Contact geometry and non-linear differential equations / Alexei Kushner, Valentin Lychagin, and Vladimir Rubtsov.

Format:

Book

Author/Creator:

Kushner, Alexei, author.

Lychagin, V. V. (Valentin Vasilʹevich), author.

Rubtsov, Vladimir, 1952- author.

Series:

Encyclopedia of mathematics and its applications ; v. 101.

Encyclopedia of mathematics and its applications ; volume 101

Language:

English

Subjects (All):

Contact manifolds.

Differential equations, Nonlinear.

Physical Description:

1 online resource (xxi, 496 pages) : digital, PDF file(s).

Other Title:

Contact Geometry & Nonlinear Differential Equations

Place of Publication:

Cambridge : Cambridge University Press, 2007.

Language Note:

English

Summary:

Methods from contact and symplectic geometry can be used to solve highly non-trivial nonlinear partial and ordinary differential equations without resorting to approximate numerical methods or algebraic computing software. This book explains how it's done. It combines the clarity and accessibility of an advanced textbook with the completeness of an encyclopedia. The basic ideas that Lie and Cartan developed at the end of the nineteenth century to transform solving a differential equation into a problem in geometry or algebra are here reworked in a novel and modern way. Differential equations are considered as a part of contact and symplectic geometry, so that all the machinery of Hodge-deRham calculus can be applied. In this way a wide class of equations can be tackled, including quasi-linear equations and Monge-Ampere equations (which play an important role in modern theoretical physics and meteorology).

Contents:

Symmetries and integrals

Distributions

Ordinary differential equations

Model differential equations and the lie superposition principle

Symplectic algebra

Linear algebra of symplectic vector spaces

Exterior algebra on symplectic vector spaces

A symplectic classification of exterior 2-forms in dimension 4

Symplectic classification of exterior 2-forms

Classification of exterior 3-forms on a six-dimensional symplectic space

Monge-Ampère equations

Symplectic manifolds

Contact manifolds

Symmetries and contact transformations of Monge-Ampère equations

Conservation laws

Monge-Ampère equations on two-dimensional manifolds and geometric structures

Systems of first-order partial differential equations on two-dimensional manifolds

Applications

Non-linear acoustics

Non-linear thermal conductivity

Meteorology applications

Classification of Monge-Ampère equations

Classification of symplectic MAOs on two-dimensional manifolds

Classification of symplectic MAEs on two-dimensional manifolds

Contact classification of MAEs on two-dimensional manifolds

Symplectic classification of MAEs on three-dimensional manifolds.

Notes:

Title from publisher's bibliographic system (viewed on 05 Oct 2015).

Includes bibliographical references and index.

ISBN:

1-139-88308-9

0-511-88975-5

1-107-38393-5

1-107-38744-2

1-107-39036-2

0-511-73514-6

1-107-39877-0

1-107-39516-X

OCLC:

776976645