Long-time asymptotic solution structure of Camassa-Holm equation subject to an initial condition with non-zero reflection coefficient of the scattering data

TL;DR

This paper develops a high-accuracy numerical scheme to analyze the long-time behavior of solutions to the Camassa-Holm equation with specific initial conditions, confirming theoretical asymptotics and connecting solutions to Painlevé transcendents.

Contribution

It introduces a symplecticity-preserving finite difference scheme for the Camassa-Holm equation and numerically verifies its long-time asymptotics and relation to Painlevé equations.

Findings

Accurate depiction of long-time solution regions with distinct behaviors.

Numerical confirmation of Painlevé transcendents describing transition zones.

Demonstrated connection between finite difference solutions and Painlevé II solutions.

Abstract

In this article we numerically revisit the long-time solution behavior of the Camassa-Holm equation. The finite difference solution of this integrable equation is sought subject to the newly derived initial condition with Delta-function potential. Our underlying strategy of deriving a numerical phase accurate finite difference scheme in time domain is to reduce the numerical dispersion error through minimization of the derived discrepancy between the numerical and exact modified wavenumbers. Additionally, to achieve the goal of conserving Hamiltonians in the completely integrable equation of current interest, a symplecticity-preserving time-stepping scheme is developed. Based on the solutions computed from the temporally symplecticity-preserving and the spatially wavenumber-preserving scheme, the long-time asymptotic CH solution characters can be accurately depicted in distinct regions of the space-time domain featuring with their own quantitatively very different solution behaviors. We also aim to numerically confirm that in the two transition zones their long-time asymptotics can indeed be described in terms of the theoretically derived Painlev\'e transcendents. Another attempt of this study is to numerically exhibit a close connection between the presently predicted finite-difference solution and the solution of the Painlev\'e ordinary differential equation of type II in two different transition zones.