Existence and uniqueness of the global conservative solutions for the generalized Camassa-Holm equation with dual-power nonlinearities

TL;DR

This paper proves the existence and uniqueness of global conservative solutions for a generalized Camassa-Holm equation with dual-power nonlinearities by transforming it into a semi-linear system and analyzing auxiliary variables.

Contribution

It introduces a new variable transformation and constructs auxiliary variables to establish the global existence and uniqueness of conservative solutions for the equation.

Findings

Global conservative solutions exist and are unique.

Transformation into semi-linear system facilitates analysis.

Auxiliary variables satisfy a semi-linear system with unique solutions.

Abstract

In this paper, we investigate the global conservative solutions to the generalized Camassa-Holm equation with dual-power nonlinearities. By introducing a new set of variables, we transform the original equation into an equivalent semi-linear system, which allows us to establish the global existence of conservative solutions. Furthermore, for a given global conservative solution, we construct some auxiliary variables tailored to its specific structure and demonstrate that they satisfy a semi-linear system with a unique solution, thereby deriving the uniqueness of conservative solutions to the original equation.