Lobachevskii Journal of Mathematics

, Volume 39, Issue 9, pp 1305–1314 | Cite as

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

  • D. Blázquez-SanzEmail author
  • J. M. Conde Martín
Part 2. Special issue “Actual Problems of Algebra and Analysis” Editors: A. M. Elizarov and E. K. Lipachev


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.

Keywords and phrases

Korteweg-de Vries equation Lie symmetries symmetry reduction 


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Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Universidad Nacional de ColombiaSede MedellínColombia
  2. 2.Universidad San Francisco de QuitoQuitoEcuador

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