Introduction to the mathematical physics of nonlinear waves / Minoru Fujimoto.

Format:

Book

Author/Creator:

Fujimoto, Minoru, author.

Series:

IOP Ebooks Series

Language:

English

Subjects (All):

Mathematical physics.

Nonlinear theories.

Nonlinear waves.

Physical Description:

1 online resource (181 pages)

Edition:

Second edition.

Place of Publication:

Bristol, England : IOP Publishing, [2021]

Summary:

The book covers the fundamental properties of nonlinear waves, dealing with both theory and experiment. The aim is to emphasize established tools and introduce new methods underpinning important new developments in this field, especially as applied to solid state materials, now including superconductivity.

Contents:

Intro

Notes on the second edition

Preface to the first edition

Acknowledgements

Author biography

Minoru Fujimoto

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

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

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

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

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

Reference

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

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

References.

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

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

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

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

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

Notes:

Description based on publisher supplied metadata and other sources.

Description based on print version record.

Includes bibliographical references.

ISBN:

9780750346122

0750346124

OCLC:

1429739770