Abstract

This article is concerned with a conjecture in [B. Dubrovin, Comm. Math. Phys., 267 (2006), pp. 117–139] on the formation of dispersive shocks in a class of Hamiltonian dispersive regularizations of the quasi-linear transport equation. The regularizations are characterized by two arbitrary functions of one variable, where the condition of integrability implies that one of these functions must not vanish. It is shown numerically for a large class of equations that the local behavior of their solution near the point of gradient catastrophe for the transport equation is described by a special solution of a Painlevé-type equation. This local description holds also for solutions to equations where blowup can occur in finite time. Furthermore, it is shown that a solution of the dispersive equations away from the point of gradient catastrophe is approximated by a solution of the transport equation with the same initial data, modulo terms of order $\epsilon^2$, where $\epsilon^2$ is the small dispersion parameter. Corrections up to order $\epsilon^4$ are obtained and tested numerically.

MSC codes

  1. 65M70
  2. 65L05
  3. 65M20

Keywords

  1. generalized Korteweg–de Vries equations
  2. Kawahara equations
  3. dispersive shocks
  4. multiscale analysis

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Published In

cover image SIAM Journal on Applied Mathematics
SIAM Journal on Applied Mathematics
Pages: 983 - 1008
ISSN (online): 1095-712X

History

Submitted: 4 January 2011
Accepted: 28 March 2011
Published online: 7 July 2011

MSC codes

  1. 65M70
  2. 65L05
  3. 65M20

Keywords

  1. generalized Korteweg–de Vries equations
  2. Kawahara equations
  3. dispersive shocks
  4. multiscale analysis

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