L. Hadzievski

M. Stepic

M.M. ?kori?

National Institute for Fusion Science (NIFS-Repository)

§ 8. Modulation Instability in Two-Dimensional Nonlinear ShcrodingerLattices= fJn,o sin2 (knn) + sin 2(kmm)n-l 2 m-l 2N M-2: 2:J n,m [cos(knn )cos(kmm )-1] (5)n-lm-l(2)(4)20401,,=0.5p=3a)0.51.51.0Fig. 1. The instability threshold as a function of the size ofthe 2D lattice with long-range interactions.1.5Fig. 2. Growth rates of the instability of the array solitonfor 2D lattice with long-range interactions.The instability occurs for 0/ < 0, which leads to thefollowing instability thresholdk 2).,2 >-+2L(N,M). (6)2The application of the variational method on the stabilityproblem of the array soliton solution leads to the nextinstability condition)., > A = (7)e Jjand the growth rate structurer' (I') = 1'(1- ); I' = :: (8)The graphic illustrations of the results (8) and (9) are shownin Fig. 1. and Fig. 2.References1) Yu. B. Gaididei, S. F. Mingaleev, P. L. Christiansenand K. 0. Rasmussen, Phys. Rev. E 55 (5), (1997) 6141.2) M. Stepic, Lj. Had ievski and M. Skoric, Phys. Rev. E65 (2), (2002) 026604.(1)whereJ 1i"-T1 = 1_' -IPr -rFor the lattice with periodic boundary conditionsimposed on the discrete dimensions we can consider a set oflattice independent stationary solutions of Eq. (1), in a form1./J r = f (z) exp o..?t. We shall restrict our stability study totwo particularly simple and most frequently studiedstationary solutions; the first one is a uniform, continuouswave (CW) solution few = )., / .J2 ,while the second one is anarray soliton (AS), given by las = A/ cosh }"z .For the case of CW solution the stability analysis givesthe following dispersion relation0/ = (k 2 + - 2A? +The problem of modulation instability of continuouswave and array soliton solutions within the framework of atwo-dimensional continuum-discrete nonlinear Schrodingerlattice model which accounts for dispersion and long-rangeinteractions between elements, is investigated. The linearstability analysis based on an energetic principle and avariational approach, which were originally developed forthe continuum nonlinear Schrodinger model, is proposed.Regions of instability are identified and analyticalexpressions for the corresponding thresholds and the growthrate spectra are calculated.The basic mathematical model describing two-dimensional lattice with nonlocal nonlinear interactingelements in anomalous dispersion regime has a form ofcontinuum-discrete nonlinear Schrodinger equation. a'l/Jr a2'l/Jr 'J.It hi. 12 ( ) 0+_, _Ir-rl'l/Jr'-'l/Jr = ,Z r ToOT)where r = (n,m,O), (n = 1,2,3...N; m =1,2,3 ..M) is thediscrete lattice vector in a x-y plane, z is the spatialcontinuous coordinate along the lattice elements and1./J - = 1/J is the wave function into the (n;m)-th latticer n,melement. The nonlocal interaction term J1r'_rl describes along-range isotropic coupling between lattice elements anddepends on the distance between interacting elements. Thisinteraction model is quite general and enables amathematical modeling of a variety of discrete dispersivephysical systems with long-range interactions. The wellknown interaction model for ID DNLS lattice model with apower law dependence on the distance between interactingelements was originally proposed in [1]. In our case, forCDNLS model (1) with regularly spaced 2D lattice withinterelement distance equal to 1, the power law dependencecan be written in a form1Hadzievski, L., Stepic, M. (Vinca Institute of NuclearSciences, Belgrade, Yugoslavia)Skoric, M.M.329

Modulation Instability in Two-Dimensional Nonlinear Shcr?dinger Lattices

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