Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 2 (2006), 088, 17 pages      nlin.SI/0612019
Contribution to the Vadim Kuznetsov Memorial Issue

Solvable Nonlinear Evolution PDEs in Multidimensional Space

Francesco Calogero a and Matteo Sommacal b, c
a) Dipartimento di Fisica, Università di Roma ''La Sapienza'', Istituto Nazionale di Fisica Nucleare, Sezione di Roma, P.le Aldo Moro 2, 00185 Rome, Italy
b) Laboratoire J.-L. Lions, Université Pierre et Marie Curie, Paris VI, 175 Rue du Chevaleret, 75013 Paris, France (until October 30th, 2006)
c) Dipartimento di Matematica, Università di Perugia, Via Vanvitelli 1, 06123 Perugia, Italy (from November 1st, 2006)

Received October 31, 2006; Published online December 08, 2006; Some typos corrected February 26, 2007

A class of solvable (systems of) nonlinear evolution PDEs in multidimensional space is discussed. We focus on a rotation-invariant system of PDEs of Schrödinger type and on a relativistically-invariant system of PDEs of Klein-Gordon type. Isochronous variants of these evolution PDEs are also considered.

Key words: nonlinear evolution PDEs in multidimensions; solvable PDEs; NLS-like equations; nonlinear Klein-Gordon-like equations; isochronicity.

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  1. Calogero F., Motion of poles and zeros of special solutions of nonlinear and linear partial differential equations, and related "solvable" many body problems, Nuovo Cimento B, 1978, V.43, 177-241.
  2. Calogero F., A class of C-integrable PDEs in multidimensions, Inverse Problems, 1994, V.10, 1231-1234.
  3. Calogero F., Classical many-body problems amenable to exact treatments, Lecture Notes in Physics Monograph, Vol. 66, Berlin Heidelberg, Springer-Verlag, 2001.
  4. Calogero F., Isochronous systems, Monograph, 200 pages, to appear.
  5. Erdélyi A. (Editor), Higher transcendental functions, Vol. II, New York, McGraw-Hill, 1953.
  6. Gómez-Ullate D., Sommacal M., Periods of the goldfish many-body problem, J. Nonlinear Math. Phys., 2005, V.12, suppl. 1, 351-362.
  7. Mariani M., Calogero F., Isochronous PDEs, Theor. Math. Phys., 2005, V.68, 958-968.

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