We study numerically solutions to the Korteweg-de Vries and Camassa-Holm equation close to the breakup of the corresponding solution to the dispersionless equation. The solutions are compared with the properly rescaled numerical solution to a fourth order ordinary differential equation, the second member of the Painlevé I hierarchy. It is shown that this solution gives a valid asymptotic description of the solutions close to breakup. We present a detailed analysis of the situation and compare the Korteweg-de Vries solution quantitatively with asymptotic solutions obtained via the solution of the Hopf and the Whitham equations. We give a qualitative analysis for the Camassa-Holm equation.
Numerical study of a multiscale expansion of KdV and Camassa-Holm equation / Grava, Tamara; Klein, C.. - 458:(2008), pp. 81-98. (Intervento presentato al convegno Conference on integrable systems, random matrices, and applications in honor of Percy Deift's 60th birthday, May 22-26, 2006, Courant Institute of Mathematical Sciences, New York University, New York tenutosi a New York, USA nel 22-26 Maggio 2006).
Numerical study of a multiscale expansion of KdV and Camassa-Holm equation
Grava, Tamara;
2008-01-01
Abstract
We study numerically solutions to the Korteweg-de Vries and Camassa-Holm equation close to the breakup of the corresponding solution to the dispersionless equation. The solutions are compared with the properly rescaled numerical solution to a fourth order ordinary differential equation, the second member of the Painlevé I hierarchy. It is shown that this solution gives a valid asymptotic description of the solutions close to breakup. We present a detailed analysis of the situation and compare the Korteweg-de Vries solution quantitatively with asymptotic solutions obtained via the solution of the Hopf and the Whitham equations. We give a qualitative analysis for the Camassa-Holm equation.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
