Construction of Special Solutions for Nonintegrable Systems
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The Painlev´e test is very useful to construct not only the Laurent series solutions of systems of nonlinear ordinary differential equations but also the elliptic and trigonmetric 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 H´enonHeiles 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 eqution in the form of the Laurent or Puiseux series we use the Painlev´e 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 fivedimensional gravitational model with a scalar field to demonstrate this.
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TY - JOUR AU - Sergey Yu. Vernov PY - 2006 DA - 2006/02/01 TI - Construction of Special Solutions for Nonintegrable Systems JO - Journal of Nonlinear Mathematical Physics SP - 50 EP - 63 VL - 13 IS - 1 SN - 1776-0852 UR - https://doi.org/10.2991/jnmp.2006.13.1.5 DO - 10.2991/jnmp.2006.13.1.5 ID - Vernov2006 ER -