Applications of Lie groups to differential equations. 2nd ed. (English) Zbl 0785.58003

Olver, Peter J.

Graduate Texts in Mathematics. 107. New York: Springer-Verlag. xxviii, 513 p. (1993).

This is the second Springer-Verlag edition. The first Springer-Verlag edition from 1986 was very thoroughly and positively reviewed in (1986; Zbl 0588.22001). Meanwhile in 1989 there was published the Russian translation (edited by A. B. Shabat) by Mir, Moscow (1989; Zbl 0743.58003).
From the author’s Preface to the second Springer-Verlag edition.: “The one substantial addition to the second edition is a short presentation of the calculus of pseudo-differential operators and their use in Shabat’s theory of formal symmetries, which provides a powerful, algorithmic method for determining the integrability of evolution equations”.

Reviewer: Antonín Vaněček (Praha)

MSC:

58-02	Research exposition (monographs, survey articles) pertaining to global analysis
35-02	Research exposition (monographs, survey articles) pertaining to partial differential equations
58J70	Invariance and symmetry properties for PDEs on manifolds
35A30	Geometric theory, characteristics, transformations in context of PDEs
35Q53	KdV equations (Korteweg-de Vries equations)
58J40	Pseudodifferential and Fourier integral operators on manifolds
35K05	Heat equation
35S05	Pseudodifferential operators as generalizations of partial differential operators
22E70	Applications of Lie groups to the sciences; explicit representations

Keywords:

classical equations of mathematical physics; partial differential equations; symmetry groups; ordinary differential equations; Euler ideal fluid equation; algebraic manipulation; Korteweg-de Vries equation; dimensional analysis; conservation laws; calculus of variations; nonlinear wave equation; Burgers’ equation; sine-Gordon equation; Hamiltonian systems; symplectic structure; evolution equations; MACSYMA; symmetries of differential equations; Lie groups; applications; pseudo-differential operators; evolution equations

Citations:

Zbl 0588.22001; Zbl 0743.58003

Software:

MACSYMA