Coupled sine-Gordon Equations Research Articles

The New G-Double-Laplace Transforms and One-Dimensional Coupled Sine-Gordon Equations

This paper establishes a novel technique, which is called the G-double-Laplace transform. This technique is an extension of the generalized Laplace transform. We study its properties with examples and various theorems related to the G-double-Laplace transform that have been addressed and proven. Finally, we apply the G-double-Laplace transform decomposition method to solve the nonlinear sine-Gordon and coupled sine-Gordon equations. This method is a combination of the G-double-Laplace transform and decomposition method. In addition, some examples are examined to establish the accuracy and effectiveness of this technique.

Two linear energy-preserving compact finite difference schemes for coupled nonlinear wave equations

In this paper, we propose and analyze two highly efficient compact finite difference schemes for coupled nonlinear wave equations containing coupled sine-Gordon equations and coupled Klein-Gordon equations. To construct energy-preserving, high-order accurate and linear numerical methods, we first utilize the scalar auxiliary variable (SAV) approach and introduce three auxiliary functions to rewrite the original problem as a new equivalent system. Then we make use of the compact finite difference method and the Crank-Nicolson method to propose an efficient fully-discrete scheme (SAV-CFD-CN). The modified energy conservation and the convergence of the SAV-CFD-CN scheme are proved in detail, which has fourth-order convergence in space and second-order convergence in time. In order to preserve the discrete energy of original system, we further combine Lagrange multiplier approach, compact finite difference method and the Crank-Nicolson method to propose the second fully-discrete scheme (LM-CFD-CN). The proposed two schemes are high-order accurate, linear and highly efficient, only four symmetric positive definite systems with constant coefficients are required to be solved at each time level. Numerical experiments for the coupled nonlinear wave equations are given to confirm theoretical findings.

Implementation of DRBEM for coupled sine-Gordon equations

In this study, two-dimensional, time dependent, nonlinear, coupled sine-Gordon equations (CSGE) is solved using the dual reciprocity boundary element method (DRBEM). The DRBEM provides to transform the domain integral, caused by the inhomogeneous terms, into the boundary integral by approximating with thin plate spline. DRBEM formulation is derived using the fundamental solution of the modified Helmholtz equations (MHE)’s. In order to write the original equations in the form of MHE, time dervatives are expanded using finite difference. The principle purpose of this study is to show that DRBEM is a successful alternative for the solution of CSGE. Method is tested with the examples which have the analitical solution. The study also includes the solutions of other examples in the literature. Numerical experiments show that obtained results are in good agreement with other studies where the problem has been solved before and thus DRBEM is effective and accurate for the solution of CSGE.

Fully decoupled, linear, and energy-preserving GSAV difference schemes for the nonlocal coupled sine-Gordon equations in multiple dimensions

Fully decoupled, linear, and energy-preserving GSAV difference schemes for the nonlocal coupled sine-Gordon equations in multiple dimensions

High-order compact difference methods for solving two-dimensional nonlinear wave equations

<abstract><p>Nonlinear wave equations are widely used in many areas of science and engineering. This paper proposes two high-order compact (HOC) difference schemes with convergence orders of $ O\left({{\tau ^4} + h_x^4 + h_y^4} \right) $ that can be used to solve nonlinear wave equations. The first scheme is a nonlinear compact difference scheme with three temporal levels. After calculating the second-order spatial derivatives of the previous time level using the Padé scheme, numerical solutions of the next time level are obtained through repeated iterations. The second scheme is a three-level linearized compact difference scheme. Unlike the first scheme, iterations are not required and it obtains numerical solutions through an explicit calculation. The two proposed schemes are applied to solutions of the coupled sine-Gordon equations. Finally, some numerical experiments are presented to confirm the effectiveness and accuracy of the proposed schemes.</p></abstract>

Energy-preserving Du Fort-Frankel difference schemes for solving sine-Gordon equation and coupled sine-Gordon equations

Energy-preserving Du Fort-Frankel difference schemes for solving sine-Gordon equation and coupled sine-Gordon equations

A class of weighted energy-preserving Du Fort–Frankel difference schemes for solving sine-Gordon-type equations

Recently, invariant energy-quadratization methods (IEQMs) have been introduced by Xiaofeng Yang’s group to develop linear and energy-dissipation-preserving methods for nonlinear energy-dissipation systems. Following their work, two auxiliary functions are firstly introduced to rewrite the sine-Gordon equation (SGE) and coupled sine-Gordon equations (CSGEs) into equivalent systems, respectively. Then, two energy-preserving Du Fort–Frankel-type finite difference methods (EP-DFFT-FDMs) have been suggested for them, respectively. By using the discrete energy methods, the discrete energy conservative laws and convergence rates in the H1-norm have been derived, rigorously. It is worth mentioning that the proposed discrete energy is an approximation to the exact energy of the continuous problem. As hx=O(Δt), hy=O(Δt) and parameter λ≥1/4, the current methods are stable in the H1-norm because numerical solutions obtained by them are bounded in the H1-norm. What is more, as parameter λ≥1/4, the current methods are unconditionally stable in the L2-norm because numerical solutions obtained by them are uniformly bounded in the L2-norm. Moreover, our methods are explicit, and very easy to be implemented. However, a shortcoming of the current methods is that they are conditionally consistent. Namely, Δthx and Δthy tend to zero as time step Δt, spatial mesh sizes hx in x-direction and hy in y-direction tend to zero. Numerical findings support the correctness of theoretical analyses and the performance of the algorithms.

Local Fractional Homotopy Perturbation Method for Solving Coupled Sine-Gordon Equations in Fractal Domain

In this paper, the coupled local fractional sine-Gordon equations are studied in the range of local fractional derivative theory. The study of exact solutions of nonlinear coupled systems is of great significance for understanding complex physical phenomena in reality. The main method used in this paper is the local fractional homotopy perturbation method, which is used to analyze the exact traveling wave solutions of generalized nonlinear systems defined on the Cantor set in the fractal domain. The fractal wave with fractal dimension ε=ln2/ln3 is numerically simulated. Through numerical simulation, we find that the obtained solutions are of great significance to explain some practical physical problems.

Double Sumudu Transform Iterative Method for One-Dimensional Nonlinear Coupled Sine-Gordon Equation

In this paper, the combined double Sumudu transform with iterative method is successfully implemented to obtain the approximate analytical solution of the one-dimensional coupled nonlinear sine-Gordon equation (NLSGE) subject to the appropriate initial and boundary conditions which cannot be solved by applying double Sumudu transform only. The solution of the nonlinear part of this equation was solved by a successive iterative method, the proposed technique has the advantage of producing an exact solution, and it is easily applied to the given problems analytically. Two test problems from mathematical physics were taken to show the liability, accuracy, convergence, and efficiency of the proposed method. Furthermore, the results indicate that the introduced method is promising for solving other types of systems of NLPDEs.

Dynamics of coupled sine–Gordon equations: Inductively stacked long Josephson junctions with heterogeneous drives

The article presents the systematic study of the stacked long Josephson junctions in presence of different magnetic inductances. The coupled sine–Gordon equation along with a phase-shift formation is considered to study localized modes in stacks. The asymptotic analysis is applied to investigate analytically the nonlinear amplitude equations by using the spatially continuous and discrete spectrums. It is observed that in the absence of external drives, the system decays exponentially in the case of both strong and weak magnetic inductance, where the synchronized oscillations are obtained. However, in the presence of AC-drive, the system oscillates synchronically and goes to a stable state after a long time. It is revealed that, when the parametric drives are applied, the system decays exponentially in the existence of small driving amplitude. However, when the driving amplitude is large enough, the system oscillates for an infinitely long time.

Solving Two Dimensional Coupled Burger's Equations and Sine-Gordon Equation Using El-Zaki Transform-Variational Iteration Method

In this paper, the combined form of the Elzaki transform and variation iteration method is implemented efficiently in finding the analytical and numerical solutions of the two-dimensional nonlinear coupled Burger's partial differential equations and sine-Gordon partial differential equation. The obtained solutions were compared to the exact solutions and other existing methods. Illustrative examples show the efficiency and the power of the used method.

The studies of the linearly modified energy-preserving finite difference methods applied to solve two-dimensional nonlinear coupled wave equations

In this paper, four linearly energy-preserving finite difference methods (EP-FDMs) are designed for two-dimensional (2D) nonlinear coupled sine-Gordon equations (CSGEs) and coupled Klein-Gordon equations (CKGEs) using the invariant energy quadratization method (IEQM). The 1st EP-FDM is designed by first introducing two auxiliary functions to rewrite the original problems into the new system only including the 1st-order temporal derivatives, and then applying Crank-Nicolson (C-N) method and 2nd-order centered difference methods for the discretizations of temporal and spatial derivatives, respectively. The 2nd EP-FDM is directly devised based on the uses of 2nd-order centered difference methods to approximate 2nd-order temporal and spatial derivatives. The 1st and 2nd EP-FDMs need numerical solutions of the algebraic system with variable coefficient matrices at each time level. By modifying the 2nd EP-FDM, the 3rd EP-FDM, which is implemented by computing the system of algebraic equations with constant coefficient matrices at each time level, is developed. Finally, an energy-preserving alternating direction implicit (ADI) finite difference method (EP-ADI-FDM) is established by a combination of ADI method with the 3rd EP-FDM. By using the discrete energy method, it is shown that they are all uniquely solvable, and their solutions have a convergent rate of ${\mathscr{O}}(\varDelta t^{2}+{h^{2}_{x}}+{h^{2}_{y}})$ in H1-norm and satisfy the discrete conservative laws. Numerical results show the efficiency and accuracy of them.

A Numerical Scheme Based on the Chebyshev Functions to Find Approximate Solutions of the Coupled Nonlinear Sine-Gordon Equations with Fractional Variable Orders

In this article, a numerical method based on the shifted Chebyshev functions for the numerical approximation of the coupled nonlinear variable-order fractional sine-Gordon equations is shown. The variable-order fractional derivative is considered in the sense of Caputo-Prabhakar. To solve the problem, first, we obtain the operational matrix of the Caputo-Prabhakar fractional derivative of shifted Chebyshev polynomials. Then, this matrix and collocation method are used to reduce the solution of the nonlinear coupled variable-order fractional sine-Gordon equations to a system of algebraic equations which is technically simpler for handling. Convergence and error analysis are examined. Finally, some examples are given to test the proposed numerical method to illustrate the accuracy and efficiency of the proposed method.

A study on the compatibility of the generalized Kudryashov method to determine wave solutions

In this article, we establish solitary wave solutions to the Estevez-Mansfield-Clarkson (EMC) equation and the coupled sine-Gordon equations which are model equations to analyze the formation of shapes in liquid drops, surfaces of negative constant curvature, etc. through contriving the generalized Kudryashov method. The extracted results introduce several types’ solitary waves, such as the kink soliton, bell-shape soliton, compacton, singular soliton, peakon and other sort of soliton for distinct valuation of the unknown parameters. The achieved analytic solutions are interpreted in details and their 2D and 3D graphs are sketched. The obtained solutions and the physical structures explain the soliton phenomenon and reproduce the dynamic properties of the front of the travelling wave deformation generated in the dispersive media. It shows that the generalized Kudryashov method is powerful, compatible and might be used in further works to found novel solutions for other types of nonlinear evolution equations ascending in physical science and engineering.

Dynamics of solitons to the coupled sine-Gordon equation in nonlinear optics

This paper employs the principle of undetermined coefficients to establish the hyperbolic and trigonometric function solutions of the coupled sine-Gordon equation (CSGE) which describes the propagation of an optical pulse in fiber waveguide. Lie point symmetry of the CSGE is derived. Previously, it was noticed that the concept of nonlinear self-adjointness (NSA) was not applied on the equation under consideration. Here, we apply the concept of NSA to find an explicit form of the differential substitution. By means of the obtained substitution, we establish a new variant of conserved vectors by a new conservation theorem.

Double energy profile of pBR322 plasmid

A small circular DNA - plasmid pBR322, is considered as a complex dynamic system where nonlinear conformational perturbations which are often named open states or kinks, can arise and propagate. To describe the internal dynamics of the plasmid we use mathematical model consisting of two coupled sine-Gordon equations that in the average field approximation are transformed to two sine-Gordon independent equations with renormalized coefficients. The first equation describes angular oscillations of nitrous bases of the main chain. The second equation describes angular oscillations of nitrous bases in the complementary chain. As a result, two types of kink-like solutions have been obtained. One type kinks were the solutions of the first equation, and the other kinks were the solutions of the second equation. We calculated the main characteristics of the kink motion, including the time dependences of the kink velocity, coordinates, and total energy. These calculations were performed at the initial velocity equal to 1881 m/s which was chosen to avoid reflections from energy barriers corresponding to CDS-1 and CDS-2. The movement of the kinks was investigated by the method of the double energy profile. The maximum complete set of the DNA dynamic parameters was used to calculate the double profile. To calculate the velocity, energy and trajectory of the kinks, the block method was used. The results obtained made it possible to explain in which region of the plasmid the formation of a transcription bubble is most likely, as well to understand in which direction the bubble will move and the transcription process will go.

Localized modes inPT-symmetric sine-Gordon couplers with phase shift

• We study both analytically and numerically the nonlinear localized modes for waveguide arrays with PT -symmetry characterized by the balanced gain and loss parameters. • To investigate the nonlinear modes in PT -symmetry, the system of coupled sine-Gordon equations, together with a variety of nonlinear terms, are considered. • We applied asymptotic analysis together multiple-scale expansion, and obtained analytical amplitude equations for the optical modes, including the fundamental and excited modes. • The nonlinear localized modes are obtained by considering the discrete and spatially continuous spectrum, that is phonon mode. • It is noted that, the obtained amplitude equations of oscillations are affected by the strength of cross-coupling, that is, when the system is coupled weakly, the amplitude in the lossy side is decreased continuously, while on the gain side is increased for a long time. • It is, further, noted that, in the presence of strong coupling parameters, when the damping is present, the rate of blow and decay re-balance waveguides, and the system oscillates for a long period. • We have also discussed the waveguides for sinusoidal and quadratic coupling. • It is also observed that, when the waveguide is coupled strongly, the interaction in kink and anti-kink oscillation form a breather. • We conclude that such breather formation can also be feasible if it is on the amplified field of the PT -symmetric system. • The analytical results are compared with obtained numerical simulations in the numerical section, where good agreements are obtained. In this article, we address the nonlinear localized modes for waveguide arrays with parity-time ( PT )-symmetry characterized by the balanced gain and loss parameters. To investigate the PT -symmetric nonlinear modes, the system of coupled sine-Gordon equations together with a variety of nonlinear terms are considered. We applied asymptotic analysis together with multiple-scale expansion and obtained analytical amplitude equations for the optical modes, including the fundamental and excited modes. The nonlinear localized modes are obtained by considering the discrete and spatially continuous spectrum. It is noted that the obtained amplitude equations of oscillations are affected by the strength of cross-coupling, that is, when the system is coupled weakly, the amplitude in the lossy side decreases continuously while it increases on the gain side of the waveguide for a long time. It is further noted that in the presence of strong and sinusoidally coupling when the damping is present, the rate of blow and decay re-balance the waveguides, and the system oscillates for a long period. It is observed that, in the case of quadratic coupling, the amplitude in the lossy side decreases while the attenuated field increases. It is also observed that when the waveguide is coupled strongly, the interaction in kink and anti-kink oscillations form a breather. The analytical results are compared with the obtained numerical simulations in the numerical section, where good agreements are obtained.

Mesoscopic Simulation of the Two-Component System of Coupled Sine-Gordon Equations with Lattice Boltzmann Method.

In this paper, a new lattice Boltzmann model for the two-component system of coupled sine-Gordon equations is presented by using the coupled mesoscopic Boltzmann equations. Via the Chapman-Enskog multiscale expansion, the macroscopical governing evolution system can be recovered correctly by selecting suitable discrete equilibrium distribution functions and the amending functions. The mesoscopic model has been validated by several related issues where analytic solutions are available. The experimental results show that the numerical results are consistent with the analytic solutions. From the mesoscopic point of view, the present approach provides a new way for studying the complex nonlinear partial differential equations arising in natural nonlinear phenomena of engineering and science.

Numerical Simulation of the Coupled Sine-Gordon Equations via a Linearized and Decoupled Compact ADI Method

This article focuses on the development and analysis of a high-order compact (HOC) alternating direction implicit (ADI) method for solving a two-dimensional (2D) coupled sine-Gordon equations, which combines fourth-order compact difference for the approximations of the second spatial derivatives, Crank–Nicolson (C–N) method for temporal discretization and the approximation factorization of difference operators. By the discrete energy analysis method, it is shown that their solutions converge to the exact solutions with an order of two in time and four in space in L2-, H1, and -norms. Meanwhile, it is pointed out that the present method can be slightly adapted to the numerical solutions of other types of sine-Gordon equations, for example, sine-Gordon equation, doubled sine-Gordon equation, and damped wave equations. Finally, numerical findings testify the computational efficiency of the algorithms.

Large Deviation Principle for Stochastic Thermoelastic System Coupled Sine-Gordon Equation Driven by Lévy Noise

This work discusses the application of large deviation principle for a stochastic nonlinear thermoelastic system coupled Sine-Gordon equation driven by Levy noise. This is done using weak convergence approach formulated by Dupuis and Ellis wherein Laplace principle and large deviation principle are considered equivalent under specific space settings. Initially, the compactness of solution of the control system corresponding to the original system is proved. The proof is completed by ensuring that the solutions of the controlled stochastic system weakly converges to that of the controlled deterministic system with the aid of the estimate of the solution obtained prior.