Semiclassical limit for generalized KdV equations before the gradient catastrophe

TL;DR

This paper investigates the semiclassical limit of generalized KdV equations with Sobolev initial data, showing convergence to the Hopf equation and providing asymptotic expansions before the gradient catastrophe.

Contribution

It establishes convergence in Sobolev spaces and derives asymptotic expansions for solutions of generalized KdV equations prior to gradient catastrophe.

Findings

Solutions converge to the Hopf equation in $H^s$

Asymptotic expansions exist in powers of the semiclassical parameter

Results extend to higher order linearities in KdV equations

Abstract

We study the semiclassical limit of the (generalised) KdV equation, for initial data with Sobolev regularity, before the time of the gradient catastrophe of the limit conservation law. In particular, we show that in the semiclassical limit the solution of the KdV equation: i) converges in $H^s$ to the solution of the Hopf equation, provided the initial data belongs to $H^s$, ii) admits an asymptotic expansion in powers of the semiclassical parameter, if the initial data belongs to the Schwartz class. The result is also generalized to KdV equations with higher order linearities.