| We consider a linearly coupled system of nonlinear Schrodinger equations (CNLSE) in one space dimension, arbitrary discrete dimension d, and power nonlinearity with exponent {dollar}2sigma + 1.{dollar} Such systems arise in nonlinear optics, where the cubic case {dollar}(sigma = 1){dollar} models an array of coupled optical fiber waveguides. These systems are like the nonlinear Schrodinger equations (NLS) in {dollar}Rsp{lcub}d + 1{rcub}{dollar} with d dimensions discretized. Many of the CNLSE results are shown to have their analog in the Discrete Self-Trapping equation, a system of ordinary differential equations also known as the Discrete Nonlinear Schrodinger equation, or Discrete Self-Trapping Equation.; We construct "ground state" solutions by solving the minimization problem {dollar}Isb{lcub}nu{rcub} = min{lcub}{lcub}cal H{rcub}:{lcub}cal P{rcub} = nu{rcub}{dollar} where {dollar}{lcub}cal H{rcub}{dollar} is the hamiltonian and {dollar}{lcub}cal P{rcub}{dollar} is the {dollar}Lsp2{dollar} norm squared. The Concentration Compactness Principle of P. L. Lions is used to show that ground states exist when {dollar}Isb{lcub}nu{rcub} < 0.{dollar} For a finite system, ground states exist for all values of the {dollar}Lsp2{dollar} constraint {dollar}{lcub}cal P{rcub} = nu,{dollar} regardless of the sign of {dollar}Isb{lcub}nu{rcub}.{dollar} On infinite systems with discrete Laplacian coupling, there are interesting threshold phenomena: there is a critical nonlinearity {dollar}sigmasb{lcub}*{rcub}(d){dollar} such that for {dollar}0 < sigma < sigmasb{lcub}*{rcub}(d){dollar} ground states exist for all values {dollar}nu > 0.{dollar} For {dollar}sigmasb{lcub}*{rcub}(d)lesigma < 2{dollar} there are "excitation thresholds" {dollar}nusb{lcub}c{rcub},{dollar} depending on d and {dollar}sigma,{dollar} such that ground states only exist for {dollar}nu > nusb{lcub}c{rcub}.{dollar} This is proved via a relation found between the threshold and optimal constants in continuous-discrete interpolation inequalities of Sobolev-Nirenberg-Gagliardo type. For the DST, the critical nonlinearity is found exactly to be {dollar}sigmasb{lcub}*{rcub} = 2/d.{dollar}; Various other results are collected about ground states. The case of periodic arrays with discrete laplacian coupling is considered. For small power, the ground state on the periodic array is the "uniform state", which has a single NLS soliton on each fiber. Bifurcations for these uniform states have been reported in the optics literature. We rigorously derive these bifurcations and show that the uniform solution loses stability at the bifurcation point. |
