Solitary Waves of Nonlinear Nonintegrable Equations

Abstract

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 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 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 GinzburgLandau equations, and to the Kuramoto-Sivashinsky equation.

Construction of Solutions for Nonintegrable Systems with the Help of the Painlevé Test

“…In such a way we obtain solutions only as formal series, but for some nonintegrable systems, for example, the generalized Hénon-Heiles system [4], the convergence of the Laurent-and psi-series solutions has been proved. Such solutions also assist to find the elliptic solutions [5].…”

“…These methods use results of the Painlevé test, but don't use the obtained Laurent-series solutions. In 2003 R. Conte and M. Musette [5] have proposed the method, which uses such solutions.…”

Construction of Special Solutions for Nonintegrable Systems

“…These methods (at least, some of them) use information about the dominant behavior of the initial system solutions in the neighbourhood of their singular points, but do not use the Laurent series representations of them. In [29] a new method has been developed as an alternative way to construct elliptic and elementary solutions and it has been shown that the use of the Laurent series solutions allows to extend the set of solutions [10] or, on the contrary, to prove non-existence of elliptic solutions [21,42]. The Laurent series solutions give the information about the global behavior of the differential system and assist in looking not only for global solutions but also for first integrals [24].…”

Доказательство Отсутствия Эллиптических Решений Кубического Комплексного Уравнения Гинзбурга - Ландау

“…В то же время для физических приложений оказывается достаточным знание частных решений с за-данными свойствами, например периодических или с требуемыми асимптотиками. В настоящее время активно развиваются методы построения частных решений в виде элементарных (вырожденных эллиптических) и эллиптических функций [1]- [12] (см. также монографию [13] и библиографию в ней).…”

“…также монографию [13] и библиографию в ней). Некоторые из этих методов предназначены для поиска только эллиптических решений [2], [5], [9], другие -толь-ко решений в терминах элементарных (вырожденных эллиптических) функций [3], [4], [11], третьи позволяют искать и те, и другие [1], [6]- [8], [10], [12]. Отметим, что и эллиптические, и вырожденные эллиптические функции обладают двумя свой-ствами, используемыми при поиске решений указанными методами.…”