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Books > Science & Mathematics > Mathematics > Calculus & mathematical analysis > Differential equations
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The Painleve Handbook (Paperback, 2008 ed.)
Loot Price: R3,116
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The Painleve Handbook (Paperback, 2008 ed.)
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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.
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