E. Fermi described the results obtained with J. Pasta and S. Ulam as “a little discovery”. He referred to an observation of near recurrence in numerical simulations of 32 coupled nonlinear ODE on the MANIAC I computer at Los Alamos. Previously due to works of Poincaré, Peierls, Fermi (!) and others, physicists tend to believe that weakly coupled nonlinear systems will exhibit that ergodic behaviour deemed necessary for an approach to equilibrium. The pioneering work by Fermi, Pasta, Ulam (FPU) inspired Kruskal and Zabusky for their discovery of soliton and, in general, initiated the era of nonlinear science. FPU lattices (FPUL) were and now are the object of enormous numerical and analytical researches. May be, now it is time for collecting and generalization of these results.
The book by Pankov is devoted to rigorous results about time periodic oscillations and travelling waves in FPU lattices (FPUL), obtained on the basis of variational methods. Similar results for chains of oscillators (CO) are also described briefly. The book is aimed at researches in the field of nonlinear dynamics. It is assumed that the reader is familiar with nonlinear analysis and variational methods. The book is divided into 4 chapters: infinite lattice systems; time periodic oscillations; travelling waves (chapters 3 and 4). Four appendices are also included. Their aim is to remind basic facts about functional spaces, concentration compactness, critical points and finite difference.
The first part describes general properties of equations that govern dynamics w.r.t. FPUL and CO, in particular, well-posedness of the Cauchy problem.. The subject of the second part is the existence of time periodic solutions in FPUL and CO. The author deals with regular multiatomic lattices. The third part is devoted to the study of travelling waves with prescribed speed in monoatomic FPUL. Both periodic and solitary waves are studied, and solitary waves are treated as limit case of periodic ones when the wavelength goes to infinity. The fourth part contains detail descriptions of the approach proposed by G. Friesecka and J. A. D. Wattis [Commun. Math. Phys., 161, No. 2, 391–418 (1994; Zbl 0807.35121)], as well as results concerning exponential decay of solitary waves and travelling waves in CO. Chapters 1,2,4 end with a special section devoted to various comments and open problems.
This well-written book is a reader-friendly and good-organized research monograph in the field of nonlinear science. It can be highly recommended for experts in ODE, PDE, and nonlinear physics.