On a novel integrable generalization of the nonlinear Schr\"odinger equation

TL;DR

This paper introduces a new integrable generalization of the nonlinear Schrödinger equation, explores its mathematical structure, derives solutions, and analyzes soliton behavior, expanding understanding of integrable systems.

Contribution

It presents a novel integrable NLS generalization, derives its conservation laws, Lax pair, and soliton solutions, advancing the theory of integrable nonlinear equations.

Findings

Derived conservation laws for the new equation

Established a Lax pair for integrability

Solved the initial value problem and analyzed solitons

Abstract

We consider an integrable generalization of the nonlinear Schr\"odinger (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.