Lie symmetry analysis of (2+1)-dimensional KdV equations with variable coefficients

ABSTRACT

In this paper, symmetry groups are used to obtain symmetry reductions of (2+1)-dimensional KdV equations with variable coefficients. Despite the fact that these equations emerge in a nonlocal form, by using suitable transformations, they can be written as systems of partial differential equations, and in potential form, as fourth-order partial differential equations. We show that the point symmetries of the potential equation involve a large number of arbitrary functions. Moreover, these symmetries are used to transform the fourth-order partial differential equations into (1+1)-dimensional fourth-order differential equations. Furthermore, we have determined all two-dimensional solvable symmetry subalgebras, under certain restrictions, which the potential equation admits. Finally, by way of example, taking into account a two-dimensional abelian subalgebra, we obtain a direct reduction of the potential equation to an ordinary differential equation.

KEYWORDS:

• Variable-coefficient KdV equations
• solvable symmetry algebras
• symmetries
• symmetry reductions

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 35C07
• 35Q53

Acknowledgements

We warmly thank the referees for their valuable comments and suggestions. The authors gratefully acknowledge Dr. Stephen Anco from Brock University for his expert guidance and help during his visit to Universidad de Cádiz.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

R. de la Rosa http://orcid.org/0000-0002-9357-9167

E. Recio http://orcid.org/0000-0001-6630-4574

T.M. Garrido http://orcid.org/0000-0002-9536-1065

M.S. Bruzón http://orcid.org/0000-0002-3599-6106

Additional information

Funding

The authors also acknowledge the financial support from Universidad de Cádiz and they express their sincerest gratitude to the Plan Propio de Investigación de la Universidad de Cádiz.