Localized waves in inhomogeneous media

Abstract

This review examines the localization of one-dimensional nonlinear waves in an inhomogeneous multiphase medium. Particular attention is devoted to the localization of two types of waves, namely, solitary waves (domains) and switching waves that are the separation boundaries between the corresponding phases (domain walls). The localized state of such waves on both point and slowly-varying (in space) inhomogeneities is investigated. It is shown that several types of waves can become localized on inhomogeneities, and variation of external parameters may be accompanied by abrupt transitions between different types of localized waves. The stability of waves localized on inhomogeneities is examined together with various hysteresis phenomena that may occur in an inhomogeneous medium. The general results presented in the first part of the review are illustrated by examples of different physical systems, including superconductors, normal metals, semiconductors, plasmas, and chemical-reaction waves. CONTENTS 1. Introduction 19 2. Nonlinear quasistationary waves 20 A. Waves in a homogeneous medium. B. Waves in an inhomogeneous medium (qualitative analysis). C. Exactly solvable model. D. Dynamics of localized waves. 3. Localized static waves (general approach) 29 A. Static distributions. B. Stability of localized waves. 4. Examples 30 A. Resistive domains in superconductors. B. Optical discharge in gases. C. Temperature-electric domains in normal metals. D. Chemical-reaction waves. E. Superconductor exposed to laser radiation. F. Dielectric-metal phase transition. G. Dissipative structures in an inhomogeneous medium. 5. Conclusions 39 References 39

Undular bore transition in bi-directional conservative wave dynamics

“…The second example is the non-integrable system describing fully nonlinear flows in a two-temperature collisionless plasma. The decay of an initial discontinuity problem for this system has been studied numerically in [27]. In both cases our analytical results are in complete agreement with the results of previous anlytical/numerical studies.…”

“…Now we note that although construction of the solution to the generalised GP problem in the upper XTplane can be a very involved task for a non-integrable case, when the Riemann invariants for the modulation system (8) are not available, this problem can be easily solved in 'non-physical' domain T < 0. Indeed, there is a unique three-parametric solution to the gas dynamic system (24) in the lower XT-half-plane satisfying the initial conditions (26) and subject to inequalities (27). This solution is a centred expansion fan (in −X, −Tcoordinates) given by the expressions…”

“…Analytical studies of the decay of an initial discontinuity problem for integrable dispersive wave equations (see for instance [10,20,23]) as well as direct numerical simulations for non-integrable systems [27] show that the asymptotic solution to the decay of an arbitrary initial discontinuity generally consists of three constant states separated by two expanding waves: centred rarefaction wave(s) and/or undular bore(s), which is quite natural taking into account the "two-wave" nature of the system (1). One of the possible decay patterns is shown in Fig.…”

“…The obtained transition conditions include a relationship between the admissible values of the afore-mentioned constant states ρ 1,2 , u 1,2 (this relationship has been proposed earlier by Gurevich and Meshcherkin [27] for "dissipationless shocks" in plasma using intuitive physical arguments) and two ordinary differential equations defining the boundaries of the expanding nonlinear oscillatory zone. These ordinary differential equations are obtained in a general form and are easily integrated in particular instances.…”

Асимптотика специального решения уравнения Абеля, связанного с особенностью сборки. II. Большие значения параметра $t$

“…В [1] описана связь этого решения u(x, t) с решениями многих дифференци-альных уравнений в частных производных, моделирующих разные процессы в так называемых плавно неоднородных средах [3]- [5]. Отметим актуальность данного решения u(x, t) и для исследования таких процессов, моделируемых посредством обыкновенных дифференциальных уравнений.…”