KnE Engineering | International Conference on Basic Sciences and Its Applications (ICBSA-2018) | pages: 40–50

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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]

iφn˙=εφn+12φn+φn1+Fφn+1,φn,φn1,

where φnφnt is a wave function at time t+ and site n , φn˙ represents the derivative of the function φn with respect to t, ε>0 represents coupling constant and F is a nonlinear term that has several forms:

  • Ablowitz-Ladik (AL)

  • Cubic

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 ε0 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. [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:

iφn˙=εφn+12φn+φn1+α2φn2φn+1+φn1+(1α)φn2φn.

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 o¨ 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 [11]. Let u(x,t) with t0 and for every xΩ, satisfy partial differential equations in the form

ut=fvx,u,ux,uxx+fnv(x,u,ux,uxx),

where fv denotes a variational term and fnv denotes a nonvariational term, i.e there is a function F(x,u,ux ) such that

fvx,u,ux,uxx=FuddxFux.

In the variational case (that is by assuming that fnv 0), the stationary solutions of (6) are indeed extrema of the functional (called Lagrangian)

𝐿:=ΩF(x,u,ux)𝑑𝑥.

Suppose a variational solution can be written in the form u=U(x,A1,...,An) with a finite number of parameters A1,A2,...,An . The result of integration (7) using such a variational solution is called effective Lagrangian ( L eff ). Thus the extreme value for effective Lagrangian satisfies

L eff Ai=ΩUAiddxFUxFU𝑑𝑥,i=1,2,,n.

Suppose now that the parameters are time-dependent functions, denoted as Ait. Thus from equation (5) we obtain the following relationship

FuddxFux=j=1nUAjA˙jfnv(x,u,ux,uxx),

where A˙j=dAjdt . Upon substituting equation (9) into equation (8), then for every i we obtain

L eff Ai=ΩUAifnvj=1nUAjA˙j𝑑𝑥.

Suppose

Mij=ΩUAiUAj𝑑𝑥,

then equation (10) for each i=1,2,,n can be rewritten as follows

j=1nMijA˙j=L eff Ai+ΩUAifnvdx.

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

j=1nMijA˙j=L eff Ai+mΦUmAifnv.

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, φnt can be written in the form

φnt=Qneit,

where Qn is a time-independent function. Next, by substituting equation (14) into (4), we obtain the following stationary equation

εQn+12Qn+Qn1Qn+α2Qn2Qn+1+Qn1+(1α)Qn2Qn=0.

In general, solutions for Qn are complex valued. However, in this paper we are only consider the real-valued solution. Therefore, equation (15) can be simplified to

εQn+12Qn+Qn1Qn+α2Qn2Qn+1+Qn1+(1α)Qn3=0.

Equation (16) can be written as follows:

fv+fnv=0,

where

fv=εQn+12Qn+Qn1Qn+(1α)Qn3,

and

fnv=α2Qn2Qn+1+Qn1.

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

L=n=εQn1Qn+121+2εQn21α4Qn4.

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:

Qn=whiteAeη(n1),n1whiteB,n=0,1whiteAeη(n2),n2

where A , B and η 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

whiteL eff =121e4η1(B4+4εABB4αB4e4η+2B2e4η+2e2ηA22B22εB2white4εA2e3η+2A2e4ηA4e4η4εABe4η+4e2ηA2ε4A2eηεwhite+B4αe4η+2εB2e4η+A4αe4η+4εA2e4η).

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

𝒜1+𝒜2=0,
1=0,
𝒞1+𝒞2=0,

where

𝒜1=1e4η12εB+2e2ηA4εAe3η+2Ae4η2A3e4η2εBe4η+4Ae2ηε
𝒜2=1e4η14Aeηε+2A3αe4η+4Aεe4η+A3αeη+A2αBe4ηA2αB+A3αe3η
1=2B3+2B2εA+3B3α+2εB+αB2A
𝒞1=1(e4η1)2eηA22ε6e2ηε+4eηε+8e3ηε6εe4η+2eη+4e3η
𝒞2=1(e4η1)2eηA22A2e3η+2e5η+2A2e3ηα2e6ηε+4e5ηε+αA2e4η+e2ηA2α

Due to complexity of the calculation, solutions for parameters A,B, and η 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 ε=0.01;0.02;0.05, and intepolation parameter α=0.01;0.44.

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

fig-1.jpg
Figure 1
Comparison between numerical (circle markers) and variational (cross markers) solutions of the intersite soliton for some parameter values.

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

RnQ=εQn+12Qn+Qn1Qn+α2Qn2Qn+1+Qn1+(1α)Qn3.

Note that if Q is an exact solution, then RnQ will be zero for every n . Thus a variational solution will approach the exact solution if RnQ0 for every n .

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 Q* is a variational solution of intersite solitonof the stationary AL-cubic DNLS equation (16) which is expressed by ansatz (21), where the variational parameters η,A,B satisfy the equations (23), (24) and (25). Then there is ε0,K>0 such that for all ε(0,ε0) , equation (16) has a unique solution Q that satisfies

QQ*l2Kε2.

<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 η can be obtained by substituting Qn=Ce-ηn , where C is a non-zero constant, into the linear part of equation (16), that is

εQn+12Qn+Qn1Qn=0,

which yields

ε2eη+eη=2c+1η= arccosh 2ε+12ε.

Taylor expansion of eη at ε0 is given by

eη=ε2ε2+5ε314ε4+𝒪C5.

Let us assume that parameters A and B can be written in the following form of expansion

A=a0+a1ε+a2ε2+a3ε3+𝒪(ε4),
B=b0+b1ε+b2ε2+b3ε3+𝒪(ε4),

where ai and bi 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 ε .

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

Rn(Q)=whiteεAeηn+22Aeηn+1+AeηnAeηn+1+12αA2eηn+12whiteAeηn+2+Aeηn+(1α)A3eηn+13,n2whiteεB2εA+εAeηA+12αA2B+12αA2eη+A3A3α,n=1whiteεB2εA+εAeηA+12αA2B+12αA2eη+A3A3α,n=2whiteεAeηn12Aeηn2+Aeηn3Aeηn2+12αA2eηn22whiteAeηn1+Aeη(n3)+(1α)A3eηn23,n3

Upon substituting expansions (30), (31), (32) into equation (33), one can obtain for α0 as follows

R0,1Q=0,
R1,±2Q=𝒪(ε2),
Rn3Q=𝒪εn3,
Rn3Q=𝒪εn+2.

One can check that sequence {RiQ}i=1 converges to 0. Therefore R12+R22+ is convergent or k=1Rk2<. This explains that RiQ is in l2 space with norm R(Q)l2=R12+R22+ . Thus from (34), we have

R(Q)l2=𝒪(ε2) as ε,α0.

Moreover, let us suppose S=0,1 , e0=,0,0,0,1,0,0,0,,e1=,0,0,0,1,0,0,,,σ0=C , and σ1=C for C>0 . For the case of intersite discrete soliton, it can be shown that the variational solution Q* expressed by ansatz (21) satisfies the relationship

lim ε,α0Q*(σ0e0+σ1e1)l2=0.

Based on Chong et al. [10], there is ε0,K>0 and the unique solution Q of stationary AL-cubic DNLS equation (16) with ε(0,ε0) such that

QQ*l2Kε2.

Since the exact solution is unknown, in practice the quantity of Q-Q*l2 can be replaced by

error =Q num Q*,

where Q num is the numerical solution. As an illustration, suppose the validation of the VA results will be checked for parameter value α=0.001. Plot of error (38) in varied ε for such α is given in Fig. 2.

fig-2.jpg
Figure 2
The error between variational and numerical solutions in varied ε for α=0.001 .

From Fig. 2, it can be seen that the error gets bigger as ε increases. To find out the function of error curve, we perform the best power fit, which gives fε=1.2997ε2.0032. Note this result satisfies equation (27) in Proposition 1. This confirms that the VA solution for intersite soliton using ansatz (21) is valid.

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.

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