From Stationary Phase to Steepest Descent

TL;DR

This paper clarifies the differences between nonlinear stationary phase and steepest descent methods, emphasizing the role of steepest descent contours, primarily using the nonlinear Schrödinger equation as a model.

Contribution

It provides a detailed comparison of nonlinear stationary phase and steepest descent techniques, highlighting the significance of steepest descent contours in solving nonlinear problems.

Findings

Distinction between nonlinear stationary phase and steepest descent methods

Importance of steepest descent contours in nonlinear analysis

Application to nonlinear Schrödinger and KdV equations

Abstract

Our aim here is to clarify the distinction between the nonlinear-stationary-phase idea and the nonlinear-steepest-descent idea, stressing the importance of actual steepest-descent contours in some problems. We mostly use the nonlinear Schr\"odinger equation as our working model, but we also digress to the KdV equation at some point. This is a slightly revised version of a review paper that will appear in the forthcoming volume (Contemporary Mathematics, AMS) honoring Percy Deift.