Nonlinear Klein-Gordon Research Articles

Numerical Solution of the Nonlinear Klein-Gordon Equation Using Multiquadric Quasi-interpolation Scheme

This paper's purpose is to provide a numerical scheme to approximate solutions of the nonlinear Klein-Gordon equation by applying the multiquadric quasi-interpolation scheme and the integrated radial basis function network scheme. Our scheme uses θ-weighted scheme for discretization of the temporal derivative and the integrated form of the multiquadric quasi-interpolation scheme for approximation of the unknown function and its spatial derivative. To confirm the accuracy and ability of the presented scheme, this scheme is applied on some test experiments and the numerical results have been compared with the exact solutions. Furthermore, it should be emphasized that with the presently available computing power, it has become possible to develop realistic mathematical models for the complicated problems in science and engineering. The mathematical description of various processes such as the nonlinear Klein-Gordon equation occurring in mathematical physics leads to a nonlinear partial differential equation. However, the mathematical model is only the first step towards the solution of the problem under consideration. The development of the well-documented, robust and reliable numerical tech- nique for handing the mathematical model under consideration is the next step in the solution of the problem. This second step is at least as important as the first one. Therefore, the robustness, the efficiency and the reliability of the numerical technique have to be checked carefully.

Integrated High Accuracy Multiquadric Quasi-interpolation Scheme For Solving The Nonlinear Klein-gordon Equation

A collocation scheme based on the use of the multiquadric quasi-interpolation operator 2 W L , integrated radial basis function networks (IRBFNs) method and three order finite difference method is applied to the nonlinear Klein-Gordon equation. In the present scheme, the three order finite difference method is used to discretize the temporal derivative and the integrated form of the multiquadric quasi-interpolation scheme is used to approximate the unknown function and its spatial derivatives. Several numerical experiments are provided to show the efficiency and the accuracy of the given method.

A Riccati-Bernoulli sub-ODE method for nonlinear partial differential equations and its application

The Riccati-Bernoulli sub-ODE method is firstly proposed to construct exact traveling wave solutions, solitary wave solutions, and peaked wave solutions for nonlinear partial differential equations. A Backlund transformation of the Riccati-Bernoulli equation is given. By using a traveling wave transformation and the Riccati-Bernoulli equation, nonlinear partial differential equations can be converted into a set of algebraic equations. Exact solutions of nonlinear partial differential equations can be obtained by solving a set of algebraic equations. By applying the Riccati-Bernoulli sub-ODE method to the Eckhaus equation, the nonlinear fractional Klein-Gordon equation, the generalized Ostrovsky equation, and the generalized Zakharov-Kuznetsov-Burgers equation, traveling solutions, solitary wave solutions, and peaked wave solutions are obtained directly. Applying a Backlund transformation of the Riccati-Bernoulli equation, an infinite sequence of solutions of the above equations is obtained. The proposed method provides a powerful and simple mathematical tool for solving some nonlinear partial differential equations in mathematical physics.

Trudinger–Moser inequality on the whole plane with the exact growth condition

Trudinger-Moser inequality is a substitute to the (forbidden) critical Sobolev embedding, namely the case where the scaling corresponds to L^{\infty} . It is well known that the original form of the inequality with the sharp exponent (proved by Moser) fails on the whole plane, but a few modified versions are available. We prove a precised version of the latter, giving necessary and sufficient conditions for the boundedness, as well as for the compactness, in terms of the growth and decay of the nonlinear function. It is tightly related to the ground state of the nonlinear Schrödinger equation (or the nonlinear Klein-Gordon equation), for which the range of the time phase (or the mass constant) as well as the energy is given by the best constant of the inequality.

Paramagnetic colloidal ribbons in a precessing magnetic field.

We investigate the dynamics of a kink in a damped parametrically driven nonlinear Klein-Gordon equation. We show by using a method of averaging that, in the high-frequency limit, the kink moves in an effective potential and is driven by an effective constant force. We demonstrate that the shape of the solitary wave can be controlled via the frequency and the eccentricity of the modulation. This is in accordance with the experimental results reported in a recent paper [Casic et al., Phys. Rev. Lett. 110, 168302 (2013)], where the dynamic self-assembly and propulsion of a ribbon formed from paramagnetic colloids in a time-dependent magnetic field has been studied.

Approximate Solution of Non-Linear Fractional Klein-Gordon Equation Using Spectral Collocation Method

In this paper, we implement the spectral collocation method with the help of the Legendre poly-nomials for solving the non-linear Fractional (Caputo sense) Klein-Gordon Equation (FKGE). We present an approximate formula of the fractional derivative. The Legendre collocation method is used to reduce FKGE to the solution of system of ODEs which is solved by using finite difference method. The results of applying the proposed method to the non-linear FKGE show the simplicity and the efficiency of the proposed method.

Nonlinear Stability of Periodic Traveling Wave Solutions for(n +1)-Dimensional Coupled Nonlinear Klein-Gordon Equations

We study the existence and orbital stability of smooth periodic traveling waves solutions of the(n +1)-dimensional coupled nonlinear Klein-Gordon equations. Such a system occurs in quantum mechanics, fluid mechanics, and optical fiber communication. Inspired by Angulo Pava’s results (2007), and by applying the stability theory established by Grillakis et al. (1987), we prove the existence of periodic traveling waves solutions and obtain the orbital stability of the solutions to this system.

Numerical Approximation of Nonlinear Klein-Gordon Equation Using an Element-Free Approach

Numerical approximation of nonlinear Klein-Gordon (KG) equation with quadratic and cubic nonlinearity is performed using the element-free improved moving least squares Ritz (IMLS-Ritz) method. A regular arrangement of nodes is employed in this study for the numerical integration to compute the system equation. A functional formulation for the KG equation is established and discretized by the Ritz minimization procedure. Newmark’s integration scheme combined with an iterative technique is applied to the resulting nonlinear system equations. The effectiveness and efficiency of the IMLS-Ritz method for the KG equation have been testified through convergence analyses and comparison study between the present results and the exact solutions.

Relativistic superfluidity and vorticity from the nonlinear Klein-Gordon equation

We investigate superfluidity, and the mechanism for creation of quantized vortices, in the relativistic regime. The general framework is a nonlinear Klein-Gordon equation in curved spacetime for a complex scalar field, whose phase dynamics gives rise to superfluidity. The mechanisms discussed are local inertial forces (Coriolis and centrifugal), and current-current interaction with an external source. The primary application is to cosmology, but we also discuss the reduction to the non-relativistic nonlinear Schr\"{o}dinger equation, which is widely used in describing superfluidity and vorticity in liquid helium and cold-trapped atomic gases.

Temperature-dependent thermal conductivities of one-dimensional nonlinear Klein-Gordon lattices with a soft on-site potential.

The temperature-dependent thermal conductivities of one-dimensional nonlinear Klein-Gordon lattices with soft on-site potential (soft-KG) are investigated systematically. Similarly to the previously studied hard-KG lattices, the existence of renormalized phonons is also confirmed in soft-KG lattices. In particular, the temperature dependence of the renormalized phonon frequency predicted by a classical field theory is verified by detailed numerical simulations. However, the thermal conductivities of soft-KG lattices exhibit the opposite trend in temperature dependence in comparison with those of hard-KG lattices. The interesting thing is that the temperature-dependent thermal conductivities of both soft- and hard-KG lattices can be interpreted in the same framework of effective phonon theory. According to the effective phonon theory, the exponents of the power-law dependence of the thermal conductivities as a function of temperature are only determined by the exponents of the soft or hard on-site potentials. These theoretical predictions are consistently verified very well by extensive numerical simulations.

On localized long-lived three-dimensional solutions of the nonlinear Klein-Gordon equation with a fractional power potential

The existence of long-lived (t ∼ 1000) stable spherically symmetric solutions in the form of pulsons has been numerically revealed for a certain class of Klein-Gordon equations. Their average amplitude of oscillations and the frequency of fast oscillation mode do not change during the entire time of calculation.

Exact Solution of Klein Gordon Equation via Homotopy Perturbation Sumudu Transform Method

We apply the proposed method (NHPSTM) which is the combination of new homotopy perturbation methodand Sumudu transform to solve analytical linear and nonlinear KleinGordon equations. The proposed method finds the solution without any discretization or restrictive assumptions and avoids the round-off errors. The fact that the proposed technique solves nonlinear problems without using Adomian’s polynomials can be considered as a clear advantage of this new method over the decomposition method. Obtained results reveal that the proposed method is very efficient, simple and can be applied to other nonlinear problems arising in mathematical physics and engineering.

Kink topology control by high-frequency external forces in nonlinear Klein-Gordon models.

A method of averaging is applied to study the dynamics of a kink in the damped double sine-Gordon equation driven by both external (nonparametric) and parametric periodic forces at high frequencies. This theoretical approach leads to the study of a double sine-Gordon equation with an effective potential and an effective additive force. Direct numerical simulations show how the appearance of two connected π kinks and of an individual π kink can be controlled via the frequency. An anomalous negative mobility phenomenon is also predicted by theory and confirmed by simulations of the original equation.

The application of modified Homotopy Analysis Method for Solving Linear and Non-Linear inhomogeneous Klein-Gordon Equations

The application of modified Homotopy Analysis Method for Solving Linear and Non-Linear inhomogeneous Klein-Gordon Equations

A numerical scheme based on differential quadrature method for numerical simulation of nonlinear Klein-Gordon equation

Purpose – The purpose of this paper is to propose a numerical scheme based on forward finite difference, quasi-linearisation process and polynomial differential quadrature method to find the numerical solutions of nonlinear Klein-Gordon equation with Dirichlet and Neumann boundary condition. Design/methodology/approach – In first step, time derivative is discretised by forward difference method. Then, quasi-linearisation process is used to tackle the non-linearity in the equation. Finally, fully discretisation by differential quadrature method (DQM) leads to a system of linear equations which is solved by Gauss-elimination method. Findings – The accuracy of the proposed method is demonstrated by several test examples. The numerical results are found to be in good agreement with the exact solutions and the numerical solutions exist in literature. The proposed scheme can be expended for multidimensional problems. Originality/value – The main advantage of the present scheme is that the scheme gives very accurate and similar results to the exact solutions by choosing less number of grid points. Secondly, the scheme gives better accuracy than (Dehghan and Shokri, 2009; Pekmen and Tezer-Sezgin, 2012) by choosing less number of grid points and big time step length. Also, the scheme can be extended for multidimensional problems.

On Time-Splitting Pseudospectral Discretization for Nonlinear Klein-Gordon Equation in Nonrelativistic Limit Regime

AbstractIn this work, we are concerned with a time-splitting Fourier pseudospectral (TSFP) discretization for the Klein-Gordon (KG) equation, involving a dimensionless parameterɛ ∊(0,1]. In the nonrelativistic limit regime, the smallɛproduces high oscillations in exact solutions with wavelength of(ɛ2) in time. The key idea behind the TSFP is to apply a time-splitting integrator to an equivalent first-order system in time, with both the nonlinear and linear subproblems exactly integrable in time and, respectively, Fourier frequency spaces. The method is fully explicit and time reversible. Moreover, we establish rigorously the optimal error bounds of a second-order TSFP for fixedɛ=(1), thanks to an observation that the scheme coincides with a type of trigonometric integrator. As the second task, numerical studies are carried out, with special efforts made to applying the TSFP in the nonrelativistic limit regime, which are geared towards understanding its temporal resolution capacity and meshing strategy for(ɛ2)-oscillatory solutions when 0 <ɛ« 1. It suggests that the method has uniform spectral accuracy in space, and an asymptotic(ɛ−2Δt2) temporal discretization error bound (Δtrefers to time step). On the other hand, the temporal error bounds for most trigonometric integrators, such as the well-established Gautschi-type integrator in, are(ɛ−4Δt2). Thus, our method offers much better approximations than the Gautschi-type integrator in the highly oscillatory regime. These results, either rigorous or numerical, are valid for a splitting scheme applied to the classical relativistic NLS reformulation as well.

Spatially localized structures and oscillons in atomic Bose-Einstein condensates confined in optical lattices

We consider the problem of formation of small-amplitude spatially localized oscillatory structures for atomic Bose-Einstein condensates confined in two- and three-dimensional optical lattices, respectively. Our approach is based on applying the regions with different signs of atomic effective masses where an atomic system exhibits effective hyperbolic dispersion within the first Brillouin zone. By using the $\mathbf{kp}$ method we have demonstrated mapping of the initial Gross-Pitaevskii equation on nonlinear Klein-Gordon and/or Ginzburg-Landau-Higgs equations, which is inherent in matter fields within ${\ensuremath{\phi}}^{4}$-field theories. Formation of breatherlike oscillating localized states---atomic oscillons---as well as kink-shaped states have been predicted in this case. Apart from classical field theories atomic field oscillons occurring in finite lattice structures possess a critical number of particles for their formation. The obtained results pave the way to simulating some analogues of fundamental cosmological processes occurring during our Universe's evolution and to modeling nonlinear hyperbolic metamaterials with condensed matter (atomic) systems.

Multi-Solitary Waves for the Nonlinear Klein-Gordon Equation

We consider the nonlinear Klein-Gordon equation in ℝ d . We call multi-solitary waves a solution behaving at large time as a sum of boosted standing waves. Our main result is the existence of such multi-solitary waves, provided the composing boosted standing waves are stable. It is obtained by solving the equation backward in time around a sequence of approximate multi-solitary waves and showing convergence to a solution with the desired property. The main ingredients of the proof are finite speed of propagation, variational characterizations of the profiles, modulation theory and energy estimates.

Multi-solitons for nonlinear Klein–Gordon equations

AbstractIn this paper, we consider the existence of multi-soliton structures for the nonlinear Klein–Gordon (NLKG) equation in $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathbb{R}^{1+d}$. We prove that, independently of the unstable character of NLKG solitons, it is possible to construct a $N$-soliton family of solutions to the NLKG equation, of dimension $2N$, globally well defined in the energy space $H^1\times L^2$ for all large positive times. The method of proof involves the generalization of previous works on supercritical Nonlinear Schrödinger (NLS) and generalized Korteweg–de Vries (gKdV) equations by Martel, Merle, and the first author [R. Côte, Y. Martel and F. Merle, Rev. Mat. Iberoam. 27 (1) (2011), 273–302] to the wave case, where we replace the unstable mode associated to the linear NLKG operator by two generalized directions that are controlled without appealing to modulation theory. As a byproduct, we generalize the linear theory described in Grillakis, Shatah, and Strauss [J. Funct. Anal. 74 (1) (1987), 160–197] and Duyckaerts and Merle [Int. Math. Res. Pap. IMRP (2008), Art ID rpn002] to the case of boosted solitons, and provide new solutions to be studied using the recent work of Nakanishi and Schlag [Zurich Lectures in Advanced Mathematics, vi+253 pp (European Mathematical Society (EMS), Zürich, 2011)] theory.

Approximate ILM dynamics in DNA models

We investigate the existence of Intrinsic Localized Modes (ILMs) in nonlinear one-dimensional KleinGordon chains. We use the Lagrangian averaging approach parameterizing ILM by several slow-varying variables, and apply the averaging directly in the action principle. Our preliminary studies yield results for ILM dynamics in accordance with those obtained by other methods.