One-dimensional Nonlinear Klein-Gordon Equation Research Articles

Solitary waves in electro-mechanical lattices

We introduce and study both analytically and numerically a class of microelectromechanical chains aiming to turn them into transmission lines of solitons. Mathematically, their analysis reduces to the study of a spatially one-dimensional nonlinear Klein–Gordon equation with a model dependent onsite nonlinearity induced by the electrical forces. Since the basic solitons appear to be unstable for most of the force regimes, we introduce a stabilizing algorithm and demonstrate that it enables a stable and persisting propagation of solitons. Among other fascinating nonlinear formations induced by the presented models, we mention the “meson”: a stable square shaped pulse with sharp fronts that expands with a sonic speed, and “flatons”: flat-top solitons of arbitrary width.

Soliton dynamics for the 1D NLKG equation with symmetry and in the absence of internal modes

We consider the dynamics of even solutions of the one-dimensional nonlinear Klein-Gordon equation $\partial_t^2 \phi - \partial_x^2 \phi + \phi - |\phi|^{2\alpha} \phi =0$ for $\alpha>1$, in the vicinity of the unstable soliton $Q$. Our main result is that stability in the energy space $H^1(\mathbb R)\times L^2(\mathbb R)$ implies asymptotic stability in a local energy norm. In particular, there exists a Lipschitz graph of initial data leading to stable and asymptotically stable trajectories. The condition $\alpha>1$ corresponds to cases where the linearized operator around $Q$ has no resonance and no internal mode. Recall that the case $\alpha>2$ is treated in Krieger-Nakanishi-Schlag using Strichartz and other local dispersive estimates. Since these tools are not available for low power nonlinearities, our approach is based on virial type estimates and the particular structure of the linearized operator observed in Chang-Gustafson-Nakanishi-Tsai.

Conditional stability of multi-solitons for the 1D NLKG equation with double power nonlinearity

We consider the one-dimensional nonlinear Klein-Gordon equation with a double power focusing-defocusing nonlinearity∂t2u−∂x2u+u−|u|p−1u+|u|q−1u=0,on[0,∞)×R, where 1<q<p<∞. The main result states the stability in the energy space H1(R)×L2(R) of the sums of decoupled solitary waves with different speeds, up to the natural instabilities. The proof is inspired by the techniques developed for the generalized Korteweg-de Vries equation and the nonlinear Schrödinger equation in a similar context by Martel, Merle and Tsai [14,15]. However, the adaptation of this strategy to a wave-type equation requires the introduction of a new energy functional adapted to the Lorentz transform.

Stability of standing waves for a nonlinear Klein–Gordon equation with delta potentials

In this paper, we study local well-posedness and orbital stability of standing waves for a singularly perturbed one-dimensional nonlinear Klein–Gordon equation. We first establish local well-posedness of the Cauchy problem by a fixed point argument. Unlike the unperturbed case, a noteworthy difficulty here arises from the possible non-unitarity of the semigroup generating the corresponding linear evolution. We then show that the equation is Hamiltonian and we establish several stability/instability results for its standing waves. Our analysis relies on a detailed study of the spectral properties of the linearization of the equation, and on the well-known ‘slope condition’ for orbital stability.

Exact solution of nonlinear Klein-Gordon equations with quadratic nonlinearity by modified Adomian decomposition method

In this paper, we used the modified Adomian decomposition method (ADM) to obtain exact solution to Nonlinear Klein-Gordon equation (NK-GE) with quadratic nonlinearity. The paper contains an introduction and the concept of modified ADM for a generalized three-dimensional NK-GE. And, we applied this concept to obtain exact solution to two one-dimensional NK-GE with quadratic nonlinearity. The modified method is based on Taylors series expansion of the source term and implementation on any computer algebra software (Maple, Mathematica, etc) is simple. We discovered that the results of the examples considered are the same as the series solution of those obtained by using any known analytical method. Furthemore, we depicted our findings in three-dimensional surface and contour plots.

Exact solution of nonlinear Klein-Gordon equations with quadratic nonlinearity by modified Adomian decomposition method

In this paper, we used the modified Adomian decomposition method (ADM) to obtain exact solution to Nonlinear Klein-Gordon equation (NK-GE) with quadratic nonlinearity. The paper contains an introduction and the concept of modified ADM for a generalized three-dimensional NK-GE. And, we applied this concept to obtain exact solution to two one-dimensional NK-GE with quadratic nonlinearity. The modified method is based on Taylors series expansion of the source term and implementation on any computer algebra software (Maple, Mathematica, etc) is simple. We discovered that the results of the examples considered are the same as the series solution of those obtained by using any known analytical method. Furthemore, we depicted our findings in three-dimensional surface and contour plots.

Almost global existence for the nonlinear Klein-Gordon equation in the nonrelativistic limit

We study the one-dimensional nonlinear Klein-Gordon equation with a convolution potential, and we prove that solutions with small Hs norm remain small for long times. The result is uniform with respect to c ≥ 1, which however has to belong to a set of large measure.

Approximate Solution of Nonlinear Klein-Gordon Equation Using Sobolev Gradients

The nonlinear Klein-Gordon equation (KGE) models many nonlinear phenomena. In this paper, we propose a scheme for numerical approximation of solutions of the one-dimensional nonlinear KGE. A common approach to find a solution of a nonlinear system is to first linearize the equations by successive substitution or the Newton iteration method and then solve a linear least squares problem. Here, we show that it can be advantageous to form a sum of squared residuals of the nonlinear problem and then find a zero of the gradient. Our scheme is based on the Sobolev gradient method for solving a nonlinear least square problem directly. The numerical results are compared with Lattice Boltzmann Method (LBM). TheL2,L∞, and Root-Mean-Square (RMS) values indicate better accuracy of the proposed method with less computational effort.

Fourth-order compact solution of the nonlinear Klein-Gordon equation

In this work we propose a fourth-order compact method for solving the one-dimensional nonlinear Klein-Gordon equation. We apply a compact finite difference approximation of fourth-order for discretizing spatial derivative and a fourth-order A-stable diagonally-implicit Runge-Kutta-Nystrom (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. Numerical results obtained from solving several problems possessing periodic, kinks, single and double-soliton waves 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 these problems.

Numerical solution of the nonlinear Klein–Gordon equation using radial basis functions

The nonlinear Klein–Gordon equation is used to model many nonlinear phenomena. In this paper, we propose a numerical scheme to solve the one-dimensional nonlinear Klein–Gordon equation with quadratic and cubic nonlinearity. Our scheme uses the collocation points and approximates the solution using Thin Plate Splines (TPS) radial basis functions (RBF). The implementation of the method is simple as finite difference methods. The results of numerical experiments are presented, and are compared with analytical solutions to confirm the good accuracy of the presented scheme.

Split local absorbing conditions for one-dimensional nonlinear Klein-Gordon equation on unbounded domain

The numerical solution of the one-dimensional nonlinear Klein-Gordon equation on an unbounded domain is studied in this paper. Split local absorbing boundary (SLAB) conditions are obtained by the operator splitting method, then the original problem is reduced to an initial boundary value problem on a bounded computational domain, which can be solved by the finite difference method. Several numerical examples are provided to show the advantages and effectiveness of the given method, and some interesting collision behaviors are also observed.

Convergence Analysis of the Finite Difference Solution for the Nonlinear Klein-Gordon Equation

This paper is devoted to drive the matrix algebraic equation for the one-dimensional nonlinear Klein-Gordon equation which is obtained from using the implicit finite difference method. The convergence analysis of the solution is discussed. Numerical computations are conducted and the solutions are stable and convergent when the sine function is used as an initial condition.

Universal family of kink-bearing models reconstructed from a Pöschl-Teller scattering potential.

We present a continuous family of kink-bearing potentials ${\mathit{V}}^{(\mathrm{\ensuremath{\Lambda}})}$(cphi) for a one-dimensional nonlinear Klein-Gordon equation. The form of these potentials has been reconstructed from the P\"oschl-Teller potential ${\mathit{U}}^{(\mathrm{\ensuremath{\Lambda}})}$(x), which has been assumed to describe the interaction between kinks and low-amplitude harmonic solutions (phonons). Shapes of kinks, their creation energy, the low-temperature free energy of a soliton gas, and other properties of the model are discussed. A physical interpretation of the characteristic parameter \ensuremath{\Lambda} (&gt;0) of the investigated model is related to the total number of kink-phonon bound states and to their highest frequency. For special values of \ensuremath{\Lambda} the presented model can be reduced to the well-known sine-Gordon, ${\mathrm{\ensuremath{\varphi}}}^{4}$, Eshelby, double-quadratic, and, in part, to double-sine-Gordon models. Properties of kinks with three or four bound states are discussed in detail. We suggest that the model can be applied to quasi-one-dimensional solids.

Numerical computations of Liapunov exponents for a discretized one-dimensional nonlinear Klein-Gordon equation

We use the method of computing numerically the maximal Liapunov characteristics exponent in order to test the stochasticity of a particular model of coupled oscillators, describing a discretized one-dimensional nonlinear Klein-Gordon equation. Such a model was studied, from a different point of view by Fucitoet al. and by Buteraet al. The result is that a transition to stochasticity occurs when one passes from low energies to higher energies, and furthermore that the stochasticity decreases, tending to zero, at very high energies.