Stationary states of the two-dimensional nonlinear Schrödinger model with disorder

Abstract

Solitonlike excitations in the presence of disorder in the two-dimensional cubic nonlinear Schrödinger equation are analyzed. The continuum as well as the discrete problem are analyzed. In the continuum model, otherwise unstable excitations are stabilized in the presence of disorder. In the discrete model, the disorder is found to leave the narrow excitations unaffected. Our results suggest that the disorder provides a possibility to control the spatial extent of the stable excitations in the continuum system. ͓S0163-1829͑98͒06429-7͔

Interplay of disorder and nonlinearity in Klein-Gordon models: Immobile kinks

“…It must be emphasized that for NLS models the conclusion does not depend on the averaging procedure: one can equally perform averaging either on absolute ground states 6 or over all local minima of the effective random potential with equal weights. 7,8 We show that it is not the case for the Klein-Gordon models: their properties exhibit strong dependence on the assumptions as to the statistical distribution of kinks over the minima of the effective random potential. For the purely dynamical problem these statistics are left beyond consideration and should be thus imposed as an additional assumption.…”

“…The importance of this conclusion resides in its prediction that the disorder can stabilize otherwise unstable solitons in 2D and 3D NLS models. Quite recently this prediction was borne out numerically 8 for the 2D case. It must be emphasized that for NLS models the conclusion does not depend on the averaging procedure: one can equally perform averaging either on absolute ground states 6 or over all local minima of the effective random potential with equal weights.…”

“…These two localization mechanisms are competitive to some extent; taken together, the nonlinearity and disorder may lead to a number of qualitatively new effects, namely, the transmission coefficient tending to zero with increasing the system length according to a power law 3,4 rather than exponentially (which would be otherwise the property of a linear system); there can arise a multistability [3][4][5] in the wave transmission through a disordered slab; excitations in highly nonlinear or multidimensional nonlinear Schrödinger (NLS) systems (which would either disperse or collapse otherwise) can be stabilized by disorder. [6][7][8] In the present paper we study the static properties of a one-kink solution (or, equivalently, of a diluted kink gas) of disordered Klein-Gordon models where the disorder is assumed to be a δ-correlated Gaussian spatial noise. The disorder of the kind is akin, for example, to Josephson junctions, where it is caused by the fluctuations of the gap between two superconductor plates.…”

Nonlinear Localized Modes in 2D Photonic Crystals and Waveguides

“…However, such modes in unbounded 2D NLS models are always unstable and either collapse or spread out [34]. In fact, they can be stabilized by some external forces (e.g., due to interactions with boundaries or disorder [37]), but in this case the excitations are pinned and cannot be used for energy or signal transfer.…”

Dynamics of a nonlocal discrete Gross-Pitaevskii equation with defects

“…In recent years, the interplay between nonlinearity, discreteness, and disorder (i.e., small random impurities or defects) has been the subject of intensive theoretical and experimental investigations [1][2][3][4][5][6][7][8][9][10][11]. The competition between nonlinearity, discreteness, and disorder can induce rich phenomena and plays a crucial role in nonlinear discreteness systems such as Anderson localization [12] and disorder-induced inhibition of transportation [13][14][15][16].…”