# On the generalized nonlinear Camassa-Holm equation

## TL;DR

This paper investigates a generalized nonlinear Camassa-Holm equation with variable coefficients, demonstrating control of dispersive terms through weighted energy and establishing solution existence and uniqueness via iterative methods.

## Contribution

It introduces a framework for controlling dispersive effects in a generalized equation with variable coefficients and proves well-posedness using Picard iteration.

## Key findings

- Control of higher order dispersive terms using weighted energy functions
- Existence and uniqueness of solutions established
- Applicable to equations with time- and space-dependent coefficients

## Abstract

In this paper, a generalized nonlinear Camassa-Holm equation with time- and space-dependent coefficients is considered. We show that the control of the higher order dispersive term is possible by using an adequate weight function to define the energy. The existence and uniqueness of solutions are obtained via a Picard iterative method.