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List of Portraits | |
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Preface | |
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Different classes of solitons | |
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Introduction | |
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Nontopological solitons: the Korteweg-de Vries equation | |
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The discovery | |
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The solutions of the KdV equation | |
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Conservation rules | |
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Nonlinear electrical lines | |
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Blood pressure waves | |
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Internal waves in oceanography | |
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Generality of the KdV equation | |
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Topological solitons: the sine-Gordon equation | |
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A simple mechanical example: the chain of coupled pendula | |
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Solutions of the sine-Gordon equation | |
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Long Josephson junctions | |
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Other examples of topological solitons | |
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Envelope solitons and nonlinear localisation: the nonlinear Schrodinger equation | |
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Nonlinear waves in the pendulum chain: the NLS equation | |
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Properties of the nonlinear Schrodinger equation | |
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Conservation laws | |
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Noether's theorem | |
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Nonlinear electrical lines | |
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Solitons in optical fibres | |
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Self-focusing in optics: the NLS equation in two space dimensions | |
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Conclusion | |
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The modelling process: ion acoustic waves in a plasma | |
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Introduction | |
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The plasma | |
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Study of the linear dynamics | |
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Nonlinear study | |
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Derivation of the nonlinear Schrodinger equation | |
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Experimental observations | |
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Discussion | |
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Mathematical methods for the study of solitons | |
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Introduction | |
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Linearisation around the soliton solution | |
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Spectrum of the excitations around a sine-Gordon soliton | |
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Application: perturbation of a soliton | |
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Spectrum of the excitations around a [phi superscript 4] soliton | |
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Collective coordinate method | |
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Sine-Gordon soliton interacting with an impurity: effective Lagrangian method | |
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Improving the method with a second collective coordinate | |
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The inverse-scattering transform | |
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Inverse scattering transform for the KdV equation | |
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The inverse scattering transform: a 'nonlinear Fourier analysis' | |
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Examples in solid state and atomic physics | |
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Introduction | |
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The Fermi-Pasta-Ulam problem | |
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The physical question | |
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Fermi, Pasta and Ulam: the characters | |
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The solution of the FPU problem | |
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Kruskal and Zabusky: the pioneers | |
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FPU and the Japanese School | |
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A simple model for dislocations in crystals | |
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Plastic deformation of crystals | |
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A one-dimensional model: the Frenkel-Kontorova model | |
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Continuum limit approximation: the sine-Gordon equation | |
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Are dislocations solitons? | |
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Applications | |
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Ferroelectric domain walls | |
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Ferroelectric materials | |
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A one-dimensional ferroelectric model | |
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Structure of the domain walls in the continuum limit approximation | |
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Dielectric response of a ferroelectric material | |
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Thermodynamics of a nonlinear system | |
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Incommensurate phases | |
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Examples in solid state physics | |
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A one-dimensional model for incommensurate phases: the Frenkel-Kontorova model | |
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Commensurate phases | |
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The commensurate-incommensurate transition | |
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Structure of the incommensurate phase | |
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Calculation of [delta subscript c] | |
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Phase diagram | |
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Dynamics of the incommensurate phase | |
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Formation of the discommensurations | |
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Conclusion | |
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Solitons in magnetic systems | |
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Ferromagnetism and antiferromagnetism | |
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Equations for the dynamics of a spin chain | |
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Magnons and solitons | |
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Validity of the sine-Gordon approximation | |
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Solitons in antiferromagnetic spin chains | |
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Conclusion | |
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Solitons in conducting polymers | |
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Materials | |
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The physical model of polyacetylene | |
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The ground state of polyacetylene | |
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The excited state of polyacetylene: the soliton solution | |
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Mechanism of electric conduction in conducting polyacetylene | |
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An experimental test for the presence of solitons | |
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The other nonlinear excitations of polyacetylene | |
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Solitons in Bose-Einstein condensates | |
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Introduction | |
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Theoretical description of a condensate | |
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Magnetic traps | |
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Dynamic properties | |
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Soliton solutions | |
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Nonlinear excitations in biological molecules | |
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Introduction | |
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Energy localisation and transfer in proteins | |
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The mechanism proposed by Davydov | |
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The Davydov equations | |
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Does the Davydov soliton exist? | |
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A model physical system: the acetanilide crystal | |
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Nonlinear dynamics and statistical physics of DNA | |
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A simple DNA model | |
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Nonlinear dynamics of DNA | |
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Statistical physics of DNA thermal denaturation | |
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Stability of a domain wall: another approach to denaturation | |
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Conclusion: Physical solitons: do they exist? | |
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Appendices | |
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Derivation of the KdV equation for surface hydrodynamic waves | |
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Basic equations and boundary conditions | |
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Mathematical formulation of the problem | |
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The linear limit | |
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The nonlinear equation in shallow water | |
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Mechanics of a continuous medium | |
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Lagrangian formalism | |
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Hamiltonian formalism | |
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Coherent states of a harmonic oscillator | |
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References | |
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Index | |