Stability of Algebraic Solitons for Nonlinear Schrödinger Equations of Derivative Type: Variational Approach

Abstract

We consider the following nonlinear Schrödinger equation of derivative type: (1) i∂tu + ∂2x u + i|u|2∂xu + b|u|4u = 0, (t, x) ∈ R×R, b ∈ R. If b = 0, this equation is a gauge equivalent form of well-known derivative nonlinear Schrödinger (DNLS) equation. The equation (1) can be considered as a generalized equation of (DNLS) while preserving both L2-criticality and Hamiltonian structure. If b > -3/16, the equation (1) has algebraically decaying solitons, which we call algebraic solitons, as well as exponentially decaying solitons. In this paper we study stability properties of solitons for (1) by variational approach and prove that if b < 0, all solitons including algebraic solitons are stable in the energy space. The stability of algebraic solitons gives the counterpart of the previous instability result for the case b > 0.