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-deVries 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-deVries 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
10.4.1 Two-level atom in an intense radiation field
10.4.2 Scattering of intense radiation
10.4.3 Sine-Gordon limit
10.5 Periodic lattices
10.5.1 Toda's lattice
10.5.2 Aperiodic transitions by pseudopotentials
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