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Symmetry reductions and exact solutions of shallow water wave equations. (English) Zbl 0835.35006
Summary: We study symmetry reductions and exact solutions of the shallow water wave (SWW) equation $$u_{xxxt}+ \alpha u_x u_{xt}+ \beta u_t u_{xx}- u_{xt}- u_{xx}= 0,\tag1$$ where $\alpha$ and $\beta$ are arbitrary, nonzero, constants, which is derivable using the so-called Boussinesq approximation. In this paper, a catalogue of symmetry reductions is obtained using the classical Lie method and the nonclassical method due to {\it G. W. Bluman} and {\it J. D. Cole} [J. Math. Mech. 18, 1025-1042 (1969; Zbl 0187.035)]. The classical Lie method yields symmetry reductions of (1) expressible in terms of the first, third and fifth Painlevé tarnscendents and Weierstrass elliptic functions. The nonclassical method yields a plethora of exact solutions of (1) with $\alpha= \beta$ which possess a rich variety of qualitative behaviours. These solutions are 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 bilinear method. Further, we show that there is an analogous nonlinear superposition of solutions for two $(2+ 1)$-dimensional generalizations of the SWW equation (1) with $\alpha= \beta$. This yields solutions expressible as the sum of two solutions of the Korteweg-de Vries equation.

##### MSC:
 35A30 Geometric theory for PDE, characteristics, transformations 37J35 Completely integrable systems, topological structure of phase space, integration methods 37K10 Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies 35Q35 PDEs in connection with fluid mechanics 58J70 Invariance and symmetry properties
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