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1993 | Buch

Poincare-Invariant Nonlinear Scalar Equations Kapitel lesen Erstes Kapitel lesen

Symmetry Analysis and Exact Solutions of Equations of Nonlinear Mathematical Physics

verfasst von: W. I. Fushchich, W. M. Shtelen, N. I. Serov

Verlag: Springer Netherlands

Buchreihe : Mathematics and Its Applications

Enthalten in: Professional Book Archive

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Über dieses Buch

by spin or (spin s = 1/2) field equations is emphasized because their solutions can be used for constructing solutions of other field equations insofar as fields with any spin may be constructed from spin s = 1/2 fields. A brief account of the main ideas of the book is presented in the Introduction. The book is largely based on the authors' works [55-109, 176-189, 13-16, 7*-14*,23*, 24*] carried out in the Institute of Mathematics, Academy of Sciences of the Ukraine. References to other sources is not intended to imply completeness. As a rule, only those works used directly are cited. The authors wish to express their gratitude to Academician Yu.A. Mitropoi­ sky, and to Academician of Academy of Sciences of the Ukraine O.S. Parasyuk, for basic support and stimulation over the course of many years; to our cowork­ ers in the Department of Applied Studies, LA. Egorchenko, R.Z. Zhdanov, A.G. Nikitin, LV. Revenko, V.L Lagno, and I.M. Tsifra for assistance with the manuscript.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Poincare-Invariant Nonlinear Scalar Equations
Abstract
In the present chapter we describe the first- and second-order n-dimensional nonlinear PDEs which are invariant under the groups \(\widetilde P\left( {1,1 - n} \right),\widetilde P\left( {1,n} \right)\) . We investigate local and tangent symmetry of the relativistic Hamilton equation, of the nonlinear d’Alembert equation, of the Euler-Lagrange-Born-Infeld equation, the Monge-Ampere equation, and some other PDEs. For this purpose the Lie method has been used with the exception of Sec. 1.3, where the symmetry of the polywave equation is investigated by the operator method expounded in Sec. 5.5.
W. I. Fushchich, W. M. Shtelen, N. I. Serov
Chapter 2. Systems of Poincare-invariant Nonlinear PDEs
Abstract
In the present chapter we will consider systems of nonlinear PDEs that are invariant under the Poincare group P(1,3), extended Poincare group \(\tilde P\left( {1,3} \right)\) and conformal group C(1,3). Ansatze for spinor fields will be constructed. The formula of generating solutions (GS) by conformal transformations is obtained for fields of arbitrary spin. Wide classes of exact solutions of nonlinear generalizations of the Dirac equation are found as well as solutions of the quantum electrodynamics nonlinear equations, coupled nonlinear equations for vector and scalar fields, the Yang-Mills equations, and some others.
W. I. Fushchich, W. M. Shtelen, N. I. Serov
Chapter 3. Euclid and Galilei Groups and Nonlinear PDEs for Scalar Fields
Abstract
In the present chapter we describe a wide class of nonlinear PDEs for scalar fields invariant under Euclid, Galilei, or larger groups. For some of such equations we construct multiparameter families of exact solutions.
W. I. Fushchich, W. M. Shtelen, N. I. Serov
Chapter 4. Systems of PDEs Invariant Under Galilei Group
Abstract
In the present chapter we consider linear and nonlinear systems of PDEs invariant under various representations of the Galilei group and its generalizations (such as the extended Galilei group, the Schrödinger group). Sets of Sch(1,3)- and G(1,3)-nonequivalent ansatze are constructed. A wide class of linear and nonlinear Sch(1,3)-invariant systems of PDEs is described. Lame equations are studied: superalgebra of symmetry is found and a Galilei-invariant generalization is constructed. Gas dynamics and Navier-Stokes equations are considered. Exact solutions of some enumerated above equations are found.
W. I. Fushchich, W. M. Shtelen, N. I. Serov
Chapter 5. Some Special Questions
Abstract
In the present chapter we consider some problems tightly connected with group-algebraic investigations such as: finding nonlocal transformations to linearize a given nonlinear PDE, symmetry analysis of the three-body problem, calculating final transformations generated by non-Lie symmetry operators, and studying symmetry of integrodifferential equations. Here we introduce the concept of conditional invariance, and study non-Lie symmetry of quasi-relativistic generalization of the Schrödinger equation, Galilean invariance of Maxwell’s equations, solutions of the Schrödinger equation invariant under the non-Lie Lorentz algebra. Finally, in the concluding topic we introduce the concept of approximate symmetry.
W. I. Fushchich, W. M. Shtelen, N. I. Serov
Backmatter
Metadaten
Titel
Symmetry Analysis and Exact Solutions of Equations of Nonlinear Mathematical Physics
verfasst von
W. I. Fushchich
W. M. Shtelen
N. I. Serov
Copyright-Jahr
1993
Verlag
Springer Netherlands
Electronic ISBN
978-94-017-3198-0
Print ISBN
978-90-481-4244-6
DOI
https://doi.org/10.1007/978-94-017-3198-0