Two-dimensional sine-Gordon Equation Research Articles

Superconvergence analysis of an energy-conserving Galerkin FEM for the two-dimensional sine-Gordon equation

In this paper, the superconvergence error analysis of an energy-conserving Galerkin fully discrete scheme is proposed and investigated for the two-dimensional sine-Gordon equation. The unique solvability of the numerical scheme as well as the energy conservation are studied firstly. Then, based on the special property of the bilinear element on the rectangular mesh and the superclose estimate between interpolation and Ritz projection of the exact solution in H 1 -norm, the global superconvergence result in H 1 -norm is obtained in terms of a post-processing technique. Finally, numerical results are provided to confirm the energy conservation and superconvergence properties.

A SPLINE-BASED DIFFERENTIAL QUADRATURE APPROACH TO SOLVE SINE-GORDON EQUATION IN ONE AND TWO DIMENSION

This paper endeavors to compute the soliton solution for the sine-Gordon (SG) equation using a hybrid numerical method. The present method has been developed using a modified trigonometric B-spline function with the differential quadrature method (DQM). The method is capable of reducing the partial differential equation into a system of ordinary differential equations which is further stimulated with SSP-RK43, which is a form of the Runge–Kutta method. The present method has been established for its applicability by the numerical simulation of one-soliton and interaction of two solitons for one- and two-dimensional SG equation. The error norms have been calculated and the obtained results are found to approach the exact solutions. The stability of the method is demonstrated with the help of eigenvalues. The results are found encouraging and thus the method can be implemented to solve the similar nonlinear partial differential equations.

Kink–antikink stripe interactions in the two-dimensional sine–Gordon equation

The main focus of the present work is to study quasi-one-dimensional kink–antikink stripes embedded in the two-dimensional sine–Gordon equation. Using variational techniques, we reduce the interaction dynamics between a kink and an antikink stripe on their respective time and space dependent widths and locations. The resulting reduced system of coupled equations is found to accurately describe the width and undulation dynamics of a single kink stripe as well as that of two interacting ones. As an aside, we also discuss two related topics: the computational identification of the kink center and its numerical implications and alternative perturbative and multiple scales approaches to the transverse direction induced dynamics for a single kink stripe in the two-dimensional realm.

Analytical Solution of Two-Dimensional Sine-Gordon Equation

In this paper, the reduced differential transform method (RDTM) is successfully implemented for solving two-dimensional nonlinear sine-Gordon equations subject to appropriate initial conditions. Some lemmas which help us to solve the governing problem using the proposed method are proved. This scheme has the advantage of generating an analytical approximate solution or exact solution in a convergent power series form with conveniently determinable components. The method considers the use of the appropriate initial conditions and finds the solution without any discretization, transformation, or restrictive assumptions. The accuracy and efficiency of the proposed method are demonstrated by four of our test problems, and solution behavior of the test problems is presented using tables and graphs. Further, the numerical results are found to be in a good agreement with the exact solutions and the numerical solutions that are available in literature. We have showed the convergence of the proposed method. Also, the obtained results reveal that the introduced method is promising for solving other types of nonlinear partial differential equations (NLPDEs) in the fields of science and engineering.

Breather stripes and radial breathers of the two-dimensional sine-Gordon equation

We revisit the problem of transverse instability of a 2D breather stripe of the sine-Gordon (sG) equation. A numerically computed Floquet spectrum of the stripe is compared to analytical predictions developed by means of multiple-scale perturbation theory showing good agreement in the long-wavelength limit. By means of direct simulations, it is found that the instability leads to a breakup of the quasi-1D breather in a chain of interacting 2D radial breathers that appear to be fairly robust in the dynamics. The stability and dynamics of radial breathers in a finite domain are studied in detail by means of numerical methods. Different families of such solutions are identified. They develop small-amplitude spatially oscillating tails (“nanoptera”) through a resonance of higher-order breather’s harmonics with linear modes (“phonons”) belonging to the continuous spectrum. These results demonstrate the ability of the 2D sG model within our finite domain computations to localize energy in long-lived, self-trapped breathing excitations.

Stability of bubble-like fluxons in disk-shaped Josephson junctions in the presence of a coaxial dipole current.

We investigate analytically and numerically the stability of bubble-like fluxons in disk-shaped heterogeneous Josephson junctions. Using ring solitons as a model of bubble fluxons in the two-dimensional sine-Gordon equation, we show that the insertion of coaxial dipole currents prevents their collapse. We characterize the onset of instability by introducing a single parameter that couples the radius of the bubble fluxon with the properties of the injected current. For different combinations of parameters, we report the formation of stable oscillating bubbles, the emergence of internal modes, and bubble breakup due to internal mode instability. We show that the critical germ depends on the ratio between its radius and the steepness of the wall separating the different phases in the system. If the steepness of the wall is increased (decreased), the critical radius decreases (increases). Our theoretical findings are in good agreement with numerical simulations.

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.

BIFURCATION ANALYSIS AND EXACT TRAVELING WAVE SOLUTIONS FOR A GENERIC TWO-DIMENSIONAL SINE-GORDON EQUATION IN NONLINEAR OPTICS

We focus on investigating a generic two-dimensional sine-Gordon equation in nonlinear optics. Based on a viable transformation, the bifurcation analysis of the equation is carried out in this paper. The phase portraits are given and different kinds of traveling wave solutions are obtained. The analytical results are also numerically simulated.

Structure-preserving algorithms for the two-dimensional sine-Gordon equation with Neumann boundary conditions

This paper presents two kinds of strategies to construct structure-preserving algorithms with homogeneous Neumann boundary conditions for the sine-Gordon equation, while most existing structure-preserving algorithms are only valid for zero or periodic boundary conditions. The first strategy is based on the conventional second-order central difference quotient but with a cell-centered grid, while the other is established on the regular grid but incorporated with summation by parts (SBP) operators. Both the methodologies can provide conservative semi-discretizations with different forms of Hamiltonian structures and the discrete energy. However, utilizing the existing SBP formulas, schemes obtained by the second strategy can directly achieve higher-order accuracy while it is not obvious for schemes based on the cell-centered grid to make accuracy improved easily. Further combining the implicit midpoint method and the scalar auxiliary variable (SAV) approach, we construct symplectic integrators and linearly implicit energy-preserving schemes for the two-dimensional sine-Gordon equation, respectively. Extensive numerical experiments demonstrate their effectiveness with the homogeneous Neumann boundary conditions.

Semi-analytical exponential RKN integrators for efficiently solving high-dimensional nonlinear wave equations based on FFT techniques

In the light of the analytical expression of solutions to nonlinear wave equations (Wu et al. (2015)), using FFT techniques in this paper, we present a class of semi-analytical exponential/extended Runge–Kutta–Nyström (ERKN) integrators, which can nearly preserve the spatial continuity as well as the oscillation of the nonlinear waves equations. The standard ERKN methods require, in every time step, the computation of the matrix–vector product whose computational complexity in terms of basic multiplication is O(N2), once a direct calculation procedure is implemented. To achieve the near preservation of spatial continuity in the implementation of ERKN integrators, the number of spatial mesh grids N is required to be large enough, which may result in the massive calculation in the magnitude of O(N2) for each time step and the possible overflow of memory once a direct calculation procedure is implemented. To avoid this potential obstacle in applications, we design and analyse two novel algorithms which are incorporated with the Fast Fourier Transform (FFT) in the implementation of ERKN integrators. It is shown that the two algorithms reduce the magnitude of calculation amount from O(N2) to O(NlogN) in terms of basic multiplication. Meanwhile, they can greatly decrease the occupied memory, which allow the sufficient mesh grids to nearly preserve the spatial continuity especially for high-dimensional wave equations. Finally, numerical experiments including a two-dimensional sine–Gordon equation are carried out, and the numerical results soundly support the superiority and potentiality of the new algorithms.

On spatio-temporal dynamics of sine-Gordon soliton in nonlinear non-homogeneous media using fully implicit spectral element scheme

One- and two-dimensional sine-Gordon equation in non-homogeneous media is considered. Sine-Gordon equation exhibits soliton-like solution whose existence and behaviour in non-homogeneous media is studied. Various non-homogeneous media are discussed which include nonlinear smooth as well as discontinuous density variations. Time-dependent continuous and discontinuous density variations which can model geodynamical processes like lithospheric deformation, fault dynamics, etc. are also discussed. In one dimension, dynamics of kink which are topological soliton is studied under such density variations, whereas in two dimensions, circular-ring soliton dynamics has been scrutinized. The governing sine-Gordon equation is solved numerically using higher order Legendre polynomial-based spectral element method. Spectral stability analysis of the numerical scheme shows the strong dependence of a critical time step not only on density value but also on the nature of the distribution. It is shown that for smooth density variation the critical time step reduces drastically which makes explicit scheme very expensive. To reduce the computational cost, unconditionally stable fully implicit spectral element scheme is proposed which is also energy conservative.

Elliptic function solutions and travelling wave solutions of nonlinear Dodd-Bullough-Mikhailov, two-dimensional Sine-Gordon and coupled Schrödinger-KdV dynamical models

In this article, we construct the traveling wave and elliptic function solutions of some special nonlinear evolution equations which are arising in mathematical physics, solid-state physics, fluid flow, fluid dynamics, nonlinear optics, electromagnetic waves, quantum field theory etc. We employed modified extended direct algebraic method to construct the traveling wave and elliptic function solutions of Dodd-Bullough-Mikhailov equation, two-dimensional Sine-Gordon equation and coupled Schrödinger-KdV equation. The obtained analytical solutions in various form of each equation have different physical structures which are also presented graphically. The advantage of the current method that is simple, direct, elementary and concise. This method can be employed with a wider applicability for handling several other types of nonlinear wave equations.

Higher order scheme for two-dimensional inhomogeneous sine-Gordon equation with impulsive forcing

In this paper higher order scheme is presented for two-dimensional sine-Gordon equation. Higher order Legendre spectral element method is used for space discretization which is basically a domain decomposition method and it retains all the advantages of spectral and finite element methods. Spectral stability analysis is performed for both homogeneous and inhomogeneous sine-Gordon equations which gives implicit expressions of critical time step. Various test cases are solved which shows the robustness and accuracy of the proposed scheme. Moreover, experimental order of convergence is obtained which gives optimal convergence rate. It is also shown that the proposed scheme exhibit conservation of energy for undamped sine-Gordon equation. In realistic scenario defects are predominant in nature. sine-Gordon equation with defect can be effectively modeled by additional impulsive forcing term. In the second part of this paper, proposed higher order scheme is used to solve inhomogeneous sine-Gordon equation with impulsive loading. Both constant as well as time-dependent strength of impulsive forcing are used. Soliton solution to sine-Gordon equation is analyzed under the action of such forcing. Different Dirac delta representations are discussed which can accurately replicate behavior of impulsive loading. This is important because, dynamics of soliton behavior strongly depends on the impulsive force representation. Various conclusions are made based on this study.

Single-user MIMO system, Painlevé transcendents, and double scaling

In this paper, we study a particular Painlevé V (denoted PV) that arises from multi-input-multi-output wireless communication systems. Such PV appears through its intimate relation with the Hankel determinant that describes the moment generating function (MGF) of the Shannon capacity. This originates through the multiplication of the Laguerre weight or the gamma density xαe−x, x > 0, for α > −1 by (1 + x/t)λ with t > 0 a scaling parameter. Here the λ parameter “generates” the Shannon capacity; see Chen, Y. and McKay, M. R. [IEEE Trans. Inf. Theory 58, 4594–4634 (2012)]. It was found that the MGF has an integral representation as a functional of y(t) and y′(t), where y(t) satisfies the “classical form” of PV. In this paper, we consider the situation where n, the number of transmit antennas, (or the size of the random matrix), tends to infinity and the signal-to-noise ratio, P, tends to infinity such that s = 4n2/P is finite. Under such double scaling, the MGF, effectively an infinite determinant, has an integral representation in terms of a “lesser” PIII. We also consider the situations where α=k+1/2,k∈N, and α ∈ {0, 1, 2, …}, λ ∈ {1, 2, …}, linking the relevant quantity to a solution of the two-dimensional sine-Gordon equation in radial coordinates and a certain discrete Painlevé-II. From the large n asymptotic of the orthogonal polynomials, which appears naturally, we obtain the double scaled MGF for small and large s, together with the constant term in the large s expansion. With the aid of these, we derive a number of cumulants and find that the capacity distribution function is non-Gaussian.

Space–time Legendre–Gauss–Lobatto collocation method for two-dimensional generalized sine-Gordon equation

In this paper, we present a space–time Legendre–Gauss–Lobatto (LGL) collocation method for solving the generalized two-dimensional sine-Gordon equation with nonhomogeneous Dirichlet boundary conditions. The proposed method is based on the LGL collocation method to discretize in space, then use the LGL collocation method or block LGL collocation method to discretize in time. Our formulation has high-order accuracy in both space and time. We introduce a new H1 projection for the error analysis of collocation method for nonhomogeneous Dirichlet boundary conditions. This projection differs from the classical H1 projection. The new H1 projection is identically equal to Lagrange interpolation on boundary. The approximation properties of the new H1 projection are obtain. We derive error bounds in both discrete L2 and H1 norms for the (spatially) semidiscrete formulation, the analysis is based on new H1 projection results.

Coupling of Adaptive Element Free Galerkin Method with Variational Multiscale Method for Two-Dimensional Sine-Gordon Equation

In this paper, a coupling of adaptive element free Galerkin method with variational multiscale method is used to solve sine-Gordon equation in two-dimensional for the first time. Meshfree method is used where no mesh regeneration is needed comparing to finite element method. Therefore, this property facilitates the insertion of nodes which is triggered by the adaptive refinement procedure. Additional new nodes will be inserted at the high gradient regions to improve the numerical solutions. The adaptive analysis such as the refinement criteria and refinement strategy will be shown as well as the development of the modified moving least squares approximation. The performance of the proposed method is validated by solving two numerical problems. The first problem is two-dimensional large localized gradient problem with available analytical solution and the second problem is sine-Gordon equation. Numerical results proved that this method can obtain higher accuracy results compared with the conventional element free Galerkin method.

Space–time spectral method for the two-dimensional generalized sine-Gordon equation

In this paper, we propose a space–time spectral method for solving the generalized two-dimensional sine-Gordon equation with nonhomogeneous Dirichlet boundary conditions and initial conditions. The proposed method is based on the Legendre–Galerkin spectral method in space and the spectral collocation method (or block spectral collocation method) in time. We derive a priori error estimates in both L2 and H1 norms for the semidiscrete formulation. Our formulation has spectral accuracy in both space and time. Our numerical results confirm the exponential convergence of the proposed method in both space and time.

Slope modulation of ring waves governed by two-dimensional sine–Gordon equation

A theory of slope modulation of waves governed by the two-dimensional sine–Gordon equation is proposed. A large time asymptotic solution describing the slope modulation of trains of ring kinks is obtained. The comparison with the numerical solution of two-dimensional sine–Gordon equation shows excellent agreement.

Numerical solution of Klein–Gordon and sine-Gordon equations by meshless method of lines

Abstract We investigate numerical solution of the one dimensional nonlinear Klein–Gordon and two-dimensional sine-Gordon equations by meshless method of lines using radial basis functions. Results are compared with some earlier work showing the efficiency of the applied method. Salient feature of this method is that it does not require a mesh in the problem domain. Multiquadric and Gaussian are used as radial basis functions, which use a shape parameter. Choice of the shape parameter is still an open problem. We explore optimal value of the shape parameter without applying any extra treatment. For multiquadric, eigenvalue stability is studied without enforcing the boundary conditions whereas for Gaussian, the boundary conditions are enforced.

A moving least square reproducing polynomial meshless method

Interest in meshless methods has grown rapidly in recent years in solving boundary value problems (BVPs) arising in science and engineering. In this paper, we present the moving least square radial reproducing polynomial (MLSRRP) meshless method as a generalization of the moving least square reproducing kernel particle method (MLSRKPM). The proposed method is established upon the extension of the MLSRKPM basis by using the radial basis functions. Some important properties of the shape functions are discussed. An interpolation error estimate is given to assess the convergence rate of the approximation. Also, for some class of time-dependent partial differential equations, the error estimate is acquired. The efficiency of the present method is examined by several test problems. The studied method is applied to the parabolic two-dimensional transient heat conduction equation and the hyperbolic two-dimensional sine-Gordon equation which are discretized by the aid of the meshless local Petrov–Galerkin (MLPG) method.