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.
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(Submitted by M. A. Malakhaltsev)
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Blázquez-Sanz, D., Conde Martín, J.M. Symmetry Reduction and Soliton-Like Solutions for the Generalized Korteweg-De Vries Equation. Lobachevskii J Math 39, 1305–1314 (2018). https://doi.org/10.1134/S1995080218090366
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DOI: https://doi.org/10.1134/S1995080218090366