Semi classical limit for a NLS with potential

Abstract

This paper is dedicated to the semiclassical limit of t the nonlinear focusing Schrödinger equation (NLS) with a potential , i\e\partial_t u^{\e}+\frac{\e^2}{2}\lap u^{\e}-V(x)u^{\e}+|u^{\e}|^{2\sigma}u^{\e}=0 with initial data in the form Q\left(\frac{x-x_0}{\e}\right)e^{i\frac{x.v_0}{\e}}, where $Q$ is the ground state of the associated unscaled elliptic problem. Using a refined version of the method introduced in \cite{BJ} by J. C. Bronski, R.L. Jerrard, we prove that, up to a time-dependent phase shift, the initial shape is conserved with parameters that are transported by the classical flow of the classical Hamiltonian $H(t,x)=\frac{|\xi|^2}{2}+V( x)$. This gives, in particular, a complete description of the dynamics of the time-dependent Wigner measure associated to the family of solutions