#### 1. Introduction

One of equations that is often studied in both the theory aspect and in the context of its application is the discrete nonlinear Schrödinger (DNLS) equation. This is because the equation models many important phenomena, such as an array of nonlinear optical waveguides etched onto a semiconductor material (AlGaAs) [1], matter wave dynamics in Bose–Einstein condensates trapped in optical lattices and molecular biology (modeling the DNA double strand) [2].

The most interesting feature about the DNLS equation is the existence of soliton. Soliton is a localized solution that has properties: it maintains its shape and propagates at a constant speed even after collision [3]. In the context of its application, Tagg [4] described the use of solitons in fiber optic communication systems which provide highly accurate signal transmission over extremely long distance. This is very important in the development of future communication technology.

The general form of the DNLS equation is given by [2]

where
*F* is a nonlinear term that has several forms:

In 1975-1976, Ablowitz and Ladik [5] showed that the DNLS equation with nonlinear term (2) is integrable, while equation (3) is not integrable. For non-integrable equations, an analytic approach is needed to approximate the solution. One of the methods which is well known and has been long used to approximate solutions (including the localized states) of a nonlinear evolution equation is the so-called variational approximation (VA). Formulation of this method is based on theory of Lagrangian and Hamiltonian mechanics (see, e.g., [6]). The success of this method depends heavily on the trial function (ansatz) used in approaching the desired solution.

VA methods have been used in various equations, including in determining soliton solution in the cubic DNLS equation (3). Aceves et al [7] used the VA method to approximate the onsite soliton solution (i.e centred on a lattice site). In addition, VA has been also applied to approximate the intersite soliton solution (i.e centred between two adjacent lattice sites) with symmetrical configuration by Cuevas et al [8]. Furthermore, Kaup et al [9] developed the VA formulation to approximate the asymmetric intersite soliton solution. The ansatz function used in [8] and [9] applies for the case

The results of VA have been confirmed its validation through numerical comparisons for certain parameter values. To justify rigorous VA validation, Chong et al. [10] have developed a theorem that can be used as a tool of validation of the VA results. Chong et al then confirmed that a trial function for solitons with more parameters many provides a more accurate approximation.

In this paper, the VA method will be applied to determine intersite soliton of the following equation:

Equation (4) can be viewed as an interpolation of the Ablowitz-Ladik DNLS equation (when α = 1) and the cubic DNLS equation (when α = 0). Equation (4) is then called the Ablowitz-Ladik-cubic discrete nonlinear Schr

#### 2. The Formulation of Variational Approximation

In this section we describe the formulation of variational approximation method. This is referred from reference [11]. Let

where

In the variational case (that is by assuming that

Suppose a variational solution can be written in the form

Suppose now that the parameters are time-dependent functions, denoted as

where

Suppose

then equation (10) for each

For discrete system in space, equation (12) can be changed analogously to be

In brief, the systematic steps of the VA method for both variational and nonvariational cases are given as follows:

• Formulate the Lagrangian of the variational part of the governing equation.

• Propose a reasonable trial function (ansatz) which contains a finite number of parameters (called variational parameters).

• Substitute the proposed ansatz into the Lagrangian and evaluate the resulting sums (for discrete systems) or integrations (for continuous system).

• Find the critical points of the variational parameters by solving the corresponding system (12) [for continuous system] or system (13) [for discrete system].

#### 3. Variational Approximation of AL-cubic DNLS Equation

By performing the separation of variables,

where

In general, solutions for

Equation (16) can be written as follows:

where

and

Next, we are ready to apply the VA method. The first step is to determine the Lagrangian formula for the variational part (

The second step is to select the appropriate ansatz function. In this paper, we are interested to find the approximation of the intersite soliton. Thus, the following ansatz function can be selected:

where

The third step is to substitute ansatz (21) into equation (20) and then evaluate the resulting sum. This gives the effective Lagrangian as follows

By substituting the effective Lagrangian (22), ansatz (21) and the nonvariational part (19) into equation (13), then the following system of equations is obtained

where

Due to complexity of the calculation, solutions for parameters

#### 4. The VA Results and Comparisons with Numerics

In this section, we compare the results of variational approximations with the corresponding numerical calculations. In this case, the numerical solution for the intersite soliton of (16) can be determined using the Newton-Raphson method where the VA solutions can be used as the initial guess. For illustrative example, in Figure 1 is shown a comparison between two intersite soliton solutions obtained from numerics and VA for coupling constant

Fig. 1 also gives the solutions for varational parameters A, B, and

#### 5. Validation of VA Results

Validation of VA results for discrete solitons in the stationary equation (16) is based on the justification formulated by Chong et al. [10]. To measure the accuracy of variational solution of equation (16), define its residual as

Note that if

Validation of the VA for the intersite soliton of the stationary AL-cubic DNLS equation (16) is given in the following propositions.

<statement> <title>Proposition 1</title>

</statement>

*Suppose *
* is a variational solution of intersite solitonof the stationary*
*AL-cubic DNLS equation (16) which is expressed by ansatz (21), where the variational parameters *
* satisfy the equations (23), (24) and (25). Then there is *
* such that for all*
*,*
*equation (16)*
*has a unique solution Q that satisfies*

<statement> <title>Proof </title>

</statement>

Note that the rate of exponential decay of discrete soliton follows from the linear theory of difference equations. Therefore, the solution for parameter

which yields

Taylor expansion of

Let us assume that parameters

where

Next, substitution of ansatz (21) into equation (26) gives

Upon substituting expansions (30), (31), (32) into equation (33), one can obtain for

One can check that sequence

Moreover, let us suppose

Based on Chong et al. [10], there is

Since the exact solution is unknown, in practice the quantity of

where

From Fig. 2, it can be seen that the error gets bigger as

#### 6. Conclusion

Variational approximation (VA) developed for stationary intersite soliton in the AL-cubic DNLS equation gives very good results for small coupling constant and small interpolation parameter. Following reference [10], we also show that the obtained VA solutions are valid.