Introduction to the mathematical physics of nonlinear waves

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PRELIMS.pdf Notes on the second edition Preface to the first edition Acknowledgements Author biography Minoru Fujimoto CH001.pdf Chapter 1 Nonlinearity and elliptic functions in classical mechanics 1.1 A pendulum 1.1.1 Oscillations 1.1.2 Vertical rotation 1.2 Vibration by a nonlinear spring force 1.3 Hyperbolic and elliptic functions 1.3.1 Definitions 1.3.2 Differentiation 1.3.3 Reverse functions cn-1anddn-1 1.4 A jumping rope 1.4.1 The shape 1.4.2 Periodicity of Jacobi’s sn-function 1.5 Variation principle 1.6 Buckling of an elastic rod Exercise References CH002.pdf Chapter 2 Wave propagation, singularities, and boundary conditions 2.1 Elastic waves along a linear string in infinite length 2.1.1 Phase and amplitude of propagation 2.1.2 Energy flow 2.1.3 Scattering by an oscillator 2.2 Microwave transmission 2.3 Wave equations 2.3.1 Schrödinger’s equation 2.3.2 Two-dimensional free waves in heterogeneous water 2.3.3 Eckart’s potential 2.4 Sound propagation in air 2.5 Asymptotic approximation in air space Exercises References CH003.pdf Chapter 3 Order variables for structural phase transitions 3.1 Symmetry group in crystals 3.2 Solitons and the Ising model for pseudospin correlations 3.2.1 Pseudospin correlations 3.2.2 Soliton correlations in pseudospin clusters 3.3 Macroscopic views of structural phase transitions 3.3.1 Landau’s mean field theory 3.3.2 The Curie–Weiss law of susceptibility 3.3.3 Critical fluctuations 3.3.4 Entropy production at critical temperatures 3.4 Observing critical anomalies 3.4.1 Amplitude anomalies 3.4.2 Frequency scanning of phase anomalies Exercises References CH004.pdf Chapter 4 Soft modes of lattice displacements 4.1 The Lyddane–Sachs–Teller relation 4.2 Soft modes in perovskite oxides 4.3 Dynamics of soft modes 4.4 Soft-mode frequency in modulated crystals 4.5 Optical studies on symmetry changes at critical temperature 4.5.1 Cochran’s model of ferroelectric transitions 4.5.2 Symmetry change at transition temperatures Exercises References CH005.pdf Chapter 5 Nonlinearity development in crystals: Korteweg–deVries’ equation for collective order variables and the complex potential 5.1 The Korteweg–deVries equation 5.1.1 Timescale for developing nonlinearity 5.1.2 The Korteweg–deVries equation 5.2 Thermal solution for the Weiss potential 5.3 Condensate pinning by the Weiss potential 5.4 Nonlinear waves and complex lattice potentials 5.4.1 Longitudinal waves of collective order variables 5.4.2 Transversal component and directional change of σ(ϕ) 5.4.3 Finite crystals and the domain structure 5.5 The complex lattice potential 5.6 Isothermal phase transition and entropy production Exercises Reference CH006.pdf Chapter 6 Soliton mobility in time–temperature conversion for thermal processes: Riccati’s theorem 6.1 Bargmann’s theorem 6.1.1 One-soliton solution 6.1.2 Two-soliton solutions 6.2 Riccati’s theorem and the modified Korteweg–deVries equation 6.2.1 Riccati’s theorem 6.2.2 Modified Korteweg–deVries equations in a conservative system 6.3 Soliton mobility studied by computational analysis Exercises References CH007.pdf Chapter 7 Toda’s lattice of correlation potentials 7.1 The Toda soliton lattice 7.1.1 Dual chains of condensates 7.1.2 Toda’s correlation potentials 7.1.3 Propagation in Toda’s soliton lattice 7.2 Developing nonlinearity 7.2.1 Matrix operators for Toda’s correlation potentials 7.2.2 Finite periodic lattice 7.3 Conversion to Korteweg–deVries’ lattice potential Exercises References CH008.pdf Chapter 8 Scattering theory of the soliton lattice 8.1 Elemental waves 8.1.1 Critical fluctuations 8.1.2 Matrix form for nonlinear development 8.2 Scattering theory: dissipation, reflection, and transmission 8.2.1 Elemental waves 8.2.2 Reflection and transmission of two-component waves 8.2.3 Singularity of reflection and transmission 8.3 Method of inverse scattering 8.4 Entropy production from soliton potentials Reference CH009.pdf Chapter 9 Pseudopotentials and sine-Gordon equation: topological correlations in domain structure 9.1 Pseudopotentials in mesoscopic phases 9.2 The sine-Gordon equation 9.3 Phase solitons in adiabatic processes 9.4 The Bäcklund transformation and domain boundaries 9.5 Computational studies of the Bäcklund transformation Exercise References CH010.pdf Chapter 10 Trigonal structural transitions: domain stability in topological order 10.1 The sine-Gordon equation 10.2 Observing adiabatic fluctuations 10.3 Toda’s theory of domain stability 10.4 Kac’s theory of nonlinearity for domain disorder 10.5 Domain separation and thermal and quasi-adiabatic transitions 10.6 Mesoscopic domains in topological disorder Exercise References CH011.pdf Chapter 11 Soliton theory of superconducting transitions 11.1 The Meissner effect and Fröhlich’s proposal 11.2 Magnetic images of Fröhlich’s interaction 11.3 The Cooper pair and persistent current 11.4 Critical temperatures and energy gap in superconducting transitions 11.5 Anderson’s theory of superconducting phase transitions 11.6 Cuprate-layer structure and the Cooper pair 11.7 Meissner’s effect in cuprate-layers and metallic hydrogen sulfide H3S Exercises References CH012.pdf Chapter 12 Irreducible thermodynamics of superconducting phase transitions 12.1 Superconducting phase transition 12.1.1 Meissner’s diamagnetism and the persistent current 12.1.2 Thermodynamics of a superconducting transition 12.2 Electromagnetic properties of superconductors 12.2.1 Persistent current 12.2.2 Penetration depth 12.2.3 London’s gauge function and magnetic flux quantization 12.3 The Ginzburg–Landau equation for superconducting phase transitions 12.4 Field theory of superconducting transitions 12.4.1 Bardeen–Cooper–Schrieffer’s ground states 12.4.2 Superconducting state at finite temperatures Exercises References