Symmetry Reductions and Exact Solutions of Shallow Water Wave Equations

摘要

In this paper we study symmetly reductions and exact solutions of the shallow water wave (SWW) equation uxxxt+αuxuxt+βutuxx-uxt-uxx=0, where αand β 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=2β was discussed by Ablowitz, Kaup, Newell and Segur (Stud. Appl. Math, 53 (194), 249). who showed that this case was solvable by inverse scattering through second-order linear problem. This case and the case α=β were studied by Hirota and Satsum; (J. Phys, Soc, Japan, 40 (1976), 611) using Hirota's bi-linear technique, Further, the case α=β is solvable by inverse scattering through a third-order linear problem. In this paper, a catalogue of symmetry reductions is obtained using the classical lie method and the nonclassical method due to Bluman and Cole (J. Math Mech, 18 (1969), 1025). The classical Lie method yields symmetry redu(ions of (1) expressible in terms of the first, third and fifth Painleve transcendents and Weierstras: elliptic functions. The nonclassical method yields a plethora of exace solutions of (1) with α=β which possess a rich variety of qualitative behaviours. these solutions all like a two-soliton solution for t<0 but differ radically for t>0 and may be viewed as a nonlinear superposition of two solitions, one travelling to the left with abitrary speed and the other to the right with equal and opposite speed. These families of solutions have important implications with regard to the numerical analysis of SWW and suggests that solving (1) numerically could pose some fundamental difficulties. In particular, one would not be able to distinguish the solutions in an initial-value problem since an exponentially small change in the initial conditions can result in completely different qualiative behaviours. We compare the two-soliton solutions obtained using the nonclassical method to those obtained using the singular manifold method and Hirota's bi-linear mehtod. Further, we show that there is an analogous nonlinear superposition of solutions for two (2+1) dimensional generalisations of the SWW Equation (1) with α=β. This yields solutions expressible as the sum of two solutions of the Korteweg-de Vries equation.