Potential well theory for the derivative nonlinear Schrödinger equation

Abstract

We consider the following nonlinear Schrödinger equation of derivative type:

$i{\partial }_{t}u+{\partial }_x^{2}u+i|u{|}^2{\partial }_{x}u+b|u{|}^{4}u=0,\left(t,x\right)\in ℝ\times ℝ,b\in ℝ.$

If $b=0$, this equation is known as a standard derivative nonlinear Schrödinger equation(DNLS), which is mass-critical and completely integrable. The equation above 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 $u_0\in H^1\left(ℝ\right)$ satisfies the mass condition , the corresponding solution is global and bounded. In this paper we first establish the mass condition on the equation above for general $b\in ℝ$, which corresponds exactly to the $4\pi$-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 wells generated by solitons. In particular, our results for DNLS give a characterization of both the $4\pi$-mass condition and algebraic solitons.