 # Numerical meshless solution of high-dimensional sine-Gordon equations via Fourier HDMR-HC approximation

In this paper, an implicit time stepping meshless scheme is proposed to find the numerical solution of high-dimensional sine-Gordon equations (SGEs) by combining the high dimensional model representation (HDMR) and the Fourier hyperbolic cross (HC) approximation. To ensure the sparseness of the relevant coefficient matrices of the implicit time stepping scheme, the whole domain is first divided into a set of subdomains, and the relevant derivatives in high-dimension can be separately approximated by the Fourier HDMR-HC approximation in each subdomain. The proposed method allows for stable large time-steps and a relatively small number of nodes with satisfactory accuracy. The numerical examples show that the proposed method is very attractive for simulating the high-dimensional SGEs.

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## 1 Introduction

The sine-Gordon equation (SGE) is a nonlinear hyperbolic partial differential equation (PDE) involving the d’Alembert operator and the sine of the unknown function, and the SGE plays an important role in many mathematical physics applications. It was originally introduced by

Bour (1862) and rediscovered by Frenkel and Kontorova (1939). Further details about the background and applications of the SGE can be found in Aero et al. (2009); Drazin and Johnson (1989); Scott et al. (1973); Taleei and Dehghan (2014). In this paper, we focus on developing an effective means for the numerical solution of the SGE in an arbitrary number of dimensions. A -dimensional SGE generally takes the form:

 utt(x,t)+βut(x,t)=Δu(x,t)−ψ(x)sin(u(x,t)), (1) x=(x1,x2,⋯,xn)∈Ω, t>0,

where is a positive integer, is the Laplacian operator in spatial dimensions, . The initial conditions associated with Eq.(1) are given by

 u(x,0)=v1(x), x∈Ω (2) ut(x,0)=v2(x), x∈Ω, (3)

and the Neumann boundary conditions are

 ∂u∂l(x,t)=w(x,t), x∈Γ, t>0 (4)

where denotes the (typically exterior) normal to the boundary of the domain, and is the boundary of , i.e . The real parameter weights the dissipative term. When , Eq.(1) reduces to an undamped SGE in spatial variables, while when , the damped SGE is obtained. The function can be interpreted as a Josephson current density, while and in Eqs.(2) and (3) represent wave modes or the kink and velocity, respectively.

The -dimensional SGE first appeared in a strictly mathematical context in differential geometry regarding the theory of surfaces of constant curvature Lamb (1971). Moreover, it is well known that the -dimensional SGE arises in many important systems, such as the Thirring model, the Coulomb gas system and the ferromagnetic model, etc. Minnhagen (1985, 1987); Ni et al. (1990). Because of its wide applications, the -dimensional SGE has been studied with a variety of numerical methods, including finite difference methods (FDM) and finite element methods (FEM), etc. Argyris and Haase (1987). Recently, additional solution methods have been proposed including collocation Dehghan and Shokri (2008a); Lakestani and Dehghan (2010), the boundary integral approach Dehghan and Mirzaei (2008a); Dehghan and Ghesmati (2010a), and a combination of the finite difference with the the diagonally implicit Runge-Kutta-Nyström (DIRKN) method Mohebbi and Dehghan (2010), etc.

There is recent interest in the SGE in higher dimensions. As Barone et al. (1971) pointed out, Eq.(1) also has been applied in many branches of physics for the and cases. The exact solutions for the undamped SGE in higher dimensions have been obtained by Hirota’s method Hirota (1973), Lamb’s method Zagrodzinsky (1979), the Bäcklund transformation Christiansen and Olsen (1979) and Painlevé transcendents Kaliappan and Lakshmanan (1979), etc. Moreover, numerical solutions for the -dimensional undamped SGE have been proposed by Christiansen and Lomdahl (1981) using a generalized leapfrog method, Guo et al. (1986) using two finite difference schemes, Argyris et al. (1991) using finite elements. Xin (2000) studied the SGE as an asymptotic reduction of the two level dissipationless Maxwell-Bloch system, Sheng et al. (2005) presented a numerical method with a split cosine scheme, and Bratsos (2005) used a three-time level fourth-order explicit finite difference scheme to solve the undamped SGE. Following a similar approach, Bratsos (2007) transformed the SGE to a second-order initial value problem with the help of the method of lines. Numerical approaches for the damped SGE were proposed by Nakajima et al. (1974) who considered dimensionless loss factors and unitless normalized bias, and Gorria et al. (2004) investigated nonlinear wave propagation in a planar wave guide consisting of two rectangular regions joined by a bent domain of constant curvature using as a model of the kink solution to the SGE. Additionally, Dehghan and Mirzaei (2008b) developed the dual reciprocity boundary element method for both the undamped and damped -dimensional SGE, and Jiwari et al. (2012b) obtained a numerical scheme based on a polynomial differential quadrature method.

Although the SGE is nonintegrable except for , some properties and exact solutions for the -dimensional SGE have been obtained by various methods. Kobayashi and Izutsu (1976) extensively studied the exact traveling wave solutions of the SGE in the field of theoretical physics. Many additional mathematical methods have been proposed for finding traveling wave solutions of the SGE. Feng (2004) applied the Painlevé analysis to the study of an approximate SGE and its traveling solitary wave solution in -dimensional space. With the help of exact solutions to the cubic nonlinear Klein-Gordon fields, Lou et al. (2005) studied the exact solutions for the -dimensional SGE. Adopting and , de la Hoz and Vadillo (2012) generalized the exact soliton solution for the -dimensional SGE:

 (5)

where . By adopting the proper ansatz, more general solutions can be obtained for the multi-dimensional SGE, including the three-dimensional case allowing for non-constant Aero et al. (2009). Obtaining the exact solution for the general SGE would be ideal, but unfortunately it is very difficult for practical engineering problems that are usually complex in nature. Despite numerical methods commonly used in many types of linear and nonlinear PDEs, de la Hoz and Vadillo (2012) remarked that there were no references to the numerical treatment of the SGE for dimensions larger than three, which motivated them to propose a numerical method for the -dimensional SGE based on using operational matrices.

Many standard numerical methods for solving PDEs are widely used in engineering, but they usually require the construction and update of a mesh, which is an inherent disadvantage. In order to overcome these difficulties, recently the meshless numerical method has attracted attention. This method can establish a system of algebraic equations over the entire problem domain without using a predefined mesh. Rather, a set of scattered nodes, called field nodes, are used within the problem domain as well as on the boundaries of the domain Liu and Gu (2005). The meshless method does not require a priori

information about the relationship between the nodes for the interpolation or approximation of unknown functions over the field of variables

Liu and Gu (2005). Employing the meshless method, Dehghan and Shokri (2008a)

studied the one-dimensional nonlinear SGE and used Thin Plate Spline Radial Basis Functions (TPS-RBF) to approximate the solution, and they also applied the TPS-RBF method to both the Klein-Gordon equation

Dehghan and Shokri (2009) and the two-dimensional SGE Dehghan and Shokri (2008b). A series of meshless approaches have been presented Asgari and Hosseini (2013); Dehghan and Ghesmati (2010b); Jiang and Wang (2012); Jiwari et al. (2012a); Karamanli and Mugan (2013); Mirzaei and Dehghan (2009, 2010); Pekmen and Tezer-Sezgin (2012); Shao and Wu (2014); Taleei and Dehghan (2014). Moreover, since the nodal distribution for most existing meshless methods is preassigned, Xu et al. (2015) proposed a numerical two-step meshless method for soliton-like structures based on the optimal sampling density of kernel interpolation.

In dealing with high-dimensional PDEs, obtaining good quality approximate solutions is a difficult problem because of the so-called ‘curse of dimensionality’. High dimensional model representation (HDMR)

Li et al. (2001); Rabitz and Alis (1999, 2000); Rabitz et al. (1998); Sobol (1993)

provides a viable approach based on the fact that high-dimensional functions often can be efficiently expressed as sums of low-dimensional functions. The HDMR decomposition is also well known in statistics as the ANOVA (analysis of variance) decomposition

Efron and Stein (1981); Fisher (1925); Griebel and Holtz (2010); Stone (1994). In recent years, the HDMR decomposition has been under rapid development becoming an important tool for understanding high-dimensional functions Griebel and Holtz (2010); Griebel et al. (2010, 2013); Li et al. (2001); Luo et al. (2014a, b); Rabitz and Alis (1999, 2000); Rabitz et al. (1998); Sobol (1993). In this paper, we will use a HDMR decomposition in conjunction with the Fourier hyperbolic cross (HC) approximation Luo et al. (2017).

The remainder of the paper is organized as follows: Section 2 presents a function approximation method using HDMR-HC. Then a new meshless numerical scheme is proposed in Section 3 for solving the -dimensional SGE using the HDMR-HC approximation. In Section 4, we provide several examples with a comparative numerical error analysis. Section 5 summarizes the relevant results.

## 2 Function approximation using HDMR-HC and the partition of unity

### 2.1 HDMR-HC approximation

Let , and be a -variate function which is -periodic in each variable, where is the -torus given by Luo et al. (2017)

 Tnλ:=[−12λ,12λ)n

and the function space is defined by

 Ws1(Tnλ)={f∈L(Tnλ):∂rf∈L(Tnλ), |r|∞⩽s} (6)

with the norm

 ∥f∥Ws1(Tnλ)=∑0⩽|r|∞⩽s∥∂rf∥L(Tnλ)

where denotes a -dimensional multi-index with the norm

 |r|∞:=max1⩽j⩽nrj (7)

and the -th order mixed derivative is given by

 ∂rf:=∂|r|1f∂xr11⋯∂xrnn (8)

We consider the multivariate Fourier series of

 f(x)∼∑m∈Zncme2πiλm⋅x, x∈Tnλ (9)

where the Fourier coefficients are defined by

 cm=cm(f)=λn∫Tnλf(x)e−2πiλm⋅xdx, m∈Zn.

For a nonempty set , let

 Λj1,⋯,jv:= {m∈Zn:ml≠0 ∀ l∈{j1,⋯,jv} and ml=0 ∀ l∉{j1,⋯,jv}, l∈{1,⋯,n}}

and

 Λkλj1,⋯,jv:={m∈Λj1,⋯,jv:kλ⩽(2π)vλj1⋯λjv|mj1⋯mjv|

Note that is the set consisting of the

-dimensional zero vector. Then

can be decomposed into the following form

 Zn=Λ0+∑jΛj+∑j1

and we refer to this as a HDMR decomposition of ; then a multiple Fourier series can be decomposed with an HDMR structure

 ∑m∈Zncme2πiλm⋅x= c0+∞∑kλ=1⎛⎜ ⎜⎝∑j∑m∈Λkλjcme2πiλm⋅x (11)

If let

 ΛKλ=Λ0+Kλ∑kλ=1(∑jΛkλj+∑j1

that is,

 ΛKλ:={m∈Zn:n∏j=1max{2πλ|mj|,1}⩽Kλ}, (13)

then we define the Fourier HDMR-HC partial sum up to -th order Luo et al. (2017)

 (14)

Suppose is the number of points , which depends on both and , and

 Fλ,m(x)=e2πiλm⋅x

then Eq.(14) can be rewritten as

 SKλ(x,f)=M∑m=1cmFλ,m(x), x∈Tnλ (15)

Moreover, it follows that, for any , and , the bound

 |∂rSKλ(x,f)−∂rf(x)|⩽Cn,λ,s,p,γ,δ∥f∥Ws1(Tnλ)K−(s−γ−p−1)λ,  |r|∞=p

holds pointwise almost everywhere, where the constant depends on and .

From the definition of , we find that is smaller as becomes larger. The convergence rate of the Fourier HDMR-HC partial sums is closely related to the value of . In particular, for a given accuracy, when is large, generally is well approximated by a low order truncated HDMR, and this is the basic starting point of the Fourier HDMR-HC approximation. For example, if there is a function of variables, and suppose and the desired accuracy is , then a -th order Fourier HDMR-HC partial sum is just a nd order truncated HDMR of the function when . Therefore, we expect that a low order truncated HDMR-HC can be used to effectively capture the behavior of a high-dimensional function and its derivatives.

### 2.2 Partition of unity

In this subsection, we will discuss how to approximate a function via HDMR-HC. Suppose is a function defined on , where . First, we divide the domain into subdomains, denoted as , thus . In each subdomain , let be the centre of the region, and let denote all nodes in , i.e., . Further, suppose can be represented as , then we have

 uj(x)=Jλ(x−xj)u(x)=Jju(x), x∈Ωj (16)

where

 Ωj⊂supp Jλ(x−xj)   and   supp Jλ(x)⊂Tnλ (17)

is the characteristic function of

satisfying

 D∑j=1Jj=1, ∀x∈Ω⊆Rn (18)

From Eq.(15), at any point can be approximated as

 (19)

where with the same for all nodes and is the -th unknown coefficient. Utilizing the values of these nodes, there are equations with one for each node, then we have following matrix form

 Uj=Fjcj (20)

where

 Uj=[u1,u2,⋯,uMj]T

is the vector of function values at the nodes, and

 cj=[c1,c2,⋯,cMj]T

is the vector of undetermined coefficients with

 Fj=⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣Fλ,11Fλ,12⋯Fλ,1MjFλ,21Fλ,22⋯Fλ,2Mj⋮⋮⋮Fλ,Mj1Fλ,Mj2⋯Fλ,MjMj⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦

where . Suppose exists (i.e., this condition can always be satisfied Powell (1992); Wendland (1998)), then can be obtained by solving Eq.(20), i.e.

 cj=F−1jUj (21)

From Eq.(16), we have

 Uj=JjU (22)

where . Then

 cj=F−1jJjU (23)

Substitute both Eq.(21) and Eq.(22) back into Eq.(19), we have an approximation of function :

 (24)

Furthermore, from Eqs.(16) and (18) we have

 u(x)=D∑j=1Jju(x)=D∑j=1uj(x), x∈Ω (25)

Then at any point can be formally approximated as

 ^u(x) (26) =ΦT(x)U

## 3 Solution for (n+1)-dimensional SGEs

Now, we present the numerical scheme for solving the -dimensional SGE based on using the Fourier HDMR-HC approximation. Suppose the approximated function of the field function , () is formally denoted as

 (27)

where is the number of field nodes used in the selected domain, and is the vector that collects the true nodal function values for these field nodes, and is the centre of this selected region. Further, the derivatives of at any point can be approximated as

 (28)

where denotes one element of .

In this paper, the time derivatives are approximated by the time-stepping method and we have the following approximation:

 ∂2u∂t2≈1τ2[u(k+1)−2u(k)+u(k−1)] (29)
 ∂u∂t≈12τ[u(k+1)−u(k−1)] (30)

where is the time step, and is the approximate value of at , . Moreover the Crank-Nicolson scheme is used to approximate at three respective times as

 u(x,t)≈13[u(k+1)+u(k)+u(k−1)] (31)

To manage the nonlinearity, a Quasilinearization Method (QLM) is adopted. The QLM is very effective for dealing with the nonlinear aspects of the SGE and other PDEs. In this fashion the nonlinear term in Eq. (1) can be represented as

 sin(u)=sin(u(k))+(u(k+1)−u(k))cos(u(k)) (32)

Thus, Eq.(1) can be discretized as

 13Δu(k+1)−(η+μ+φ(k))u(k+1) (33) = −13Δu(k)−(2μ+φ(k))u(k)+ψ(x)sin(u(k)) −13Δu(k−1)−(η−μ)u(k−1)

where , and .

Suppose field nodes are denoted as , where is determined by both and . Then from Eq.(26) we have an approximation of the field variable according to the HDMR-HC approximation

 ^u(x)=ΦT(x)U=N∑j=1ϕj(x)uj (34)

where depends on and . The derivatives of can be approximated as

 Δ^u(x)=ΔΦT(x)U (35)

Therefore, for any point , the approximation in Eq.(33) can be written as

 N∑j=1[13Δϕj(xi)−(η+μ+φ(k))ϕj(xi)]u(k+1)j (36) = N∑j=1[−13Δϕj(xi)−(2μ+φ(k))ϕj(xi)]u(k)j +N∑j=1[−13Δϕj(xi)−(η−μ)ϕj(xi)]u(k−1)j +ψ(xi)sin(u(k))

Let , , and , , , , then Eq.(36) can be re-written as

 N∑j=1E(k)iju(k+1)j=N∑j=1G(k)iju(k)j+N∑j=1Hiju(k−1)j+C(k)i (37)

For all field nodes we have following matrix form:

 E′(k)^u′(k+1)=G′(k)^u′(k)+H′^u′(k−1)+C′(k) (38)

where .

In using a meshless strong method to solve the PDE, the solution can be unstable if there is a derivative boundary condition, so the fictitious points method is used to impose derivative boundary conditions Liu and Gu (2005). Suppose there are nodes on the boundary, then along the derivative boundaries, another fictitious points are added outside of the domain. Two sets of equations are established at each derivative boundary node: one for the derivative boundary condition, and the other for the governing equation. With the

, added into the system, then Eq.(36) can be re-written as

 N+Nb∑j=1E(k)iju(k+1)j=N+Nb∑j=1G(k)iju(k)j+N+Nb∑j=1Hiju(k−1)j+C(k)i (39)

and for a node at , that is, on the derivative boundary, the derivative boundary conditions have the form

 N+Nb∑j=1∂∂lϕj(xib)u(k+1)j=−N+Nb∑j=1∂∂lϕj(xib)u(k)j−N+Nb∑j=1∂∂lϕj(xib)(k−1)+3w(xib,t) (40)

Assembling Eqs.(39) and (40) for the corresponding nodes, the discretized global system equation becomes

 E(k)^u(k+1)=G(k)^u(k)+H^u(k−1)+C(k) (41)

where , , and are matrices, , . At the first time level, i.e. , we adopt the following:

 ^u(0)=v1 (42)

and

 ^u(−1)=^u(1)−2τv2 (43)

where and are the initial conditions for all nodes introduced in Eqs.(2) and (3).

## 4 Numerical experiments

In this section, the proposed meshless numerical scheme is applied to several examples to show the efficiency and accuracy for the -dimensional SGE. As mentioned in the previous section, to approximate the time derivatives we use a finite difference method, so an iterative scheme is employed to reach the final time . In order to test the performance of the numerical solution, we use the error and root-mean-square (RMS) error norms defined as

 L∞=∥∥f(xi)−^f(xi)∥∥∞=max1⩽i⩽N∣∣f(xi)−^f(xi)∣∣ (44)

and

 RMS= ⎷1NN∑i=1∣∣f(xi)−^f(xi)∣∣2 (45)

where is the number of nodes, is the exact solution, and is the numerical solution. To assess both the stability and the solution accuracy, we compute the condition number of the system matrix, which is defined as

 κ(F)=∥F∥∥F−1∥ (46)

where depends on the parameter and the number of nodes .

### 4.1 Test problem for a (2+1)-dimensional SGE

The test problem for a -dimensional SGE has the following form Djidjeli et al. (1995); Dehghan and Shokri (2008b); Jiwari et al. (2012b)

 utt=Δu−sin(u),  (x,y)∈Ω, t>0, (47)

where , and the initial conditions are

 v1(x,y)=4tan−1[exp(x+y)], (x,y)∈Ωv2(x,y)=−4exp(x+y)1+exp(2x+2y), (x,y)∈Ω. (48)

and the Neumann boundary condition is

 w(x,y,t)=4exp(x+y−t)1+exp(2x+2y−2t), (x,y)∈∂Ω, t>0 (49)

The analytic solution of this problem is:

 u(x,y,t)=4tan−1[exp(x+y−t)], (x,y)∈Ω (50)

In this example, both the proposed HDMR-HC meshless method and the radial basis function (RBF) method in Dehghan and Shokri (2008b) are used to numerically solve the equation. Since the field nodes of the RBF method in Dehghan and Shokri (2008b) is the Sobol sequence with , then we adopt the same total number of field nodes for the HDMR-HC meshless method (). The time step is set to . The results of the two different measures of error are presented in Table 1. We see that with the same number of field nodes, the errors of the proposed HDMR-HC meshless method are smaller than those of the RBF method. The condition numbers at particular times are also listed in Table 1.

### 4.2 Test problem for a (5+1)-dimensional SGE

To further test the proposed HDMR-HC scheme, we choose a -dimensional example, which involves all the implementation issues explained in the previous subsection. The exact solution has the form of Eq.(5)

where , , and

 ai={√2/2,i=1,2,31/2,i=4,5 (51)

The initial conditions are

 v1(x)=4arctan[Cexp(5∑i=1aixi)], x∈Ω, (52) v2(x)=−4bCexp(∑ni=1aixi)1+C2exp(∑ni=12aixi), x∈Ω,

and the Neumann boundary conditions are

 wj(x,t)=4ajCexp(∑5i=1aixi−bt)1+C2exp(∑5i=12aixi−2bt), j=1,2,⋯,5, x∈∂Ω, t>0 (53)

Both the proposed HDMR-HC method and the RBF method are used to solve the equation. The Sobol sequence is chosen as the field nodes with a total number of for the HDMR-HC method ( and we use the HDMR approximation up to order ) and for the RBF method. In this case the time step is chosen as . Table 2 presents , RMS errors and the condition number at some selected times .

## 5 Conclusions

In this paper, we propose a new meshless solution method for high-dimensional sine-Gordon equations. First, we present a function approximation using the HDMR-HC decomposition. Then we divided the whole domain into several subdomains with the help of the partition of unity, and obtain a function approximation at any random point in each subdomain. Hence, we develop a numerical procedure for the high-dimensional SGEs by a meshless strong solution method. The time-stepping method is used to approximate the time derivatives of SGEs, and a quasilinearization scheme is performed to treat the nonlinearity of the equation. Finally, to demonstrate the accuracy of the proposed method with two numerical experiments. The examples suggest that the proposed procedure is attractive for solving high-dimensional SGEs.

###### Acknowledgements.
The authors X.X. and X.L. acknowledge support from the National Science Foundation (Grant No. CHE-1763198), and H.R. acknowledges support from the Templeton Foundation (Grant No. 52265).

## References

• Aero et al. (2009) Aero E, Bulygin A, Pavlov Y (2009) Solutions of the three-dimensional sine-Gordon equation. Theoretical and Mathematical Physics 158:313–319
• Argyris and Haase (1987) Argyris J, Haase M (1987) An engineer’s guide to soliton phenomena, Application of the finite element method. Computer Methods in Applied Mechanics and Engineering 61:71–122
• Argyris et al. (1991) Argyris J, Haase M, Heinrich J (1991) Finite element approximation to two-dimensional sine-Gordon solitons. Computer Methods in Applied Mechanics and Engineering 86:1–26
• Asgari and Hosseini (2013) Asgari Z, Hosseini S (2013) Numerical solution of two-dimensional sine-Gordon and MBE models using Fourier spectral and high order explicit time stepping methods. Computer Physics Communications 184:565–572
• Barone et al. (1971) Barone A, Esposito F, Magee C, Scott A (1971) Theory and applications of the sine-Gordon equation. La Rivista del Nuovo Cimento 1(2):227–267
• Bour (1862) Bour E (1862) Théorie de la déformation des surfaces. J Ecole Imperiale Polytechnique 19:1–48
• Bratsos (2005) Bratsos A (2005) An explicit numerical scheme for the sine-Gordon equation in dimensions. Appl Numer Anal Comput Math 2:189–211
• Bratsos (2007) Bratsos A (2007) The solution of the two-dimensional sine-Gordon equation using the method of lines. Journal of Computational and Applied Mathematics 206:251–277
• Christiansen and Lomdahl (1981) Christiansen P, Lomdahl P (1981) Numercal solution of dimensional sine-Gordon solitons. Physica D 2:482–494
• Christiansen and Olsen (1979) Christiansen P, Olsen O (1979) On dynamical two-dimensional solutions to the sine-Gordon equation. Z Angew Math Mech 56:30–31
• Dehghan and Ghesmati (2010a) Dehghan M, Ghesmati A (2010a) Application of the dual reciprocity boundary integral equation technique to solve the nonlinear Klein-Gordon equation. Computer Physics Communications 181(8):1410–1418
• Dehghan and Ghesmati (2010b) Dehghan M, Ghesmati A (2010b) Numerical simulation of two-dimensional sine-gordon solitons via a local weak meshless technique based on the radial point interpolation method (RPIM). Computer Physics Communications 181:772–786
• Dehghan and Mirzaei (2008a) Dehghan M, Mirzaei D (2008a) The boundary integral equation approach for numerical solution of the one-dimensional sine-Gordon equation. Numerical Methods for Partial Differential Equations 24:1405–1415
• Dehghan and Mirzaei (2008b) Dehghan M, Mirzaei D (2008b) The dual reciprocity boundary element method (DRBEM) for two-dimensional sine-Gordon equation. Computer Methods in Applied Mechanics and Engineering 197:476–486
• Dehghan and Shokri (2008a) Dehghan M, Shokri A (2008a) A numerical method for one-dimensional nonlinear sine-Gordon equation collocation and radial basis functions. Numerical Methods for Partial Differential Equations 24:687–698
• Dehghan and Shokri (2008b) Dehghan M, Shokri A (2008b) A numerical method for solution of the two-dimensional sine-Gordon equation using the radial basis functions. Mathematics and Computers in Simulation 79(3):700–715
• Dehghan and Shokri (2009) Dehghan M, Shokri A (2009) Numerical solution of the nonlinear Klein-Gordon equation using radial basis functions. Journal of Computational and Applied Mathematics 230(2):400–410
• Djidjeli et al. (1995) Djidjeli K, Price W, Twizell E (1995) Numerical solutions of a damped sine-Gordon equation in two space variables. Journal of Engineering Mathmatics 29:347–369
• Drazin and Johnson (1989) Drazin P, Johnson R (1989) Solitons: An Introduction. Cambridge University Press
• Efron and Stein (1981)

Efron B, Stein C (1981) The jackknife estimate of variance. The Annals of Statistics 9(3):586–596

• Feng (2004) Feng Z (2004) An approximate sine-Gordon equation and its traveling wave solution in -dimensional space. Applied Mathematics and Computation 152:597–610
• Fisher (1925) Fisher R (1925) Statistical methods for research workers. Genesis Publishing Pvt Ltd
• Frenkel and Kontorova (1939) Frenkel J, Kontorova T (1939) On the theory of plastic deformation and twinning. Journal of Physics (USSR) 1:137–149
• Gorria et al. (2004) Gorria C, Gaididei Y, Soerensen M, Christiansen P, Caputo J (2004) Kink propagation and trapping in a two-dimensional curved Josephson junction. Physical Review B 69:1–10
• Griebel and Holtz (2010) Griebel M, Holtz M (2010) Dimension-wise integration of high-dimensional functions with applications to finance. Journal of Complexity 26:455–489
• Griebel et al. (2010) Griebel M, Kuo F, Sloan I (2010) The smoothing effect of the ANOVA decomposition. Journal of Complexity 26:523–551
• Griebel et al. (2013) Griebel M, Kuo F, Sloan I (2013) The smoothing effect of integration in and the ANOVA decomposition. Mathematics of Computation 82(281):383–400
• Guo et al. (1986) Guo B, Rodriguez P, Vazquez L (1986) Numerical solution of the sine-Gordon equation. Applied Mathematics and Computation 18:1–14
• Hirota (1973) Hirota R (1973) Exact three-soliton solution of the two-dimensional sine-Gordon equation. Journal of the Physical Society of Japan 35:15–66
• de la Hoz and Vadillo (2012) de la Hoz F, Vadillo F (2012) Numerical simulation of the -dimensional sine-Gordon equation via operational matrices. Computer Physics Communications 183:864–879
• Jiang and Wang (2012) Jiang Z, Wang R (2012) Numerical solution of one-dimensional Sine-Gordon equation using high accuracy multiquadric quasi-interpolation. Applied Mathematics and Computation 218(15):7711–7716
• Jiwari et al. (2012a) Jiwari R, Pandit S, Mittal R (2012a) A differential quadrature algorithm to solve the two dimensional linear hyperbolic telegraph equation with Dirichlet and Neumann boundary conditions. Applied Mathematics and Computation 218(13):7279–7294
• Jiwari et al. (2012b) Jiwari R, Pandit S, Mittal R (2012b) Numerical simulation of two-dimensional sine-Gordon solitons by differential quadrature method. Computer Physics Communications 183:600–616
• Kaliappan and Lakshmanan (1979) Kaliappan P, Lakshmanan M (1979) Kadomtsev-Petviashvili and two-dimensional sine-Gordon equations: reduction to Painlevé transcendents. Phys A: Math Gen 249:1–23
• Karamanli and Mugan (2013) Karamanli A, Mugan A (2013) Strong form meshless implementation of taylor series method. Applied Math ematics and Computati on 219:9069–9080
• Kobayashi and Izutsu (1976) Kobayashi K, Izutsu M (1976) Exact solution on the -dimensional sine-Gordon equation. Journal of the Physical Society of Japan 41(3):1091–1092
• Lakestani and Dehghan (2010) Lakestani M, Dehghan M (2010) Collocation and finite difference-collocation methods for the solution of nonlinear Klein-Gordon equation. Computer Physics Communications 181(8):1392–1401
• Lamb (1971) Lamb G (1971) Analytical descriptions of ultrashort optical pulse propagation in a resonant medium. Reviews of Modern Physics 43:99–124
• Li et al. (2001) Li G, Rosenthal C, Rabitz H (2001) High dimensional model representations. The Journal of Physical Chemistry A 105(33):7765–7777
• Liu and Gu (2005) Liu G, Gu Y (2005) An Introduction to Meshfree Methods and Their Programing. Springer
• Lou et al. (2005) Lou S, Hu H, Tang X (2005) Interactions among periodic waves and solitary waves of the -dimensional sine-gordon field. Physical Review E 71(3):036604
• Luo et al. (2014a) Luo X, Lu Z, Xu X (2014a) Non-parametric kernel estimation for the ANOVA decomposition and sensitivity analysis. Reliability Engineering and System Safety 130:140–148
• Luo et al. (2014b) Luo X, Lu Z, Xu X (2014b) Reproducing kernel technique for high dimensional model representations (HDMR). Computer Physics Communications 185(12):3099–3108
• Luo et al. (2017) Luo X, Xu X, Rabitz H (2017) On the fundamental conjecture of HDMR: a Fourier analysis approach. Journal of Mathematical Chemistry 55:632–660
• Minnhagen (1985) Minnhagen P (1985) Nonuniversal jumps and the Kosterlitz-Thouless transition. Physical Review Letters 54(21):2351–2354
• Minnhagen (1987) Minnhagen P (1987) The two-dimensional Coulomb gas, vortex unbinding, and superfluid-superconducting films. Review of Modern Physics 59:1001–1066
• Mirzaei and Dehghan (2009) Mirzaei D, Dehghan M (2009) Implementation of meshless LBIE method to the 2D non-linear SG problem. International Journal for Numerical Methods in Engineering 79(13):1662–1682
• Mirzaei and Dehghan (2010) Mirzaei D, Dehghan M (2010) Meshless local Petrov-Galerkin (MLPG) approximation to the two dimensional sine-Gordon equation. Journal of Computational and Applied Mathematics 233(10):2737–2754
• Mohebbi and Dehghan (2010) Mohebbi A, Dehghan M (2010) High-order solution of one-dimensional sine-Gordon equation using compact finite difference and DIRKN methods. Mathematical and Computer Modelling 51:537–549
• Nakajima et al. (1974) Nakajima K, Onodera Y, Nakamura T, Sato R (1974) Numerical analysis of vortex motion on Josephson structures. Journal of Applied Physics 45:4095–4099
• Ni et al. (1990) Ni G, Lou S, Chen S, Lee H (1990) Two-dimensional Coulomb gas studied in the sine-Gordon formulation. Physical Review B: Condensed Matter 41(10):6947–6957
• Pekmen and Tezer-Sezgin (2012) Pekmen B, Tezer-Sezgin M (2012) Differential quadrature solution of nonlinear Klein¨cGordon and sine-Gordon equations. Computer Physics Communications 183(8):1702–1713
• Powell (1992) Powell M (1992) Advanced in Numerical Analysis, Oxford Univercity Press, chap Theory of radial basis function approximation in 1990, pp 303–322
• Rabitz and Alis (1999) Rabitz H, Alis Ö (1999) General foundations of high-dimensional model representations. Journal of Mathematical Chemistry 25(2-3):197–233
• Rabitz and Alis (2000) Rabitz H, Alis Ö (2000) Sensitivity analysis, John Wiley and Sons Publishers, chap Managing the tyranny of parameters in mathematical modeling of physical systems, pp 199–223
• Rabitz et al. (1998) Rabitz H, Alis Ö, Shorter J, Shim K (1998) Efficient input-output model representations. Computer Physics Communications 117(1-2):11–20
• Scott et al. (1973) Scott A, Chu F, McLaughlin D (1973) The soliton: A new concept in applied science. Proceedings of the IEEE 61(10):1443–1483
• Shao and Wu (2014) Shao W, Wu X (2014) The numerical solution of the nonlinear Klein-Gordon and Sine-Gordon equations using the Chebyshev tau meshless method. Computer Physics Communications 185:1399–1409
• Sheng et al. (2005) Sheng Q, Khaliq A, Voss D (2005) Numerical simulation of two-dimensional sine-Gordon solitons via a split cosine scheme. Mathematics and Computers in Simulation 68:355–373
• Sobol (1993) Sobol I (1993) Sensitivity estimates for nonlinear mathematical models. Mathematical Modeling and Computational Experiment 1(4):407–414, english translation
• Stone (1994)

Stone C (1994) The use of polynomial splines and their tensor products in multivariate function estimation. The Annals of Statistics 22(1):118–171

• Taleei and Dehghan (2014) Taleei A, Dehghan M (2014) A pseudo-spectral method that uses an overlapping multidomain technique for the numerical solution of sine-Gordon equation in one and two spatial dimensions. Mathematical Methods in the Applied Sciences 37(13):1909–1923
• Wendland (1998) Wendland H (1998) Error estimates for interpolation by compactly supported radial basis functions of minimal degree. J Approx Theory 93:258–396
• Xin (2000) Xin J (2000) Modeling light bullets with the two-dimensional sine-Gordon equation. Physica D 135:345–368
• Xu et al. (2015) Xu X, Lu Z, Luo X (2015) A numerical meshless method of soliton-like structures model via an optimal sampling density based kernel interpolation. Computer Physics Communications 192:12–22
• Zagrodzinsky (1979) Zagrodzinsky J (1979) Particular solutions of the sine-gordon equation in dimensions. Physics Letters A 72:287–286