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

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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

First available in Project Euclid: 30 April 2013

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]


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

Export citation