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By Mark J. Ablowitz (auth.), Peter J. Olver, David H. Sattinger (eds.)
This IMA quantity in arithmetic and its purposes SOLITONS IN PHYSICS, arithmetic, AND NONLINEAR OPTICS relies at the court cases of 2 workshops that have been a vital part of the 1988-89 IMA application on NONLINEAR WAVES. The workshops focussed at the major components of the idea of solitons and at the functions of solitons in physics, biology and engineering, with a different focus on nonlinear optics. We thank the Coordinating Committee: James Glimm, Daniel Joseph, Barbara Keyfitz, An Majda, Alan Newell, Peter Olver, David Sattinger and David Schaeffer for drew making plans and imposing the stimulating year-long software. We in particular thank the Workshop Organizers for Solitons in Physics and arithmetic, Alan Newell, Peter Olver, and David Sattinger, and for Nonlinear Optics and Plasma Physics, David Kaup and Yuji Kodama for his or her efforts in bringing jointly some of the significant figures in these learn fields within which solitons in physics, arithmetic, and nonlinear optics theories are used. A vner Friedman Willard Miller, Jr. PREFACE This quantity comprises a number of the lectures given at workshops, "Solitons in Physics and arithmetic" and "Solitons in Nonlinear Optics and Plasma Physics" held throughout the 1988-89 LM. A. 12 months on Nonlinear Waves. for the reason that their discovery via Kruskal and Zabusky within the early 1960's, solitons have had a profound influence on many fields, starting from engineering and physics to algebraic geometry.
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Csch [V7r /4,),l r Sill a o] sm(vro) , [1 - ! cos a o] sin /30 . 2 • 2 csch [V7r/4,),lr slllao] cos(vro) . 4,),1 sm a o 56 These functions have two families of simultaneous simple zeroes, whose existence is a necessary condition for the occurrence of Arnold diffusion. Physically, this diffusion means that the polarization state transfers back and forth among the nonlinear modes of the system in an erratic manner. 3. 1. Definition and reduction to the sphere. We now restrict attention to the problem of a single travelling-wave optical pulse.
A. I. YOBLONSKII, Diff. , 3, 264 (1967). S. J. ABLOWITZ, J. Math. , 23, 2033 (1982). A. LUKASHEVICH, Diff. , 7, 1124 (1971). I. GROMAK, Diff. , 14, 2131 (1978). A. LUKASHEVICH, Diff. , 3, 771 (1967). 1. GROMAK, Diff. Eq's. 11,285 (1975). J. ABLOWITZ, H. SEGUR, SIAM Stud. Appl. Math. (Studies 4) (1981). G. KREIN, Usp, Mat. Nauk. 13, 3 (1958). [lOb] I. G. KREIN, Usp. Mat. Nauk, 13, 2 (1958). [11] R. BEALS, R. COIFMAN, Commun. Pure and Applied Math, 87, 39-90 (1984). [12] H. J. ABLOWITZ, Proc. E. V.
We analyze the problem of two counterpropagating optical laser beams in a slightly nonlinear medium from the point of view of Hamiltonian systems; the one-beam subproblem is also investigated as a special case. We are interested in these systems as integrable dynamical systems which undergo chaotic behavior under various types of perturbations. The phase space for the two-beam problem is C2 X C2 when we restrict to the regime of travellingwave solutions. We use the method of reduction for Hamiltonian systems invariant under oneparameter symmetry groups to demonstrate that the phase space reduces to the two-sphere 8 2 and is therefore completely integrable.