Nonlinear Waves in Inhomogeneous and Hereditary Media

This booklet presents a study of one-dimensional waves in solids which can be modelled by nonlinear wave equations of different types. The factorization method is the main tool in this analysis. It allows for an exact or at least asymp­ totic decomposition of the wave(s) under consideration in terms of first order multipliers. Chapter 1 provides a general introduction. It presents some well-known results on characteristics, Riemann invariants, simple waves, etc. The main result of Chap. 1 is Theorem 1.3.2. (Sect. 1.3.2) which establishes the possibility of exact factorization of the nonlinear wave equation EPa(a) 1 EPa _ 0 Ij(l-u- x2 with constant coefficients. This theorem permits one to construct further factor­ izations of more complicated wave equations which the reader will meet in the following chapters. Chapter 2 is devoted to short wave processes in inhomogeneous media, the main result being the uniform asymptotic factorization of nonlinear wave equa­ tions with variable coefficients and the description of corresponding single-wave processes without the usual assumption of a small wave amplitude.

1. Nonlinear Waves in Homogeneous Media.- 1.1 Preliminaries.- 1.2 Nonlinear Hyperbolic Equations of the First Order.- 1.3 Exact Factorization of the Nonlinear Wave Equation with Constant Coefficients.- 1.4 Shock-Wave in a Simple System.- 1.5 The Shock-Wave in a Simple System (Continuation).- 2. Nonlinear Short Waves of Finite Amplitude in Inhomogeneous Media.- 2.1 Asymptotic Factorization of the Nonlinear Wave Equation with a Variable Coefficient.- 2.2 When is the Factorization Exact?.- 2.3 Asymptotic Factorization of the General Nonlinear Wave Equation with Variable Coefficients.- 2.4 Evolution of Maximal Amplitude of the Stress Wave.- 2.5 Propagation of a Stress Wave in a Homogeneous Nonlinear Elastic Rod Located in the Gravity Field.- 3. Nonlinear Waves in Media with Memory.- 3.1 Hereditary Elasticity.- 3.2 Small Quadratic Nonlinearity.- 3.3 Continuous Stationary Profile Waves and Nonzero Solutions of Homogeneous Integral Volterra Equations.- 3.4 Stationary Profile Shock-Waves and Self-Coordinated Integral Volterra Equations.- 3.5 Waves Tending to a Stationary Profile.- 3.6 Nonstationary Waves Analog of the Landau-Whitham Formula.- 3.7 General Nonlinearity. Further Factorization Theorems for Nonlinear Wave Equations with Memory.- 3.8 Nonstationary Waves for an Exponential Memory Function.- 3.9 Reflection of a Wave from the Boundary Between Linear Elastic and Nonlinear Hereditary Media.- 3.10 The Exactly Factorizable Linear Wave Equation with Memory and a Variable Coefficient.- References.