As far as we know, there exists not any report that the POD method is utilized to simplify the classical FD scheme for the 2D **generalized** **nonlinear** **Sine**-**Gordon** **equation**. Therefore, in this work, we extend the approaches in [22-25] to the 2D **generalized** **nonlinear** **Sine**-**Gordon** **equation**, employing the POD technique to build a reduced-order extrapolating finite difference iterative (ROEFDI) scheme containing very few unknowns but having high enough accuracy. Especially, we are going to analyze the stability and convergence of the ROEFDI solutions by theoretical analysis and verify the feasibility and effectiveness of the ROEFDI scheme via numerical simulations.

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Wehave developed successfully introduce the homogeneous balance method and ob- tained wider classes of exact traveling wave solutions for the **generalized** **sine**–**Gordon** **equation** by using this binary method. This implies that our method is more powerful and effective in finding the exact solutions of NLEEs in mathematical physics.We hope this method can be more effectively used to solve many **nonlinear** partial differential equations in applied mathematics, engineering and mathematical physics.

u with respect to the variable t; u xx is the two-order partial derivative of the u about the independent variable x. Subsequently, Zhu [4] considered the following problem: u tt − α u t − u xx + λ g ( sin u ) = f x t ( ) , (where Ω is a bounded domain of R 3 ) and he proved the existence of the global solution of the **equation**. For more research on the global solutions and global attractors of Kirchhoff and **sine**-**Gordon** equations, we refer the reader to [5]-[11].

sense in the algebra G. Therefore the algebra G is a very convenient one to find and study solutions of **nonlinear** differential equations with singular data and coefficients. The paper is placed in the framework of algebras of **generalized** functions introduced by Colombeau in [4, 5]. Note also several examples have been studied by many authors in [12], [15, 16, 17] [13,16,17]. In particular, the authors [18] [18] processing the **nonlinear** wave with a data u|{t < 0} = 0. In this paper, we study the **Sine**- **Gordon** **equation** which a **nonlinear** wave, but in this time with conditions initial are distribution. The paper is organized as follows. In section 2, we recall the theory of Colombeau. Section 3, we proved the existence and uniqueness of solution in the algebra of Colombeau. The association with the classical solution is established in Section 4

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It is known the conservative scheme is better than the nonconservative ones. Zhang et al. 1 point out that the nonconservative scheme may easily show **nonlinear** blow up. In 2 Li and Vu-Quoc said “. . . in some areas, the ability to preserve some invariant properties of the original diﬀerential **equation** is a criterion to judge the success of a numerical simulation”. In 3–11 , some conservative finite diﬀerence schemes were used for a system of the **generalized** **nonlinear** Schr ¨odinger equations, Regularized long wave RLW equations, **Sine**-**Gordon** **equation**, Klein-**Gordon** **equation**, Zakharov equations, Rosenau **equation**, respectively. Numerical results of all the schemes are very good. Hence, we propose a new conservative diﬀerence scheme for the general Rosenau-RLW **equation**, which simulates conservative laws 1.4 and 1.5 at the same time. The outline of the paper is as follows. In Section 2, a **nonlinear** diﬀerence scheme is proposed and corresponding convergence and stability of the scheme are proved. In Section 3, some numerical experiments are shown.

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Abstract. In this paper, we used the modified Adomian decomposition method (ADM) to obtain exact solution to **Nonlinear** Klein-**Gordon** **equation** (NK-GE) with quadratic nonlinearity. The paper contains an introduction and the concept of modified ADM for a **generalized** three-dimensional NK-GE. And, we applied this concept to obtain exact solution to two one-dimensional NK-GE with quadratic nonlinearity. The modified method is based on Taylors series expansion of the source term and implementation on any computer algebra software (Maple, Mathematica, etc) is simple. We discovered that the results of the examples considered are the same as the series solution of those obtained by using any known analytical method. Furthemore, we depicted our findings in three- dimensional surface and contour plots.

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INTERNATIONAL JOURNAL OF ADVANCES IN ENGINEERING RESEARCH (25)And so on, where μ is a constant. The positive integer M in Eq.(24)can be determined by considering the homogeneous balance between the highest-order derivatives and **nonlinear** terms appearing in Eq.(23) If M is equal to a fractional or negative number, we can take the following transformations [4].

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negative real part can not be explained on the basis of the Breit-Wigner formula. Their occurence can be avoided by an additional breaking of parity (see discussion in [33]). The restriction on the parameters makes the model somewhat unattractive as this limitation eliminates the analogue of the entire breather sector which is present in the **sine**-**Gordon** model, such also that in the trigonometric limit one only obtains the soliton-antisoliton sector of that model, instead of a theory with a richer particle content. For this reason, the arguments outlined in the introduction and the fact that the constraint does not yield any Tachyon free theory anyhow, we relax here the restriction on ν. The the poles

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The element D is the third 4-D constant of integration, z0 is arbitrary constant and the function Sn[...] is the Jaco- bi elliptic function. Above proposition gives us the general analytical solution for the KleinGordon **equation** in terms of the 4-D commutative hypercomplex variable Z. It is the complete solution for the ODE form in as much as we have integrated twice and have a solution including two arbitrary constants of integration.

When the metal received energy, the atoms vibrated; but their electrons re- mained in the “metal grid”, vibrating in unison; that is , collectively , and pro- ducing a certain “ note ”, possibly associated with a particular type of energy [67]. To observe the behavior of the energy as vibratory notes in the grid, Fermi, Pasta and Ulam curiously prepared a model of five musical notes (as it is used during the Therapeutic Acupunctural Resonance ) looking for how they interacted with the grid. The **nonlinear** behavior of the experience transformed the grid into an ideal field for solitons : when a vibrational frequency lost energy, the solitonic interconnections allowed another to begin to gain it from the others, agglome- rating successively in each modality [66].

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f x = **equation**. Among the methods, iteration methods are very popular and are used by many researchers. Bisection method, fixed point iteration, secant method and … [1] – [3] and [8] – [10] are among various methods used for solving these problems.

and ω = ω( e x, e t). The level surfaces of θ and φ can be recognized as waves with two distinct slowly varying wave speeds. The moving breather will be periodic in θ with local wave number ωk/µ, local frequency ω 2 /µ and local phase velocity ω/k. The short scale φ is required to model the second slowly varying wave speed k/ω. The moving breather is not periodic in φ; however, the solution is exponentially small outside a short interval. These definitions are a generalization of the definition of the local wave number and local frequency for a strongly **nonlinear** wave train (see, for example, [36]), this formulation being based on the Lorentz invariance of the unperturbed problem. If k = 0, then we recover the fast scales for the stationary breather except that θ t = − ω. The definitions

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such as various forms of Kudryashov approach, exponential rational function tech- nique, simple hyperbolic ansatzes [13–22], the fractional form of the **Sine**-**Gordon** **equation** method is implemented to both equations to derive exact solutions in traveling wave forms. Before constructing the solutions, some preliminaries and basic properties of conformable derivative are given below. A brief summary of the method is also given in the next sections.

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The simulation was carried out using **sine** **Gordon** **equation** for the one-dimensional Frenkel-Kontorova dislocation model. In the Frenkel-Kontorova model, the atoms above the glide plane are material points connected by springs of rigidity « k », and the atoms under the glide plane (substrate) are described by a sinusoidal potential. The Frenkel-Kontorova model is discrete and this is its ad- vantage over other models. Within the framework of the chosen model, the mo- tion of dislocations is described by the **sine** **Gordon** **equation**.

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This paper studied the BME with log law nonlinearity. The 1-soliton solution was obtained by the ansatz method. This solution is also known as Gausson in the context of **nonlinear** optics. A couple of conserved quantities are also obtained using the Gausson solution. The constraint relation also fell out naturally from the solution. An exact solution with the IMD perturbation term taken into account is also derived.

describes, for example, structural phase transitions in ferroelectric and ferromagnetic materials, topological excitations in quasi one dimensional system like biological macromolecules and hydrogen chains, or polymers, etc. Its simplest localized solutions so-called ”kinks” which are related to the motion of the aforementioned topological excitations, e.g., domain walls in second order phase transitions, or polymerization mismatches. A more realistic modeling of physical situation in condensed matter physics often requires the inclusion of perturbations of different types like thermal noise and time or spatial dependent potential fluctuations [ 8 ]. The **equation** was first proposed by Aubry , Krumhansl and Schrieffer in 1975 and 1976, to describe displacive and order-disorder transitions in solids, mainly magnetic compounds [9 ]. Manna and Merle (1997) used multiple- sale perturbation theory. They showed that a **nonlinear** (quadratic) Klein – **Gordon** type **equation** substitutes in a short- wave analysis the ubiquitous Korteweg-de Vries **equation** of long-wave approach. Dmitriev et. al. (2006) discussed some discrete equations free of the peierls-Nabarro barrier and identified for them the full space of available static solutions, including those derived recently in physics but not limited to them [1].

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The infinite sets of polynomial conserved densities which have been found for the Korteweg-de Vries equation, the modified Korteweg-de Vries equation, the Sine-Gordon equation, and the c[r]

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Finally we explore how the degenerate solutions may be obtained within the context of Hirota’s direct method [31]. The key idea of this solution procedure is to convert the original **nonlinear** equations into bilinear forms, which can be solved systematically. When parameterizing φ(x, t) = 2i ln[g(x, t)/f (x, t)] the **sine**-**Gordon** **equation** (1.1) was found [6, 31] to be equivalent to the two equations

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strained optimal control problem. An alternating direction implicit (ADI) Legendre spec- tral element method for the two-dimensional Schrodinger **equation** is developed in [8], and the optimal H 1 error estimate for the linear case is given. The aim of [9] is the Lagrange–Galerkin spectral element method for solving two-dimensional shallow water equations. The authors of [10] considered the numerical approximation of the acoustic wave **equation** by the spectral element method based on the Gauss–Lobatto–Legendre quadrature formulas and ﬁnite diﬀerence Newmark’s explicit time advancing schemes. A modiﬁed set of basis functions for use with spectral element methods is presented in [11] for solving a mixed elliptic boundary value problem. These basis functions are constructed so that the axial conditions along a plane or axis of symmetry are satisﬁed identically. A numerical spectral element method for the computation of ﬂuid ﬂows governed by the incompressible Euler equations in a complex geometry is presented in [12]. Zhuang and Chen [13] used this method to solve biharmonic equations. In [14], the authors used the spectral element method with least-square formulation for parabolic interface problems. Ai et al. [15] used fully diagonalized Legendre spectral element methods using Sobolev orthogonal/biorthogonal basis functions for solving second-order elliptic boundary value problems. A Legendre spectral element formulation of an improved time-splitting method is developed for the natural convection heat transfer problem in a square cavity by Wang and Qin [16].

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