Welcome to Loot.co.za!
Sign in / Register |Wishlists & Gift Vouchers |Help | Advanced search
|
Your cart is empty |
|||
Showing 1 - 5 of 5 matches in All Departments
This book, now in its second edition, introduces the singularity analysis of differential and difference equations via the Painleve test and shows how Painleve analysis provides a powerful algorithmic approach to building explicit solutions to nonlinear ordinary and partial differential equations. It is illustrated with integrable equations such as the nonlinear Schroedinger equation, the Korteweg-de Vries equation, Henon-Heiles type Hamiltonians, and numerous physically relevant examples such as the Kuramoto-Sivashinsky equation, the Kolmogorov-Petrovski-Piskunov equation, and mainly the cubic and quintic Ginzburg-Landau equations. Extensively revised, updated, and expanded, this new edition includes: recent insights from Nevanlinna theory and analysis on both the cubic and quintic Ginzburg-Landau equations; a close look at physical problems involving the sixth Painleve function; and an overview of new results since the book's original publication with special focus on finite difference equations. The book features tutorials, appendices, and comprehensive references, and will appeal to graduate students and researchers in both mathematics and the physical sciences.
Many physical phenomena are described by nonlinear evolution
equation. Those that are integrable provide various mathematical
methods, presented by experts in this tutorial book, to find
special analytic solutions to both integrable and partially
integrable equations. The direct method to build solutions includes
the analysis of singularities a la Painleve, Lie symmetries leaving
the equation invariant, extension of the Hirota method,
construction of the nonlinear superposition formula. The main
inverse method described here relies on the bi-hamiltonian
structure of integrable equations. The book also presents some
extension to equations with discrete independent and dependent
variables.
This book, now in its second edition, introduces the singularity analysis of differential and difference equations via the Painleve test and shows how Painleve analysis provides a powerful algorithmic approach to building explicit solutions to nonlinear ordinary and partial differential equations. It is illustrated with integrable equations such as the nonlinear Schroedinger equation, the Korteweg-de Vries equation, Henon-Heiles type Hamiltonians, and numerous physically relevant examples such as the Kuramoto-Sivashinsky equation, the Kolmogorov-Petrovski-Piskunov equation, and mainly the cubic and quintic Ginzburg-Landau equations. Extensively revised, updated, and expanded, this new edition includes: recent insights from Nevanlinna theory and analysis on both the cubic and quintic Ginzburg-Landau equations; a close look at physical problems involving the sixth Painleve function; and an overview of new results since the book's original publication with special focus on finite difference equations. The book features tutorials, appendices, and comprehensive references, and will appeal to graduate students and researchers in both mathematics and the physical sciences.
Nonlinear differential or difference equations are encountered not only in mathematics, but also in many areas of physics (evolution equations, propagation of a signal in an optical fiber), chemistry (reaction-diffusion systems), and biology (competition of species). 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 Painleve test. If the equation under study passes the Painleve test, the equation is presumed integrable. If on the contrary the test fails, the system is nonintegrable or even chaotic, but it may still be possible to find solutions. The examples chosen to illustrate these methods are mostly taken from physics. These include on the integrable side the nonlinear Schroedinger equation (continuous and discrete), the Korteweg-de Vries equation, the Henon-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), the Lorenz model of atmospheric circulation and the Bianchi IX cosmological model. Written at a graduate level, the book contains tutorial text as well as detailed examples and the state of the art on some current research.
Many physical phenomena are described by nonlinear evolution
equation. Those that are integrable provide various mathematical
methods, presented by experts in this tutorial book, to find
special analytic solutions to both integrable and partially
integrable equations. The direct method to build solutions includes
the analysis of singularities a la Painleve, Lie symmetries leaving
the equation invariant, extension of the Hirota method,
construction of the nonlinear superposition formula. The main
inverse method described here relies on the bi-hamiltonian
structure of integrable equations. The book also presents some
extension to equations with discrete independent and dependent
variables.
|
You may like...
|