March 23, 2001
A way of quantizing weakly nonlinear lattices is proposed. It is based on introducing "pseudo-field" operators. In the new formalism quantum envelope solitons together with phonons are regarded as elementary quasi-particles making up boson gas. In the classical limit the excitations corresponding to frequencies above linear cut-off frequency are reduced to conventional envelope solitons. The approach allows one to identify the quantum soliton which is localized in space and understand existence of a narrow soliton frequency band.
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December 22, 2022
In recent years, there has been considerable interest in the study of wave propagation in nonlinear photonic lattices. The interplay between nonlinearity and periodicity has led researchers to manipulate light and discover new and interesting phenomena such as new classes of localized modes, usually referred to as solitons, and novel surface states that propagate robustly. A field where both nonlinearity and periodicity arises naturally is nonlinear optics. But there are othe...
October 13, 2004
We analyze finite temperature effects in the generation of bright solitons in condensates in optical lattices. We show that even in the presence of strong phase fluctuations solitonic structures with well defined phase profile can be created. We propose a novel family of variational functions which describe well the properties of these solitons and account for the non-linear effects in the band structure. We discuss also the mobility and collisions of these localized wave pac...
October 11, 2010
This article offers a comprehensive survey of results obtained for solitons and complex nonlinear wave patterns supported by purely nonlinear lattices (NLs), which represent a spatially periodic modulation of the local strength and sign of the nonlinearity, and their combinations with linear lattices. A majority of the results obtained, thus far, in this field and reviewed in this article are theoretical. Nevertheless, relevant experimental settings are surveyed too, with emp...
July 21, 2009
The phenomenon of dynamical localization of matter wave solitons in optical lattices is first demonstrated and the conditions for its existence are discussed. In addition to the trapping linear periodic potential we use a periodic modulation of the nonlinearity in space to eliminate nonexistence regions of gap-solitons in reciprocal space. We show that when this condition is achieved, the observation of dynamical localization in true nonlinear regime becomes possible. The res...
May 17, 2004
We consider several effects of the matter wave dynamics which can be observed in Bose-Einstein condensates embedded into optical lattices. For low-density condensates we derive approximate evolution equations, the form of which depends on relation among the main spatial scales of the system. Reduction of the Gross-Pitaevskii equation to a lattice model (the tight-binding approximation) is also presented. Within the framework of the obtained models we consider modulational ins...
April 19, 1999
Group velocity and group velocity dispersion for a wave packet in vectorial discrete Klein-Gordon models are obtained by an expansion, based on perturbation theory, of the linear system giving the dispersion relation and the normal modes. We show how to map this expansion on the Multiple Scale Expansion in the real space and how to find Non Linear Schr\"odinger small amplitude solutions when a nonlinear one site potential balances the group velocity dispersion effect.
February 18, 2004
It is shown that matter solitons can be effectively managed by means of smooth variations of parameters of optical lattices in which the condensate is loaded. The phenomenon is based on the effect of lattice modulations on the carrier wave transporting the soliton and that is why is well understood in terms of the effective mass approach, where a particular spatial configuration of the band structure is of primary importance. Linear, parabolic, and spatially localized modulat...
February 16, 2011
We investigate theoretically soliton excitations and dynamics of their formation in strongly correlated systems of ultracold bosonic atoms in two and three dimensional optical lattices. We derive equations of nonlinear hydrodynamics in the regime of strong interactions and incommensurate fillings, when atoms can be treated as hard core bosons. When parameters change in one direction only we obtain Korteweg-de Vries type equation away from half-filling and modified KdV equatio...
March 6, 2011
We study the gap solitons and nonlinear Bloch waves of interacting bosons in one-dimensional optical lattices, taking into account the interaction from the weak to the strong limits. It is shown that composition relation between the gap solitons and nonlinear Bloch waves exists for the whole span of the interaction strength. The linear stability analysis indicates that the gap solitons are stable when their energies are near the bottom of the linear Bloch band gap. By increas...
August 29, 2012
Nonlinear dynamics of wave packets in PT-symmetric optical lattices near the phase-transition point are analytically studied. A nonlinear Klein-Gordon equation is derived for the envelope of these wave packets. A variety of novel phenomena known to exist in this envelope equation are shown to also exist in the full equation including wave blowup, periodic bound states and solitary wave solutions.