TY - JOUR

T1 - Higher-order lattice diffraction

T2 - Solitons in the discrete NLS equation with next-nearest-neighbor interactions

AU - Kevrekidis, P. G.

AU - Malomed, B. A.

AU - Saxena, A.

AU - Bishop, A. R.

AU - Frantzeskakis, D. J.

N1 - Funding Information:
BAM acknowledges hospitality of the Center for Nonlinear Studies at Los Alamos National Laboratory. PGK acknowledges the hospitality of the Center for Nonlinear Studies of Los Alamos National Laboratory, as well as partial support from the University of Massachusetts through a Faculty Research Grant, from the Clay Mathematics Institute through a Special Project Prize Fellowship, from the Eppley Foundation for Research and from the National Science Foundation through DMS-0204585. Work at Los Alamos is supported by the US Department of Energy, under contract W-7405-ENG-36.

PY - 2003/9/1

Y1 - 2003/9/1

N2 - We introduce a general model of a one-dimensional dynamical lattice with the on-site cubic nonlinearity and both nearest-neighbor (NN) and next-nearest-neighbor (NNN) linear interactions between lattice sites (i.e., a generalized discrete nonlinear Schrödinger equation). Unlike some previously considered cases, our model includes a complex coefficient of the NNN interaction, which applies, e.g., to optical waveguide arrays, where fields in adjacent cores may be phase shifted. The application to optical arrays is especially important in the case in which the effective lattice diffraction coefficient generated by the NN coupling is close to zero, which may be achieved by means of the so-called diffraction management. Three types of fundamental solitons are considered: site-centered and intersite-centered ones, and twisted localized modes (TLMs). It is found that, with the increase of the imaginary part of the NNN coupling constant, site-centered solitons lose their stability, but then regain it. The instability region disappears if the real part of the NNN coupling constant is negative and sufficiently large. If the site-centered soliton is unstable, it rearranges itself into a quasi-periodic (in time) breathing soliton. Intersite-centered solitons cannot be fully stabilized by the NNN interactions. TLM solitons are stable in a limited parametric region, then they become unstable, and eventually disappear. Direct simulations of the evolution of the intersite-centered solitons and unstable TLMs show that the instability reshapes them into site-centered solitons with intrinsic vibrations (breathers).

AB - We introduce a general model of a one-dimensional dynamical lattice with the on-site cubic nonlinearity and both nearest-neighbor (NN) and next-nearest-neighbor (NNN) linear interactions between lattice sites (i.e., a generalized discrete nonlinear Schrödinger equation). Unlike some previously considered cases, our model includes a complex coefficient of the NNN interaction, which applies, e.g., to optical waveguide arrays, where fields in adjacent cores may be phase shifted. The application to optical arrays is especially important in the case in which the effective lattice diffraction coefficient generated by the NN coupling is close to zero, which may be achieved by means of the so-called diffraction management. Three types of fundamental solitons are considered: site-centered and intersite-centered ones, and twisted localized modes (TLMs). It is found that, with the increase of the imaginary part of the NNN coupling constant, site-centered solitons lose their stability, but then regain it. The instability region disappears if the real part of the NNN coupling constant is negative and sufficiently large. If the site-centered soliton is unstable, it rearranges itself into a quasi-periodic (in time) breathing soliton. Intersite-centered solitons cannot be fully stabilized by the NNN interactions. TLM solitons are stable in a limited parametric region, then they become unstable, and eventually disappear. Direct simulations of the evolution of the intersite-centered solitons and unstable TLMs show that the instability reshapes them into site-centered solitons with intrinsic vibrations (breathers).

KW - Discrete nonlinear Schrödinger equation

KW - Next-nearest-neighbor interactions

KW - Twisted localized modes

UR - http://www.scopus.com/inward/record.url?scp=0042976626&partnerID=8YFLogxK

U2 - 10.1016/S0167-2789(03)00178-7

DO - 10.1016/S0167-2789(03)00178-7

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AN - SCOPUS:0042976626

SN - 0167-2789

VL - 183

SP - 87

EP - 101

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

IS - 1-2

ER -