Numerical study of blow-up in solutions to generalized Kadomtsev-Petviashvili equations

Christian Klein; Ralf Peter

doi:10.3934/dcdsb.2014.19.1689

Article Contents

2014, Volume 19, Issue 6: 1689-1717. Doi: 10.3934/dcdsb.2014.19.1689

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Numerical study of blow-up in solutions to generalized Kadomtsev-Petviashvili equations

Christian Klein^1, and
Ralf Peter^1,

1.
Institut de Mathématiques de Bourgogne, Université de Bourgogne, 9 avenue Alain Savary, 21078 Dijon Cedex

Received: November 30, 2013

Revised: February 28, 2014

Published: July 2014

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Abstract

We present a numerical study of solutions to the generalized Kadomtsev-Petviashvili equations with critical and supercritical nonlinearity for localized initial data with a single minimum and single maximum. In the cases with blow-up, we use a dynamic rescaling to identify the type of the singularity. We present the first discussion of the observed blow-up scenarios. We show that the blow-up in solutions to the $L_{2}$ critical generalized Kadomtsev-Petviashvili I case is similar to what is known for the $L_{2}$ critical generalized Korteweg-de Vries equation. No blow-up is observed for solutions to the generalized Kadomtsev-Petviashvili II equations for $n\leq2$.

Keywords:

Generalized Kadomtsev-Petviasvili equations,
blow-up,
numerical approaches,
dynamic rescaling.

Mathematics Subject Classification: Primary: 35Q53, 35B44; Secondary: 35B65, 65M20.

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