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.
2008
Integrable systems and random matrices : in honor of Percy Deift
458
81
98
978-0-8218-4240-9
978-0-8218-8137-8
http://dx.doi.org/10.1090/conm/458
https://arxiv.org/abs/math-ph/0702038
American Mathematical Society
Grava, Tamara; Klein, C.
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/15494
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus ND
  • ???jsp.display-item.citation.isi??? 16
social impact