Exact Solution for Nonintegrable Systems

Main ideas

In many areas of physics, in particular, in gravitation and quantum field theory, to solve the physical problem one has to find solutions of nonlinear ordinary differential equations or systems of such equations. In many cases it is not possible to solve this problem completely, i.e. to integrate the system and find the general solution in the form of quadratures. On the other hand, for construction of a physical model knowledge of only special solutions with some required properties, for example, periodicity, is often sufficient.

One of directions of my investigations is the search of special solutions for nonintegrable systems, i.e. systems, which general solutions can not be obtained by the known methods. At the same time there exist methods, which allow to find special solutions of systems of ordinary differential equations in the form of elementary or elliptic functions, reducing the initial system to a system of nonlinear algebraic equations. The main problem of such methods is the complexity of the obtaining algebraic system.

It has been proved, that using the Groebner basis one can solve any polynomial system of algebraic equations by finite number of steps, but time of calculation and the required size of computer memory allow to solve only the simplest systems. In 2003 new method for search of special solutions of ordinary differential equations has been presented (R. Conte, M. Musette, arXiv:nlin.PS/0302051). The given method allows to transform the initial nonlinear differential equation to an infinite system of linear equations. This method uses the solutions in the form of Laurent series for the initial systems of differential equations. Directions of my activity are an automatization of the described method with the help of computer algebra systems (the packages of procedures have been written in Maple and REDUCE), its generalization on multivalued solutions, representable in the form of Puiseux series, and search of solution for some physically important systems.

The Painleve 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 Henon-Heiles system that the use of the Laurent series solutions of the initial differential equation assists to solve the obtained algebraic system.

Main results

In two nonintegrable cases of the generalized Henon-Heiles system new three-parameter solutions in the form of the Laurent series with nonzero domains of convergence.

The generalized Henon-Heiles system with an additional nonpolynomial term has been considered. In two nonintegrable cases new two-parameter solutions have been obtained in terms of elliptic functions.

The procedure of the search of Laurent series solutions 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 Painleve test.

The cubic complex one-dimensional Ginzburg-Landau equation has been considered. Using the Hone's method, based on the use of the Laurent-series solutions and the residue theorem, we have proved that this equation has neither elliptic standing wave nor elliptic travelling wave solutions. This result amplifies the Hone's result, that this equation has no elliptic travelling wave solutions.

The Conte-Musette method has been modified for the search of only elliptic solutions to systems of differential equations. A key idea of this a priory restriction is to simplify calculations by means of the use of a few Laurent series solutions instead of one and the use of the residue theorem. The application of our approach to the quintic complex one-dimensional Ginzburg-Landau equation (CGLE5) allows to find elliptic solutions in the wave form. We also find restrictions on coefficients, which are necessary conditions for the existence of elliptic solutions for the CGLE5.

Main publications

• S.Yu. Vernov,On elliptic solutions of the quintic complex one-dimensional Ginzburg-Landau equation, J. Phys. A: Math. Theor. 40 (2007) 9833-9844; arXiv:nlin/0602060
• S.Yu. Vernov, On elliptic solutions of the cubic complex one-dimensional Ginzburg-Landau equation , Theor. Math. Phys. 146 (2006) 131-139; arXiv:nlin/0503009
• S.Yu. Vernov, Construction of Special Solutions for Nonintegrable Systems , J. Nonlin. Math. Phys. 13 (2006) 50-63; arXiv:astro-ph/0502356
• E.I. Timoshkova and S.Yu. Vernov, On two nonintegrable cases of the generalized Henon-Heiles system with an additional nonpolynomial term , Phys. Atom. Nucl. 68 (2005) 1947-1955, arXiv:math-ph/0402049
• S.Yu. Vernov, Construction of solutions for the generalized Henon-Heiles system with the help of the Painleve test , Theor. Math. Phys. 135 (2003) 792-801 arXiv:math-ph/0209063
• E.I. Timoshkova and S.Yu. Vernov, The Painleve analysis and construction of solutions for the generalized Henon-Heiles system, ASP Conference Series 316, "Order and Chaos in Stellar and Planetary Systems", 2004, pp. 28-33
• S.Yu. Vernov, Construction of Three-parameter Solutions for the Generalized Henon-Heiles system with the help of the Painleve Test, chapter in the book "Mathematical Physics Frontiers", F. Columbus (Ed.), Nova Science Publishers, New York, 2004, pp. 123-140