LOcalized modes on an Ablowitz-Ladik nonlinear impurity

This integrable version support moving, nonlinear, spatially-localized excitations in the form of lattice solitons, found through the use of the inverse scattering transform method. The AL equations constitute a starting point for many studies on the interplay of disorder, nonlinearity and discreteness. For instance, when examining the effects of disorder, a wellknown approach is to assume a perturbative approach and try to compute the evolution of the soliton parameters [3]. When the scale of the disorder is high, this approach is no longer tenable and one must resort to numerical schemes. On the other hand, when nonlinearity is large, the spatial soliton profile is well localized in space, meaning that only a small number of sites around the soliton center are effectively nonlinear. The system then looks very similar to a linear system containing a small cluster of nonlinear sites, or even a single nonlinear impurity. This simplified system is now amenable to exact mathematical treatment, and the influence of other potentially competing effects, such as dimensionality, boundary effects, noise, etc., can be more easily studied without losing the essential physics. This approach has been successfully used for the DNLS equation [4],

where it was predicted that, for a semi-infinite nonlinear chain, the presence of a surface would increase the amount of nonlinearity required to form a localized surface mode. This was subsequently observed in later studies [5,6]. When used for the two-dimensional semiinfinite square lattice, this procedure predicted that this time, the presence of a boundary would decrease the minimum nonlinearity needed to create a surface localized mode [7]. This was later found to be the case [8].

In this Letter, we introduce a novel type of nonlinear defect in a one-dimensional discrete chain, this time using the framework of the AL equation (1).

We consider a one-dimensional array of linear sites, containing a single, Ablowitz-Ladik impurity located at site n 0 . In the tight-binding framework, the evolution equation for the amplitude is given by

where C n is the complex amplitude at site n, V is the nearest-neighbor coupling coefficient, and µ is the Ablowitz-Ladik (AL) parameter. We will be interested in stationary-state solutions of the form C n (t) = C n exp(iωt). This leads to the system of equations:

From Eq.(3) it can be easily proven that the norm

is a conserved quantity, where the prime in the sum indicates that the sum is carried out over all sites, excepting the impurity site, n = n 0 . We normalize the time to τ = V t and the probability amplitude to φ n = C n / √ N . With these definitions, Eq.( 3) simplifies to

where ν ≡ N µ/V . The normalization condition becomes

The equation for the stationary state, acquires now its dimensionless form:

where, β ≡ ω/V .

We will focus on two special cases, (i) Impurity in the “bulk” and (ii) “surface” impurity.

Impurity in the “bulk”: In this case, -∞ < n < ∞ and without loss of generality, we choose n 0 = 0. We pose a solution in the form φ n = A ξ |n| , where 0 < |ξ| < 1. After inserting this ansatz into Eq.( 8), one obtains β = 2ξ(1 + νA 2 ) and β = ξ + (1/ξ). After solving for ξ, one obtains

On the other hand, from the normalization condition, Eq.( 7), one obtains the relation The top (bottom) row shows the unstaggered (staggered) versions of the mode.

After combining these last two equations, one obtains ξ = ±[2 exp(ν -1) -1] -1/2 , and A = ((exp(ν -1) -1)/ν) 1/2 , which implies

The dimensionless bound state energy is

As can be seen from Eq.( 11), a localized bound state is possible provided ν > ν c = 1, and for a given ν, there is an unstaggered (staggered) version of the bound state for β > 2 (< -2). Fig. 1 shows a couple of profiles φ n and their staggered versions, for two different dimensionless nonlinearity parameters ν. In Fig. 2 we show ξ and the bound state energy as a function of nonlinearity. Standard linear stability analysis reveals that this stationary localized state is stable.

An interesting feature arises when we consider the dynamical excitation of a localized state. In this case, one considers Eq.( 6) for a highly localized initial condition, chosen as φ n (0) = δ n,0 (exp(ν) -1)/ν. This choice corresponds to the one that saturates the normalization condition, Eq.( 7). Examination of the ensuing dynamics reveals that at low nonlinearity values, the excitation tends to diffract across the array, while for higher nonlinearities, it tends to selftrap at the impurity site, with a high-amplitude oscillation, as Fig. 3 clearly shows. The magnitude and frequency of these oscillations increase as the nonlinear

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