This book provides an up-to-date overview of mathematical theories and research results on solitons, presenting related mathematical methods and applications as well as numerical experiments. Different types of soliton equations are covered along with their dynamical behaviors and applications from physics, making the book an essential reference for researchers and graduate students in applied mathematics and physics.

Contents
Introduction
Inverse scattering transform
Asymptotic behavior to initial value problems for some integrable evolution nonlinear equations
Interaction of solitons and its asymptotic properties
Hirota method
Bäcklund transformations and the infinitely many conservation laws
Multi-dimensional solitons and their stability
Numerical computation methods for some nonlinear evolution equations
The geometric theory of solitons
Global existence and blow up for the nonlinear evolution equations
The soliton movements of elementary particles in nonlinear quantum field
The theory of soliton movement of superconductive features
The soliton movements in condensed state systemsontents

• Contents 6
• 1. Introduction 10
• 2. Inverse scattering transform 22
• 3. Asymptotic behavior to initial value problems for some integrable evolution nonlinear equations 88
• 4. Interaction of solitons and its asymptotic properties 114
• 5. Hirota method 136
• 6. Bäcklund transformations and the infinitely many conservation laws 152
• 7. Multi-dimensional solitons and their stability 180
• 8. Numerical computation methods for some nonlinear evolution equations 200
• 9. The geometric theory of solitons 218
• 10. Global existence and blow up for the nonlinear evolution equations 232
• 11. The soliton movements of elementary particles in nonlinear quantum field 266
• 12. The theory of soliton movement of superconductive features 298
• 13. The soliton movements in condensed state systems 338
• Bibliography 370