Symmetry Reductions and Exact Solutions of Shallow Water Wave Equations
| dc.creator | Clarkson, Peter A. | |
| dc.creator | Mansfield, Elizabeth L. | |
| dc.date | 1994-09-23 | |
| dc.date.accessioned | 2026-07-07T09:17:46Z | |
| dc.date.available | 2026-07-07T09:17:46Z | |
| dc.description | In this paper we study symmetry reductions and exact solutions of the shallow water wave (SWW) equation $$u_{xxxt} + αu_x u_{xt} + βu_t u_{xx} - u_{xt} - u_{xx} = 0,\eqno(1)$$ 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 $u_x=U$, have been discussed in the literature. The case $α=2β$ was discussed by Ablowitz, Kaup, Newell and Segur [{\it Stud.\ Appl.\ Math.}, {\bf53} (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 [{\it J.\ Phys.\ Soc.\ Japan}, {\bf40} (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 [{\it J.\ Math.\ Mech.\/}, {\bf 18} (1969) 1025]. The classical Lie method yields symmetry reductions of (1) expressible in terms of the first, third and fifth \p\ 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. | |
| dc.description | Tex file 19 pages, figures available from author | |
| dc.identifier | https://arxiv.org/abs/solv-int/9409003 | |
| dc.identifier | http://arxiv.org/abs/solv-int/9409003 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/153790 | |
| dc.subject | Exactly Solvable and Integrable Systems | |
| dc.title | Symmetry Reductions and Exact Solutions of Shallow Water Wave Equations | |
| dc.type | text |