Multi-phase computations of the semiclassical limit of the Schrödinger equation and related problems: Whitham vs Wigner

Abstract

We present and compare two different techniques to obtain the multi-phase solutions for the Schrödinger equation in the semiclassical limit. The first is Whitham’s averaging method, which gives the modulation equations governing the evolution of multi-phase solutions. The second is the Wigner transform, a convenient tool to derive the semiclassical limit equation in the phase space—the Vlasov equation—for the linear Schrödinger equation. Motivated by the linear superposition principle, we derive and prove the multi-phase ansatz for the Wigner function by the stationary phase method, and then use the ansatz to close the moment equations of the Vlasov equation and obtain the multi-phase equations in the physical space. We show that the multi-phase equations so derived agree with those derived by Whitham’s averaging method, which can be proved using different arguments. Generic way of obtaining and computing the multi-phase equations by the Wigner function is given, and kinetic schemes are introduced to solve the multi-phase equations. The numerical schemes, based on multiphase closure of the Liouville equation, are purely Eulerian and only operate in the physical space. Several numerical examples are given to explore the validity of this approach. Similar studies are conducted for the linearized Korteweg–de Vries equation and the linear wave equation.