Soliton: Difference between revisions
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Solitons are travelling wave solutions of [[Completely integrable|completely integrable]] equations. The term soliton is also used to describe travelling wave or solitary wave solutions of non-integrable equations. For example, assuming <math>u(t,x) = f(x- ct)</math> is a solution of the [[KdV Equation| KdV equation]] leads to an ODE of the form <math>-c f + {\frac{d}{dx}} | Solitons are travelling wave solutions of [[Completely integrable|completely integrable]] equations. The term soliton is also used to describe travelling wave or solitary wave solutions of non-integrable equations. For example, assuming <math>u(t,x) = f(x- ct)</math> is a solution of the [[KdV Equation| KdV equation]] leads to an ODE of the form <math>-c f + {\frac{d^2}{dx^2}} f + f^2 = 0</math>. This equation can be explicitly solved in terms of <math>\sech</math>. Other equations, such as focusing [[NLS|nonlinear Schrodinger equations]] also have Solitons are remarkably robust. For certain equations, Solitons have been shown to be [[Orbital Stability|orbitally stable]] and [[Asymptotic stability|asymptotically stable]]. | ||
[[Category:Concept]] [[Category:Airy]] | [[Category:Concept]] [[Category:Airy]] | ||
Revision as of 01:38, 31 July 2006
Solitons are travelling wave solutions of completely integrable equations. The term soliton is also used to describe travelling wave or solitary wave solutions of non-integrable equations. For example, assuming is a solution of the KdV equation leads to an ODE of the form . This equation can be explicitly solved in terms of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \sech} . Other equations, such as focusing nonlinear Schrodinger equations also have Solitons are remarkably robust. For certain equations, Solitons have been shown to be orbitally stable and asymptotically stable.
