WKB analysis of nonelliptic nonlinear Schrödinger equations

Abstract

We justify the WKB analysis for generalized nonlinear Schrödinger equations (NLS), including the hyperbolic NLS and the Davey-Stewartson II system. Since the leading order system in this analysis is not hyperbolic, we work with analytic regularity, with a radius of analyticity decaying with time, in order to obtain better energy estimates. This provides qualitative information regarding equations for which global well-posedness in Sobolev spaces is widely open.

On the hyperbolic nonlinear Schrödinger equations

“…Long-time behavior of the local H 1 (R 3 ) solutions to the cubic nonelliptic cubic Schrödinger equations in three spatial dimensions. Remark 2.2 A WKB analysis is justified in [10] for (2.4) and related equations. Since, contrary to the classical NLS equation, the leading order system in this analysis is not hyperbolic, one needs to work in analytic classes.…”