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Hylomorphic solitons. (English) Zbl 1205.35040

Summary: This paper is devoted to the study of solitary waves and solitons whose existence is related to the ratio energy/charge. These solitary waves are called hylomorphic. This class includes the Q-balls, which are spherically symmetric solutions of the nonlinear Klein-Gordon equation, as well as solitary waves and vortices which occur, by the same mechanism, in the nonlinear Schrödinger equation and in gauge theories. Mainly we will be interested in the very general principles which are at the base of the existence of solitons such as the Variational Principle, the Invariance Principle, the Noether’s theorem, the Hamilton-Jacobi theory etc. We give a general definition of hylomorphic solitons and an interpretation of their nature (swarm interpretation) which is very helpful in understanding their behavior. We apply these ideas to the Nonlinear Schrödinger Equation and to the Nonlinear Klein-Gordon Equation respectively.

MSC:

35C08 Soliton solutions
35A15 Variational methods applied to PDEs
35Q51 Soliton equations
35Q55 NLS equations (nonlinear Schrödinger equations)
70S10 Symmetries and conservation laws in mechanics of particles and systems
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