Description |
1 online resource (xvii, 250 pages) |
Series |
Springer Monographs in Mathematics, 1439-7382 |
|
Springer monographs in mathematics.
|
Contents |
880-01 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 |
|
880-01/(S Machine generated contents note: 1. General Principles -- 1.1. Variational Principle -- 1.2. Invariance Principle -- 1.2.1. Poincare Invariance -- 1.2.2. Galileo Invariance -- 1.2.3. Gauge Invariance -- 1.3. Conservation Laws -- 1.3.1. Noether's Theorem -- 1.3.2. Main Constants of Motion -- 1.4. Hamilton-Jacobi Theory -- 2. Solitary Waves and Solitons: Abstract Theory -- 2.1. Solitary Waves and Solitons -- 2.1.1. Definitions -- 2.1.2. Solitons and Symmetry -- 2.1.3. Hylomorphic Solitons and Minimizers -- 2.2. Existence Results of Hylomorphic Solitons -- 2.2.1. Abstract Framework -- 2.2.2. Statement of the Abstract Existence Theorems -- 2.2.3. Minimization Result in the Positive Energy Case -- 2.2.4. Minimization Result in the Positive Charge Case -- 2.2.5. Stability Result -- 2.3. Structure of Hylomorphic Solitons -- 2.3.1. Meaning of Hylenic Ratio -- 2.3.2. Swarm Interpretation of Hylomorphic Solitons -- 3. Nonlinear Schrodinger Equation -- 3.1. General Features of NS -- 3.1.1. Constants of Motion of NS -- 3.1.2. Swarm Interpretation of NS -- 3.2. Existence Results for NS -- 3.2.1. Existence of Solitary Waves -- 3.2.2. Existence of Solitons -- 3.2.3. Splitting and Coercivity -- 3.2.4. Analysis of the Hylenic Ratio -- 3.2.5. Symmetry, Travelling Solitary Waves and Solitons in NS -- 3.3. Dynamics of Solitons in NS -- 3.3.1. Rescaling the Soliton -- 3.3.2. Statement of the Problem and Main Results -- 3.3.3. Analysis of the Energy and Momentum of the Soliton -- 3.3.4. Definition of the Soliton -- 3.3.5. Equation of Dynamics of the Soliton -- 3.3.6. Analysis of the Concentration Point of the Soliton -- 3.3.7. Definition of the Density ρε -- 3.3.8. Dynamics of the Soliton -- 4. Nonlinear Klein-Gordon Equation -- 4.1. General Features of NKG -- 4.1.1. Constants of Motion of NKG -- 4.1.2. Swarm Interpretation of NKG -- 4.2. Existence Results for NKG -- 4.2.1. Existence of Solitary Waves -- 4.2.2. Existence of Solitons -- 4.2.3. Coercivity -- 4.2.4. Analysis of the Hylenic Ratio for NKG -- 4.2.5. Symmetry, Travelling Solitary Waves and Solitons in NKG -- 4.3. Dynamical Properties of Solitary Waves in NKG -- 4.3.1. Space Contraction and Time Dilation of Solitary Waves -- 4.3.2. Mass of Solitary Waves -- 4.3.3. Einstein Equation -- 4.3.4. Energy-Momentum 4-Vector -- 4.3.5. Remarks on Fields and Particles -- 5. Nonlinear Klein-Gordon-Maxwell Equations -- 5.1. General Feature of the Klein-Gordon-Maxwell Equations -- 5.1.1. Maxwell Equations in Empty Space -- 5.1.2. Gauge Theories -- 5.1.3. Maxwell Equations and Matter -- 5.1.4. Constants of Motion of NKGM -- 5.1.5. Swarm Interpretation of NKGM -- 5.2. NKGM as a Dynamical System -- 5.2.1. Modified Lagrangian -- 5.2.2. Hamiltonian Formulation -- 5.2.3. Phase Space of NKGM -- 5.3. Existence of Charged Solitons for NKGM -- 5.3.1. Statement of the Existence Result -- 5.3.2. Working with the Gauge Invariant Variables -- 5.3.3. Proof of the Existence Result -- 6. Nonlinear Schrodinger-Maxwell Equations -- 6.1. General Feature of the Schrodinger-Maxwell Equations -- 6.1.1. Construction of the Nonlinear Schrodinger-Maxwell Equations -- 6.1.2. Energy for NSM -- 6.2. NSM as a Dynamical System -- 6.2.1. Modified Lagrangian -- 6.2.2. Phase Space of NSM -- 6.3. Existence of Charged Solitons for NSM -- 6.3.1. Gauge Invariant Variables and the Splitting Property -- 6.3.2. Analysis of the Hylenic Ratio for NSM -- 7. Nonlinear Beam Equation -- 7.1. General Feature -- 7.2. Existence of Solitons -- 7.2.1. Coercivity and Splitting Property -- 7.2.2. Analysis of the Hylenic Ratio -- 8. Vortices -- 8.1. Vortices for the Nonlinear Schrodinger Equation -- 8.1.1. Statement of the Results -- 8.1.2. Proof of the Main Result -- 8.1.3. Analysis of Hylenic Ratio for Vortices -- 8.1.4. Solutions in the Sense of Distribution -- 8.2. Vortices for the Nonlinear Klein-Gordon Equation -- 8.3. Vortices for Nonlinear Klein-Gordon-Maxwell Equations -- 8.3.1. Main Existence Result -- Appendix -- A.1. Some Inequalities -- A.2. Pohozhev-Derrick Theorem -- A.3. Existence Result for an Elliptic Equation |
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 |
Analysis |
mathematische natuurkunde |
|
mathematical physics |
|
wiskunde |
|
mathematics |
|
partial differential equations |
|
analyse |
|
analysis |
|
Mathematics (General) |
|
Wiskunde (algemeen) |
Bibliography |
Includes bibliographical references |
Notes |
Online resource; title from PDF title page (SpringerLink, viewed November 14, 2014) |
Subject |
Differential equations, Nonlinear.
|
|
Nonlinear wave equations.
|
|
Variational inequalities (Mathematics)
|
|
MATHEMATICS -- Calculus.
|
|
MATHEMATICS -- Mathematical Analysis.
|
|
Differential equations, Nonlinear
|
|
Nonlinear wave equations
|
|
Variational inequalities (Mathematics)
|
Form |
Electronic book
|
Author |
Fortunato, D. (Donato), author.
|
ISBN |
9783319069142 |
|
3319069144 |
|
3319069136 |
|
9783319069135 |
|
9783319069159 |
|
3319069152 |
|
9783319361222 |
|
3319361228 |
|