Orbital Stability of Soliton for the Derivative Nonlinear Schr\"odinger Equation in the $L^2$ Space

Abstract

In this paper, we establish the orbital stability of the 1-soliton solution for the derivative nonlinear Schr\"odinger equation under perturbations in $L^2(\mathbb{R})$. We demonstrate this stability by utilizing the B\"acklund transformation associated with the Lax pair and by applying the first conservation quantity in $L^2(\mathbb{R}).$