Solitons in Physics, Mathematics, and Nonlinear Optics by Mark J. Ablowitz (auth.), Peter J. Olver, David H. Sattinger

By Mark J. Ablowitz (auth.), Peter J. Olver, David H. Sattinger (eds.)

This IMA quantity in arithmetic and its functions SOLITONS IN PHYSICS, arithmetic, AND NONLINEAR OPTICS relies at the complaints of 2 workshops that have been an essential component of the 1988-89 IMA application on NONLINEAR WAVES. The workshops focussed at the major elements of the speculation of solitons and at the purposes of solitons in physics, biology and engineering, with a distinct 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 particularly 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 a number of the significant figures in these examine fields within which solitons in physics, arithmetic, and nonlinear optics theories are used. A vner Friedman Willard Miller, Jr. PREFACE This quantity contains the various lectures given at workshops, "Solitons in Physics and arithmetic" and "Solitons in Nonlinear Optics and Plasma Physics" held through the 1988-89 LM. A. yr on Nonlinear Waves. due to the fact that their discovery via Kruskal and Zabusky within the early 1960's, solitons have had a profound impression on many fields, starting from engineering and physics to algebraic geometry.

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2. 1. Definition and reduction to 52. The two-beam problem for two counterpropagating travelling optical wave pulses is described by the following Hamiltonian function and equations of motion defined on C 2 x C 2 : The Poisson bracket is canonical; as is usual for Hamiltonian systems, the travellingwave evolution of a dynamical quantity F is determined by OF/aT = {F, H}, where T is the travelling-wave variable. In the above equations, the dependent variables e and e represent the electric field amplitudes associated with each of the beams and both are complex two-vectors taking values in C 2 • The quantity X(3) is the third order susceptibility tensor parametrizing the nonlinear polarizability of the medium · (3) (3)* d (3) (3) (3) d- d t an d ven·f· ymg th· e mvoIutJons Xijkl = Xjilk an Xijkl = Xljki = Xikjl.

I. 1. ,~ ~·' ~, L•• .. , I •,1\1 I ,~ ... 11\11 . ,. \1 I . ,~ I ~~ . L L L Figure 8 Non-soliton collision with radius 5. , • .. I I • 21 Figure 9 - Non-soliton collision with radius 3. This type of collision is known as a bound state. e. particles which repeat after a fixed period of time with a given displacement (or set of displacements). The FRT is an excellent vehicle in order to find algorithms for computing periodic particles. Indeed Papatheodorou and Fokas [13] used the FRT in order to construct one periodic particles and Keiser [15] developed algorithms for higher periodic particles.

Math. Phys. 21, 715 (1980). J. ABLOWITZ, A. RAMANI, H. SEGUR, J. Math. Phys. 21, 1006 (1980). [5] A. RAMANI, B. GRAMMATICOS AND T. BOFNTIS, The Painleve Property and Singularity Analysis of Integrable and Non-Integrable Systems, Preprint (1988). [6] M. D. KRUSKAL, private communication. [7] [8a] M. WEISS, M. TABOR AND J. CARNEVALE, J. Math. Phys. 24, 522 (1973). A. ERUGIN, Dok!. Akad. Nark. BSSR 2 (1958). A. LUKASHEVICH, Diff. , 3, 994 (1967). A. LUKASHEVICH, Diff. , 1, 731 (1965). I. GROMAK, Dilf.

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