On a novel integrable generalization of the nonlinear Schrödinger equation

Abstract

We consider an integrable generalization of the nonlinear Schrödinger (NLS) equation that was recently derived by one of the authors using bi-Hamiltonian methods. This equation is related to the NLS equation in the same way that the Camassa Holm equation is related to the KdV equation. In this paper we: (a) Use the bi-Hamiltonian structure to write down the first few conservation laws. (b) Derive a Lax pair. (c) Use the Lax pair to solve the initial value problem. (d) Analyze solitons.

New Exact Solutions of the New Hamiltonian Amplitude-Equation and Fokas Lenells Equation

“…In optics, the FL equation models the propagation of nonlinear light pulses in monomode optical fibers when certain higher order nonlinear effects are taken into consideration [36]. The complete integrability of the FL equation has been exhibited by using the inverse scattering transform (IST) method [37]. Especially, a Lax pair and a few conservation laws related to it have been found clearly using the bi-Hamiltonian structure and the multisoliton solutions have been obtained by using the dressing method [38].…”

Integrable twofold hierarchy of perturbed equations and application to optical soliton dynamics

“…We show that these linear equations can be nonlinearized in a few simple steps to uniquely recover the full set of integrable equations. The first step is to note that linear equations (4), (5), and (6) fix the corresponding scaling dimension…”

“…On the other hand, in addition to the well-known Camassa-Holms and Degasperis-Procesi equations, some other unusual new integrable equations were recently discovered, namely, the KdV6 [4], LF DNLS [5], mKdV-SG [10], and higher mKdV [11] equations. We realized that these uncommon equations are merely particular cases of general integrable perturbations of the AKNS-Zakharov-Shabat and the KN systems [6]- [8].…”

“…Bur it appears that many more undiscovered chapters with significant promise of implementation remain unexplored in this well-investigated subject. One such area is the nonholonomic deformation of integrable systems [2], which leads to integrable perturbations of known integrable systems with a twofold hierarchy and rich soliton dynamics [3] and also to discovering new equations such as integrable perturbations of the KdV (mKdV), sine-Gordon (SG), nonlinear Schrödinger (NLS), and the derivative NLS (DNLS) equations, and to obtaining some recently proposed equations, for example, the KdV6 [4] and Lennels-Fokas (LF) [5] equations, and so on. Moreover, using nonholomorphic deformations allows explaining the integrability of these systems and obtaining exact soliton solutions [3], [6]- [8].…”

Algebro-Geometric Solutions and Their Reductions for the Fokas-Lenells Hierarchy

“…Related results can also be directly applied to (1.1) and (1.3) since the existence of these simple transformations among them. It is shown that (1.4)/(1.5) is a completely integrable nonlinear partial differential equation possessing Lax pair, bi-Hamiltonian structure, and soliton solutions [5,[16][17][18]. One of the most remarkable feature of the FL equation is that it possesses various kinds of exact solutions such as solitons, breathers, etc..…”