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The book analyzes the existence of solitons, namely of finite energy solutions of field equations which exhibit stability properties. The book is divided in two parts. In the first part, the authors give an abstract definition of solitary wave and soliton and we develop an abstract existence theory for hylomorphic solitons, namely for those solitons which minimize the energy for a given charge. In the second part, the authors apply this theory to prove the existence of hylomorphic solitons for some classes of field equations (nonlinear Klein-Gordon-Maxwell equations, nonlinear Schrödinger-Maxwell equations, nonlinear beam equation,..). The abstract theory is sufficiently flexible to be applied to other situations, like the existence of vortices. The books is addressed to Mathematicians and Physicists.
List of contents
1 The general principles.- 2 Solitary waves and solitons: abstract theory.- 3 The nonlinear Schrödinger equation.- 4 The nonlinear Klein-Gordon equation.- 5 The Nonlinear Klein-Gordon-Maxwell equations.- 6 The nonlinear Schrödinger-Maxwell equations.- 7 The nonlinear beam equation.- 8 Vortices.- 9 Appendix.
Summary
The book analyzes the existence of solitons, namely of finite energy solutions of field equations which exhibit stability properties. The book is divided in two parts. In the first part, the authors give an abstract definition of solitary wave and soliton and we develop an abstract existence theory for hylomorphic solitons, namely for those solitons which minimize the energy for a given charge. In the second part, the authors apply this theory to prove the existence of hylomorphic solitons for some classes of field equations (nonlinear Klein-Gordon-Maxwell equations, nonlinear Schrödinger-Maxwell equations, nonlinear beam equation,..). The abstract theory is sufficiently flexible to be applied to other situations, like the existence of vortices. The books is addressed to Mathematicians and Physicists.
Additional text
“This book is a useful, enlightening, and thought-provoking
exposition of the research and findings over the last 35 years of using and
implementing variational techniques to better understand the dynamics of
nonlinear field equations. These reviewers would highly recommend this text be
on the shelf of any applied mathematician/physicist who uses variational
techniques to analyze nonlinear field equations.” (David J. Kaup and Thomas
Keith Vogel, Mathematical Reviews, November, 2015)
Report
"This book is a useful, enlightening, and thought-provoking exposition of the research and findings over the last 35 years of using and implementing variational techniques to better understand the dynamics of nonlinear field equations. These reviewers would highly recommend this text be on the shelf of any applied mathematician/physicist who uses variational techniques to analyze nonlinear field equations." (David J. Kaup and Thomas Keith Vogel, Mathematical Reviews, November, 2015)
