Lattice representation and dark solitons of the Fokas-Lenells equation

which appears in physical applications in the form

FLE is known (see [1]) to be an integrable equation describing the first negative flow of the integrable hierarchy associated with the derivative nonlinear Schrödinger (DNLS) equation. In papers [2,3] the authors derived the FLE from the Maxwell equations for the propagation of femtosecond pulses in optical fibers, established its bi-Hamiltonian structure, elaborated the inverse scattering transform scheme for the case of zero boundary conditions and obtained the N-bright soliton solutions. The aim of this paper is two-fold. Firstly, I want to derive the N-dark soliton solutions. Secondly, I am going to establish the relationships between the FLE and other integrable models: the nonlinear Schrödinger (NLS) equation, the Merola-Ragnisco-Tu (MRT) equations [4] and the Ablowitz-Ladik (AL) model [5]. It this paper I present the commuting pair of the Bäcklund transformations (section 2) which I use in section 3 to obtain the lattice representation of the FLE. After bilinearizing the lattice equations and establishing their relationship with the AL hierarchy I use the N-soliton solutions of the latter to obtain the N-soliton solutions for the FLE (section 4). Finally, in section 5 I discuss the links between the DNLS, NLS, MRT and AL equations.

The method of this work is based on two transformations, T and Ť , given by T :

and

These transformations possess two important properties:

• Transformations T and Ť send solutions of the FLE to solutions of the FLE.

• Transformations T and Ť are inverse when applied to solutions of the FLE.

To prove the first statement consider the quantities ∆[u, v] and ∆[u, v] which are the left-hand sides of equations (1):

It can be shown by direct calculations that if pairs (u, v) and (û, v) are related by (3), then

From these equations one can easily see that if u and v are solutions of the FLE,

= 0 which means that û and v also satisfy the FLE. In a similar way one can show that ∆ and ∆ calculated for the Ť -transformed functions are given by

Thus the identities ∆

which proves the fact that transformations T and Ť send solutions of the FLE to solutions of the FLE.

To demonstrate that T

consider the results of joint action of T -and Ť -transformations:

where û, v, ǔ and v are defined in (3), (4). Using definitions (4) one can obtain from (12)

In a similar way definitions (13) and (3) lead to

These identities together with (10) prove (11) In what follows I will use the symbol T for both transformations, T = T ,

bearing in mind that T -transformations are used only with solutions of the FLE.

Using the transformations discussed in the previous section I introduce (instead of single solution of the FLE, u and v) an infinite sequence of solutions u n and v n defined by iteration of T -transformations:

In terms of u n and v n the definitions of transformations T and T -1 , equations ( 3) and ( 4), can be written as

and

It is easy to see that the above equations can be regrouped in two systems

These equations are not new: they belong to the MRT hierarchy that has been introduced, in the framework of the reduction technique for Poisson-Nijenhuis structures, in [4] as an integrable lattice hierarchy, whose continuum limit is the AKNS hierarchy. In [4] the authors demonstrated that this hierarchy is endowed with a canonical Poisson structure and admits a vector generalisation. They solved the associated spectral problem and explicity contructed action-angle variables through the r-matrix approach. In this work I show that equations (20) and ( 21), as well as the whole MRT hierarchy are closely related to another discrete version of the AKNS equations, namely the famous AL hierarchy. To this end I bilinearize equations (20) and (21) by introducing the τ -functions ρ n , σ n and τ n related by

as follows:

By straightforward algebra one ca obtain that

where

and D x and D y are the Hirota’s bilinear operators defined as

One can easily see that equations ( 20) and ( 21) can be satisfied by imposing the conditions

Thus, instead of system (20) and (21) one can solve the bilinear system given by

Equations ( 36)-(41) belong to the AL hierarchy that has been introduced in [5]. Indeed, in terms of the functions

equations ( 36)-(41) together with (22) become

and

equations that describe the simplest flows of the AL hierarchy and that can be viewed as superposition of the famous discrete NLS and discrete modified KdV equations. That means that one can use the already known solutions of the AL hierarchy to construct the ones for the MRT equation and hence for the FLE.

In the appendix one can find dark-soliton solutions of the generalized AL equations [6], the system which is a generalization of ( 36)-(41). Using these results solutions of the latter can be written as

where

and

Here A(x, y) is a square matrix with the elements

L and R are diagonal matrice

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