TY - JOUR

T1 - Symmetry Reduction and Soliton-Like Solutions for the Generalized Korteweg-De Vries Equation

AU - Blázquez-Sanz, D.

AU - Conde Martín, J. M.

N1 - Publisher Copyright:
© 2018, Pleiades Publishing, Ltd.

PY - 2018/11/1

Y1 - 2018/11/1

N2 - 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.

AB - 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.

KW - Korteweg-de Vries equation

KW - Lie symmetries

KW - symmetry reduction

UR - http://www.scopus.com/inward/record.url?scp=85059607333&partnerID=8YFLogxK

U2 - 10.1134/S1995080218090366

DO - 10.1134/S1995080218090366

M3 - Artículo

AN - SCOPUS:85059607333

SN - 1995-0802

VL - 39

SP - 1305

EP - 1314

JO - Lobachevskii Journal of Mathematics

JF - Lobachevskii Journal of Mathematics

IS - 9

ER -