Bulletin of the American Mathematical Society

MathSciNet review: 1567661
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Book Information:

Author: Peter J. Olver
Title: Applications of Lie groups to differential equations
Additional book information: Graduate Texts in Mathematics, Volume 107, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1986, xxvi + 497 pp., $54.00. ISBN 0-387-96250-6.

1.

S. Lie, Über die Integration durch bestimmte Integrale von einer Klasse linear partieller Differentialgleichungen, Arch. Math. 6 (1881), 328-368; also Gesammelte Abhandlungen, vol. III, B. G. Teubner, Leipzig and H. Aschehoug & Co., Kristiania, 1922 (Johnson Reprint Corporation, New York, London, 1973), 492-523.

F. Schwarz, Automatically determining symmetries of partial differential equations, Computing 34 (1985), no. 2, 91–106 (English, with German summary). MR 793075, DOI 10.1007/BF02259838

L. V. Ovsiannikov, Group analysis of differential equations, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1982. Translated from the Russian by Y. Chapovsky; Translation edited by William F. Ames. MR 668703

G. W. Bluman and J. D. Cole, Similarity methods for differential equations, Applied Mathematical Sciences, Vol. 13, Springer-Verlag, New York-Heidelberg, 1974. MR 0460846

Garrett Birkhoff, Hydrodynamics. A Study in Logic, Fact, and Similitude, Princeton University Press, Princeton, N. J., 1950. MR 0038180

Willard Miller Jr., Symmetry and separation of variables, Encyclopedia of Mathematics and its Applications, Vol. 4, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1977. With a foreword by Richard Askey. MR 0460751

7.

E. Noether, Invariante Variationsprobleme, Nachr. König. Gessell. Wissen. Göttingen, Math. -Phys. K1. (1918), 235-257.

Robert Leonard Anderson, Sukeyuki Kumei, and Carl E. Wulfman, Generalization of the concept of invariance of differential equations. Results of applications to some Schrödinger equations, Phys. Rev. Lett. 28 (1972), no. 15, 988–991. MR 398309, DOI 10.1103/PhysRevLett.28.988

Peter J. Olver, Evolution equations possessing infinitely many symmetries, J. Mathematical Phys. 18 (1977), no. 6, 1212–1215. MR 521611, DOI 10.1063/1.523393

Peter J. Olver, Applications of Lie groups to differential equations, Lecture Notes, Oxford University, Mathematical Institute, Oxford, 1980. MR 673378

V. Arnold, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits, Ann. Inst. Fourier (Grenoble) 16 (1966), no. fasc. 1, 319–361 (French). MR 202082

V. I. Arnol′d, The Hamiltonian nature of the Euler equations in the dynamics of a rigid body and of an ideal fluid, Uspehi Mat. Nauk 24 (1969), no. 3 (147), 225–226 (Russian). MR 0277163

Clifford S. Gardner, Korteweg-de Vries equation and generalizations. IV. The Korteweg-de Vries equation as a Hamiltonian system, J. Mathematical Phys. 12 (1971), 1548–1551. MR 286402, DOI 10.1063/1.1665772

1.

S. Lie, Über die Integration durch bestimmte Integrale von einer Klasse linear partieller Differentialgleichungen, Arch. Math. 6 (1881), 328-368; also Gesammelte Abhandlungen, vol. III, B. G. Teubner, Leipzig and H. Aschehoug & Co., Kristiania, 1922 (Johnson Reprint Corporation, New York, London, 1973), 492-523.

2.

F. Schwarz, Automatically determining symmetries of partial differential equations, Computing 34 (1985), 91-106. MR 0793075

3.

L. V. Ovsiannikov, Group analysis of differential equations, Academic Press, New York, 1982. MR 668703

4.

G. W. Bluman and J. D. Cole, Similarity methods for differential equations, Springer-Verlag, New York, Berlin, 1974. MR 460846

5.

G. Birkhoff, Hydrodynamics-A study in logic, fact and similitude, 1st ed., Princeton Univ. Press, Princeton, 1950. MR 38180

6.

W. Miller, Jr., Symmetry and separation of variables, Addison-Wesley, Reading, Mass., 1971. MR 460751

7.

E. Noether, Invariante Variationsprobleme, Nachr. König. Gessell. Wissen. Göttingen, Math. -Phys. K1. (1918), 235-257.

8.

R. L. Anderson, S. Kumei, and C. E. Wulfman, Generalizations of the concept of invariance of differential equations. Results of applications to some Schrödinger equations, Phys. Rev. Lett. 28 (1972), 988-991. MR 398309

9.

P. J. Olver, Evolution equations possessing infinitely many symmetries, J. Math. Phys. 18 (1977), 1212-1215. MR 521611

10.

P. J. Olver, Application of Lie groups to differential equations, Mathematical Institute, Oxford, June, 1980. MR 673378

11.

V. I. Arnol'd, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits, Ann. Inst. Fourier Grenoble 16 (1966), 319-361. MR 202082

12.

V. I. Arnol'd, The Hamiltonian nature of the Euler equations in the dynamics of a rigid body and an ideal fluid, Uspekhi Mat. Nauk 24 (1969), 225-226 (Russian). MR 277163

13.

C. S. Gardner, Korteweg-de Vries equation and generalizations. IV, The Korteweg-de Vries equation as a Hamiltonian system, J. Math. Phys. 12 (1971), 1548-1551. MR 286402