Symmetry Reduction and Soliton-Like Solutions for the Generalized Korteweg-De Vries Equation

D. Blázquez-Sanz, J. M. Conde Martín

Research output: Contribution to journalArticlepeer-review

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Abstract

We analyze the gKdV equation, a generalized version of Korteweg-de Vries with an arbitrary function f(u). In general, for a function f(u) the Lie algebra of symmetries of gKdV is the 2-dimensional Lie algebra of translations of the plane xt. This implies the existence of plane wave solutions. Indeed, for some specific values of f(u) the equation gKdV admits a Lie algebra of symmetries of dimension grater than 2. We compute the similarity reductions corresponding to these exceptional symmetries. We prove that the gKdV equation has soliton-like solutions under some general assumptions, and we find a closed formula for the plane wave solutions, that are of hyperbolic secant type.

Original languageEnglish
Pages (from-to)1305-1314
Number of pages10
JournalLobachevskii Journal of Mathematics
Volume39
Issue number9
DOIs
StatePublished - 1 Nov 2018

Keywords

  • Korteweg-de Vries equation
  • Lie symmetries
  • symmetry reduction

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