(2+2)-Dimensional Discrete Soliton Equations and Integrable Coupling System

In this paper, we extend a （2＋2）-dimensional continuous zero curvature equation to （2＋2）-dimensional discrete zero curvature equation, then a new （2＋2）-dimensional cubic Volterra lattice hierarchy is obtained. Fhrthermore, the integrable coupling systems of the （2＋2）-dimensional cubic Volterra lattice hierarchy and the generalized Toda lattice soliton equations are presented by using a Lie algebraic system sl（4）.

Conservation Laws of a Discrete Soliton System

In this paper, we obtain an infinite number of conservation laws for a discrete soliton system by using a solvable generalized Riccati equation.

Transmission, reflection, scattering, and trapping of traveling discrete solitons by C and V point defects

We study the interactions of moving discrete solitons in waveguide arrays with two types of point defects that are constructed by varying either the local linear coupling or local waveguide propagation constant at the center of the waveguide array. A broad discrete soliton is kicked toward the defect and interacts with it. Transmission, reflection, scattering, and trapping during the interaction between the soliton and the defect occur depending on the parameters. The detailed behavior of the soliton dynamics is analyzed numerically. A transmission window in the parameter domain is found and the behavior of this window for different parameters is studied. The dynamics of the soliton in the transmission window is found to have chaotic features under certain circumstances and the causes of these phenomena are identified and discussed.

Discrete solitons in azimuthally modulated Bessel lattices:An introduction to solitons in quasi-periodic structure

The investigation of discrete solitons in quasi-periodic structure,namely azimuthally modulated Bessel lattices imprinted in photorefractive crystal,is introduced.It is shown that the discrete solitons centralize more energy in the internal layers than the Bessel lattice and moreover,the effect of centralization of discrete solitons in focusing media is stronger than that in defocusing media.The discrete solitons are unstable in some propagation constant windows and they are absolutely stable when the propagation constant is large enough.The stable solitons perform long-distance and periodic oscillation of intensity and shape under the perturbation of intrinsic excitation.

Vector Lattice Vortex Solitons

Two-dimensional vector vortex solitons in harmonic optical lattices are investigated. The stability properties of such solitons are closely connected to the lattice depth V0. For small V0, vector vortex solitons with the total zero-angular momentum are more stable than those with the total nonzero-angular momentum, while for large V0, this case is inversed. If V0 is large enough, both the types of such solitons are stable.

On some classes of two-dimensional local models in discrete two-dimensional monatomic FPU lattice with cubic and quartic potential

This paper discusses the two-dimensional discrete monatomic Fermi- Pasta-Ulam lattice, by using the method of multiple-scale and the quasi-discreteness approach. By taking into account the interaction between the atoms in the lattice and their nearest neighbours, it obtains some classes of two-dimensional local models as follows： two-dimensional bright and dark discrete soliton trains, two-dimensional bright and dark line discrete breathers, and two-dimensional bright and dark discrete breather.

Discrete spatial solitons formed in periodically poled lithium niobate by electro-optical effect

We report the numerical observation of discrete spatial solitons in a periodically poled lithium niobate waveguide array by applying an electrical field through electro-optical effect. We show that discrete spatial soliton can be controlled by applied voltage in the periodically poled lithium niobate.

Cross-symmetry breaking of two-component discrete dipolar matter-wave solitons

We study the spontaneous symmetry breaking of dipolar Bose-Einstein condensates trapped in stacks of two-well systems, which may be effectively built as one-dimensional trapping lattices sliced by a repelling laser sheet. If the potential wells are sufficiently deep, the system is modeled by coupled discrete Gross-Pitaevskii equations with nonlocal self- and cross-interaction terms representing dipole-dipole interactions. When the dipoles are not polarized perpendicular or parallel to the lattice, the cross- interaction is asymmetric, replacing the familiar symmetric two-component solitons with a new species of cross-symmetric or -asymmetric ones. The orientation of the dipole moments and the interwell hopping rate strongly affect the shapes of the discrete two-component solitons as well as the characteristics of the cross-symmetry breaking and the associated phase transition. The sub- and super-critical types of cross-symmetry breaking can be controlled by either the hopping rate between the components or the total norm of the solitons. The effect of the interplay between the contact nonlinearity and the dipole angle on the cross-symmetry breaking is also discussed.

TWO INTEGRABLE COUPLINGS FOR A LATTICE HIERARCHY BY THE KRONECKER PRODUCT

It is shown that the Kronecker product can be applied to construct integrable couplings for discrete systems. In this paper, using this method, we derive two integrable couplings for a lattice hierarchy.