Optical Solitons: Theoretical Challenges and Industrial by A. Hasegawa (auth.), V. E. Zakharov, S. Wabnitz (eds.)

By A. Hasegawa (auth.), V. E. Zakharov, S. Wabnitz (eds.)

This booklet provides an outline of modern theoretical and experimental advances within the box of optical solitons, starting from the mathematical foundations of integrability concept to the quickly evolving expertise of fiber soliton-based telecommunication platforms. the topics coated within the booklet should be generally grouped into 4 major different types: optical soliton thought, fiber soliton telecommunications, optical soliton new release equipment, and all-optical details processing through spatial solitons. This publication may still supply a fascinating reference either for postgraduate scholars beginning their learn within the box, and researchers actively fascinated about nonlinear optics and optical communications.

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Additional info for Optical Solitons: Theoretical Challenges and Industrial Perspectives: Les Houches Workshop, September 28 – October 2, 1998

Example text

F(u- v)u,,. f 2 . G. B. l i3f- ;g + (cg'.! + ag + tS)j f = const and hence could be integrated. Summing up, we remark that above table presents in very compact form the basics of the classification of integrable generalizations of nonlinear Schrodinger equation coupled system (Cf [2)). Although the theory discussed below is general (Cf [4). ) 2. , v) = h(U, v) i{u, v)vx + _(i(u, v) = h('u, ~~·) (20) the lattice notations as follows (21) That yields (22) where Comparison of these distinct formulae (22) for qx results in totally discrete lattice equations (23) which provide compatibility condition for (22).

The three-wave system can be used as such an example. For this system the linearized operator represents a product of two (3 x 3)-matrix Schrodinger operators to which the oscillation theorem cannot be applied. We discuss this situation in detail for solitons describing a bound state of the fundamental frequency and its second harmonics. , the Lyapunov method and linear analysis, it is possible to get a complete answer. By means of this combined method we consider how to study the influence of phase mismatch on soliton stability [13].

4. Nonlinear stability Next we demonstrate how Lyapunov stability can be established for solitons in this three-wave system. As for the NLS solitons, for demonstration of the Hamiltonian boundedness we shall use the imbedding theorems. The difference with the NLSE case, consists, first of all, in that, instead of one complex field '1/J, now we have three fields '¢1, '¢2 and '¢3 , while, instead of cubic nonlinearity, we have a quadratic one. ,p;w~•a,,p,dnx] ''' where i 1 , > 0 and the tensors wf13 are assumed positive definite.

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