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This book highlights the methods to engineer dissipative and magnetic nonlinear waves propagating in nonlinear systems. In the first part of the book, the authors present methodologically mathematical models of nonlinear waves propagating in one- and two-dimensional nonlinear transmission networks without/with dissipative elements. Based on these models, the authors investigate the generation and the transmission of nonlinear modulated waves, in general, and solitary waves, in particular, in networks under consideration. In the second part of the book, the authors develop basic theoretical results for the dynamics matter-wave and magnetic-wave solitons of nonlinear systems and of Bose–Einstein condensates trapped in external potentials, combined with the time-modulated nonlinearity. The models treated here are based on one-, two-, and three-component non-autonomous Gross–Pitaevskii equations. Based on the Heisenberg model of spin–spin interactions, the authors also investigate thedynamics of magnetization in ferromagnet with or without spin-transfer torque. This research book is suitable for physicists, mathematicians, engineers, and graduate students in physics, mathematics, and network and information engineering.

The Motivation. With intensified use of mathematical ideas, the methods and techniques of the various sciences and those for the solution of practical problems demand of the mathematician not only greater readi­ ness for extra-mathematical applications but also more comprehensive orientations within mathematics. In applications, it is frequently less important to draw the most far-reaching conclusions from a single mathe­ matical idea than to cover a subject or problem area tentatively by a proper'variety'of mathematical theories. To do this the mathematician must be familiar with the shared as weIl as specific features of differ­ ent mathematical approaches, and must have experience with their inter­ connections. The Atiyah-Singer Index Formula,'one of the deepest and hardest results in mathematics','probably has wider ramifications in topology and analysis than any other single result'(F. Hirzebruch) and offers perhaps a particularly fitting example for such an introduction to'Mathematics': In spi te of i ts difficulty and immensely rich interrela­ tions, the realm of the Index Formula can be delimited, and thus its ideas and methods can be made accessible to students in their middle • semesters. In fact, the Atiyah-Singer Index Formula has become progressively'easier'and'more transparent'over the years. The discovery of deeper and more comprehensive applications (see Chapter 111. 4) brought with it, not only a vigorous exploration of its methods particularly in the many­ facetted and always new presentations of the material by M. F.