The Schwarzian Korteweg de Vries equation in (2 1) dimensions

Abstract

In this paper, the complete Lie group classification of a (2 + 1)-dimensional integrable Schwarzian Korteweg–de Vries equation is obtained. The reduction to systems of partial differential equations in (1 + 1) dimension is derived from the optimal system of subalgebras. The invariance study of these systems leads to second-order ODEs. These ODEs provide several classes of solutions; all of them are expressible in terms of known functions, some of them are expressible in terms of the second and third Painleve transcendents. The corresponding solutions of the (2 + 1)-dimensional equation involve up to three arbitrary smooth functions. They even appear in the form ρ(z)f(x + (t)). Consequently, the solutions exhibit a rich variety of qualitative behaviour. Indeed, by making appropriate choices for the arbitrary functions, we are able to exhibit large families of solitary waves, coherent structures and different types of bound states.