Symmetry reductions and exact solutions of shallow water wave equations

摘要: 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 s are arbitrary, nonzero, constants, which is derivable using so-called Boussinesq approximation. Two special cases equation, or equivalent nonlocal obtained by setting u x U, have been discussed in literature. The case 2s was Ablowitz, Kaup, Newell Segur (Stud. Appl. Math., 53 (1974), 249), who showed that solvable inverse scattering through a second-order linear problem.This were studied Hirota Satsuma (J. Phys. Soc. Japan, 40 (1976), 611) Hirota’s bi-linear technique. Further, third-order problem.