Interdependence between the laurent-series and elliptic solutions of nonintegrable systems

Authors:
S. Yu. Vernov
Affiliations:
Skobeltsyn Institute of Nuclear Physics, Moscow State University, Moscow, Russia
Venue:
CASC'05 Proceedings of the 8th international conference on Computer Algebra in Scientific Computing
Year:
2005

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Abstract

The standard methods for the search for the elliptic solutions consist of two independent steps: transformation of a nonlinear polynomial differential equation into a nonlinear algebraic system and the search for solutions of the obtained system. It has been demonstrated by the example of the generalized Hénon–Heiles system that the use of the Laurent-series solutions of the initial differential equation assists to solve the obtained algebraic system and, thereby, simplifies the search for elliptic solutions. This procedure has been automatized with the help of the computer algebra systems Maple and REDUCE. The Laurent-series solutions also assist to solve the inverse problem: to prove the non-existence of elliptic solutions. Using the Hone's method based on the use the Laurent-series solutions and the residue theorem, we have proved that the cubic complex one-dimensional Ginzburg–Landau equation has neither elliptic standing wave nor elliptic travelling wave solutions. To find solutions of the initial differential equation in the form of the Laurent series we use the Painlevé test.