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) , matter wave dynamics in Bose–Einstein condensates trapped in optical lattices and molecular biology (modeling the DNA double strand) .
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 . In the context of its application, Tagg  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 
where is a wave function at time and site , represents the derivative of the function with respect to represents coupling constant and F is a nonlinear term that has several forms:
In 1975-1976, Ablowitz and Ladik  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., ). 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  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 . Furthermore, Kaup et al  developed the VA formulation to approximate the asymmetric intersite soliton solution. The ansatz function used in  and  applies for the case or known as the anti-continuum limit.
The results of VA have been confirmed its validation through numerical comparisons for certain parameter values. To justify rigorous VA validation, Chong et al.  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 dinger (AL-cubic DNLS) equation.
2. The Formulation of Variational Approximation
In this section we describe the formulation of variational approximation method. This is referred from reference . Let with and for every satisfy partial differential equations in the form
where denotes a variational term and denotes a nonvariational term, i.e there is a function ) such that
In the variational case (that is by assuming that 0), the stationary solutions of (6) are indeed extrema of the functional (called Lagrangian)
Suppose a variational solution can be written in the form with a finite number of parameters . The result of integration (7) using such a variational solution is called effective Lagrangian ( ). Thus the extreme value for effective Lagrangian satisfies
Suppose now that the parameters are time-dependent functions, denoted as Thus from equation (5) we obtain the following relationship
where . Upon substituting equation (9) into equation (8), then for every we obtain
then equation (10) for each can be rewritten as follows
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, can be written in the form
where is a time-independent function. Next, by substituting equation (14) into (4), we obtain the following stationary equation
In general, solutions for are complex valued. However, in this paper we are only consider the real-valued solution. Therefore, equation (15) can be simplified to
Equation (16) can be written as follows:
Next, we are ready to apply the VA method. The first step is to determine the Lagrangian formula for the variational part ( , which is given by
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 , and are real valued variational parameters.
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
Due to complexity of the calculation, solutions for parameters and for given ε and α in the above system can be determined numerically using the Newton-Raphson method.
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 and intepolation parameter
Fig. 1 also gives the solutions for varational parameters A, B, and which are obtained by solving the system of equations (23)-(25) numerically for given ε and α. From the figure we can observe that the VA soliton solutions and the numerical solutions have a very good agreement for some parameter values 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. . To measure the accuracy of variational solution of equation (16), define its residual as
Note that if is an exact solution, then will be zero for every . Thus a variational solution will approach the exact solution if for every .
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>
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>
Note that the rate of exponential decay of discrete soliton follows from the linear theory of difference equations. Therefore, the solution for parameter can be obtained by substituting , where is a non-zero constant, into the linear part of equation (16), that is
Taylor expansion of at is given by
Let us assume that parameters and can be written in the following form of expansion
where and are coefficients that will be determined its values by substituting equations (30), (31) and (32) into equations (23) and (24), and then collect the resulting terms in successive powers of .
Next, substitution of ansatz (21) into equation (26) gives
Upon substituting expansions (30), (31), (32) into equation (33), one can obtain for as follows
One can check that sequence converges to 0. Therefore is convergent or This explains that is in space with norm . Thus from (34), we have
Moreover, let us suppose , , and for . For the case of intersite discrete soliton, it can be shown that the variational solution expressed by ansatz (21) satisfies the relationship
Based on Chong et al. , there is and the unique solution of stationary AL-cubic DNLS equation (16) with such that
Since the exact solution is unknown, in practice the quantity of can be replaced by
where is the numerical solution. As an illustration, suppose the validation of the VA results will be checked for parameter value Plot of error (38) in varied for such is given in Fig. 2.
From Fig. 2, it can be seen that the error gets bigger as increases. To find out the function of error curve, we perform the best power fit, which gives Note this result satisfies equation (27) in Proposition 1. This confirms that the VA solution for intersite soliton using ansatz (21) is valid.
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 , we also show that the obtained VA solutions are valid.