Growth fluctuations in a class of deposition models

We compute the growth fluctuations in equilibrium of a wide class of deposition models. These models also serve as general frame to several nearest-neighbor particle jump processes, e.g. the simple exclusion or the zero range process, where our result turns to current fluctuations of the particles. We use martingale technique and coupling methods to show that, rescaled by time, the variance of the growth as seen by a deterministic moving observer has the form |V-C|*D, where V and C is the speed of the observer and the second class particle, respectively, and D is a constant connected to the equilibrium distribution of the model. Our main result is a generalization of Ferrari and Fontes' result for simple exclusion process. Law of large numbers and central limit theorem are also proven. We need some properties of the motion of the second class particle, which are known for simple exclusion and are partly known for zero range processes, and which are proven here for a type of deposition models and also for a type of zero range processes.Comment: A minor mistake in lemma 5.1 is now correcte

Exact Solution of the Zakharov-Shabat Scattering Problem for Doubly-Truncated Multi-Soliton Potentials

Recent studies have revealed that multi-soliton solutions of the nonlinear Schr\"odinger equation, as carriers of information, offer a promising solution to the problem of nonlinear signal distortions in fiber optic channels. In any nonlinear Fourier transform based transmission methodology seeking to modulate the discrete spectrum of the multi-solitons, choice of an appropriate windowing function is an important design issue on account of the unbounded support of such signals. Here, we consider the rectangle function as the windowing function for the multi-solitonic signal and provide the exact solution of the associated Zakharov-Shabat scattering problem for the windowed/doubly-truncated multi-soliton potential. This method further allows us to avoid prohibitive numerical computations normally required in order to accurately quantify the effect of time-domain windowing on the nonlinear Fourier spectrum of the multi-solitonic signals. The method devised in this work also applies to general type of signals and may prove to be a useful tool in the theoretical analysis of such systems.Comment: The manuscript is revised for submission to PRE. Also, some typos have been correcte

The Orbit Bundle Picture of Cotangent Bundle Reduction

Cotangent bundle reduction theory is a basic and well developed subject in which one performs symplectic reduction on cotangent bundles. One starts with a (free and proper) action of a Lie group G on a configuration manifold Q, considers its natural cotangent lift to T*Q and then one seeks realizations of the corresponding symplectic or Poisson reduced space. We further develop this theory by explicitly identifying the symplectic leaves of the Poisson manifold T^*Q/G, decomposed as a Whitney sum bundle, T^*⊕(Q/G)g^* over Q/G. The splitting arises naturally from a choice of connection on the G-principal bundle Q → Q/G. The symplectic leaves are computed and a formula for the reduced symplectic form is found