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We study the periodic cubic derivative non-linear Schr\"odinger equation (dNLS) and the (focussing) quintic non-linear Schr\"odinger equation (NLS). These are both $L^2$ critical dispersive models, which exhibit threshold type behavior,…
Solitary waves in one-dimensional periodic media are discussed employing the nonlinear Schr\"odinger equation with a spatially periodic potential as a model. This equation admits two families of gap solitons that bifurcate from the edges of…
An analysis of discrete systems is important for understanding of various physical processes, such as excitations in crystal lattices and molecular chains, the light propagation in waveguide arrays, and the dynamics of Bose-condensate…
We study the nonlinear Schr\"odinger equation arising in dipolar Bose-Einstein condensate in the unstable regime. Two cases are studied: the first when the system is free, the second when gradually a trapping potential is added. In both…
Discrete bright breathers are well known phenomena. They are localized excitations that consist of a few excited oscillators in a lattice and the rest of them having very small amplitude or none. In this paper we are interested in the…
Discrete solitons of the discrete nonlinear Schr\"odinger (dNLS) equation become compactly supported in the anti-continuum limit of the zero coupling between lattice sites. Eigenvalues of the linearization of the dNLS equation at the…
We derive a class of discrete nonlinear Schr{\"o}dinger (DNLS) equations for general polynomial nonlinearity whose stationary solutions can be found from a reduced two-point algebraic problem. It is demonstrated that the derived class of…
Recently, using a numerical surface cooling approach, we have shown that highly energetic discrete breathers (DB) can form in the stiffest parts of nonlinear network models of large protein structures. In the present study, using an…
We provide evidence of an extremely slow thermalization occurring in the Discrete NonLinear Schr\"odinger (DNLS) model. At variance with many similar processes encountered in statistical mechanics - typically ascribed to the presence of…
We investigate the properties of discrete breathers in a Bose-Einstein condensate with two- and three-body interactions in optical lattice. In the tight-binding approximation the Gross-Pitaevskii equation with periodic potential for the…
Nonlinear classical Hamiltonian lattices exhibit generic solutions in the form of discrete breathers. These solutions are time-periodic and (typically exponentially) localized in space. The lattices exhibit discrete translational symmetry.…
We construct Darboux transformation of a coupled generalized nonlinear Schr\"odinger (CGNLS) equations and obtain exact analytic expressions of breather and rogue wave solutions. We also formulate the conditions for isolating these…
We study the statistical mechanics of the one-dimensional discrete nonlinear Schr\"odinger (DNLS) equation with saturable nonlinearity. Our study represents an extension of earlier work [Phys. Rev. Lett. {\bf 84}, 3740 (2000)] regarding the…
Exact stability analysis of 1-site breathers in an NLS type model (ref. [4], see below) indicates destabilisation through a growth rate becoming positive as a stability border is crossed, while above a critical spatial decay rate…
Breathers have been experimentally and theoretically found in many physical systems -- in particular, in integrable nonlinear-wave models. A relevant problem is to study the \textit{breather gas}, which is the limit, for $N\rightarrow…
We demonstrate that stabilization of solitons of the multidimensional Schrodinger equation with a cubic nonlinearity may be achieved by a suitable periodic control of the nonlinear term. The effect of this control is to stabilize the…
The existence of compactons in the discrete nonlinear Schr\"odinger equation in the presence of fast periodic time modulations of the nonlinearity is demonstrated. In the averaged DNLS equation the resulting effective inter-well tunneling…
Coupled backward and forward wave amplitudes of an electromagnetic field propagating in a periodic and nonlinear medium at Bragg resonance are governed by the nonlinear coupled mode equations (NLCME). This system of PDEs, similar in…
The existence and stability of dissipative discrete breathers (DDBs) in rf superconducting quantum interference device (SQUID) arrays in both one and two dimensions is investigated numerically. In an rf SQUID array, the nonlinearity which…
We observe dark and bright intrinsic localized modes (ILMs) or discrete breathers (DB) experimentally and numerically in a diatomic-like electrical lattice. The generation of dark ILMs by driving a dissipative lattice with…