Related papers: On Integrability and Chaos in Discrete Systems
Exact solutions to a nonlinear Schr{\"o}dinger lattice with a saturable nonlinearity are reported. For finite lattices we find two different standing-wave-like solutions, and for an infinite lattice we find a localized soliton-like…
We present several methods, which utilize symplectic integration techniques based on two and three part operator splitting, for numerically solving the equations of motion of the disordered, discrete nonlinear Schr\"odinger (DDNLS)…
We have studied the discrete nonlinear Schroedinger equation (DNLSE) with on-site defects and periodic boundary conditions. When the array dynamics becomes chaotic, the otherwise quasiperiodic amplitude correlations between the oscillators…
Dynamics of vector dark solitons in two-component Bose-Einstein condensates is studied within the framework of the coupled one-dimensional nonlinear Schr\"odinger (NLS) equations. We consider the small amplitude limit in which the coupled…
We consider modifications of the nonlinear Schrodinger model (NLS) to look at the recently introduced concept of quasi-integrability. We show that such models possess an infinite number of quasi-conserved charges which present intriguing…
We consider the nonlinear Schr\"odinger (NLS) equation posed on the box $[0,L]^d$ with periodic boundary conditions. The aim is to describe the long-time dynamics by deriving effective equations for it when $L$ is large and the…
A discrete version of the inverse scattering method proposed by Ablowitz and Ladik is generalized to study an integrable full-discretization (discrete time and discrete space) of the coupled nonlinear Schr\"{o}dinger equations. The…
We consider a family of regularized defocusing nonlinear Schrodinger (NLS) equations proposed in the context of the cubic NLS equation with a bounded dispersion relation. The time evolution is well-posed if the black soliton is perturbed by…
In this paper we deal with the well-known nonlinear Lorenz system that describes the deterministic chaos phenomenon. We consider an interesting problem with time-varying phenomena in quantum optics. Then we establish from the motion…
We consider a class of one dimensional Vector Nonlocal Non-linear Schr\"odinger Equation (VNNLSE) in an external complex potential with time-modulated Balanced Loss-Gain(BLG) and Linear Coupling(LC) among the components of Schr\"odinger…
We start a study of various nonlinear PDEs under the effect of a modulation in time of the dispersive term. In particular in this paper we consider the modulated non-linear Schr\"odinger equation (NLS) in dimension 1 and 2 and the…
We consider the discrete and continuous vector non-linear Schrodinger (NLS) model. We focus on the case where space-like local discontinuities are present, and we are primarily interested in the time evolution on the defect point. This in…
Spectral method related to Lame equation with finite-gap potential is used to study the optical cascading equations. These equations are known not to be integrable by inverse scattering method. Due to "partial integrability" two-gap…
A new integrable nonlocal nonlinear Schroedinger (NLS) equation with clear physical motivations is proposed. This equation is obtained from a special reduction of the Manakov system, and it describes Manakov solutions whose two components…
Based on the completeness relation for the squared solutions of the Lax operator $L$ we show that a subset of nonlocal equations from the hierarchy of nonlocal nonlinear Schr\"odinger equations (NLS) is a completely integrable system. The…
The discrete symmetry's dressing chains of the nonlinear Schrodinger equation (NLS) and Davey-Stewartson equations (DS) are consider. The modified NLS (mNLS) equation and the modified DS (mDS) equations are obtained. The explicitly…
From among the waves whose dynamics are governed by the nonlinear Schr\"odinger (NLS) equation, we find a robust, spatiotemporally disordered family, in which waves initialized with increasing amplitudes, on average, over long time scales,…
We present a number of second order maps, which pass the singularity confinement test commonly used to identify integrable discrete systems, but which nevertheless are non-integrable. As a more sensitive integrability test, we propose the…
An overview is given of basic models combining discreteness in their linear parts (i.e. the models are built as dynamical lattices) and nonlinearity acting at sites of the lattices or between the sites. The considered systems include the…
Classical integrable impurities associated to high rank (gl_N) algebras are investigated. A particular prototype i.e. the vector non-linear Schr\"{o}dinger (NLS) model is chosen as an example. A systematic construction of local integrals of…