Korteweg-de Vries and nonlinear Schrödinger equations: qualitative theory.

*(English)*Zbl 0987.35001
Lecture Notes in Mathematics. 1756. Berlin: Springer. 147 p. (2001).

In this volume the author studies important nonlinear evolution equations: the generalized Korteweg-de Vries (KdVE) and the generalized nonlinear Schrödinger equation (NLSE). The topics are of qualitative nature, namely the existence of global solutions and blow-up results, stationary problems, stability of special solutions (travelling waves, soliton-like solutions, kinks, plane waves), and existence of invariant measures. These themes (esp. stability and invariant measures for KdVE and NLSE) are usually very technically and hard to explain, nevertheless the author has succeeded to give a rigorous presentation of the mathematical problems involved which is also readable not only for experts but also for students. E.g., it is worthwile to mention the draw up of the concentration-compactness method of Lions, the \(Q\)-stability of travelling wave solutions of KdVE and stationary solutions of NLSE, and the construction of invariant measures for both KdVE and NLSE. I can only recommend this book strongly for researchers and students working in this area.

Reviewer: Horst Lange (Köln)

##### MSC:

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

35Q55 | NLS equations (nonlinear Schrödinger equations) |

35Q53 | KdV equations (Korteweg-de Vries equations) |