Symmetry Reductions and Exact Solutions of ShallowWater Wave Equations

In this paper we study symmetry reductions and exact solutions of the shallow water wave (SWW) equation uxxxt + αuxuxt + βutuxx − uxt − uxx = 0, (1) where α and 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 ux = U , have been discussed in the literature. The case α = 2β 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 = β were studied by Hirota and Satsuma [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 reductions of (1) expressible in terms of the first, third and fifth Painlevé transcendents and Weierstrass elliptic functions. The nonclassical method yields a plethora of exact 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 solitons, one travelling to the left with arbitrary 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 qualitative 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 method. Further, we show that there is an analogous nonlinear superposition of solutions for two 2 + 1dimensional 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. Nonclassical symmetries and exact solutions of a shallow water wave equation 1