This book introduces the reader to methods allowing one to build explicit solutions to these equations. A prerequisite task is to investigate whether the chances of success are high or low, and this can be achieved without any a priori knowledge of the solutions, with a powerful algorithm presented in detail called the Painlevé test. If the equation under study passes the Painlevé test,the equation is presumed integrable. If the test fails, the system is nonintegrable or even chaotic, but it may still be possible to find solutions. The examples include on the integrable side the nonlinear Schrödinger equation (continuous and discrete), the Korteweg-de Vries equation, the Hénon-Heiles Hamiltonians, on the nonintegrable side the complex Ginzburg-Landau equation (encountered in optical fibers, turbulence, etc), the Kuramoto-Sivashinsky equation (phase turbulence), the Kolmogorov-Petrovski-Piskunov equation (KPP, a reaction-diffusion model). Thorough treatment of the Painlevé property and Painlevé tests of differential equations With many worked examples illustrating the methods developed Written at a level accessible to graduate students as well as experts No other book offers such a thorough treatment of the search for analytic solutions of nonlinear differential equations The methods explained are used in a wide variety of applications INDICE: From the contents 1. Introduction. 2. Singularity structure in thecomplex plane, the Painlevé test. 3. Integrating ordinary differential equations. 4. Painlevé property and Painlevé test for partial differential equations. 5. From the test to explicit solutions of PDEs. 6. Quartic Hénon-Heiles Hamiltonian. 7. Discrete nonlinear equations. 8. FAQ (Frequently asked questions).A. The classical results of Painlevé and followers. B. Brief presentation of the elliptic functions. C. Basic introduction to the Nevanlinna theory. D. More on the Painlevé transcendents. E. The bilinear operator of Hirota. F. Algorithm for computing the Laurent series.
- ISBN: 978-1-4020-8490-4
- Editorial: Springer
- Encuadernacion: Cartoné
- Páginas: 256
- Fecha Publicación: 01/05/2008
- Nº Volúmenes: 1
- Idioma: Inglés