Potential Well Theory for the Derivative Nonlinear Schrödinger Equation

Abstract

We consider the following nonlinear Schrödinger equation of derivative type: (1) i∂tu + ∂2xu + i|u|2∂xu + b|u|4u = 0, (t, x) ∈ R×R, b ∈ R. If b = 0, this equation is known as a standard derivative nonlinear Schrödinger equation (DNLS), which is mass critical and completely integrable. The equation (1) can be considered as a generalized equation of DNLS while preserving mass criticality and Hamiltonian structure. For DNLS it is known that if the initial data u0∈H1(R) satisfies the mass condition ||u0||2L2 < 4π, the corresponding solution is global and bounded. In this paper we first establish the mass condition on (1) for general b∈R, which is exactly corresponding to 4π-mass condition for DNLS, and then characterize it from the viewpoint of potential well theory. We see that the mass threshold value gives the turning point in the structure of potential well generated by solitons. In particular, our results for DNLS give a characterization of both 4π-mass condition and algebraic solitons.