Construction of Special Solutions for Nonintegrable Systems

Abstract

The Painlevé test is very useful to construct not only the Laurent series solutions of systems of nonlinear ordinary differential equations but also the elliptic and trigonometric ones. The standard methods for constructing the elliptic solutions consist of two independent steps: transformation of a nonlinear polynomial differential equation into a nonlinear algebraic system and a 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. This procedure has been automatized and generalized on some type of multivalued solutions. To find solutions of the initial differential equation in the form of the Laurent or Puiseux series we use the Painlevé test. This test can also assist to solve the inverse problem: to find the form of a polynomial potential, which corresponds to the required type of solutions. We consider the five-dimensional gravitational model with a scalar field to demonstrate this.

Elliptic solutions of the quintic complex one-dimensional Ginzburg–Landau equation

“…Another way for future investigations is the generalization of the Conte-Musette method on the case of multivalued solutions. The first results in this direction have been obtained in [19,50].…”

“…At present methods for construction of special solutions of nonintegrable systems in terms of elementary (more precisely, degenerated elliptic) and elliptic functions are actively developed [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] (see also [21] and references therein). Some of these methods are intended for the search for elliptic solutions only [11,15]; others allow us to find either solutions in terms of elementary functions only [2][3][4]20] or both types of solutions [1, 5-10, 12-14, 16-19].…”