Instability of degenerate solitons for nonlinear Schrödinger equations with derivative. (September 2022) - Fukaya, Noriyoshi; Hayashi, Masayuki
Abstract: We consider the following nonlinear Schrödinger equation with derivative: (1) i u t = − u x x − i | u | 2 u x − b | u | 4 u, ( 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 soliton profile of DNLS satisfies a certain double power elliptic equation with cubic–quintic nonlinearities. The quintic nonlinearity in (1) 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 ≥ 0, Eq. (1) has degenerate solitons whose momentum and energy are zero, and if b = 0, they are algebraic solitons. Inspired from the works (Martel and Merle, 2001; Farah et al., 2019) on instability theory of the L 2 -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, which is a small step towards understanding the dynamics around algebraic solitons of the DNLS equation.