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Gazeau, J. P.; Winternitz, P.

Symmetries of variable coefficient Korteweg-de Vries equations. (English) Zbl 0767.35077

J. Math. Phys. 33, No. 12, 4087-4102 (1992).
Summary: The Lie point symmetries of the equation \(u_ t+f(x,t)uu_ x+g(x,t)u_{xxx}=0\) are studied. The symmetry group is shown to be, at most, four dimensional, and this occurs if and only if the equation is equivalent, under local point transformations, to the KdV equation with \(f=g=1\). For nine different classes of functions \(f\) and \(g\), the symmetry group turns out to be three dimensional. Two-dimensional and one-dimensional symmetry groups occur for 11 and 15 classes of equations, respectively.

Cited in 35 Documents

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35A30 Geometric theory, characteristics, transformations in context of PDEs
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)

Keywords:

Lie point symmetries; symmetry groups
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\textit{J. P. Gazeau} and \textit{P. Winternitz}, J. Math. Phys. 33, No. 12, 4087--4102 (1992; Zbl 0767.35077)
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