Symmetry reduction and soliton-like solutions for the generalized\n Korteweg-de Vries equation

2017/01/29 by Juan Manuel Conde Martín, Martín, Juan Manuel Conde, David Blázquez-Sanz +1
Mathematics · Physics and Astronomy · #35Q53 #Advanced Mathematical Physics Problems #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #Pattern Formation and Solitons (nlin.PS)

paper · pdf · doi:10.48550/arxiv.1701.08460

openalex publication_date 2017/01/29 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28

Abstract

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

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