dorsal/arxiv
View SchemaSymmetry Reductions and Exact Solutions of Shallow Water Wave Equations
| Authors | Peter A. Clarkson, Elizabeth L. Mansfield |
|---|---|
| Categories | |
| ArXiv ID | solv-int/9409003 |
| URL | https://arxiv.org/abs/solv-int/9409003 |
Abstract
In this paper we study symmetry reductions and exact solutions of the shallow water wave (SWW) equation $$u_{xxxt} + \alpha u_x u_{xt} + \beta u_t u_{xx} - u_{xt} - u_{xx} = 0,\eqno(1)$$ where $\alpha$ and $\beta$ 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 $\alpha=2\beta$ 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 $\alpha=\beta$ were studied by Hirota and Satsuma [{\it J.\ Phys.\ Soc.\ Japan}, {\bf40} (1976) 611] using Hirota's bi-linear technique. Further the case $\alpha=\beta$ 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 $\alpha=\beta$ 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.
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"abstract": "In this paper we study symmetry reductions and exact solutions of the shallow\nwater wave (SWW) equation $$u_{xxxt} + \\alpha u_x u_{xt} + \\beta u_t u_{xx} -\nu_{xt} - u_{xx} = 0,\\eqno(1)$$ where $\\alpha$ and $\\beta$ are arbitrary,\nnonzero, constants, which is derivable using the so-called Boussinesq\napproximation. Two special cases of this equation, or the equivalent nonlocal\nequation obtained by setting $u_x=U$, have been discussed in the literature.\nThe case $\\alpha=2\\beta$ was discussed by Ablowitz, Kaup, Newell and Segur\n[{\\it Stud.\\ Appl.\\ Math.}, {\\bf53} (1974) 249], who showed that this case was\nsolvable by inverse scattering through a second order linear problem. This case\nand the case $\\alpha=\\beta$ were studied by Hirota and Satsuma [{\\it J.\\ Phys.\\\nSoc.\\ Japan}, {\\bf40} (1976) 611] using Hirota\u0027s bi-linear technique. Further\nthe case $\\alpha=\\beta$ is solvable by inverse scattering through a third order\nlinear problem. In this paper a catalogue of symmetry reductions is obtained\nusing the classical Lie method and the nonclassical method due to Bluman and\nCole [{\\it J.\\ Math.\\ Mech.\\/}, {\\bf 18} (1969) 1025]. The classical Lie method\nyields symmetry reductions of (1) expressible in terms of the first, third and\nfifth \\p\\ transcendents and Weierstrass elliptic functions. The nonclassical\nmethod yields a plethora of exact solutions of (1) with $\\alpha=\\beta$ which\npossess a rich variety of qualitative behaviours. These solutions all like a\ntwo-soliton solution for $t\u003c0$ but differ radically for $t\u003e0$ and may be viewed\nas a nonlinear superposition of two solitons, one travelling to the left with\narbitrary speed and the other to the right with equal and opposite speed.",
"arxiv_id": "solv-int/9409003",
"authors": [
"Peter A. Clarkson",
"Elizabeth L. Mansfield"
],
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"title": "Symmetry Reductions and Exact Solutions of Shallow Water Wave Equations",
"url": "https://arxiv.org/abs/solv-int/9409003"
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