Analytic solutions and Singularity formation for the Peakon b--Family equations

TL;DR

This paper proves local and global analyticity results for the $b$-family equations under certain conditions and investigates singularity formation using pseudospectral numerical methods, providing insights into the complex plane dynamics.

Contribution

It establishes the existence and uniqueness of analytic solutions for the $b$-family equations and analyzes singularity formation through numerical methods.

Findings

Unique local analytic solutions for $b$-family equations.

Global analyticity for initial data with sign-preserving momentum density.

Numerical tracking of singularity formation and decay rates in the complex plane.

Abstract

Using the Abstract Cauchy-Kowalewski Theorem we prove that the $b$-family equation admits, locally in time, a unique analytic solution. Moreover, if the initial data is real analytic and it belongs to $H^s$ with $s > 3/2$, and the momentum density $u_0 - u_{0,{xx}}$ does not change sign, we prove that the solution stays analytic globally in time, for $b\geq 1$. Using pseudospectral numerical methods, we study, also, the singularity formation for the $b$-family equations with the singularity tracking method. This method allows us to follow the process of the singularity formation in the complex plane as the singularity approaches the real axis, estimating the rate of decay of the Fourier spectrum.