Nonlinear sine-Gordon Equation Research Articles

An implicit RBF meshless approach for solving the time fractional nonlinear sine-Gordon and Klein–Gordon equations

In this paper, we propose a numerical method for the solution of time fractional nonlinear sine-Gordon equation that appears extensively in classical lattice dynamics in the continuum media limit and Klein–Gordon equation which arises in physics. In this method we first approximate the time fractional derivative of the mentioned equations by a scheme of order O(τ3−α),1<α<2 then we will use the Kansa approach to approximate the spatial derivatives. We solve the two-dimensional version of these equations using the method presented in this paper on different domains such as rectangular and non-rectangular domains. Also, we prove the unconditional stability and convergence of the time discrete scheme. We show that convergence order of the time discrete scheme is O(τ). We solve these fractional PDEs on different non-rectangular domains. The aim of this paper is to show that the meshless method based on the radial basis functions and collocation approach is also suitable for the treatment of the nonlinear time fractional PDEs. The results of numerical experiments are compared with analytical solutions to confirm the accuracy and efficiency of the presented scheme.

Numerical Solution of Nonlinear Sine-Gordon Equation by Modified Cubic B-Spline Collocation Method

Modified cubic B-spline collocation method is discussed for the numerical solution of one-dimensional nonlinear sine-Gordon equation. The method is based on collocation of modified cubic B-splines over finite elements, so we have continuity of the dependent variable and its first two derivatives throughout the solution range. The given equation is decomposed into a system of equations and modified cubic B-spline basis functions have been used for spatial variable and its derivatives, which gives results in amenable system of ordinary differential equations. The resulting system of equation has subsequently been solved by SSP-RK54 scheme. The efficacy of the proposed approach has been confirmed with numerical experiments, which shows that the results obtained are acceptable and are in good agreement with earlier studies.

The cosmology of a fundamental scalar

We observe that the standard homogeneous cosmologies, those of Minkowski, de Sitter, and anti-de Sitter, which form the matrix for the Robertson–Walker scale factor, live naturally as isolated points inside a larger family of conformally flat metrics obtained by allowing a tensor containing the information of conformal symmetry breaking to be more general. So the standard cosmological metrics are parametrically unstable in this sense, and therefore unphysical. When we pass to the stable family of perturbed metrics, we immediately encounter a scalar field, which drives the conformal expansion of the universe and which automatically obeys the non-linear sine-Gordon equation. The Lagrangian for the sine-Gordon equation is a cosine potential agreeing to the fourth order with the potential used in the approach to the generation of mass in gauge theories. Accordingly we identify our geometric scalar field—actually of the type of an abelian gauge field—with the recently discovered scalar field. There are two constants in the theory: the first, named \(m\), is positive and defines a mass scale for the universe; the second, named \(\varLambda \), is the cosmological constant. For the space–time to be everywhere non-singular, equivalently for the (strict) dominant energy condition to hold, these constants must obey the inequality \(\varLambda > m^2/4\).

Stochastic Nonlinear Thermoelastic System Coupled Sine-Gordon Equation Driven by Jump Noise

This paper considers a stochastic nonlinear thermoelastic system coupled sine-Gordon equation driven by jump noise. We first prove the existence and uniqueness of strong probabilistic solution of an initial-boundary value problem with homogeneous Dirichlet boundary conditions. Then we give an asymptotic behavior of the solution.

Nonconforming quadrilateral finite element method for a class of nonlinear sine–Gordon equations

Nonconforming quadrilateral finite element method (FEM) of the two-dimensional nonlinear sine–Gordon equation is studied for semi-discrete and Crank–Nicolson fully-discrete schemes, respectively. Firstly, we prove a special feature of a new arbitrary quadrilateral element (named modified Quasi–Wilson element), i.e., the consistency error is of order O(h2) (h denotes the mesh size) in H1-norm, which leads to optimal order error estimate and superclose result with order O(h2) for the semi-discrete scheme through a different approach from the existing literature. Secondly, because the consistency error estimate of the new modified Quasi–Wilson element can reach a staggering O(h3) order, two orders higher than that of interpolation error, the optimal order error estimates of Crank–Nicolson fully-discrete scheme are obtained on arbitrary quadrilateral meshes with Ritz projection. Moreover, a superclose result in H1-norm is presented on generalized rectangular meshes by a new technique. Thirdly, the global superconvergence results of H1-norm for both semi-discrete and fully-discrete schemes are derived on rectangular meshes with interpolated postprocessing technique. Finally, a numerical test is carried out to verify the theoretical analysis.

Identifiability for Linearized Sine-Gordon Equation

The paper presents theoretical and numerical results on the identifiability, i.e. the unique identification for the one-dimensional sine-Gordon equation. The identifiability for nonlinear sine-Gordon equation remains an open question. In this paper we establish the identifiability for a linearized sine-Gordon problem. Our method consists of a careful analysis of the Laplace and Fourier transforms of the observation of the system, conducted at a single point. Numerical results based on the best fit to data method confirm that the identification is unique for a wide choice of initial approximations for the sought test parameters. Numerical results compare the identification for the nonlinear and the linearized problems.

The Spectral Method for Solving Sine-Gordon Equation Using a New Orthogonal Polynomial

We have presented a numerical scheme for solving one-dimensional nonlinear sine-Gordon equation. We apply the spectral method with a basis of a new orthogonal polynomial which is orthogonal over the interval with weighting function one. The results show the accuracy and efficiency of the proposed method.

On the behavior of high-order compact approximations in the one-dimensional sine–Gordon equation

In this paper, a family of high-order compact finite difference methods in combination with Krylov subspace methods is used for solution of the nonlinear sine–Gordon equation. We developed numerical methods by replacing the time and space derivatives by compact finite-difference approximations. The system of resulting nonlinear finite-difference equations is solved by Krylov subspace methods. The behavior of the compact finite-difference method is analyzed for error estimate and computational cost. Numerical results are presented to verify the behavior of high-order compact approximations for stability and convergence. The accuracy and efficiency of the proposed scheme are also considered.

Linear and Nonlinear Seismic Rayleigh Waves with Damping: A Heuristic Direct Method

In this study, nonlinearity in earthquake is investigated for the propagating seismic waves instead of linear waves. Assuming the presence of nonlinear effects in this earthquake modeling, the Rayleigh waves are formed by incorporating the nonlinear sine-Gordon equation into the linear asymptotic governing equations for finding similarity reduction. The existence of reduction to the modified asymptotic governing equations is demonstrated and is consequently shown to give both the linear and nonlinear Rayleigh waves solutions. The related velocity and amplitude dependent Rayleigh waves are obtained and also the nonlinear form of Rayleigh waves. Multiple nonlinear surface displacements are identified. These nonlinear waves are shown to leave the trails of crucial surface displacements.

Nonlinear theory of waves in solid state with cardinally changing crystalline structure

A nonlinear theory of propagating periodic and nonlinear solitary waves (like kinks and solitons) related to the motion of defects in crystals and of specific periodic waves into which the former ones transform in the field of the compression stress was developed. The role of intense tension stress leading to the heavy structural rearrangement of the crystal as a result of the effect of the external stress on the interatomic potential barriers was taken into account as well. Crystals with a complex lattice consisting of two sublattices were considered. Arbitrarily large displacements of sublattices were analyzed. The nonlinear theory is based on an additional element of the translational symmetry typical for complex lattices but not introduced earlier in solid-state physics. The variational equations of macroscopic and microscopic displacements turn out to be a nonlinear generalization of the linear equations of acoustic and optical modes obtained by Carman, Born, and Huang Kun. The microscopic displacement fields are described by the nonlinear sine-Gordon equation. In the one-dimensional case, exact solutions of the nonlinear equations were found and their features were revealed. In the case of two-dimensional (2+1) fields, new methods of the exact solutions of the sine-Gordon equation were developed. They describe the interaction of the nonlinear waves with the structural inhomogeneities of solid state due to the external fields of stress and deformations.

Tension spline solution of nonlinear sine-Gordon equation

The sine-Gordon equation plays an important role in modern physics. By using spline function approximation, two implicit finite difference schemes are developed for the numerical solution of one-dimensional sine-Gordon equation. Stability analysis of the method has been given. It has been shown that by choosing the parameters suitably, we can obtain two schemes of orders $\mathcal{O}(k^{2}+k^{2}h^{2}+h^{2})$ and $\mathcal{O}(k^{2}+k^{2}h^{2}+h^{4})$ . At the end, some numerical examples are provided to demonstrate the effectiveness of the proposed schemes.

Numerical simulation of two-dimensional sine-Gordon solitons via a local weak meshless technique based on the radial point interpolation method (RPIM)

In this paper the meshless local radial point interpolation method (LRPIM) is adopted to simulate the two-dimensional nonlinear sine-Gordon (S-G) equation. The meshless LRPIM is one of the “truly meshless” methods since it does not require any background integration cells. In this case, all integrations are carried out locally over small quadrature domains of regular shapes, such as circles or squares in two dimensions and spheres or cubes in three dimensions. A technique is proposed to construct shape functions using radial basis functions. These shape functions which are constructed by point interpolation method using the radial basis functions have delta function property. The time derivatives are approximated by the time-stepping method. In order to eliminate the nonlinearity, a simple predictor–corrector scheme is performed. Numerical results are obtained for various cases involving line and ring solitons. Also the conservation of energy in undamped sine-Gordon equation is investigated.

High-order solution of one-dimensional sine–Gordon equation using compact finite difference and DIRKN methods

In this work we propose a high-order and accurate method for solving the one-dimensional nonlinear sine–Gordon equation. The proposed method is based on applying a compact finite difference scheme and the diagonally implicit Runge–Kutta–Nyström (DIRKN) method for spatial and temporal components, respectively. We apply a compact finite difference approximation of fourth order for discretizing the spatial derivative and a fourth-order A -stable DIRKN method for the time integration of the resulting nonlinear second-order system of ordinary differential equations. The proposed method has fourth-order accuracy in both space and time variables and is unconditionally stable. The results of numerical experiments show that the combination of a compact finite difference approximation of fourth order and a fourth-order A -stable DIRKN method gives an efficient algorithm for solving the one-dimensional sine–Gordon equation.

A numerical method for one-dimensional nonlinear sine-Gordon equation using multiquadric quasi-interpolation

In this paper, we use a univariate multiquadric quasi-interpolation scheme to solve the one-dimensional nonlinear sine-Gordon equation that is related to many physical phenomena. We obtain a numerical scheme by using the derivative of the quasi-interpolation to approximate the spatial derivative and a difference scheme to approximate the temporal derivative. The advantage of the obtained scheme is that the algorithm is very simple so that it is very easy to implement. The results of numerical experiments are presented and compared with analytical solutions to confirm the good accuracy of the presented scheme.

Adaptive decomposition finite difference methods for solving singular problems—A review

Decomposition, or splitting, finite difference methods have been playing an important role in the numerical solution of nonsingular differential equation problems due to their remarkable efficiency, simplicity, and flexibility in computations as compared with their peers. Although the numerical strategy is still in its infancy for solving singular differential equation problems arising from many applications, explorations of the next generation decomposition schemes associated with various kinds of adaptations can be found in many recent publications. The novel approaches have been proven to be highly effective and reliable in operations. In this article, we will focus on some of the latest developments in the area. Key comments and discussion will be devoted to two particularly interesting issues in the research, that is, direct solutions of degenerate singular reaction-diffusion equations and nonlinear sine-Gordon wave equations. Numerical experiments with simulated demonstrations will be given.

A numerical method for solution of the two-dimensional sine-Gordon equation using the radial basis functions

The nonlinear sine-Gordon equation arises in various problems in science and engineering. In this paper, we propose a numerical scheme to solve the two-dimensional damped/undamped sine-Gordon equation. The proposed scheme is based on using collocation points and approximating the solution employing the thin plate splines (TPS) radial basis function (RBF). The new scheme works in a similar fashion as finite difference methods. Numerical results are obtained for various cases involving line and ring solitons.

Computer simulation on terahertz emission from intrinsic Josephson junctions of high-Tcsuperconductors

Solving coupled nonlinear sine-Gordon equations and Maxwell equations numerically, we study the electromagnetic and superconducting properties of the single crystal of high-${T}_{c}$ superconductor ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}\mathrm{Ca}{\mathrm{Cu}}_{2}{\mathrm{O}}_{8+\ensuremath{\delta}}$ with a static magnetic field applied parallel to the $ab$ plane and a dc fed in along the $c$ axis. Cavity resonances of transverse plasma occur in the intrinsic Josephson junctions with frequencies in terahertz regime. It is revealed that the electromagnetic wave can transmit from the junctions into space. The emitted energy counted by the Poynting vector is about $400\phantom{\rule{0.3em}{0ex}}\mathrm{W}∕{\mathrm{cm}}^{2}$. The frequency as well as the energy of emission can be tuned almost continuously by the current and magnetic field.

The Bradbury Butterfly Effect in Long Josephson Junctions

In a well-known science fiction novel by Ray Bradbury, a butterfly in the pre-historic past is accidentally crushed by time travelers and this seemingly insignificant event radically changes history. We will call this the “Bradbury Butterfly” Effect (BBE). We know that such an effect occurs in Long Josephson Junctions as described by a time-dependent nonlinear sine-Gordon equation. This equation states that any alteration within the initial perturbation fundamentally changes the asymptotic state of the system. The actual manifestation of this effect is proven by a numerical simulation of the time-dependent sine-Gordon equation.

New peakon and multi-compacton solitary wave solutions of fully nonlinear sine-Gordon equation

New peakon and multi-compacton solitary wave solutions of fully nonlinear sine-Gordon equation

New peakon and multi-compacton solitary wave solutions of fully nonlinear sine-Gordon equation

In the paper, we introduce the concept of nonlinear intensity, study a fully nonlinear sine-Gordon equation SG( m , n , p ) and obtain a new type of peakon solutions and kink solutions by a direct method. Research approximate SG( m , n , p ) equation and obtain a new multi-compacton and peakon solutions of approximate SG( m , n , p ) equation. At last, we study a fully nonlinear sine-Gordon equation under high dimension.