Nonlinear sine-Gordon Equation Research Articles

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.

Numerical study of the one-dimensional coupled nonlinear sine-Gordon equations by a novel geometric meshless method

In the paper, we derive a geometric meshless method for coupled nonlinear sine-Gordon (CNSG) equations. Approximate solutions of the CNSG equations are supposed to be expressed as the moving Kriging (MK) shape functions in the space direction. Global weak form of the CNSG equations is obtained, and then, a system of ODEs in time coordinate is extracted after imposing the MK meshless method. Then the geometric integrator, namely group preserving scheme, is offered to approximate the solution of obtained system of ODEs. Stability analysis of the method is numerically investigated. Numerical experiments show that the proposed method is effective and accurate for the CNSG equations.

A stable time–space Jacobi pseudospectral method for two-dimensional sine-Gordon equation

In this article, we present the stability analysis and approximate solutions of circular ring solitary solitons wave for (2 + 1)-dimensional nonlinear sine-Gordon equation using time–space pseudospectral method. Error bounds on discrete $$L_{2}$$-norm and Sobolev norm $$(H^{p})$$ are presented. The discretization of the problem using proposed method leads a system of nonlinear equations, which are solved using Newton–Raphson method. Further, a study is carried out to investigate the effects of dissipative term in sine-Gordon equation, which presence forms circular ring solitons. To support the theoretical results, the proposed method is tested for two problems, single circular ring solitary soliton waves and the collision of four circular ring solitary soliton waves and numerical results are presented.

Numerical solution of two-dimensional nonlinear sine-Gordon equation using localized method of approximate particular solutions

Sine-Gordon equation is one of the most famous nonlinear hyperbolic partial differential equations, it arises in many science and engineering fields. In the present work, we consider the numerical solution of two dimensional sine-Gordon equation using localized method of approximate particular solutions (LMAPS), in this technique, the method of approximate particular solutions (MAPS) occurs on some local domains, that greatly reduces the size of the collection matrix, and by combining the conditional positive radial basis function (RBF) generalized thin plate splines (GTPS) with additional low-order polynomial basis to avoid selecting shape parameters during localization. This method is effective compared with other existing methods and since this method is really meshless, it can be used to solve the nonlinear model with complicated computational domains. Several numerical examples are given to demonstrate the ability and accuracy of the present approach for solving nonlinear sine-Gordon equation.

A posteriori error analysis of the local discontinuous Galerkin method for the sine–Gordon equation in one space dimension

In this paper, we present asymptotically exact a posteriori error estimators of the local discontinuous Galerkin (LDG) method for the nonlinear sine–Gordon equation in one space dimension. We apply our recent optimal L2 error estimates and superconvergence results (Baccouch, 2018) to show that the LDG errors on each element can be split into two parts. The first part is proportional to the (p+1)-degree Radau polynomial and the second part converges with order p+3∕2 in the L2-norm, when piecewise polynomials of degree at most p are used. This new result is used to construct asymptotically exact a posteriori error estimates by solving a local steady problem on each element. The proposed error estimates are computationally simple and are obtained by solving a local steady problem with no boundary conditions on each element. We further prove that, for smooth solutions, these a posteriori error estimates for the solution and its derivative, at a fixed time, converge to the exact spatial errors in the L2-norm under mesh refinement. The theoretical order of convergence is proved to be p+3∕2 while our computational results show higher O(hp+2) convergence rate. Finally, we prove that the global effectivity index converges to unity at O(h1∕2) rate. Several numerical results are presented to validate the theoretical results.

Lie point symmetries of difference equations for the nonlinear sine-Gordon equation

A procedure for obtaining the Lie point symmetries of a difference scheme for a nonlinear sine-Gordan equation is presented. The invariant discretization of the first order of accuracy difference scheme for the equation is concerned. The symmetry transformations act both on the difference scheme and lattices, and leave the solution set of the difference scheme invariant.

Meshless Singular Boundary Method for Nonlinear Sine-Gordon Equation

A meshless method based on the singular boundary method is developed for the numerical solution of the time-dependent nonlinear sine-Gordon equation with Neumann boundary condition. In this method, by using a time discrete scheme to approximate the time derivatives, the time-dependent nonlinear problem is transformed into a sequence of time-independent linear boundary value problems. Then, the singular boundary method is used to establish the system of discrete algebraic equations. The present method is meshless, integration-free, and easy to implement. Numerical examples involving line and ring solitons are given to show the performance and efficiency of the proposed method. The numerical results are found to be in good agreement with the analytical solutions and the numerical results that exist in literature.

Fully discrete spectral methods for solving time fractional nonlinear Sine–Gordon equation with smooth and non-smooth solutions

We consider the initial boundary value problem of the time fractional nonlinear Sine–Gordon equation and the fractional derivative is described in Caputo sense with the order α(1 < α < 2). Two fully discrete schemes are developed based on Legendre spectral approximation in space and finite difference discretization in time for smooth solutions and non-smooth solutions, respectively. Numerical stability and convergence are analysed. Numerical experiments for both the fully discrete schemes are presented to confirm our theoretical analysis.

A Reduced-Order Extrapolating Finite Difference Iterative Scheme for 2D Generalized Nonlinear Sine-Gordon Equation

In this study, a reduced-order extrapolating finite difference iterative (ROEFDI) scheme holding sufficiently high accuracy but containing very few degrees of freedom for the two-dimensional (2D) generalized nonlinear Sine-Gordon equation is built via the proper orthogonal decomposition. The stability and convergence of the ROEFDI solutions are analyzed. And the feasibility and effectiveness of the ROEFDI scheme are verified by numerical simulations. This means that the ROEFDI scheme is effective and feasible for finding the numerical solutions of the 2D generalized nonlinear Sine-Gordon equation.

LATTICE BOLTZMANN MODEL FOR TWO-DIMENSIONAL GENERALIZED SINE-GORDON EQUATION

The nonlinear sine-Gordon equation arises in various problems in science and engineering. In this paper, we propose a numerical model based on lattice Boltmann method to obtain the numerical solutions of two-dimensional generalized sine-Gordon equation, including damped and undamped sine-Gordon equation. By choosing properly the conservation condition between the macroscopic quantity <i>u</i>_<i>t</i> and the distribution functions and applying the Chapman-Enskog expansion, the governing equation is recovered correctly from the lattice Boltzmann equation. Moreover, the local equilibrium distribution function is obtained. The numerical results of the first three examples agree well with the analytic solutions, which indicates the lattice Boltzmann model is satisfactory and efficient. Numerical solutions for cases involving the most known from the bibliography line and ring solitons are given. Numerical experiments also show that the present scheme has a good long-time numerical behavior for the generalized sine-Gordon equation. Moreover, the model can also be applied to other two-dimensional nonlinear wave equations, such as nonlinear hyperbolic telegraph equation and Klein-Gordon equation.

High accuracy analysis of the lowest order H1-Galerkin mixed finite element method for nonlinear sine-Gordon equations

The lowest order H1-Galerkin mixed finite element method (for short MFEM) is proposed for a class of nonlinear sine-Gordon equations with the simplest bilinear rectangular element and zero order Raviart-Thomas element. Base on the interpolation operator instead of the traditional Ritz projection operator which is an indispensable tool in the traditional FEM analysis, together with mean-value technique and high accuracy analysis, the superclose properties of order O(h2)/O(h2 + τ2) in H1-norm and H(div;Ω)-norm are deduced for the semi-discrete and the fully-discrete schemes, where h, τ denote the mesh size and the time step, respectively, which improve the results in the previous literature.

Numerical solution of nonlinear sine–Gordon equation by using the modified cubic B-spline differential quadrature method

This paper deals with numerical solution of one dimensional nonlinear sine–Gordon. “Modified cubic B-spline differential quadrature method” is used to solve one dimensional nonlinear sine–Gordon. The method reduces the problem to a system of first order ordinary differential equations (ODEs). The resulting system of ODEs is solved by “an optimal five stage and fourth-order strong stability preserving Runge–Kutta (SSP-RK54) method”. Finally, the method is illustrated and compared with existing methods via numerical examples. It is found that the method not only is quite easy to implement, but also gives better results than the ones already existing in the literature.

Energy control of distributed parameter systems via speed-gradient method: case study of string and sine-Gordon benchmark models

ABSTRACTEnergy control problems are analysed for infinite dimensional systems. Benchmark linear wave equation and nonlinear sine-Gordon equation are chosen for exposition. The relatively simple case of distributed yet uniform over the space control is considered. The speed-gradient method for energy control of Hamiltonian systems proposed by A. Fradkov in 1996, has already successfully been applied to numerous nonlinear and adaptive control problems is presently developed and justified for the above partial differential equations (PDEs). An infinite dimensional version of the Krasovskii–LaSalle principle is validated for the resulting closed-loop systems. By applying this principle, the closed-loop trajectories are shown to either approach the desired energy level set or converge to a system equilibrium. The numerical study of the underlying closed-loop systems reveals reasonably fast transient processes and the feasibility of a desired energy level if initialised with a lower energy level.

Chebyshev Wavelet Quasilinearization Scheme for Coupled Nonlinear Sine-Gordon Equations

Radial basis function (RBF) has been found useful for solving coupled sine-Gordon equation with initial and boundary conditions. Though this approach produces moderate accuracy in a larger domain, it requires more grid points. In the present study, we develop an alternative numerical scheme for solving one-dimensional coupled sine-Gordon equation to improve accuracy and to reduce grid points. To achieve these objectives, we make use of a wavelet scheme and solve coupled sine-Gordon equation. Based on the numerical results from the wavelet-based scheme, we conclude that our proposed method is more efficient than the radial basic function method in terms of accuracy.

Analysis and application of the element-free Galerkin method for nonlinear sine-Gordon and generalized sinh-Gordon equations

In this paper, a numerical scheme based on the element-free Galerkin (EFG) method is proposed to find the numerical solutions of nonlinear sine-Gordon equation with Neumann boundary condition and generalized sinh-Gordon equation with Dirichlet boundary condition. In this scheme, a time stepping technique is used to approximate the time derivative terms of the given equations. Then, the penalty method is adopted to enforce the Dirichlet boundary condition and lastly, the EFG method is performed to establish the system of discrete equations. The convergence of the proposed scheme is derived theoretically and verified numerically by doing its error analysis. Numerical examples involving line and ring solitons are given to show the accuracy and efficiency of the scheme. The numerical results are in excellent agreement with the analytical solutions and/or previously reported numerical results.

A meshless method of lines using Lagrange interpolation polynomials for solving the coupled nonlinear sine-Gordon equations

In this work, meshless method of lines (MMOL) using Lagrange interpolation polynomials (LIPs) is used to solve one dimensional time dependent the coupled nonlinear sine-Gordon equations. An alternative approach called the meshless method of lines using Lagrange interpolation polynomials (MMOLLIPs) present to overcome demerit of calculation of interpolation matrix in meshless method of lines using radial basis functions (MMOL-RBFs). To test the accuracy of this new method, L2 and L∞ error norms are calculated for each test problems. Comparing the methodology with some known techniques shows that the present approach is effective, practice and powerful.

Optimal energy-conserving local discontinuous Galerkin method for the one-dimensional sine-Gordon equation

ABSTRACTThe nonlinear sine-Gordon equation arises in various problems in science and engineering. In this paper, we propose and analyze a high-order and energy-conserving local discontinuous Galerkin (LDG) method for the sine-Gordon nonlinear hyperbolic equation in one space dimension. We prove the energy-conserving property and the stability for the semi-discrete LDG method. Optimal a priori error estimates for the solution and for the auxiliary variable that approximates the first-order derivative are derived in the -norm for the semi-discrete formulation. In particular, we identify a special numerical flux and a particular projection of the initial conditions for the LDG scheme for which the -norm of the solution and its spatial derivative are of order , when piecewise polynomials of degree at most p are used. Our numerical experiments demonstrate optimal order of convergence. Several numerical results are presented to validate the theoretical analyze of the proposed algorithm. It appears that similar conclusions are valid for the two-dimensional case.

Numerical simulation of two dimensional sine-Gordon solitons using modified cubic B-spline differential quadrature method

In this paper, a modified cubic B-spline differential quadrature method (MCB-DQM) is employed for the numerical simulation of two-space dimensional nonlinear sine-Gordon equation with appropriate initial and boundary conditions. The modified cubic B-spline works as a basis function in the differential quadrature method to compute the weighting coefficients. Accordingly, two dimensional sine-Gordon equation is transformed into a system of second order ordinary differential equations (ODEs). The resultant system of ODEs is solved by employing an optimal five stage and fourth-order strong stability preserving Runge–Kutta scheme (SSP-RK54). Numerical simulation is discussed for both damped and undamped cases. Computational results are found to be in good agreement with the exact solution and other numerical results available in the literature.

The use of radial basis functions (RBFs) collocation and RBF-QR methods for solving the coupled nonlinear sine-Gordon equations

Radial basis function (RBF) approximation is an extremely powerful tool for solving various types of partial differential equations, since the method is meshless and can be spectrally accurate. A perceived practical obstacle is that the interpolation matrix becomes increasingly ill-conditioned as the RBF shape parameter becomes small, corresponding to flat RBFs. In this paper, the authors solve the one and two-dimensional time-dependent coupled sine-Gordon equations using RBFs collocation and RBF-QR methods and show how one can overcome the ill-conditioning of coefficient matrix for the small shape parameters using RBF-QR method. The main aim of the current paper is to show that the meshless techniques based on the collocation methods are also suitable for solving the system of coupled nonlinear equations especially sine-Gordon equation. Several test problems are employed and results of numerical experiments are presented and also are compared with analytical solutions. The obtained results confirm the acceptable accuracy of the new methods.

Interaction of sine-Gordon kinks and breathers with a parity-time-symmetric defect.

The scattering of kinks and low-frequency breathers of the nonlinear sine-Gordon (SG) equation on a spatially localized parity-time-symmetric perturbation (defect) with a balanced gain and loss is investigated numerically. It is demonstrated that if a kink passes the defect, it always restores its initial momentum and energy, and the only effect of the interaction with the defect is a phase shift of the kink. A kink approaching the defect from the gain side always passes, while in the opposite case it must have sufficiently large initial momentum to pass through the defect instead of being trapped in the loss region. The kink phase shift and critical velocity are calculated by means of the collective variable method. Kink-kink (kink-antikink) collisions at the defect are also briefly considered, showing how their pairwise repulsive (respectively, attractive) interaction can modify the collisional outcome of a single kink within the pair with the defect. For the breather, the result of its interaction with the defect depends strongly on the breather parameters (velocity, frequency, and initial phase) and on the defect parameters. The breather can gain some energy from the defect and as a result potentially even split into a kink-antikink pair, or it can lose a part of its energy. Interestingly, the breather translational mode is very weakly affected by the dissipative perturbation, so that a breather penetrates more easily through the defect when it comes from the lossy side, than a kink. In all studied soliton-defect interactions, the energy loss to radiation of small-amplitude extended waves is negligible.