dorsal/arxiv
View SchemaSolitary Waves and Compactons in a class of Generalized Korteweg-DeVries Equations
| Authors | Fred Cooper, Harvey Shepard, Pasquale Sodano |
|---|---|
| Categories | |
| ArXiv ID | patt-sol/9305006 |
| URL | https://arxiv.org/abs/patt-sol/9305006 |
| DOI | 10.1103/PhysRevE.48.4027 |
Abstract
We study the class of generalized Korteweg-DeVries equations derivable from the Lagrangian: $ L(l,p) = \int \left( \frac{1}{2} \vp_{x} \vp_{t} - { {(\vp_{x})^{l}} \over {l(l-1)}} + \alpha(\vp_{x})^{p} (\vp_{xx})^{2} \right) dx, $ where the usual fields $u(x,t)$ of the generalized KdV equation are defined by $u(x,t) = \vp_{x}(x,t)$. This class contains compactons, which are solitary waves with compact support, and when $l=p+2$, these solutions have the feature that their width is independent of the amplitude. We consider the Hamiltonian structure and integrability properties of this class of KdV equations. We show that many of the properties of the solitary waves and compactons are easily obtained using a variational method based on the principle of least action. Using a class of trial variational functions of the form $u(x,t) = A(t) \exp \left[-\beta (t) \left|x-q(t) \right|^{2n} \right]$ we find soliton-like solutions for all $n$, moving with fixed shape and constant velocity, $c$. We show that the velocity, mass, and energy of the variational travelling wave solutions are related by $ c = 2 r E M^{-1}$, where $ r = (p+l+2)/(p+6-l)$, independent of $n$.\newline \newline PACS numbers: 03.40.Kf, 47.20.Ky, Nb, 52.35.Sb
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"abstract": "We study the class of generalized Korteweg-DeVries equations derivable from\nthe Lagrangian: $ L(l,p) = \\int \\left( \\frac{1}{2} \\vp_{x} \\vp_{t} - {\n{(\\vp_{x})^{l}} \\over {l(l-1)}} + \\alpha(\\vp_{x})^{p} (\\vp_{xx})^{2} \\right)\ndx, $ where the usual fields $u(x,t)$ of the generalized KdV equation are\ndefined by $u(x,t) = \\vp_{x}(x,t)$. This class contains compactons, which are\nsolitary waves with compact support, and when $l=p+2$, these solutions have the\nfeature that their width is independent of the amplitude. We consider the\nHamiltonian structure and integrability properties of this class of KdV\nequations. We show that many of the properties of the solitary waves and\ncompactons are easily obtained using a variational method based on the\nprinciple of least action. Using a class of trial variational functions of the\nform $u(x,t) = A(t) \\exp \\left[-\\beta (t) \\left|x-q(t) \\right|^{2n} \\right]$ we\nfind soliton-like solutions for all $n$, moving with fixed shape and constant\nvelocity, $c$. We show that the velocity, mass, and energy of the variational\ntravelling wave solutions are related by $ c = 2 r E M^{-1}$, where $ r =\n(p+l+2)/(p+6-l)$, independent of $n$.\\newline \\newline PACS numbers: 03.40.Kf,\n47.20.Ky, Nb, 52.35.Sb",
"arxiv_id": "patt-sol/9305006",
"authors": [
"Fred Cooper",
"Harvey Shepard",
"Pasquale Sodano"
],
"categories": [
"patt-sol",
"nlin.PS"
],
"doi": "10.1103/PhysRevE.48.4027",
"title": "Solitary Waves and Compactons in a class of Generalized Korteweg-DeVries Equations",
"url": "https://arxiv.org/abs/patt-sol/9305006"
},
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