CHAPTER 1 Introduction and a Brief Review of the History During the last thirty years the theory of stability of travelling wave solutions for nonlinear evolution equations has grown into a large field that attracts the attention of both mathematicians and physicists in view of its applications to real- world nonlinear models and of the novelty of the problems. The qualitative theory of nonlinear equations includes, in particular, investigations on the well-posedness of various problems for these equations, the behavior of solutions such as blowing-up, the existence and stability of solitary wave and periodic travelling wave solutions as well as properties of the dynamical system generated by these equations. The purpose of this book is to give a self-contained presentation of some basic and detailed results concerning the existence and stability of travelling wave solu- tions. It is intended to be a new source for modern research dealing with nonlinear phenomena of dispersive type. The selection of the material is mainly related to the author’s scientific interest. There are four main topics: the existence of solitary wave and periodic travelling wave solutions, the problems of the stability of these special kinds of solutions, the applicability of the Concentration-Compactness Prin- ciple in the study of the stability of solitary wave solutions of nonlinear dispersive equations, and the instability of solitary waves. A detailed and clear explanation of every concept and method introduced is given. The exposition is accompanied by a careful selection of modern examples. The book provides information that puts the reader at the forefront of current research. Nonlinear evolution equations for modelling waves take into account both non- linearity and dispersion effects. Their birth was the discovery of the solitary wave, or great wave of translation, observed on the Edinburgh-Glasgow canal in 1834 by J. Scott Russell. The story of the first encounter of Russell with the solitary wave was reported by him to the British Association in 1844 with the name Report on Waves [246]. Fascinated with this long water wave without a change in shape, which he called the “great wave of translation, or solitary wave”, Russell made some laboratory experiments on this phenomenon, generating solitary waves by dropping a weight at one end of a water channel. He deduced empirically that the volume of water in the wave is equal to the volume displaced by the weight and that the speed c of the solitary wave and its maximum amplitude a above the free surface liquid of finite depth h satisfy the relation c2 = g(h + a), where g is the acceleration due to gravity. His description of solitary waves contra- dicted the theories of water waves according to G. B. Airy and G. G. Stokes they raised questions on the existence of Russell’s solitary waves and conjectured that such waves cannot propagate in a liquid medium without a change of form. Despite the mathematical theory, the experimental evidence in favor of solitary waves was 3 http://dx.doi.org/10.1090/surv/156/01
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