Differential and Integral Equations

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

Daniel B. Dix and William R. McKinney

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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.

Article information

Source
Differential Integral Equations, Volume 11, Number 5 (1998), 679-723.

Dates
First available in Project Euclid: 30 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.die/1367329666

Mathematical Reviews number (MathSciNet)
MR1664756

Zentralblatt MATH identifier
1007.65061

Subjects
Primary: 65M99: None of the above, but in this section
Secondary: 35B40: Asymptotic behavior of solutions 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10]

Citation

Dix, Daniel B.; McKinney, William R. Numerical computations of self-similar blow-up solutions of the generalized Korteweg-de Vries equation. Differential Integral Equations 11 (1998), no. 5, 679--723. https://projecteuclid.org/euclid.die/1367329666


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