Contents
Chapter 1 Physical Backgrounds for Some Nonlinear Evolution Equations 1
1.1 The wave equation under weak nonlinear action and KdV equation 2
1.2 Zakharov equations and the solitons in plasma 10
1.3 Landau-Lifshitz equations and the magnetized motion 19
1.4 Boussinesq equation�� Toda Lattice and Born-Infeld equation 22
1.5 2D K-P equation 26
Chapter 2 The Properties of the Solutions for Some Nonlinear Evolution Equations 29
2.1 The smooth solution for the initial-boundary value problem of nonlinear Schrdinger equation 30
2.2 The existence of the weak solution for the initial-boundary value problem of generalized Landau-Lifshitz equations 34
2.2.1 The basic estimates of the linear parabolic equations 34
2.2.2 The existence of the spin equations 36
2.2.3 The existence of the solution to the initial-boundary value problem of the generalized Landau-Lifshitz equations 39
2.3 The large time behavior for generalized KdV equation 42
2.4 The decay estimates for the weak solution of Navier-Stokes equations 60
2.5 The ��blowing up�� phenomenon for the Cauchy problem of nonlinear Schrdinger equation 71
2.6 The ��blow up�� problem for the solutions of some semi-linear parabolic hyperbolic equations 78
2.7 The smoothness of the weak solutions for Benjamin-Ono equations 93
Chapter 3 Some Results for the Studies of Some Nonlinear Evolution Equations .105
3.1 Nonlinear wave equations and nonlinear Schr.dinger equation 105
3.2 KdV equation 121
3.3 Landau-Lifshitz equation 132
Chapter 4 Similarity Solution and the Painlev�� Property for Some Nonlinear Evolution Equations 141
4.1 Classical infinitesimal transformations 142
4.2 Structure of Lie algebra for infinitesimal operator 156
4.3 Nonclassical infinitesimal transformations 158
4.4 A direct method for solving similarity solutions 163
4.5 The Painlev�� properties for some PDE 173
Chapter 5 Infinite Dimensional Dynamical Systems .182
5.1 Infinite dimensional dynamical systems 183
5.2 Some problems for infinite dimensional dynamical systems 187
5.3 Global attractor and its Hausdorff�� fractal dimensions 196
5.4 Global attractor and the bounds of Hausdorff dimensions for weak damped KdV equation 206
5.4.1 Uniform a priori estimation with respect to t 207
5.5 Global attractor and the bounds of Hausdorff dimensions for weak damped nonlinear Schr.dinger equation 217
5.5.1 Uniform a priori estimation with respect to t 218
5.5.2 Transforming to Cauchy problem of the operator 221
5.5.3 The existence of bounded absorbing set of H1 modular 224
5.5.4 The existence of bounded absorbing set of H2 modular 225
5.5.5 Nonlinear semi-group and long-time behavior 228
5.5.6 The dimension of invariant set 231
5.6 Global attractor and the bounds of Hausdorff�� fractal dimensions for damped nonlinear wave equation 238
5.6.1 Linear wave equation 238
5.6.2 Nonlinear wave equation 243
5.6.3 The maximal attractor 250
5.6.4 Dimension of the maximal attractor 253
5.6.5 Application 260
5.6.6 Non-autonomous system 265
5.7 Inertial manifold for one class of nonlinear evolution equations 269
5.8 Approximate inertial manifold 287
5.9 Nonlinear Galerkin method 296
5.10 Inertial set 323
Chapter 6 Appendix 345
6.1 Basic notation and functional space 345
6.2 Sobolev embedding theorem and interpolation formula 348
6.3 Fixed point theorem 350
Bibliography 352
Index 365