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Numerical Study of Blow up and Stability of Solutions of Generalized Kadomtsev–Petviashvili Equations

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Abstract

We first review the known mathematical results concerning the Kadomtsev–Petviashvili type equations. Then we perform numerical simulations to analyze various qualitative properties of the equations: blow-up versus long time behavior, stability and instability of solitary waves.

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Notes

  1. Note that here T=0 corresponds to the absence of surface tension. In the Fluid Mechanics community, the Bond number is often defined as the inverse of our T.

  2. We will see in Sect. 2.4 that KP type equations (in particular the classical KP I and KP II equations) do possess solutions in this class.

  3. In the case of KP II, one can use the explicit form of the fundamental solution found in Redekopp (1980).

  4. ϵ is thus the small parameter which measures the weak nonlinear and long wave effects.

  5. Some extra regularity on u 0 is actually needed.

  6. This means that ψ(x−ct,y) solves (43).

  7. Note, however, that when the BBM trick is applied to the KdV or to the KP equation with strong surface tension (Bond number greater than \(\frac{1}{3}\)) one gets an ill-posed equation, so that what is called KP I/BBM equation is merely a mathematical object without, as far as we know, modeling relevance.

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Acknowledgements

We thank M. Haragus and D. Chiron for useful discussions and hints. The second author thanks L. Molinet and N. Tzvetkov for useful suggestions and for a lengthy and fruitful joint research on KP equations. This work has been supported in part by the project FroM-PDE funded by the European Research Council through the Advanced Investigator Grant Scheme, the Conseil Régional de Bourgogne via a FABER grant, the ANR via the program ANR-09-BLAN-0117-01 and the Wolfgang Pauli Institute in Vienna.

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

    C. Klein

  2. Laboratoire de Mathématiques, UMR 8628, Université Paris-Sud et CNRS, 91405, Orsay, France

    J.-C. Saut

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  1. C. Klein
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Correspondence to C. Klein.

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Communicated by M.I. Weinstein.

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Klein, C., Saut, JC. Numerical Study of Blow up and Stability of Solutions of Generalized Kadomtsev–Petviashvili Equations. J Nonlinear Sci 22, 763–811 (2012). https://doi.org/10.1007/s00332-012-9127-4

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  • DOI: https://doi.org/10.1007/s00332-012-9127-4

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