Nonlinear Wave Propagation in Disordered Media

Abstract

We briefly review the state-of-the-art of research on nonlinear wave propagation in disordered media. The paper is intended to provide the non-specialist reader with a flavor of this active field of physics. Firstly, a general introduction to the subject is made. We describe the basic models and the ways to study disorder in connection with them. Secondly, analytical and numerical techniques suitable for this purpose are outlined. We summarize their features and comment on their respective advantages, drawbacks and applicability conditions. Thirdly, the Nonlinear Klein-Gordon and Schrbdinger equations are chosen as specific examples. We collect a number of results that are representative of the phenomena arising from the competition between nonlinearity and disorder. The review is concluded with some remarks on open questions, main current trends and possible further developments.

Length scale competition in nonlinear Klein—Gordon models: A collective coordinate approach

“…Fronts, solitons, solitary waves, breathers, or vortices are instances of such coherent structures of relevance in a plethora of applications in very different fields. One of the chief reasons that gives all these nonlinear excitations their paradigmatic character is their robustness and stability: Generally speaking, when systems supporting these structures are perturbed, the structures continue to exist, albeit with modifications in their parameters or small changes in shape (see [3,4] for reviews). This property that all these objects (approximately) retain their identity allows one to rely on them to interpret the effects of perturbations on general solutions of the corresponding models.…”

“…Collective coordinate approaches were introduced in [6,7] to describe kinks as particles (see [2,3,4,9] for a very large number of different techniques and applications of this idea). Although the original approximation was to reduce the equation of motion for the kink to an ordinary differential equation for a time dependent, collective coordinate which was identified with its center, it is being realized lately that other collective coordinates can be used instead of or in addition to the kink center.…”

Soliton pinning by long-range order in aperiodic systems

“…We show that most of these phenomena can be understood by means of simple collective coordinate arguments, with the exception of long range order effects. In the conclusion we comment on the interesting implications that our work could bring about in the field of solitons in molecular (e.g., DNA) chains.PACS numbers: 03.20.+i, 85.25.Cp, 61.44.+p The subtle interplay between nonlinearity and disorder is being laboriously unveiled throughout the past few years [1]. A rich diversity of phenomena stems from such interaction, their manifestations being found in a number of systems ranging from condensed matter physics to biophysics [2].…”

Nonlinear Physics: Integrability, Chaos and Beyond

“…Such complete integrability aspects can be further generalized to the area of quantum integrable systems, exactly solvable statistical models and so on (Wadati et al, [1989]) through Yang-Baxter relations and the quantum inverse scattering method. Also the study of perturbation of soliton systems, often leading to spatiotemporal complexity, is of great physical interest in condensed matter physics, fluid dynamics, nonlinear optics, liquid crystals and so on (Sanchez and Vazquez, [1991]; Hasegawa, [1989]; Lam and Prost, [1991]). Thus one finds the study of soliton-bearing systems is of fundamental importance in several branches of physics and natural sciences.…”