Instability of degenerate solitons for nonlinear Schrödinger equations with derivative

Abstract

We consider the following nonlinear Schrödinger equation with derivative: (1) iut = ーuxx ー i|u|2ux ー b|u|4u, (t; x)∈R × R, b∈R. If b = 0, this equation is a gauge equivalent form of the well-known derivative nonlinear Schrödinger (DNLS) equation. The equation (1) for b ≥ 0 has degenerate solitons whose momentum and energy are zero, and if b = 0, they are algebraic solitons. Inspired from the works [29, 8] on instability theory of the L2-critical generalized KdV equation, we study the instability of degenerate solitons of (1) 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, and in particular the unstable directions of algebraic solitons are detected. This is a step towards understanding the dynamics around algebraic solitons of the DNLS equation.