Introduction to the Mathematical Physics of Nonlinear Waves

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