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

Abstract

We consider the following nonlinear Schrödinger equation of derivative type: i∂tu+∂²xu+i|u|²∂xu+b|u|⁴u=0, (t, x)∈R×R, b∈R. (1) If b=0, this equation is a gauge equivalent form of well-known derivative nonlinear Schrödinger (DNLS) equation. The soliton profile of the DNLS equation 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 the additional nonlinearity is natural as a perturbation preserving soliton profiles of the DNLS equation. If b>−3/16, Eq. (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 existence of stable algebraic solitons in (1) shows an interesting mathematical example because stable algebraic solitons are not known in the context of double power NLS equations.