The interplay between disorder and nonlinearity in wave propagation is a very challenging topic for theoretical research. The reason lies in the fact that the two effects are strongly competing, and treating both of them at the same level is really a difficult enterprise. At the moment, most of the approaches either tackle with nonlinearity as a perturbation to disorder, or with disorder as a perturbation to nonlinearity. In both of the cases, the results of the analysis do not seem to contain a substantially innovative picture, as it appears to be the case in practically all the related fields of research (as nonlinear optics or Bose-Einstein condensation).
The application of spin glass theory seems to open new perspectives, if methods from the statistical mechanics of disordered systems can be applied to nonlinear waves. Such an interdisciplinary approach is developed and extensively described in a recent paper ,authored by L. Leuzzi and C. Conti, (arXiv:1009.3290) which extends previous theoretical investigations (see, e.g., these articles in this website).
The main result of this approach is the predicition of the existence of a complex landscape (i.e., many energetically equivalent states) for nonlinear waves in disordered media, and, specifically, the explicit calculation of the complexity in terms of the strength of nonlinearity for various degrees of disorder. Specific dynamic and thermodynamic phases, organized in a universal phase-diagram and with different complexity, can be identified.
All of this finds application in random photonics, nonlinear waves in random systems and Bose-Einstein condensation, including finite temperature effects. Details in arXiv:1009.3290.
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