Numerical computations of self-similar blow-up solutions of the generalized Korteweg-de Vries equation

Abstract

The structure of the blow-up in finite time of a solution of the Generalized Korteweg-de Vries equation arising from a perturbed unstable solitary wave is studied numerically. The computed solution is observed to blow-up in the $L^\infty$-norm in finite time by forming a spike of infinite height at $x=x^*$ and at $t=t^*$. Scaled coordinates are introduced to examine the detailed structure of the solution in the immediate neighborhood of the blow-up. The appropriately rescaled solution is observed to converge in these coordinates as $t\to{t^*}^-$, indicating self-similar behavior. A best-fit solution $w(\xi)$ of the nonlinear ODE satisfied by self-similar profiles is computed for the statistical data compiled from this convergence. The asymptotics at $\pm\infty$ of this solution of the ODE are studied, and found to coincide with those of solutions $w_\pm(\xi)$ of the linearized ODE as $\pm\xi\to\infty$. The self-similar part of the solution is also matched (numerically) to the part of the solution more removed from the blow-up point, showing how rapidly decaying initial data can give rise to self-similar blow-up. Heuristic explanations of how nonlinearity and dispersion cooperate to yield existence of a solution $w(\xi)$ of the ODE with the desired asymptotics as $\pm\xi\to\infty$ are discussed.