Spectral and dynamical stability of nonlinear waves
The book under review focuses on stability of the equilibria of evolution equations. It is a well-known fact that the dynamics of a nonlinear evolution equation mainly depends on (and can be understood by) the dynamics of the linear system obtained by linealization, and that this linear dynamics is fully dependent on the spectrum of the corresponding linear operator. The spectral analysis of general linear operators is provided by functional analysis, while the specific results for differential operators are given by the theory of dynamical systems. Both theoretical viewpoints are involved in this presentation. The book consists of two parts. The first part, Chapters 3--7, is about essential, absolute, and point spectra and develops nonlinear stability theory for dissipative and Hamiltonian systems. The theory is exemplified by the cases of reaction-diffusion equations, scalar viscous conservation law, nonlinear Schrödinger type equations, generalized Korteweg-de Vries equation, forced Ginzburg-Landau equation, Hamiltonian and non-Hamiltonian perturbations of the nonlinear Schrödinger equation. Particularly, the stability theory for Hamiltonian systems is generously developed and includes Krein signature and other stability indices. The second part of the book, Chapters 8--10, grants it maximum of originality compared to everything that has been written on the subject. It offers for the first time in a book format, the theory based on the Evans function, a complex-valued function of the spectral parameter whose zeros are the eigenvalues of the corresponding linear operator. The construction of the Evans function, akin in some sense to that of the Wronskian, is first given for the Sturm-Liouville problem and then, for a series of cases of increasing complexity. The connection of the Evans function with different concepts from dynamical systems, such as winding number, orientation index, or edge bifurcation, is presented in a general framework or on some particular cases. Each chapter contains exercises, bibliographic comments and additional reading information. The reviewer evaluates the book as a significant achievement and recommends it to all interested readers.
scientific article; zbMATH DE number 6151577