One-dimensional Nonlinear Klein-Gordon Research Articles

Solving One-Dimensional Non-Linear Klein-Gordon Equations via Combining Adomian Polynomials and Rohit Transform Method

In this study, one-dimensional non-linear Klein-Gordon equations are solved by applying the integral transform known as the Rohit transform method. The approximate solutions of one-dimensional non-linear Klein- Gordon equations are obtained by combining the Adomian polynomials with the Rohit transform. To show the effectiveness and performance of the Rohit transform method, five one-dimensional non-linear Klein- Gordon type equations are considered and solved. The graphs of the solutions obtained are plotted to indicate the generality and clarity of the proposed method. It can be easily verified that the proposed method yielded the results that satisfy their corresponding non-linear Klein-Gordon equations. The integral Rohit transform combined with Adomian polynomials brought progressive methodologies that offer new insights on the problems (i.e., one-dimensional non-linear Klein-Gordon) examined in the paper, distinguishing itself from existing methods and doubtlessly beginning up new research instructions. Moreover, it reduces the complexity that occurs when non-linear Klein-Gordon are solved by other methods available in the literature. It provided precise results for the specific problems discussed in the paper, surpassing the capabilities of different methods in terms of decision, constancy, or robustness to noise and disturbances.

Slow dissipation and spreading in disordered classical systems: A direct comparison between numerics and mathematical bounds.

We study the breakdown of Anderson localization in the one-dimensional nonlinear Klein-Gordon chain, a prototypical example of a disordered classical many-body system. A series of numerical works indicate that an initially localized wave packet spreads polynomially in time, while analytical studies rather suggest a much slower spreading. Here, we focus on the decorrelation time in equilibrium. On the one hand, we provide a mathematical theorem establishing that this time is larger than any inverse power law in the effective anharmonicity parameter λ, and on the other hand our numerics show that it follows a power law for a broad range of values of λ. This numerical behavior is fully consistent with the power law observed numerically in spreading experiments, and we conclude that the state-of-the-art numerics may well be unable to capture the long-time behavior of such classical disordered systems.

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.

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.

A Nekhoroshev type theorem for the nonlinear Klein-Gordon equation with potential

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

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.

Stability of discrete breathers in nonlinear Klein-Gordon type lattices with pure anharmonic couplings

We consider the discrete breathers in one-dimensional nonlinear Klein-Gordon type lattices with pure anharmonic couplings. A discrete breather in the limit of vanishing couplings, i.e., the anti-continuous limit, consists of a number of in-phase or anti-phase excited particles, separated by particles at rest. Existence of the discrete breathers is proved for weak couplings by continuation from the anti-continuous limit. We prove a theorem which determines the linear stability of the discrete breathers. The theorem shows that the stability or instability of a discrete breather depends on the phase difference and distance between the two sites in each pair of adjacent excited sites in the anti-continuous solution. It is shown that there are two types of the dependence determined by the sign of αɛ, where α and ɛ are parameters such that positive (respectively, negative) α represents hard (respectively, soft) on-site nonlinearity and positive (respectively, negative) ɛ represents attractive (respectively, repulsive) couplings.

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.

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.

Bound States in One-dimensional Klein-Gordon Systems Admitting Periodic-Kink Soliton Excitations

The theory of phonon responses to soliton-lattice (periodic-kink) excitations in one-dimensional nonlinear Klein-Gordon systems is developed with emphasis on bound states. Results for the ϕ4 model reveal exactly four nonzero frequency modes in addition to the zero-frequency (Goldstone) mode.

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.