Lie symmetries, invariant solutions, conservation laws and stability analysis of the Camassa-Holm-Nonlinear Schrödinger equation

In this work, we explore the defocusing nonlinear Schrödinger equation known as the Camassa–Holm–Nonlinear Schrödinger (CH–NLS) model, which is newly derived in the sense of deformation of hierarchies of integrable systems. This equation captures important features of nonlinear wave propagation in shallow water dynamics, plasma physics, and nonlinear optics. A comprehensive Lie symmetry analysis is carried out to determine the symmetry generators and to derive similarity reductions of the equation. These reductions enable us to construct several exact group-invariant solutions. In addition, we employ Ibragimov’s conservation theorem to derive associated conservation laws based on the identified symmetries. The study also includes an investigation of modulational instability using the framework of linear stability analysis to assess the stability properties of continuous wave solutions. The results provide analytical insights into the interplay between symmetry, conservation, and stability in the CH–NLS model, thereby contributing to the broader understanding of nonlinear dispersive wave phenomena.