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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 ±à¼­ÊÖ¼Ç