Instability of degenerate solitons for nonlinear Schrödinger equations with derivative

Abstract

We consider the following nonlinear Schrödinger equation with derivative: \begin{equation} iu_t =-u_{xx} -i |u|^{2}u_x -b|u|^4u , \quad (t,x) \in \mathbb{R}\times\mathbb{R}, \ b \in\mathbb{R}. \end{equation} If $b=0$, this equation is a gauge equivalent form of the well-known derivative nonlinear Schrödinger (DNLS) equation. The soliton profile of DNLS satisfies a certain double power elliptic equation with cubic-quintic nonlinearities. The quintic nonlinearity in our equation only affects the coefficient in front of the quintic term in the elliptic equation, so in this sense the additional nonlinearity is natural as a perturbation preserving soliton profiles of DNLS. When $b\ge 0$, the equation has degenerate solitons whose momentum and energy are zero, and if $b=0$, they are algebraic solitons. Inspired from the works on instability theory of the $L^2$-critical generalized KdV equation, we study the instability of degenerate solitons in a qualitative way, and when $b>0$, we obtain a large set of initial data yielding the instability. The arguments except one step in our proof work for the case $b=0$ in exactly the same way, which is a small step towards understanding the dynamics around algebraic solitons of the DNLS equation.

arXiv comment26 pages, slightly modified introduction

References in corpus (10)

• Global well-posedness on the derivative nonlinear Schrödinger equation revisited
• $H^s$ Bounds for the Derivative Nonlinear Schrödinger Equation
• Well-posedness for a generalized derivative nonlinear Schrödinger equation
• A sufficient condition for global existence of solutions to a generalized derivative nonlinear Schrödinger equation
• Global Existence for the Derivative Nonlinear Schrödinger Equation with Arbitrary Spectral Singularities
• Stability of multi-solitons for the derivative nonlinear Schr{ö}dinger equation
• Orbital stability of solitary waves for derivative nonlinear Schrödinger equation
• Stability of algebraic solitons for nonlinear Schrödinger equations of derivative type: variational approach
• Potential well theory for the derivative nonlinear Schrödinger equation
• Instability of the solitary wave solutions for the genenalized derivative Nonlinear Schrödinger equation in the critical frequency case