Chapter 1 Physical Backgrounds for Some Nonlinear Evolution Equations
1.1 The wave equation under weak nonlinear action and KdV equation
1.2 Zakharov equations and the solitons in plasma
1.3 Landau-Lifshitz equations and the magnetized motion
1.4 Boussinesq equation， Toda Lattice and Born-Infeld equation
1.5 2D K-P equation
Chapter 2 The Properties of the Solutions for Some Nonlinear Evolution Equations
2.1 The smooth solution for the initial-boundary value problem of nonlinear Schrdinger equation
2.2 The existence of the weak solution for the initial-boundary value problem of generalized Landau-Lifshitz equations
2.2.1 The basic estimates of the linear parabolic equations
2.2.2 The existence of the spin equations
2.2.3 The existence of the solution to the initial-boundary value problem of the generalized Landau-Lifshitz equations
2.3 The large time behavior for generalized KdV equation
2.4 The decay estimates for the weak solution of Navier-Stokes equations
2.5 The “blowing up” phenomenon for the Cauchy problem of nonlinear Schrdinger equation
2.6 The “blow up” problem for the solutions of some semi-linear parabolic hyperbolic equations
2.7 The smoothness of the weak solutions for Benjamin-Ono equations
Chapter 3 Some Results for the Studies of Some Nonlinear Evolution Equations .
3.1 Nonlinear wave equations and nonlinear Schr.dinger equation
3.2 KdV equation
3.3 Landau-Lifshitz equation
Chapter 4 Similarity Solution and the Painlevé Property for Some Nonlinear Evolution Equations
4.1 Classical infinitesimal transformations
4.2 Structure of Lie algebra for infinitesimal operator
4.3 Nonclassical infinitesimal transformations
4.4 A direct method for solving similarity solutions
4.5 The Painlevé properties for some PDE
Chapter 5 Infinite Dimensional Dynamical Systems .
5.1 Infinite dimensional dynamical systems
5.2 Some problems for infinite dimensional dynamical systems
5.3 Global attractor and its Hausdorff， fractal dimensions
5.4 Global attractor and the bounds of Hausdorff dimensions for weak damped KdV equation
5.4.1 Uniform a priori estimation with respect to t
5.5 Global attractor and the bounds of Hausdorff dimensions for weak damped nonlinear Schr.dinger equation
5.5.1 Uniform a priori estimation with respect to t
5.5.2 Transforming to Cauchy problem of the operator
5.5.3 The existence of bounded absorbing set of H1 modular
5.5.4 The existence of bounded absorbing set of H2 modular
5.5.5 Nonlinear semi-group and long-time behavior
5.5.6 The dimension of invariant set
5.6 Global attractor and the bounds of Hausdorff， fractal dimensions for damped nonlinear wave equation
5.6.1 Linear wave equation
5.6.2 Nonlinear wave equation
5.6.3 The maximal attractor
5.6.4 Dimension of the maximal attractor
5.6.5 Application
5.6.6 Non-autonomous system
5.7 Inertial manifold for one class of nonlinear evolution equations
5.8 Approximate inertial manifold
5.9 Nonlinear Galerkin method
5.10 Inertial set
Chapter 6 Appendix
6.1 Basic notation and functional space
6.2 Sobolev embedding theorem and interpolation formula
6.3 Fixed point theorem
Bibliography
Index
