dorsal/arxiv
View SchemaSymmetries of a class of Nonlinear Third Order Partial Differential Equations
| Authors | P. A. Clarkson, E. L. Mansfield, T. J. Priestley |
|---|---|
| Categories | |
| ArXiv ID | solv-int/9609004 |
| URL | https://arxiv.org/abs/solv-int/9609004 |
Abstract
In this paper we study symmetry reductions of a class of nonlinear third order partial differential equations $u_t -\epsilon u_{xxt} +2\kappa u_x= u u_{xxx} +\alpha u u_x +\beta u_x u_{xx}$ where $\epsilon$, $\kappa$, $\alpha$ and $\beta$ are arbitrary constants. Three special cases of equation (1) have appeared in the literature, up to some rescalings. In each case the equation has admitted unusual travelling wave solutions: the Fornberg-Whitham equation, for the parameters $\epsilon=1$, $\alpha=-1$, $\beta=3$ and $\kappa=\tfr12$, admits a wave of greatest height, as a peaked limiting form of the travelling wave solution; the Rosenau-Hyman equation, for the parameters $\epsilon=0$, $\alpha=1$, $\beta=3$ and $\kappa=0$, admits a ``compacton'' solitary wave solution; and the Fuchssteiner-Fokas-Camassa-Holm equation, for the parameters $\epsilon=1$, $\alpha=-3$ and $\beta=2$, has a ``peakon'' solitary wave solution. A catalogue of symmetry reductions for equation (1) is obtained using the classical Lie method and the nonclassical method due to Bluman and Cole.
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"abstract": "In this paper we study symmetry reductions of a class of nonlinear third\norder partial differential equations $u_t -\\epsilon u_{xxt} +2\\kappa u_x= u\nu_{xxx} +\\alpha u u_x +\\beta u_x u_{xx}$ where $\\epsilon$, $\\kappa$, $\\alpha$\nand $\\beta$ are arbitrary constants. Three special cases of equation (1) have\nappeared in the literature, up to some rescalings. In each case the equation\nhas admitted unusual travelling wave solutions: the Fornberg-Whitham equation,\nfor the parameters $\\epsilon=1$, $\\alpha=-1$, $\\beta=3$ and $\\kappa=\\tfr12$,\nadmits a wave of greatest height, as a peaked limiting form of the travelling\nwave solution; the Rosenau-Hyman equation, for the parameters $\\epsilon=0$,\n$\\alpha=1$, $\\beta=3$ and $\\kappa=0$, admits a ``compacton\u0027\u0027 solitary wave\nsolution; and the Fuchssteiner-Fokas-Camassa-Holm equation, for the parameters\n$\\epsilon=1$, $\\alpha=-3$ and $\\beta=2$, has a ``peakon\u0027\u0027 solitary wave\nsolution. A catalogue of symmetry reductions for equation (1) is obtained using\nthe classical Lie method and the nonclassical method due to Bluman and Cole.",
"arxiv_id": "solv-int/9609004",
"authors": [
"P. A. Clarkson",
"E. L. Mansfield",
"T. J. Priestley"
],
"categories": [
"solv-int",
"nlin.SI"
],
"title": "Symmetries of a class of Nonlinear Third Order Partial Differential Equations",
"url": "https://arxiv.org/abs/solv-int/9609004"
},
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