂f 1 ∂z 1 + f 1 f 2 ∂f 1 ∂z 1 + f 2 f 2 ∂f 2 ∂z 1 + f 2 f 2 ∂f 2 ∂z 1 f being hyperholomorphic, g hyperholomorphic is equivalent to: ( * * ) +f 1 f 1 ∂f 2 ∂z 2 + f 2 f 2 ∂f 1 ∂z 1 -f 1 f 1 ∂f 1 ∂z 1 -f 1 f 2 ∂f 2 ∂z 1 -f 1 f 2 ∂f 1 ∂z 2 -f 2 f 2 ∂f 2 ∂z 2 +f 2 ∂f 2 ∂z 1 (f 1 -f 1 ) = 0 +f 2 f 2 ∂f 1 ∂z 2 -f 1 f 2 ∂f 2 ∂z 2 -f 1 f 2 ∂f 2 ∂z 2 + f 1 ∂f 1 ∂z 2 (f 1 -f 1 ) + f 1 f 2 ∂f 1 ∂z 1 + f 1 f 2 ∂f 1 ∂z 1 +f 2 f 2 ∂f 2 ∂z 1 = 0
After conjugaison of the first equation, and using

(1)

∂f 1 ∂z 1 - ∂f 2 ∂z 2 = 0; ∂f 1 ∂z 2 + ∂f 2 ∂z 1 = 0,
we get:

( * * ) +f 2 f 2 ∂f 1 ∂z 1 +f 1 (f 1 -f 1 ) ∂f 1 ∂z 1 -f 1 f 2 ∂f 2 ∂z 1 +f 2 ∂f 2 ∂z 1 (f 1 -f 1 )-f 1 f 2 ∂f 1 ∂z 2 -f 2 f 2 ∂f 2 ∂z 2 = 0 +f 1 f 2 ∂f 1 ∂z 1 +(f 1 -f 1 )f 2 ∂f 1 ∂z 1 +f 2 f 2 ∂f 2 ∂z 1 +f 1 ∂f 1 ∂z 2 (f 1 -f 1 )+f 2 f 2 ∂f 1 ∂z 2 .-f 1 f 2 ∂f 2 ∂z 2 = 0
Assume f 1 = 0, f 2 = 0. After multiplication of the first equation by f 1 and of the second by -f 2 , and sum, we get

(f 1 -f 1 ) ∂f 1 ∂z 1 -f 2 ∂f 2 ∂z 1 -f 2 ∂f 1 ∂z 2 = 0
By an analogous process, we get:

f 2 ∂f 1 ∂z 1 + ∂f 2 ∂z 1 (f 1 -f 1 ) -f 2 ∂f 2 ∂z 2 = 0
Corollary 3.2. If f satisfies the conditions of the Proposition, the same is true for αf with α ∈ IR.

Let f = f 1 + 0j be an almost everywhere holomorphic function, then the condition (ii) of Proposition 3.1 is satisfied. Now give another example of quaternionic function satisfying the conditions of Proposition 3.1:

Proposition 3.3. Let f = f 1 + f 2 j, with f 1 = z 1 + z 1 + z 2 + z 2 + A, f 2 = -z 1 -z 1 + z 2 + z 2 + B, A, B ∈ IR, then: f and f -1 outside the zero set of f , are hyperholomorphic The null set of f = f 1 + f 2 j, for f 1 , f 2 as above, for A = B = 0, is: z 1 + z 1 + z 2 + z 2 = 0; -z 1 -z 1 + z 2 + z 2 = 0
i.e, by difference and sum:

z 1 + z 1 = 0; z 2 + z 2 = 0, i.e. x 1 = 0; x 2 = 0 in IR 4 . Proof. f 1 = z 1 + z 1 + z 2 + z 2 + A, A constant; then: (3) -f 2 ∂f 2 ∂z 1 -f 2 ∂f 1 ∂z 2 = 0, i.e. -f 2 ∂f 2 ∂z 1 -f 2 = 0, Try: f 2 real. Then ∂f 2 ∂z 1 = -1 and f 2 = -z 1 + C(z 1 , z 2 , z 2 ) = -z 1 -z 1 + C ′ (z 1 , z 2 , z 2 ), with C ′ real and ∂C ′ ∂z 1 = 0. From (4), ∂f 2 ∂z 2 = ∂C ′ ∂z 2 = 1, and C ′ = z 2 + z 2 + C"(z 1 , z 2 ), with C" real. f 2 = -z 1 -z 1 + z 2 + z 2 + C"(z 1 , z 2 )
, with C" real, and ∂C" ∂z 2 = 0, ∂C" ∂z 1 = 0.

C" being holomorphic in z 2 and z 1 is a constant B. Hence:

f 2 = -z 1 -z 1 + z 2 + z 2 + B, with B ∈ IR.
4. On the space of hyperalgebraic functions 4.1. Definition. Let U be an open neighborhood of 0 in IHI ∼ = C 2 . From now on,we will only consider the quaternionic functions f = f 1 + f 2 j having the following properties: (i) when f 1 and f 2 are not holomorphic, the set Z(f 1 ) ∩ Z(f 2 ) is discrete on U;

(ii) for every q ∈ Z(f 1 ) ∩ Z(f 2 ), J α q (.) denoting the jet of order α at q [M 66], let m i = sup

α i J α i q (f i ) = 0; m i , i = 1, 2, is finite. Define: m q = inf i m i .
Remark that, in this paper, the considered peculiar examples of quaternionic functions f satisfy: the set Z(f 1 ) ∩ Z(f 2 ) is reduced to one point and that α i = 1.

Let f = f 1 + f 2 j be a quaternonic function on U and g = g 1 + g

2 j = |f | -1 (f 1 -f 2 j) its inverse; so g 1 = |f | -1 f 1 ; g 2 = -|f | -1 f 2 , where |f | = (f 1 f 1 + f 2 f 2 ). The right inverse of g is h = h 1 + h 2 j, with h 1 = |g| -1 g 1 ; h 2 = -|g| -1 g 2 ; |g| = |f | -2 (f 1 f 1 + f 2 f 2 ) = |f | -1 ; then: h 1 = |g| -1 g 1 = f 1 , ... So the right inverse of g is f .
4.2. Definition. We will call hypermeromorphic function, on U, any almost everywhere defined hyperholomorphic function whose right inverse is hyperholomorphic almost everywhere.

Thanks to Definition 4.1, meromorphic functions in one complex varable are hypermeromorphic.

From Proposition 3.3, the set M of hypermeromorphic functions is not reduced to the space of meromorphic functions in one complex variable.

Let M 0 be the set of elements f of M described in Proposition 3.3. M 0 is an IR-vector space; it also contains f -1 and the products f * f -1 = 1. Proposition 4.1. Let f , g be two hypermeromorphic functions on U, then the following conditions are equivalent:

(i) the product f * g is hypermeremorphic;

(ii) f and g satisfy the system of PDE:

g 1 ( ∂f 1 ∂z 1 + ∂f 2 ∂z 2 ) + (f 1 -f 1 ) ∂g 1 ∂z 1 + f 2 ∂g 1 ∂z 2 -f 2 ∂g 2 ∂z 1 = 0 g 1 ( ∂f 1 ∂z 2 - ∂f 2 ∂z 1 ) + (f 1 -f 1 ) ∂g 1 ∂z 2 -f 2 ∂g 1 ∂z 1 -f 2 ∂g 2 ∂z 2 = 0 Proof. Let f = f 1 + f 2 j and g = g 1 + g 2 j two hypermeromorphic functions and f * g = f 1 g 1 -f 2 g 2 + (f 1 g 2 -f 2 g 1 )j their product, then ∂f 1 ∂z 1 - ∂f 2 ∂z 2 = 0; ∂f 1 ∂z 2 + ∂f 2
∂z 1 = 0 and..., on U and, the conditions for the product to be hyperholomorphic are:

∂(f 1 g 1 -f 2 g 2 ) ∂z 1 - ∂(f 1 g 2 -f 2 g 1 ) ∂z 2 = g 1 ( ∂f 1 ∂z 1 + ∂f 2 ∂z 2 ) + f 1 ∂g 1 ∂z 1 -f 1 ∂g 2 ∂z 2 + f 2 ∂g 1 ∂z 2 -f 2 ∂g 2 ∂z 1 = 0 ∂(f 1 g 1 -f 2 g 2 ) ∂z 2 + ∂(f 1 g 2 -f 2 g 1 ) ∂z 1 = g 1 ( ∂f 1 ∂z 2 - ∂f 2 ∂z 1 ) + f 1 ∂g 1 ∂z 2 + f 1 ∂g 2 ∂z 1 -f 2 ∂g 1 ∂z 1 -f 2 ∂g 2 ∂z 2 = 0
Corollary 4.2. Let f , g be two hypermeromorphic functions on U,whose components are real, then the following conditions are equivalent: (i) the product f * g is hypermeremorphic;

(ii) f and g satisfy te system of PDE:

g 1 ( ∂f 1 ∂z 1 + ∂f 2 ∂z 2 ) + f 2 ∂g 1 ∂z 2 -f 2 ∂g 2 ∂z 1 = 0 g 1 ( f 1 ∂z 2 - ∂f 2 ∂z 1 ) -f 2 ∂g 1 ∂z 1 -f 2 ∂g 2 ∂z 2 = 0 4.3.
Definition. We will call hyperalgebraic the hypermeromorphic functions whose sum and product are hypermeromorphic: see section 4.2.

Proposition 4.3. The set M of hypermeromorphic functions on U is a subalgebra of the algebra of quaternionic functions.

Proposition 4.4. The set A of hyperalgebraic functions on U is a "field" with only associativity of the multiplication.

5. About residue current in quaternionic analysis: particular cases.

5.1. Residue current in an open set of C. [D10a], [D10b] Let ω = g(z)dz be a meromorphic 1-form on a small enough open set 0 ∈ U ⊂ C having 0 as unique pole, with multiplicity k:

g = k l=1 a -l z l + holomorphic function Note that ω is d-closed.
Let ψ = ψ 0 dz ∈ D 1 (U) be a 1-test form. In general gψ is not integrable, but the Cauchy principal value ∂ j ∂z j δ 0 where res 0 (ω) = a -1 is the Cauchy residue. We remark that δ 0 is the integration current on the subvariety {0} of U, that D = k-1 j=0 b j ∂ j ∂z j with b j = λ j a -j where the λ j are universal constants.

V p[ω](ψ) = lim
Conversely, given the subvariety {0} and the differential operator D, then the meromorphic differential form ω is equal to gdz, up to holomorphic form; hence the residue current Res[ω] = Dδ 0 , can be constructed.

m i = 1, i = 1, 2. Let ω = df f = 1 f (df 1 + df 2 j).
We want to extend ω into a current of degree 1; first, consider the part of type (1,0). Let ψ = ψ 1 dz 1 ∧ dz 2 ∧ dz 2 + ψ 2 dz 1 ∧ dz 1 ∧ dz 2 be a test form. Define:

V p[ω](ψ) = lim ε→0 |f |≥ε (|f 1 | 2 +|f 2 | 2 ) -1 (f 1 -f 2 j)(df 1 +df 2 j)∧(ψ 1 dz 1 ∧dz 2 ∧dz 2 +ψ 2 dz 1 ∧dz 1 ∧dz 2 )
We have to prove the existence of V p[ω] at least if f is hyperholomorphic (Check the proof in the classical case where f is holomorphic in one complex variable or in two complex variables where the proof is less easy, but don't need the resolution of singularities).

(df 1 + df 2 j) ∧ (ψ 1 dz 1 ∧ dz 2 ∧ dz 2 + ψ 2 dz 1 ∧ dz 1 ∧ dz 2 ) = ∂f 1 ∂z 1 ψ 1 + ∂f 1 ∂z 2 ψ 2 -( ∂f 2 ∂z 1 ψ 1 - ∂f 2 ∂z 2 ψ 2 )j dz 1 ∧ dz 1 ∧ dz 2 ∧ dz 2
Take polar coordinates:

λ =|| q ||= (|z 1 | 2 + |z 2 | 2 )
1 2 and the spherical coordinates on λS 3 . Let dσ be the volume element on S 3 , and K a convenient universal constant, then:

dz 1 ∧ dz 1 ∧ dz 2 ∧ dz 2 = Kλdλ ∧ dσ V p[ω](ψ) = lim ε→0 |f |≥ε (|f 1 | 2 +|f 2 | 2 ) -1 (f 1 -f 2 j) ∂f 1 ∂z 1 ψ 1 + ∂f 1 ∂z 2 ψ 2 -( ∂f 2 ∂z 1 ψ 1 - ∂f 2 ∂z 2 ψ 2 )j Kλdλ∧dσ
Same result for the part of type ((0,1) of V p[ω].

We will prove the existence of V p[ω] in a particular case: 5.2.2. Particular case:

f 1 = z 1 ; f 2 = z 2 . Then: Df = 0. V p[ω](ψ) = lim ε→0 |f |≥ε (|z 1 | 2 +|z 2 | 2 ) -1 (z 1 -z 2 j)(dz 1 +dz 2 j)∧(ψ 1 dz 1 ∧dz 2 ∧dz 2 +ψ 2 dz 1 ∧dz 1 ∧dz 2 ) V p[ω](ψ) = lim ε→0 |f |≥ε (|z 1 | 2 + |z 2 | 2 ) -1 (z 1 -z 2 j)(ψ 2 dz 1 ∧ dz 1 ∧ dz 2 ∧ dz 2 j) V p[ω](ψ) = lim ε→0 λ≥ε λ -2 (z 1 -z 2 j)(Kψ 2 λdλ ∧ dσj) Same result for the part of type ((0,1) of V p[ω].
This defines a current of order 0 on IHI.

5.2.3. If f 2 = 0 and f = f 1 is holomorphic, then V p[ω](ψ) = lim ε→0 |f |≥ε df f ∧ (ψ 1 dz 1 ∧ dz 2 ∧ dz 2 + ψ 2 dz 1 ∧ dz 1 ∧ dz 2 ) = V p[ω](ψ) = lim ε→0 |f |≥ε 1 f ( ∂f ∂z 1 ψ 1 + ∂f ∂z 2 ψ 2 )dz 1 ∧ dz 1 ∧ dz 2 ∧ dz 2 5.3. Residue. 5.3.1. Assume: Z(f ) = {0}.
We want to define a current, Res[ω], and first its part of type (1, 1), on a test form ϕ = ϕ 11 dz 1 ∧ dz 1 + ϕ 12 dz 1 ∧ dz 2 + ϕ 21 dz 2 ∧ dz 1 + ϕ 22 dz 2 ∧ dz 2 , as follows:

( Take polar coordinates: λ =|| q ||= (|z 1 | 2 + |z 2 | 2 ) 1 2 and the spherical coordinates on S 3 . Let dσ the volume element on S 3 , and K j , (j = 1, 2) a convenient universal constant:

5) Res[ω](ϕ) = lim ε→0 |f |=ε (|f 1 | 2 + |f 2 | 2 ) -1 (f 1 -f 2 j)(df 1 + df 2 j)(ϕ) (df 1 + df 2 j)(ϕ) = = ∂f 1 ∂z 1 dz 1 + ∂f 1 ∂z 2 dz 2 -( ∂f 2 ∂z 1 dz 1 - ∂f 2 ∂z 2 dz 2 )j (ϕ
dz 1 ∧ dz 1 ∧ dz 2 | S 3 = K 1 dλ ∧ dσ dz 1 ∧ dz 2 ∧ dz 2 | S 3 = K 2 dλ ∧ dσ

  current, and dV p[ω] = d"V p[ω] = Res[ω] is the residue current of ω. For any test function ϕ on U, Res[ω](ϕ) = lim ǫ→0 |z|=ǫ ω ∧ ϕ Then Res[ω] = 2πi res 0 (ω)δ 0 + dB =

5. 2 .

 2 Cauchy principal value of a quaternionic 1-form. Let f be a quaternionic function on an open neighborhood U of 0 in IHI satisfying the conditions of Definition 4.1, with

  Same result for the part of any type of Res[ω].5.3.2. Particular case. f 1 = z 1 ; f 2 = z 2 . Then: Df = 0. (dz 1 + dz 2 j)(ϕ) = ϕ 22 dz 1 ∧ dz 2 ∧ dz 2 + dz 2 j ∧ ϕ 12 dz 1 ∧ dz 2 = ϕ 22 dz 1 ∧ dz 2 ∧ dz 2 + ϕ 12 dz 1 ∧ dz 2 ∧ dz 2 j Res[ω](ϕ) = lim ε→0 |f |=ε (|z 1 | 2 +|z 2 | 2 ) -1 (z 1 -z 2 j)(ϕ 22 dz 1 ∧dz 2 ∧dz 2 +ϕ 12 dz 1 ∧dz 2 ∧dz 2 j) = lim ε→0 ||f ||=ε (|z 1 | 2 + |z 2 | 2 ) -1 (z 1z 2 j)(ϕ 22ϕ 12 j)dz 1 ∧ dz 2 ∧ dz 2 ) = lim ε→0 S 3 (|z 1 | 2 + |z 2 | 2 ) -1 (z 1z 2 j)(ϕ 22ϕ 12 j)εdσ = res[ω](ϕ 22ϕ 12 j)(0) = δ 0 (ϕ 22ϕ 12 j) ||f ||=εmeans the integration on εS 3 , where S 3 is the unit 3-sphere and dσ the volume element on S 3 , res[ω], a constant playing the part of the Cauchy residue, and δ 0 the Dirac measure at 0. Same result for the part of any type of Res[ω].5.3.3. f holomorphic. If f 2 = 0 and f = f 1 is holomorphic, f 1 = z 1 , then: z 1 = ε iθ , and Res[ω](ϕ) = lim ε→0 2π 0 idθϕ 2,2 (εe iθ , z 2 )dz 2 ∧ dz 2 = 2πi[Z(z 1 )](ϕ)Same result for any holomorphic f 1 since f 1 = εe iθ .

  11 dz 1 ∧dz 1 +ϕ 12 dz 1 ∧dz 2 +ϕ 21 dz 2 ∧dz 1 +ϕ 22 dz 2 ∧dz 2 ) ∧(ϕ 21 dz 2 ∧dz 1 +ϕ 22 dz 2 ∧dz 2 )+ ∂f 1 ∂z 2 dz 2 ∧(ϕ 11 dz 1 ∧dz 1 +ϕ 12 dz 1 ∧dz 2 ) j∧(ϕ 21 dz 2 ∧dz 1 +ϕ 22 dz 2 ∧dz 2 )+ ∂f 2 ∂z 2 dz 2 j∧(ϕ 11 dz 1 ∧dz 1 +ϕ 12 dz 1 ∧dz 2) 11 )dz 1 ∧ dz 1 ∧ dz 2 + ( ∂f 1 ∂z 1 ϕ 22 -∂f 1 ∂z 2 ϕ 12 )dz 1 ∧ dz 2 ∧ dz 2 12 jdz 1 ∧ dz 2 ∧ dz 2 = (A + Bj)dz 1 ∧ dz 1 ∧ dz 2 + (C + Dj)dz 1 ∧ dz 2 ∧ dz 2

	= dz 1 -∂f 1 ∂z 1 ∂f 2 ∂z 1 dz 1 = (-∂f 1 ∂z 1 ϕ 21 + ϕ + ∂f 1 ∂z 2 ∂f 2 ∂z 1 ϕ 21 -∂f 2 ∂z 2 ϕ 11 jdz 1 ∧ dz 1 ∧ dz 2 + -	∂f 2 ∂z 1	ϕ 22 -	∂f 2 ∂z 2	ϕ


Claire David Spurious

Claire David
email: claire.david@upmc.fr

Spurious lattice solitons for linear finite difference schemes

Keywords: Numerical schemes; spurious lattice solitary waves AMS Subject Classification: 65 M06, 65M12, 65M60, 35B99 PACS Codes: 47.11.Bc, 647.35.Fg

published or not. The documents may come

Spurious lattice solitons for linear finite difference schemes

Introduction

The analysis and the control of numerical error in discretized propagation-type equations is of major importance for both theoretical analysis and practical applications. A huge amount of works has been devoted to the analysis of the numerical errors, its dynamics and its influence on the computed solution (the reader is referred to classical books, among which [START_REF] Hirsch | Numerical Computation of Internal and External Flows[END_REF][START_REF] Sengupta | Fundamentals of Computational Fluid Dynamics[END_REF]). It appears that existing works are mostly devoted to linear, one-dimensional numerical models, such as the linear advection equation

∂u ∂t + c ∂u ∂x = 0 ( 1
)
where c is a constant uniform advection velocity. A striking observation is that, despite the tremendous efforts devoted to the analysis of numerical schemes in this simple case, the full exact non-homogeneous error equation has been derived only very recently [START_REF] Sengupta | A Fourier-Laplace spectral theory of computing for non-periodic problems: signal and error propagation dynamics[END_REF]. The two sources of numerical error are the dispersive and dissipative properties of the numerical scheme, which are very often investigated in unbounded or periodic domains thanks to a spectral analysis. In previous work [START_REF] Cl | A linear dispersive mechanism for numerical error growth: spurious caustics[END_REF], we analyzed a linear dispersive mechanism which results in local error focusing, i.e. to a sudden local error burst in the L ∞ norm for polychromatic solutions, referred to as the spurious caustic phenomenon. We showed that, for some specific values of the Courant number, spurious caustics can exist for some popular finite-difference schemes.

In other works [START_REF] Cl | Structural stability of finite dispersion-relation preserving schemes[END_REF], [START_REF] Cl | Spurious solitons and structural stability of finite difference schemes for nonlinear wave equations[END_REF] we have determined classes of traveling solitary wave solutions for a differential approximation of a finite difference scheme by means of a hyperbolic ansatz. We showed that spurious solitary waves can occur in finitedifference solutions of nonlinear wave equation. The occurance of such a spurious solitary wave, which exhibits a very long life time, results in a non-vanishing numerical error for arbitrary time in unbounded numerical domain. Such a behavior is referred here to have a structural instability of the scheme, since the space of solutions spanned by the numerical scheme encompasses types of solutions (solitary waves in the present case) that are not solution of the original continuous equations.

In physics, mathematics or engineering, a spurious solution refers to a non-physical one, which bears no resemblance to real one (see, for instance, [START_REF] Pember | Numerical methods for hyperbolic conservation laws with stiff relaxation I. Spurious solutions[END_REF].) Sometimes, a spurious solution may just mean an unwelcome or unexpected artifact. To avoid an unwelcome spurious solution, one has to find out its true origin, which in general can be due to either physical and mathematical modelings, or numerical methods. Spurious solutions induced by mathematical modeling are usually due to the missing of certain analytical constraints in mathematical formulations. Yet, the most interesting spurious solutions are those that are directly related to numerical methods, i.e., spurious solutions induced by algorithm.

Analysis of spurious waves in the numerical approximation of a one dimensional advection equation has been extensively investigated over the past few years by means of Fourier Analysis, through the concept of group velocity (see [START_REF] Vichnevesky | Spurious error waves in numerical approximations of hyperbolic equations[END_REF], [START_REF] Vichnevesky | Error waves in finite element and finite difference methods for hyperbolic equations[END_REF], [START_REF] Vichnevetsky | Fourier Analysis of Numerical Approximations of Hyperbolic Equations[END_REF]). It was also the kind of technique we used in [START_REF] Cl | A linear dispersive mechanism for numerical error growth: spurious caustics[END_REF].

In the present study, we investigate a new kind of spurious solution, whichmight appear in the process of the numerical scheme, when a solution which is not the searched one appears to satisfy the recursive difference relation and emerges through the numerical computation. Following [START_REF] Craik | Wave interactions andfluidflows[END_REF], and the fact that a wide class of numerical instabilities can be interpreted as spurious triad wave interactions generated by discretization, we show that such a spurious solution could then be related to a kind of "locking" of the scheme, since it takes the place of the searched one and gives birth to a wrong solution.

We thus presently extend our previous works, in so far as we exhibit lattice solitary waves solution of the general linear finite-differenced version of the linear advection equation, rejoining the fact that there exists travelling solitary wave solutions for a differential approximation of a finite difference scheme.

So far, we would like to lay the emphasis on the fact that, contrary to most beliefs, solitary waves and solitons can not uniquely be obtained as solutions of nonlinear differential equations and as solutions of linear differential equations, as it is very well shown in the very interesting paper of C. Radhakrishnan [START_REF] Radhakrishnan | The myth about nonlinear differential equations[END_REF], where, taking the example of the Korteweg-de Vries equation, it is shown that soliton solutions need not always be the consequence of the trade-off between the nonlinear terms and the dispersive term in the nonlinear differential equation, and that even the ordinary one dimensional linear partial differential equation can produce a soliton. The author explains that solutions of both linear and nonlinear differential equations are functions which depend nonlinearly on the independent variable, and that one can construct linear as well as nonlinear differential equations from the same function, as it is the case for the linear advection equation. Thus, as it is explained, the claim that a particular physical phenomenon can be described only by a nonlinear differential equation, and not by any linear differential equation is not tenable, provided a linear differential equation with the same solution as that of the nonlinear differential equation exists, and that, incidentally, linearization is the oldest and most popular method of solving nonlinear differential equations. In the same way, in [START_REF] Liu | Solitary Wave in Linear ODE with Variable Coefficients[END_REF], Liu et al. proved the existence of solitary waves in Linear ODE with variable coefficients.

In the present paper, we consider the linear advection equation [START_REF] Hirsch | Numerical Computation of Internal and External Flows[END_REF], which happens to be obtained by linearizing the nonlinear Burgers equation. Our analysis is restricted to interior stencil, and the influence of boundary conditions will not be considered.

Test numerical schemes

A linear finite difference scheme for the Burgers equation can be written under the following general form:

F (u l m , h, τ ) = 0, (2)
where the discrete solution is denoted

u l m = u (l dx, m dt) (3) l ∈ {j -1, j, j + 1}, m ∈ {n -1, n, n + 1}, j = 1, ..., n x , n = 1, ..., n t , h, τ
denoting respectively the mesh size and time step, and σ the Courant-Friedrichs-Lewy number (cf l) coefficient, defined as σ = c τ /h.

A numerical scheme is specified by selecting appropriate expression of the linear function F in equation [START_REF] Leonard | The NIRVANA scheme applied to one-dimensional advection[END_REF].

For the sake of simplicity, the analysis will be restricted to schemes which involves at most two time levels and three grid points. The extension of the present analysis to other schemes is straightforward. For this class of schemes, the general finitedifferenced version of the linear advection equation (1) can therefore be written as follows

u j n+1 -u j n τ + j+1 ∑ k=j-1 m jk u n k = 0 ( 4
)
where m jk are coefficients related to the numerical scheme. Those coefficients are constants, and can take only three different values, that we will denote m -, m, m + , which implies that the matrix

[m jk ] 1≤j≤nx, 1≤k≤nt
is a block cyclic one, of the form:

M =      m m + 0 . . . . . . . . . 0 m -m m + 0 . . . 0 0 m -m m + 0 . . . 0 . . . . . . . . . . . . . . . . . .      (5)
If one writes it under the form:

du n j dt + j+1 ∑ k=j-1 m jk u n k = 0 ( 6
)
where the m jk denote the linear scheme coefficients, it appears as a special case of a more general equation, the Discrete Self-Trapping (DST ) equation (see, among numerous references, [START_REF] Eilbeck | The discrete self-trapping equation[END_REF], [START_REF] Eilbeck | Theory and applications of the discrete self-trapping equation[END_REF], [START_REF] Faddeev | Hamiltonian methods in the theory of solitons[END_REF]):

dA j dt + γ A 2 j A j + ε ∑ k∈I m jk A k = 0 (7)
where:

γ = 0 (8)
and where I denotes a set of integers.

Nonlinear localized modes in discrete systems exist due to the interplay between lattice coupling and nonlinearity effects. We presently aim at proving that finite difference schemes admit spatially localized modes or discrete solitons. It is a fundamental property of discrete systems, or, more generally, spatially periodic systems with a band-gap structure for the dispersion relation of linear waves, that bright and dark localized modes may appear for the same physical system. It is well known that depending upon of the parameters and the chosen initial condition the DST equation can lead either to self-trapping (i.e. local modes or solitons), or to chaos, or to a mixture of the above two behaviors (see [START_REF] Eilbeck | The discrete self-trapping equation[END_REF], [START_REF]Self Trapping of Vibrational Energy in Proteins[END_REF], [23]).

It has been shown that the DST equation is not integrable when max {n x , n t } > 2, and, thus, will admit Hamiltonian chaos [START_REF] Eilbeck | The discrete self-trapping equation[END_REF].

The classical way to obtain discrete soliton solutions (see, for instance [START_REF] Arévalo | Solitary wave solutions as a signature of the instability in the discrete nonlinear Schrdinger equation[END_REF]) is to consider a travelling wave ansatz for an envelope complex function reading as:

ϕ j (x, t) = +∞ ∑ Λ=0 ψ j (ξ) e i Λ θ (9)
where

{ ξ = x -v j t θ = δ j + ω j t ( 10
)
where v j is a velocity, δ j is a phase, ω j a constant related to the particle energy, and where Λ represents a phase shift among successive discrete sites, and, in essence, translates the spatial carrier wave number within the Brillouin zone1 . Usually, ones uses then the full Taylor expansion of the function ψ j±1 → e ±∂ j ψ j to transform eq. ( 7) into an operator equation. In the following, we choose to directly search exact solutions without this approximation.

Solitary waves solutions can be searched using a hyperbolic ansatz (see [START_REF] Feng | Solitary Wave Solutions of the Compound Burgers-Korteweg-de Vries Equation[END_REF], [START_REF] Cl | A note on "general solitary wave solutions of the Compound Burgers-Korteweg-de Vries Equation[END_REF]), as a combination of bell-profile and kink-profile waves, of the form:

U (x, t) = Np ∑ p=1 {U p tanh p [ C p (x -v t + x 0 ) ] + V p sech p [ C p (x -v t + x 0 ) ]} (11)
where N p is a positive integer, U p , V p , C p , constants; v denotes the velocity. Due to the p th exponents that lead to over-determined algebraic equations, one usually work in the case N p = 1.

Proposition 2.1. Expression [START_REF] Cl | Spurious solitons and structural stability of finite difference schemes for nonlinear wave equations[END_REF] can be obtained searching solitary wave solutions as the following infinite series:

ϕ j (x, t) = +∞ ∑ Λ=-∞ Ψ j e Λ (k j x-v j t) ( 12
)
Proof. Set:

ξ = x -v t + x 0 ( 13
)
The proof is in the case where C p ξ > 0.

One has then:

U (x, t) = Np ∑ p=1 {U p tanh p [ C p ξ] + V p sech p [ C p ξ ]} = Np ∑ p=1 { U p ( e Cp ξ -e -Cp ξ e Cp ξ + e -Cp ξ ) p + V p ( 2 e Cp ξ + e -Cp ξ ) p } = Np ∑ p=1 { U p ( 1 -e -2 Cp ξ 1 + e -2 Cp ξ ) p + V p ( 2 e -Cp ξ 1 + e -2 Cp ξ ) p } = Np ∑ p=1 { U p ( p ∑ k=0 (-1) k e -2 k Cp ξ ) + 2 p e -p Cp ξ V p } { +∞ ∑ l=0 (-1) l e -2 l Cp ξ } p (14)
The term

{ +∞ ∑ l=0 (-1) l e -2 l Cp ξ } p
, that we will call a p-Cauchy product, is obtained by applying p times the Cauchy product to the infinite series

+∞ ∑ l=0 (-1) l e -2 l Cp ξ .
Thus, this latter term can be written under the form

{ +∞ ∑ l=0 (-1) l e -2 l Cp ξ } p = +∞ ∑ m=0 c m,p e -2 m Cp ξ ( 15
)
where the c m,p denote the coefficients of the p-Cauchy product. Thus, expression [START_REF] Cl | Spurious solitons and structural stability of finite difference schemes for nonlinear wave equations[END_REF] can be written under the following form:

U (x, t) = +∞ ∑ Λ=-∞ d Λ e Λ Cp ξ (16)
We thus search for solitary waves under the form:

ϕ j (x, t) = +∞ ∑ Λ=-∞ Ψ j e Λ (k j x-v j t) ( 17
)
which appear to be a more general one. Expressions of the form (11) appear thus a a particular case of [START_REF] Radhakrishnan | The myth about nonlinear differential equations[END_REF].

Spurious lattice solitons

Since we are considering the solution at the j th node of the mesh, at the time n τ , we will have:

{ x = j h t = n τ (18)
By substituting ( 17) in ( 6), one obtains:

-Λ v j Ψ j e Λ (k j x-v j t) + j+1 ∑ k=j-1 m jk Ψ k e Λ (k k x-v k t) = 0 (19)
If, by following again [START_REF] Feng | Solitary Wave Solutions of the Compound Burgers-Korteweg-de Vries Equation[END_REF], [START_REF] Cl | A note on "general solitary wave solutions of the Compound Burgers-Korteweg-de Vries Equation[END_REF], one makes the assumption that, for any value of the integer j in {0, . . . , n x }:

{ k j = k j±1 = k v j = v j±1 = v ( 20
)
due to the fact that the characteristic scheme coefficients are constants and cyclically take the values m -, m, m + , it leads to:

-Λ v Ψ j + m -Ψ j-1 + m Ψ j + m + Ψ j+1 = 0 (21) or:
(m

-Λ v) Ψ j + m -Ψ j-1 + m + Ψ j+1 = 0 (22)
which appears to be a second-order recurrence solution with regards to ψ j , whose characteristic equation is given by:

m + r 2 + (m -Λ v) r + m -= 0 ( 23
)
The discriminant is given by:

(m -Λ v) 2 -4 m + m - (24)
If m + m -< 0, real solutions always exist. If, on the contrary, If m -m -> 0, real solutions exist if:

| m -Λ v | ≥ 2 √ | m + m -| ( 25
)
The solutions are then given by:

r = -m = Λ v ± √ (m -Λ v) 2 -4 m + m - 2 (26)
From now on, we set:

           r 1 Λ = -m + Λ v - √ (m -Λ v) 2 -4 m + m - 2 r 2 Λ = -m + Λ v + √ (m -Λ v) 2 -4 m + m - 2 (27)
which ensures, for any ψ j :

Ψ j = a (Λ) r j 1 Λ + b (Λ) r j 2 Λ ( 28
)
where a (Λ), b (Λ) are constants that will be determined using the boundary conditions.

Set:

ψ ∞ = lim n→+∞ ψ n = ψ(n x ) (29)
Since we deal with the Burgers equation, we assume that the boundary conditions are of Dirichlet type, periodic:

u(x = 0, t) = u(x = n x h, t) (30)
with an initial condition:

u(x, 0) = u 0 (x) (31)
which leads to:

ψ 0 = ψ ∞ (32) It ensures: { a (Λ) + b (Λ) = ψ ∞ a (Λ) r nx 1 Λ + b (Λ) r -nx 2 Λ = ψ ∞ ( 33
)
which yields:

           a (Λ) = ψ ∞ (1 -r nx 2 Λ ) r nx 1 Λ -r nx 2 Λ b (Λ) = ψ ∞ (r nx 1 Λ -1) r nx 1 Λ -r nx 2 Λ (34)                r 1 Λ = -1 τ + Λ v - √ ( 1 τ -Λ v ) 2 + 1-σ 2 τ 2 2 r 2 Λ = -1 τ + Λ v + √ ( 1 τ -Λ v ) 2 + 1-σ 2 τ 2 2 (41)
One has then:

           a (Λ) = ψ ∞ (1 -r nx 2 Λ ) r nx 1 Λ -r nx 2 Λ b (Λ) = ψ ∞ (r nx 1 Λ -1) r nx 1 Λ -nx 2 Λ (42)
and, for 0 ≤ j ≤ n x :

Ψ j = a r j 1 Λ + b r j 2 Λ (43)
which lead to the following spurious solution at the j th node of the mesh:

ϕ j (x = j h, t = n τ ) = +∞ ∑ Λ=-∞ Ψ j e Λ (k j h-n v τ ) = +∞ ∑ Λ=-∞ { a (Λ) r j 1Λ + b (Λ) r j 2Λ } e Λ (k j h-n v τ ) = +∞ ∑ Λ=-∞ ψ ∞ r nx 1Λ -r nx 2Λ { (1 -r nx 2Λ ) r j 1Λ + (r nx 1Λ -1) r j 2Λ } e Λ (k j h-n v τ ) (44)
Particular solutions can be obtained for a given value of the summation indice Λ.

One can also search directly search solutions of Eq. (35) under the form:

u(x, t, k) = A sech [k (x -v t)] + B tanh [k (x -v t)] (45)
which leads, at the j th node, and time step n:

A(σ + 1)sech ( k τ (j h-n vσ-1) σ ) -(σ -1)sech ( k τ (j-n v σ+1) σ ) ( A + B sinh ( k τ (j-n v σ+1) σ
))

+B(σ + 1)tanh

( k τ (j-n v σ-1) σ ) = 0 (46)
Also, when the time step number n goes towards infinity, sech(k τ (j -n v-1)) tends towards zero, and the solitary wave dispersion relation (46) tends to be satisfied, accounting for the scheme to become numerically instable. As expected, it can be noted that the solitary wave begins to become greatly unstable as the cf l number tends towards 1. A "lattice solitary wave", in the case of the Lax scheme, as a function of the cfl number σ and of the wave velocity v scheme, appearing as parasitic solutions of the correct one. Such spurious solitary waves have constant energy, and therefore the numerical error norm does not vanish at arbitrary long integration time on unbounded numerical domains.

Thanks

The author would like to thank Tingwen Huang for his useful remarks.

Figures 1 ,

 1 Figures 1, 2 respectively display a lattice solitary wave, first, for σ = 0.7, h = 0.01, v = 5, k = 5, as a function of the mesh points, and, second, as a function of the cf l number σ and of the wave velocity v. As expected, it can be noted that the solitary wave begins to become greatly unstable as the cf l number tends towards 1.

Figure 1 : 5 4

 15 Figure 1:A "lattice solitary wave", in the case of the Lax scheme, as a function of the mesh points, for σ = 0.7, h = 0.01, v = 5, k = 5

Figure 2 :

 2 Figure 2:

the Brillouin zone, or first Brillouin zone, is a uniquely defined primitive cell in reciprocal space, which boundaries are given by planes related to points on the reciprocal lattice. The importance of the Brillouin zone stems from the Bloch wave description of waves in a periodic medium, i.e. by means of their wavefunction[START_REF] Magnus | Hill's Equation[END_REF], in which it is found that the solutions can be completely characterized by their behavior in a single Brillouin zone.

Numerical example: the case of the Lax scheme

For the classical Lax scheme, the finite difference relation writes:

which is equivalent to

so that one obtains:

that we can write as:

Identification leads then to the following values for the characteristic scheme coefficients m -, m, m + :

It leads to:

since the cfl number σ satifying 0 ≤ σ ≤ 1.

The case σ = 1 refering to unstable schemes will not be considered in the present study.

Hence, the solutions of ( 23) are real ones, given for v ∈ IR by: which is satisfied for:

where B, k, v can take any values in IR.

It thus exhibits the existence of lattice solitons, related to the discrete numerical scheme, of the form

In the specific case where σ = 1, one obtains:

For A = 0, v j = 1, one has:

which occurs on the recursive calculation of the approximate solution.

It thus exhibits the existence of lattice solitons, related to the discrete numerical scheme, of the form

2. More generally, numerical simulations usually dealing with values of the time step number n ≫ 1, for A = 0, v = 1 p , p ∈ IN * , the solitary wave dispersion relation is satisfied when

It thus also exhibits the existence of lattice solitons, related to the discrete numerical scheme, of the form