Nonlinear Sciences – Pattern Formation and Solitons
Scientific paper
2004-07-09
Nonlinear Sciences
Pattern Formation and Solitons
28 pages, chapter in book "Dissipative solitons", ed. Akhmediev, to appear
Scientific paper
10.1007/10928028_15
Our goal is to find closed form analytic expressions for the solitary waves of nonlinear nonintegrable partial differential equations. The suitable methods, which can only be nonperturbative, are classified in two classes. In the first class, which includes the well known so-called truncation methods, one \textit{a priori} assumes a given class of expressions (polynomials, etc) for the unknown solution; the involved work can easily be done by hand but all solutions outside the given class are surely missed. In the second class, instead of searching an expression for the solution, one builds an intermediate, equivalent information, namely the \textit{first order} autonomous ODE satisfied by the solitary wave; in principle, no solution can be missed, but the involved work requires computer algebra. We present the application to the cubic and quintic complex one-dimensional Ginzburg-Landau equations, and to the Kuramoto-Sivashinsky equation.
Conte Robert
Musette Micheline
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