Foreword

The solitary wave phenomenon have seen, during the last few decades, the emergence of many nonlinear models and of the development of related nonlinear concepts. This has been driven by modern computer power as well as by the discovery of new mathematical techniques, which include two contrasting themes:

1. the theory of dynamical systems, most popularly associated with the study of chaos;
2. the theory of integrable systems associated, among other things, with the study of solitary waves.

It seemed important to Claire David and Zhaosheng Feng to offer scientists and engineers a kind of “handbook”, which would summarize classical and innovative techniques used to study this phenomenon, while also presenting prominent advances in this field.

The book covers at least five large classes of nonlinear evolution equations:

1. Nonlinear dispersive wave equations such as the Korteweg-de Vries equation, the cubic nonlinear Schrodinger equation, the Boussinesq equation and the KP equation.
2. Nonlinear dissipative dispersive wave equations such as the Korteweg-de Vries-Burgers equation and the complex Ginzburg-Landau equation.
3. Nonlinear convection equations such as the n-dimensional Burgers equation.
4. Nonlinear reaction diffusion equations such as the Fishers equation and the scalar bistable equation.
5. Nonlinear hyperbolic equations such as the Sine-Gordon equation.

The book provides modern methods and techniques on the existence and explicit forms of traveling wave solutions of the partial differential equations. Additionally, the book offers a very good survey on:

1. Asymptotic behaviors of solitary waves.
2. Numerical simulations of solitary waves.
3. Existence of global solutions of the Cauchy problems for the PDEs.
4. Global attractors and their Hausdorff dimensions.
5. Fourier approximation of the global attractors.
6. Singular traveling waves.

Overall, I believe the book is a valuable contribution to the mathematical society. Math professors, post-doctorals and graduate students will benefit a lot from this book.