Wave energy self-trapping by self-focusing in large molecular structures: A damped stochastic discrete nonlinear Schrödinger equation model

Wave self-focusing in molecular systems subject to thermal effects, such as thin molecular films and long biomolecules, can be modeled by stochastic versions of the discrete self-trapping equation of Eilbeck et al. [J.C. Eilbeck, P.S. Lomdahl, A.C. Scott, The discrete self-trapping equation, Physica D 16 (1985) 318–338], which gives as a continuum limit approximation the stochastic nonlinear Schrödinger equation (SNLS): NLS plus a noise term in the form of a random, time dependent potential. Previous studies directed at such SNLS approximations have indicated that the self-focusing of wave energy to highly localized states can be inhibited by phase noise (modeling thermal effects) and can be restored by phase damping (modeling heat radiation). Here, the discrete models are studied directly, with some discussion of the validity and limitations of continuum approximations. Also, as has been noted by Bang et al. [O. Bang, P.L. Christiansen, F. If, K.Ø. Rasmussen, Yu.B. Gaididei, Temperature effects in a nonlinear model of monolayer Scheibe aggregates, Phys. Rev. E 49 (1994) 4627–4636], omission of damping produces highly unphysical results. Numerical results are presented here for the first time for discrete models that include the highly nonlinear damping term, and a new numerical method is introduced for this purpose. The results in general confirm previous conjectures and observations that noise can inhibit energy self-trapping (the discrete counterpart of NLS self-focusing blow-up), while damping can reverse this and restore self-trapping. Damping is also shown to strongly stabilize the self-trapped states of the discrete models. It appears that the previously noted inhibition of nonlinear wave phenomena by noise is an artifact of models that includes the effects of heat input, but not of heat loss. c © 2006 Elsevier B.V. All rights reserved.