Waves and structures in nonlinear nondispersive media. General theory and applications to nonlinear acoustics.
The book is divided into two parts. The first part (Chapters 1-4) gives a detailed description of basic concepts and methods for analysis of nonlinear wave problems in nondispersive media, while the second part (Chapters 5-11) is devoted to applications in nonlinear acoustics. If possible, the presentation is given on the physical level of rigor, considering only hydrodynamic nonlinear waves in nondispersive media. Chapter 1 examines the solutions of thesimplest nonlinear PDEs of first order: simple wave equations (canonical form of the equation, Riemann solution, construction of the density field, Fourier transforms of density and velocity), line-growth equation of forest-fire propagation, and one-dimensional laws of gravitation (Lagrangian and Eulerian descriptions). Chapter 2 discusses generalized solutions to PDEs of first order (multistream solutions, weak solutions). The main ideas and solution methods for nonlinear PDEs of second order are presented in Chapter 3 with the help of the examples of the Kardar-Parisi-Zhang equation (arising as a regularized version of the line-growth equation, and describing the deposition of semiconductor films) and the Burgers equation (arising as a regularization of first order PDE of nonlinear acoustics). The regularization is understood here as the introduction into the equation of additional terms containing higher order derivatives to prevent the gradient catastrophe. Two sections of Chapter 3 study general solutions of the Burgers equation, demonstrating the effectivity of standard methods for investigation of nonlinear phenomena. Some modifications and generalizations of the Burgers equation admitting general solutions are discussed as model equations of gas dynamics. In Chapter 4, the authors describe some solutions of the Burgers equation for a single-scale field (evolution of one-dimensional signals) with discussion of typical scenarios of solution evolution realized under various initial conditions with special attention to strongly nonlinear regimes. They also study properties of solutions to the Burgers equation for multi-scale field evolution of complex signals with special attention to fractal signals. Chapter 5 contains a brief review of evolution of random fields satisfying the Burgers equation, with discussion of statistical properties of solutions to the Burgers equation in the limit of a vanishing diffusion coefficient. This chapter also gives a classification of different evolution regimes of the Burgers turbulence with applications to acoustics. Chapter 6 is devoted to multidimensional nonlinear 2D Kardar-Parisi-Zhang (KPZ) and 3D Burgers equations. Here, the authors examine the evolution of main perturbation types in the KPZ equation and in the multidimensional Burgers equation. The evolution forms are simple localized perturbations, periodic structures at infinite Reynolds numbers, anisotropic Burgers turbulence, perturbations with complex internal structure, and asymptotic long-time localized perturbations. The separate Section 6.4 describes the large-scale structure evolution of the Universe closely related to the known astrophysical Zeldovich approximation. Namely, here the authors derive the gravitational instability of expanding Universe from the Vlasov-Poisson equation and consider the adhesion model. Chapters 1-4 are equipped by exercises and open problems. Part II applies mathematical models and physical results of Part I to nonlinear acoustics. The authors consider here the general theory of nonlinear waves and wave structures, such as shock fronts, solitary waves, cellular multidimensional structures, and nondispersive and weakly dispersive waves. Chapter 7 contains model equations and methods of finding their exact solutions. In particular, the methods of group analysis of differential equations are used to find exact solutions of the Burgers equation. Chapter 8 is devoted to the classification of acoustic nonlinearities and methods of nonlinear acoustic diagnostics. Chapter 9 discusses interactions of strongly distorted waves containing shock fronts. Such sawtooth perturbations are formed during the wave propagation through media where nonlinearity predominates over competitive factors like dispersion, diffraction and absorption. Chapter 10 ``Self-action of spatially bounded waves containing shock waves'' is a survey of problems for waves with broad frequency spectra, whose time profiles contain discontinuities or steep shock fronts of a finite width, which is small compared with the wave period or with a characteristic length of pulse signal. A separate section studies symmetries and conservation laws for evolution equation describing beam propagation in nonlinear media. The final Chapter 11 presents the general theory of nonlinear standing waves, resonance phenomena and frequency characteristics of distributed systems applied to nonlinear acoustics, in particular, to nonlinear resonators filled by nonlinear media. An appendix contains the basic facts from the theory of generalized functions (distribution theory).
scientific article; zbMATH DE number 5961790