Open Access
1998 Numerical computations of self-similar blow-up solutions of the generalized Korteweg-de Vries equation
Daniel B. Dix, William R. McKinney
Differential Integral Equations 11(5): 679-723 (1998). DOI: 10.57262/die/1367329666


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.


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Daniel B. Dix. William R. McKinney. "Numerical computations of self-similar blow-up solutions of the generalized Korteweg-de Vries equation." Differential Integral Equations 11 (5) 679 - 723, 1998.


Published: 1998
First available in Project Euclid: 30 April 2013

zbMATH: 1007.65061
MathSciNet: MR1664756
Digital Object Identifier: 10.57262/die/1367329666

Primary: 65M99
Secondary: 35B40 , 35Q53

Rights: Copyright © 1998 Khayyam Publishing, Inc.

Vol.11 • No. 5 • 1998
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