Contents Chapter 1 Chaos for Nearly Integrable Systems 1 1.1 Direct methods of perturbation theory for solitons 1 1.2 Perturbation theory based on the inverse scattering transform 4 1.3 Motion of a soliton in a driven Sine-Gordon equation 8 1.3.1 Soliton motion of Sine-Gordon equation 8 1.3.2 Motion of a SG soliton in the fields of two waves 10 1.3.3 Stochastic dynamics of a three-dimensional bubble in a driven SG equation 11 1.3.4 SG soliton similar to the Fermi-Pasta-Ulam problem 13 1.3.5 Dynamical chaos of a breather under the action of an external field 14 1.3.6 Dynamical chaos in the SG system with parametric excitation 16 1.3.7 Stochastization of soliton lattices in the perturbed SG equation 18 1.4 Motion of the soliton of nonlinear Schr.odinger equation with damping under the action of an external field 20 1.4.1 Nonlinear Schr.odinger equation 20 1.4.2 Stochastic dynamics of NLS solitons in a periodic potential 20 1.5 Dynamical chaos of the KdV equation and the perturbation equations 23 1.5.1 Chaotic state of the cnoidal wave in the periodic inhomogeneous medium 23 1.5.2 Karamoto-Sivashinsky equation 24 Chapter 2 Some Numerical Results and Their Analysis 26 2.1 Coherent structure and numerical calculation results 27 2.2 Fundamental analysis 54 2.2.1 Connections between NLS equation and Sine-Gordon equation 54 2.2.2 Space independent fixed point 55 2.2.3 Space dependent fixed point 57 2.2.4 Integrable structure of nonlinear Schr.odinger equation 59 2.2.5 The Whisker ring of focusing nonlinear Schr.odinger equation 75 Chapter 3 Homoclinic Orbits in a Four Dimensional Model of a Perturbed Nonlinear Schr.odinger Equation 91 3.1 Dynamics and geometric structure for the unperturbed systerm 91 3.1.1 M0 and Ws(M0)TWu(M0) 93 3.1.2 The dynamics on M0 95 3.1.3 The unperturbed homoclinic orbits and their relationship to the dynamics on M0 and Ws(M0) \Wu(M0) 95 3.2 Geometric structure of the perturbed systerm 98 3.2.1 The persistence of M0;Ws(M0) and Wu(M0) under perturbation 99 3.2.2 The dynamics on M
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