Blow-up and Global Similarity Solutions for Semilinear Third-Order Dispersive PDEs

Abstract

Blow-up and global self-similar solutions of the {\em semilinear dispersion equation} \begin{equation*} u_t=u_{xxx}+\mu({|u|^{p-1}u})_{xx} \quad \mbox{in} \quad \mathbb{\textbf{R}} \times \mathbb{\textbf{R}_+}, \quad p>1, \end{equation*} with sufficiently good initial data $u(x,0)=u_0(x)$ in $\mathbb{\textbf{R}}$, for $\mu=\pm1$, are studied. This equation represents a kind of shallow water model, where, unlike the classical KdV and mKdV equations, the leading nonlinear operator is of the second order, meaning unstable, stable, and ``semi-stable" (e.g., for the term $\pm (u^2)_{xx}$). Actually, such an approximation implies that second-order diffusion-like (or forward and backward ``porous medium operators") play a leading and key role in contrast to more standard first-order ones. There are various numerical and analytical challenges in order to observe admissible profiles due to the highly oscillatory nature of the problem, in contrast to parabolic equations. The classification of the solutions governed by self-similarity is given in terms of the initial data, $p$ and $\mu$, where the numerical experiments play a key role. A reliable numerical algorithm for large step sizes, called an exponentially fitted Runge--Kutta (EFRK) method, is proposed for the corresponding second-order ODE of the first critical exponent $p=p_0=2$ and the second Painlev\'e equation related to the KdV equation. Lastly, single-point blow-up similarity solutions for nonlinear extension of the problem, \begin{equation*} u_t=({|u|^{n}u})_{xxx}\pm({|u|^{p-1}u})_{xx} \quad \mbox{in} \quad \mathbb{\textbf{R}} \times \mathbb{\textbf{R}_+}, \quad n>0 \quad \mbox{and} \quad p>n+1, \end{equation*} is very briefly studied. Although the studies on nonlinear dispersion equations have been popular in the mathematical literature for at least the last fifty years, these third-order equations were not investigated in the literature in the sense of self-similarity.

Date of Award	26 Jun 2015
Original language	English
Awarding Institution	University of Bath
Sponsors	CEC - Marie Curie
Supervisor	Chris Budd (Supervisor) & Victor Galaktionov (Supervisor)