dorsal/arxiv
View SchemaStability of Compacton Solutions of Fifth-Order Nonlinear Dispersive Equations
| Authors | Bishwajyoti Dey, Avinash Khare |
|---|---|
| Categories | |
| ArXiv ID | patt-sol/9904007 |
| URL | https://arxiv.org/abs/patt-sol/9904007 |
| DOI | 10.1088/0305-4470/33/30/305 |
| Journal | J. Phys. A33 (2000) 5335 |
Abstract
We consider fifth-order nonlinear dispersive $K(m,n,p)$ type equations to study the effect of nonlinear dispersion. Using simple scaling arguments we show, how, instead of the conventional solitary waves like solitons, the interaction of the nonlinear dispersion with nonlinear convection generates compactons - the compact solitary waves free of exponential tails. This interaction also generates many other solitary wave structures like cuspons, peakons, tipons etc. which are otherwise unattainable with linear dispersion. Various self similar solutions of these higher order nonlinear dispersive equations are also obtained using similarity transformations. Further, it is shown that, like the third-order nonlinear $K(m,n)$ equations, the fifth-order nonlinear dispersive equations also have the same four conserved quantities and further even any arbitrary odd order nonlinear dispersive $K(m,n,p...)$ type equations also have the same three (and most likely the four) conserved quantities. Finally, the stability of the compacton solutions for the fifth-order nonlinear dispersive equations are studied using linear stability analysis. From the results of the linear stability analysis it follows that, unlike solitons, all the allowed compacton solutions are stable, since the stability conditions are satisfied for arbitrary values of the nonlinear parameters.
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"abstract": "We consider fifth-order nonlinear dispersive $K(m,n,p)$ type equations to\nstudy the effect of nonlinear dispersion. Using simple scaling arguments we\nshow, how, instead of the conventional solitary waves like solitons, the\ninteraction of the nonlinear dispersion with nonlinear convection generates\ncompactons - the compact solitary waves free of exponential tails. This\ninteraction also generates many other solitary wave structures like cuspons,\npeakons, tipons etc. which are otherwise unattainable with linear dispersion.\nVarious self similar solutions of these higher order nonlinear dispersive\nequations are also obtained using similarity transformations. Further, it is\nshown that, like the third-order nonlinear $K(m,n)$ equations, the fifth-order\nnonlinear dispersive equations also have the same four conserved quantities and\nfurther even any arbitrary odd order nonlinear dispersive $K(m,n,p...)$ type\nequations also have the same three (and most likely the four) conserved\nquantities. Finally, the stability of the compacton solutions for the\nfifth-order nonlinear dispersive equations are studied using linear stability\nanalysis. From the results of the linear stability analysis it follows that,\nunlike solitons, all the allowed compacton solutions are stable, since the\nstability conditions are satisfied for arbitrary values of the nonlinear\nparameters.",
"arxiv_id": "patt-sol/9904007",
"authors": [
"Bishwajyoti Dey",
"Avinash Khare"
],
"categories": [
"patt-sol",
"nlin.PS",
"nlin.SI",
"quant-ph",
"solv-int"
],
"doi": "10.1088/0305-4470/33/30/305",
"journal_ref": "J. Phys. A33 (2000) 5335",
"title": "Stability of Compacton Solutions of Fifth-Order Nonlinear Dispersive Equations",
"url": "https://arxiv.org/abs/patt-sol/9904007"
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