Well-posedness and stability results for the Gardner equation

TL;DR

This paper establishes local and global well-posedness results for the Gardner equation in Sobolev spaces, constructs explicit solitons, and proves their orbital stability, advancing understanding of the equation's mathematical properties.

Contribution

It provides new well-posedness results for the Gardner equation in Sobolev spaces and demonstrates the orbital stability of explicit soliton solutions.

Findings

Local well-posedness in H^s(R) for s > 1/4

Global well-posedness in H^1(R) due to conservation laws

Explicit soliton solutions proven to be orbitally stable

Abstract

In this article we present local well-posedness results in the classical Sobolev space H^s(R) with s > 1/4 for the Cauchy problem of the Gardner equation, overcoming the problem of the loss of the scaling property of this equation. We also cover the energy space H^1(R) where global well-posedness follows from the conservation laws of the system. Moreover, we construct solitons of the Gardner equation explicitly and prove that, under certain conditions, this family is orbitally stable in the energy space.