Symmetry reductions and exact solutions of shallow water wave equations

Abstract

In this paper we study symmetry reductions and exact solutions of the shallow water wave (SWW) equation $$ {u_{xxxt}} + \alpha {u_x}{u_{xt}} + \beta {u_t}{u_{xx}} -{u_{xt}} -{u_{xx}} = 0,$$ where α and s are arbitrary, nonzero, constants, which is derivable using the so-called Boussinesq approximation. Two special cases of this equation, or the equivalent nonlocal equation obtained by setting u x = U, have been discussed in the literature. The case α = 2s was discussed by Ablowitz, Kaup, Newell and Segur (Stud. Appl. Math., 53 (1974), 249), who showed that this case was solvable by inverse scattering through a second-order linear problem.This case and the case α = s were studied by Hirota and Satsuma (J. Phys. Soc. Japan, 40 (1976), 611) using Hirota’s bi-linear technique. Further, the case α = s is solvable by inverse scattering through a third-order linear problem.