本书是为高年级本科生和研究生编写的，是一本关于非线性物理学的升级物理课程的教科书。尽管如此，非线性数学仍以双曲函数和椭圆函数为主导。本书共十章，用非齐次微分方程来分析非线性问题获得的相关信息，主要内容包括：经典力学中的非线性特征、波传播、奇异性与边界、孤立子与绝热势、结构相变、非线性波、散射理论、逆散射法、准静态孤子态、贝可隆变换与正弦-戈登方程等。本书还回顾了物理实践中的代表性案例，然后通过一般理论进行了应用。

1 Nonlinearity in classical mechanics
1.1 A pendulum
1.1.1 Oscillation
1.1.2 Vertical rotation
1.2 Vibration by a nonlinear spring force
1.3 A jumping rope
1.4 Hyperbolic and elliptic functions
1.4.1 Definitions
1.4.2 Differentiation
1.4.3 Reverse functions cn-1 and dn-1
1.4.4 Periodicity of Jacobi's sn-function
1.5 Variation principle
1.6 Buckling deformation of a rod
Exercise
2 Wave propagation, singularities and boundaries
2.1 Elastic waves along a linear string of infinite length
2.1.1 Phase of propagation
2.1.2 Energy flow
2.1.3 Scattering by an oscillator
2.2 Microwave transmission
2.3 Schr6dinger's equation
2.4 Scattering by the potential V(x) = Vo sech2 x
2.5 Two-dimensional waves in inhomogeneous medium
2.6 Sound propagation in air
Exercises
3 Solitons and adiabatic potentials
3.t The Korteweg-deVries equation
3.2 Steady solutions of the Korteweg~teVries equation
3.3 Developing equations of nonlinear vector waves
3.4 Bargmann's theorem
3.4.1 One-soliton solution
3.4.2 Two-soliton solution
3.5 Riccati's theorem
3.6 Properties of the Eckart potential in the soliton field
3.7 Zabusky-Kruskal's computational analysis
Exercises
4 Structural phase transitions
4.1 Initial uncertainties and transition anomalies
4.1.1 Specific heat anomalies
4.1.2 Landau's theory
4.2 Dynamical theory of collective motion
4.2.1 Longitudinal waves
4.2.2 Transverse waves
4.3 Pseudopotential and sine-Gordon equation
Exercises
5 Nonlinear waves
5.1 Elemental waves
5.2 Matrix formulation for nonlinear development
5.3 Heat dissipation of wave motion
5.4 Born-Huang transitions in crystals
5.5 Symmetry of media for the Korteweg~teVries equation
5.6 Soliton description
Exercise
6 Scattering theory
6.1 One-component waves
6.1.1 Scatterings of elemental waves
6.1.2 Singularity of a soliton potential
6.2 Two-component scatterings
6.2.1 A two-component wave
6.2.2 Reflection and transmission
6.2.3 Poles of transmission and reflection coefficients
6.2.4 Soliton potentials
6.2.5 Asymptotic expansion
Exercises
7 Method of inverse scatterings
7.1 Coherent wave packets and Marchenko's equation
7.1.1 Delta and truncated step functions for coherent wave packets
7.1.2 Fourier transforms and Marchenko's equations
7.2 Reflectionless multi-soliton potentials
7.3 Two-component systems
7.3.1 Inverse scatterings
7.3.2 Matrix method
7.3.3 Modified Korteweg-deVries equation, part 1
Exercises
8 Quasi-static soliton states
8.1 Developing the Korteweg-deVries equation
8.1.1 N onstationary states
8.1.2 Thermal perturbation
8.2 Multi-soliton potentials in unsteady states
8.3 The modified Korteweg-deVries equation, part 2
8.4 Thermodynamic instability and Breezer potentials
8.5 The third-order Schrodinger equation
Exercises
9 The Baicklund transformation and sine-Gordon equations
9.1 The Klein-Gordon equation
9.2 The Backlund transformation
9.3 The sine-Gordon equation
9.4 Numerical analysis of the sine-Gordon equation
9.5 Inverse scatterings and the Backlund transformation
9.6 Scatterings by a pseudopotential
10 Miscellaneous applications
10.1 Surface waves
10.1.1 The first approximation
10.1.2 The second approximation
10.2 Vortex motion in fluid media
10.2.1 A vortex
10.2.2 Vortex motion
10.3 Plasma oscillation
10.4 Laser light transmission through absorbing media
10.4 . l Two-level atom in an intense radiation field
10.4.2 Scattering of intense radiation
t0.4.3 Sine-Gordon limit
10.5 Periodic lattices
10.5.1 Toda's lattice
10.5.2 Aperiodic transitions by pseudopotentials
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