On the long-wave approximation of solitary waves in cylindrical coordinates

TL;DR

This paper rigorously justifies the cylindrical KdV equation as a long-wave approximation for radially symmetric solitary waves, providing error estimates and highlighting limitations in existing solitary wave solutions.

Contribution

It offers the first error estimates between solutions of the regularized Boussinesq equation and the cylindrical KdV approximation in L2 spaces, and shows certain solitary waves are not in L2, affecting modeling.

Findings

Error estimates established between true and approximate solutions.

Solitary wave solutions in the literature are not in L2 spaces.

The cylindrical KdV approximation's applicability is limited by these findings.

Abstract

We address justification and solitary wave solutions of the cylindrical KdV equation which is formally derived as a long wave approximation of radially symmetric waves in a two-dimensional nonlinear dispersive system. For a regularized Boussinesq equation, we prove error estimates between true solutions of this equation and the associated cylindrical KdV approximation in the L2-based spaces. The justification result holds in the spatial dynamics formulation of the regularized Boussinesq equation. We also prove that the class of solitary wave solutions considered previously in the literature does not contain solutions in the L2-based spaces. This presents a serious obstacle in the applicability of the cylindrical KdV equation for modeling of radially symmetric solitary waves since the long wave approximation has to be performed separately in different space-time regions.