Essays on nonlinear waves: Patterns under water; pulse propagation through random media

Abstract

This is a collection of essays on weakly and strongly nonlinear systems and possible ways of solving/interpreting them. Firstly, we study sand patterns which are often observed on sea (river) beds. One of the most common features looks like straight rolls perpendicular to the water motion. In many cases, the straight rolls are superimposed on a much longer wave so that two vastly different length scales coexist. In general, there are at least two mechanisms responsible for the growth of periodic sand waves. One is linear instability, and the other is nonlinear coupling between long waves and short waves. One novel feature of this work is to suggest that the latter can be much more important than the former one for the generation of long waves. A weakly nonlinear analysis of the corresponding physical system suggests that the nonlinear coupling leads to the growth of the longer features if the amplitude of the shorter waves has a non-zero curvature. For the case of a straight channel and a tidal shallow sea, we derive nonlinear amplitude equations governing the dynamics of the main features. Estimates based on these equations are consistent with measurements. Secondly, we consider strongly nonlinear systems with randomness. The phenomenon of self-induced transparency (SIT) is reinterpreted in the context of competition between randomness, nonlinearity and dispersion. The problem is then shown to be isomorphic to a problem of the nonlinear Schroedinger (NLS) type with a random (in space) potential. It is proven that the SIT result continues to hold when the uniform medium of inhomogeneously broadened two-level atoms is replaced by a series of intervals in each of which the frequency mismatch is randomly chosen from some distribution. The exact solution of this problem suggests that nonlinearity can improve the transparency of the medium. Also, the small amplitude, almost monochromatic limit of SIT is taken and results in an envelope equation which is an exactly integrable combination of NLS and a modified SIT equation. Some generalizations are made to describe a broad class of integrable systems which combine randomness, nonlinearity and dispersion.