Soliton Solutions of Nonlinear and Nonlocal Sine-Gordon Equation Involving Riesz Space Fractional Derivative

1 Introduction

The classical sine-Gordon equation (SGE) [1] is one of the basic equations of modern nonlinear wave theory, and it arises in many different areas of physics such as nonlinear optics, Josephson junction theory, field theory, and theory of lattices [2]. In these applications, the SGE provides the simplest nonlinear description of physical phenomena in different configurations. The theory, methods of solutions, and applications of the celebrated fractional SGE are discussed in great detail in two recent books [3, 4]. Special attention is also paid to soliton, antisoliton solutions, and a remarkable new mode that propagates in a two-level atomic system. In order to further emphasis on the analysis of one-soliton and two-soliton solitary wave solutions, it may be referred to [5].

The more adequate modelling can be prevailed corresponding to generalisation of classical SGE. In particular, taking into account the nonlocal effects, such as long-rang interactions of particles, complex law of medium dispersion or curvilinear geometry of the initial boundary problem, classical SGE results in nonlocal generalisation of SGE.

In this article, we consider the nonlocal generalisation of SGE proposed in [6] as follows:

(1)utt − DRxαu + sinu = 0, (1)

where the nonlocal operator DRxα is the Riesz space fractional derivative, 1 ≤ α ≤ 2.

These similar types of evolution equation (2) arise in various interesting problems of nonlocal Josephson electrodynamics. These problems were introduced in [7–12], among these one of the basic model equations is

(2)utt − H[ux] + sinu = 0, (2)

where H is the Hilbert transform, given by

H[ϕ] ≡ 1πv.p.∫−∞∞ϕ(ξ)ξ − xdξ,

and the integral is understood in the Cauchy principal value sense. The evolution equation (2) was an object of study in a series of papers [7, 8, 11, 13, 14] available in open literature. Other nonlocal SGEs were considered in [15, 16].

In this article, the derived approximate solutions are based on modified homotopy analysis method with the Fourier transform. In this present study, we employ a new technique such as applying the Fourier transform followed by homotopy analysis method. This new technique enables derivation of the approximate solutions for the nonlocal fractional SGE (1). To the best possible information of the author, there exists no other approximation technique for solving nonlocal fractional SGE.

2 Riesz Fractional Derivative and Integration

In this section, we briefly present some important definitions, namely the Riesz fractional derivative, left Riemann–Liouville derivative, right Riemann–Liouville derivative, and Riesz fractional integral, which are used later in this article.

Definition 1. The left and right Riemann–Liouville fractional derivatives of a function f(x) of order α(n − 1<α<n) are defined as [9, 10]

(3)D−∞xαf(x) = 1Γ(n − α)∂n∂xn∫−∞x(x − ξ)n − 1 − αf(ξ)dξ,     (3)

(4)Dx∞αf(x) = (−1)nΓ(n − α)∂n∂xn∫x∞(ξ − x)n − 1 − αf(ξ)dξ.     (4)

Definition 2. The Riesz fractional derivative of a function f(x) is defined as [9, 10]

(5)∂α∂|x|αf(x) = −12cos(απ2)[D−∞xαf(x) + Dx+∞αf(x)].  (5)

Definition 3. The fractional Riesz integral of a function f(x) is defined as [9, 10]

IRZαf(x) = 12cos(απ2)[I+α + I−α]f(x) = 12Γ(α)cos(απ2) ∫−∞+∞|x − ξ|α − 1f(ξ)dξ,   α > 0, α ≠ 1,3, 5, ….

3 Basic Idea of Modified Homotopy Analysis Method with Fourier Transform (MHAM-FT)

In this article, we apply the HAM [17–20] to the discussed problem. To show the basic idea, let us consider the following factional differential equation:

(6)N[u(x, t)] = 0, (6)

where N is a nonlinear differential operator containing Riesz fractional derivative defined in (3), x and t denote independent variables, and u(x, t) is an unknown function. For simplicity, we ignore all boundary or initial conditions that can be treated in the similar way.

Then, applying the Fourier transform and using (5), we can reduce fractional differential equation (6) to the following Fourier-transformed differential equation:

(7)N[u^(k, t)] = 0, (7)

where u^(k, t) is the Fourier transform of u(x, t).

By means of the HAM, one first constructs the zeroth-order deformation equation of (7) as

(8)(1 − p)L[ϕ(k, t; p) − u^0(k, t)] = pℏN[ϕ(k, t; p)], (8)

where L is an auxiliary linear operator, ϕ(k, t; p) is an unknown function, u^0(k, t) is an initial guess of u^(k, t), ℏ ≠ 0 is an auxiliary parameter, and p∈[0, 1] is the embedding parameter. For the sake of convenience, the expression in nonlinear operator form has been modified in HAM. In this modified homotopy analysis method, nonlinear term appeared in expression for nonlinear operator form has been expanded using Adomian type of polynomials as ∑n = 0∞Anpn [21].

Obviously, when p=0 and p=1, we have

(9)ϕ(k, t; 0) = u^0(k, t), ϕ(k, t; 1) = u^(k, t), (9)

respectively. Thus, as p increases from 0 to 1, the solution ϕ(k, t; p) varies from the initial guess u^0(k, t) to the solution u^(k, t). Expanding ϕ(x, t; p) in the Taylor series with respect to the embedding parameter p, we have

(10)ϕ(k, t; p) = u^0(k, t) + ∑m = 1+∞pmu^m(k, t), (10)

where u^m(k, t) = 1m!∂m∂pmϕ(k, t; p)|p = 0.

The convergence of the series (10) depends upon the auxiliary parameter ℏ. If it is convergent at p=1, we have

u^(k, t) = u^0(k, t) + ∑m = 1+∞u^m(k, t),

which must be one of the solutions of the original nonlinear equation, as proved in [17].

Differentiating the zeroth-order deformation equation (8) m times with respect to p and then setting p=0 and finally dividing them by m!, we obtain the following m^th-order deformation equation:

(11)L[u^m(k, t) − χmu^m − 1(k, t)] = ℏℜm(u^0, u^1, …, u^m − 1), (11)

where

ℜm(u^0, u^1, ..., u^m − 1) = 1(m − 1)!∂m − 1N[ϕ(k, t; p)]∂pm − 1|p = 0

and

(12)χm = {1,m > 1,0,m ≤ 1. (12)

It should be noted that u^m(k, t) for m ≥ 1 is governed by the linear equation (11), which can be solved by symbolic computational software. Then, by applying inverse Fourier transformation, we can get u_m(x, t).

In the present analysis, for reducing Riesz space fractional differential equation to ordinary differential equation, we applied here the Fourier transform. In this MHAM-FT, we applied inverse Fourier transform for getting the solution of Riesz space fractional differential equation.

4 Implementation of the MHAM-FT Method for Approximate Solution of Nonlocal Fractional Sine-Gordon Equation (SGE)

In this section, we first consider two examples for the application of MHAM-FT for the solution of nonlocal fractional SGE (1).

Example 1. In this example, we shall find approximate solution of the nonlocal fractional SGE (1) with given initial conditions [22–24]

(13)u(x, 0) = 0, ut(x, 0) = 4sechx, (13)

Then, by applying the Fourier transform and using (5) on (1) and (13), we get

(14)u^tt(k, t) + |k|αu^(k, t) + ℱ(sinu) = 0 (14)

with initial conditions

(15)u^(k, 0) = 0, u^t(k, 0) = 22πsech(kπ2), (15)

where ℱ denotes the Fourier transform and k is the transform parameter for the Fourier transform.

Expanding ϕ(k, t; p) in the Taylor series with respect to p, we have

(16)ϕ(k, t; p) = u^0(k, t) + ∑m = 1+∞pmu^m(k, t), (16)

where

u^m(k, t) = 1m!∂mϕ(k, t; p)∂pm|p = 0.

To obtain the approximate solution of the fractional SGE in (14), we choose the linear operator

(17)L[ϕ(k, t; p)] = ϕtt(k, t; p). (17)

From (7), we define a nonlinear operator as

(18)N[ϕ(k, t; p)] = ϕtt(k, t; p) + |k|αϕ(k, t; p) + ℱ(sin(ϕ(k, t; p))), (18)

where the nonlinear term sin(ϕ(k, t; p)) is expanded in Adomian-like polynomial.

The nonlinear term sin(ϕ(k, t; p)) has been taken as

sin(ϕ(k, t; p)) = ∑n = 0∞pnAn,

where An = (1/n!)∂n∂pn(sin(u^0(k, t) + ∑m = 1+∞pmu^m(k, t)))p = 0, n ≥ 0.

Using (8), we construct the so-called zeroth-order deformation equation

(19)(1 − p)L[ϕ(k, t; p) − u^0(k, t)] = pℏN[ϕ(k, t; p)]. (19)

Obviously, when p=0 and p=1, (19) yields

ϕ(k, t; 0) = u^0(k, t); ϕ(k, t; 1) = u^(k, t).

Therefore, as the embedding parameter p increases from 0 to 1, ϕ(k, t; p) varies from the initial guess to the exact solution u^(k, t).

If the auxiliary linear operator, the initial guess, and the auxiliary parameter ℏ are so properly chosen, the above series in (16) converges at p=1 and we obtain

(20)u^(k, t) = ϕ(k, t; 1) = u^0(k, t) + ∑m = 1+∞u^m(k, t). (20)

According to (11), we have the m^th-order deformation equation:

(21)L[u^m(k, t) − χmu^m − 1(k, t)] = ℏℜm(u^0, u^1, …, u^m − 1), m ≥ 1, (21)

where

(22)ℜm(u^0, u^1, …, u^m − 1) = 1(m − 1)!∂m−1∂pm−1N[ϕ(k, t; p)]|p = 0 = ∂2u^m − 1(k, t; p)∂t2 + |k|αu^m − 1(k, t; p) + ℱ(Am − 1). (22)

Now, the solution of the m^th-order deformation equation (21) for m ≥ 1 becomes

(23)u^m(k, t) = χmu^m − 1(k, t) + ℏL−1[ℜm(u^0, u^1, …, u^m − 1)]. (23)

From (23), we have the following equations:

(24)u^0(k, t) = u^(k, 0) + tu^t(k, 0),u^1(k, t) = ℏL−1(∂2u^0(k, t;p)∂t2 + |k|αu^0(k, t;p) + ℱ(A0)),u^2(k, t) = u^1(k, t) + ℏL−1(∂2u^1(k, t;p)∂t2 + |k|αu^1(k, t;p)+ ℱ(A1)),u^3(k, t) = u^2(k, t) + ℏL−1(∂2u^2(k, t;p)∂t2 + |k|αu^2(k, t;p) + ℱ(A2)), (24)

and so on.

But here for the sake of efficient computation for nonlinear term, the above scheme in (24) has been modified in the following way:

(25)u^0(k, t) = u^(k, 0),u^1(k, t) = tu^t(k, 0) + ℏL−1(∂2u^0(k, t; p)∂t2 + |k|αu^0(k, t; p)+ ℱ(A0)),u^2(k, t) = ℏL−1(∂2u^1(k, t; p)∂t2 + |k|αu^1(k, t; p) + ℱ(A1)),u^3(k, t) = u^2(k, t) +ℏL−1(∂2u^2(k, t; p)∂t2 + |k|αu^2(k, t; p)+ ℱ(A2)),u^4(k, t) = u^3(k, t) + ℏL−1(∂2u^3(k, t; p)∂t2 + |k|αu^3(k, t; p)+ ℱ(A3)), (25)

and so on.

By putting the initial conditions in (15) into (25) and solving them, we now successively obtain

(26)u^0(k, t) = 0, (26)

(27)u^1(k, t) = 22πtsech(kπ2), (27)

(28)u^2(k, t) = ℏ(132πt3sech(kπ2) + 132πt3|k|αsech(kπ2)), (28)

and so on.

Then, by applying the inverse Fourier transform of (26–28), we have

u0(x, t) = 0,u1(x, t) = 4tsechx,u2(x, t) = 13t3ℏ(2sechx + 2−απ−1 − αΓ(1 + α)(ζ(1 + α,π − 2ix4π) + ζ(1 + α,π + 2ix4π)−ζ(1 + α, 34 − ix2π) − ζ(1 + α, 34 + ix2π))),

and so on, where ζ(s, a) = ∑k = 0∞1/(k + a)s is the Hurwitz zeta function, which is a generalisation of the Riemann zeta function ζ(s) and also known as the generalised zeta function.

In this manner, the other components of the homotopy series can be easily obtained by which u(x, t) can be evaluated in a series form as

(29)u(x, t) = u0(x, t) + u1(x, t) + u2(x, t) + ⋯ =4tsechx + 13t3ℏ(2sechx + 2−απ−1 − αΓ(1 + α) (ζ(1 + α,π − 2ix4π) + ζ(1 + α,π + 2ix4π) −ζ(1 + α, 34 − ix2π) − ζ(1 + α, 34 + ix2π))) + ⋯. (29)

Example 2. In this case, we shall find approximate solution of the nonlocal fractional SGE (1) with given initial conditions [25–27]

(30)u(x, 0) = π + εcos(μx), ut(x, 0) = 0. (30)

Then, by applying the Fourier transform and using (5) on (1) and (30), we get

(31)u^tt(k, t) + |k|αu^(k, t) + ℱ(sinu) = 0 (31)

with initial conditions

(32)u^(k, 0) = 2π3/2δ(k) + π2εδ(k − μ) + π2εδ(k + μ), u^t(k, 0) = 0, (32)

where ℱ denotes the Fourier transform, k is the transform parameter for the Fourier transform, and δ(.) denotes the Dirac delta function.

Analogous to arguments as discussed in Example 1, we may obtain the following equations:

(33)u^0(k, t) = 2π3/2δ(k),u^1(k, t) = π2εδ(k − μ) + π2εδ(k + μ)  + ℏL−1(∂2u^0(k, t; p)∂t2+|k|αu^0(k, t; p)+ℱ(A0)),u^2(k, t) = ℏL−1(∂2u^1(k, t; p)∂t2 + |k|αu^1(k, t; p) + ℱ(A1)),u^3(k, t) = u^2(k, t) + ℏL−1(∂2u^2(k, t; p)∂t2 + |k|αu^2(k, t; p)+ ℱ(A1)),u^4(k, t) = u^3(k, t) + ℏL−1(∂2u^3(k, t; p)∂t2 + |k|αu^3(k, t; p)+ ℱ(A3)), (33)

and so on.

Solving (33), we now successively obtain

(34)u^0(k, t) = 2π3/2δ(k), (34)

(35)u^1(k, t) = π2εδ(k − μ) + π2εδ(k + μ), (35)

(36)u^2(k, t) = ℏ(−12π2t2εδ(k − μ) + 12π2t2ε|k|αδ(k − μ) − 12π2t2εδ(k + μ) + 12π2t2ε|k|αδ(k + μ)), (36)

and so on.

Then, by applying the inverse Fourier transform of (34–36), we have

u0(x, t) = π,u1(x, t) = εcos(μx),u2(x, t) = 12t2εℏ(−1 + μα)cos(μx),u3(x, t) = 124t2εℏ(−1 + μα)(12 − (−12 + t2)ℏ + t2ℏμα)cos(μx),

and so on.

In this manner, the other components of the homotopy series can be easily obtained by which u(x, t) can be evaluated in a series form as

(37)u(x, t) = u0(x, t) + u1(x, t) + u2(x, t) + ⋯ = 124(24π + ε(24 + 12t2ℏ(2 + ℏ)(−1 + μα) + t4ℏ2(−1 + μα)2)cos(μx)) + ⋯. (37)

5 The After Treatment Technique

The Padé approximation may be used to enable us to increase the radius of convergence of the series. This method can be used for analytic continuation of a series for extending the radius of convergence. A Padé approximant is the ratio of two polynomials constructed from the coefficients of the Maclaurin series expansion of a function. Given a function f(t) expanded in a Maclaurin series f(t) = ∑n = 0∞cntn, we can use the coefficients of the series to represent the function by a ratio of two polynomials denoted by [L/M] and called the Padé approximant, i.e.

(38)[LM] = PL(t)QM(t), (38)

where P_L(t) is a polynomial of degree at most L, and Q_M(t) is a polynomial of degree at most M. The polynomials P_L(t) and Q_M(t) have no common factors. Such rational fractions are known to have remarkable properties of analytic continuation. Even though the series has a finite region of convergence, we can obtain the limit of the function as t→∞ if L=M.

In the case of Example 2, u(x, t) can be evaluated in a series form as

(39)u(x, t) = 124(24π + ε(24 + 12t2ℏ(2 + ℏ)(−1 + μα) + t4ℏ2(−1 + μα)2)cos(μx)). (39)

Putting x=0.05, ℏ=−1, ε=0.01, μ = 12, and α=2 and applying the Padé approximant [5/5] to (39), we obtain

(40)u(0.05, t) = (3.15158 − 0.066294 t2 + 0.00072717 t41 − 0.021828 t2 + 0.00021501 t4). (40)

6 The ℏ-Curve and Numerical Simulations for MHAM-FT Method and Discussions

As pointed out by Liao [17] in general, by means of the so-called ℏ-curve, it is straightforward to choose a proper value of ℏ, which ensures that the solution series is convergent.

To investigate the influence of ℏ on the solution series, we plot the so-called ℏ-curve of partial derivatives of u(x, t) at (0, 0) obtained from the sixth-order MHAM-FT solutions as shown in Figure 1. In this way, it is found that our series converges when ℏ=−1.

Figure 1:

The ℏ-curve for partial derivatives of u(x, t) at (0, 0) for the sixth-order MHAM-FT solution when α=2.

In this present numerical experiment, (29) obtained by MHAM-FT has been used to draw the graphs as shown in Figure 2 for α=1.75. The numerical solutions of the Riesz fractional SGE in (1) have been shown in Figure 2 with the help of third-order approximation for the homotopy series solution of u(x, t), when ℏ=−1.

Figure 2:

(a) The MHAM-FT method solution for u(x, t) and (b) corresponding solution for u(x, t) when t=0.4.

In this present analysis, (37) obtained by MHAM-FT has been used to draw the graphs as shown in Figure 3 for fractional order value α=1.75. The numerical solutions of fractional SGE (1) have been shown in Figure 3 with the help of sixth-order approximation for the homotopy series solution of u(x, t), when ℏ=−1.

Figure 3:

(a) The numerical results for u(x, t) obtained by MHAM-FT for (a) ε=0.001, (b) ε=0.05, (c) ε=0.1, and (d) ε=1.0.

In order to examine the numerical results obtained by the proposed method, both Examples 1 and 2 have been solved by a numerical method involving the Chebyshev polynomial. The comparison of the approximate solutions for fractional SGE (1) given in Examples 1 and 2 has been exhibited in Tables 1 and 4, which are constructed using the results obtained by MHAM and the Chebyshev polynomial at different values of x and t taking α=1.75 and 1.5, respectively. Similarly, Tables 2 and 5 show the comparison of absolute errors for classical SGE given in Examples 1 and 2, respectively. To show the accuracy of proposed MHAM method over Chebyshev polynomials, L₂ and L_∞ error norms for classical order SGE given in Examples 1 and 2 have been presented in Tables 3 and 6, respectively. Agreement between present numerical results obtained by MHAM with Chebyshev polynomials and exact solutions appears very satisfactory through illustrations in Table 1–6. Figure 4 demonstrates graphical comparison of the numerical solutions for u(0.05, t) obtained by MHAM-FT and Padé approximation with regard to the exact solution for Example 1.

Figure 4:

Graphical comparison of the numerical solutions u(0.05, t) obtained by MHAM-FT and the Padé approximation with regard to the exact solution for Example 1.

7 Conclusion

In this article, a new semi-numerical technique MHAM-FT method has been proposed to obtain the approximate solution of nonlocal fractional SGE. The fractional SGE with nonlocal Riesz derivative operator has been first time solved by MHAM-FT method in order to justify applicability of the above method. The approximate solutions obtained by MHAM-FT provide us with a convenient way to control the convergence of approximate series solution and solve the problem without any need for discretisation of the variables. To control the convergence of the solution, we can choose the proper values of ℏ; in this article, we choose ℏ=−1. In order to examine the numerical results obtained by the proposed method, both Examples 1 and 2 have been solved by a numerical method involving the Chebyshev polynomial. To show the accuracy of proposed MHAM method over Chebyshev polynomials, L₂ and L_∞ error norms for classical order SGE given in Examples 1 and 2 have been presented in Tables 3 and 6, respectively. Agreement between present numerical results obtained by MHAM with Chebyshev polynomials and exact solutions appears very satisfactory through illustrations in Tables 1–6. The proposed MHAM-FT method is very simple and efficient for solving nonlinear fractional SGE with nonlocal Riesz derivative operator.