Gordon Equation Research Articles

Nonrelativistic and relativistic bound state solutions of the molecular Tietz potential via the improved asymptotic iteration method

We have obtained the approximate analytical solutions of the relativistic and nonrelativistic molecular Tietz potential using the improved asymptotic iteration method. By approximating the centrifugal term through the Greene–Aldrich approximation scheme, we have obtained the energy eigenvalues and wave functions for all orbital quantum numbers [Formula: see text]. Where necessary, we made comparison with the result obtained previously in the literature. The relative closeness of the two results reveal the accuracy of the method presented in this study. We proceed further to obtain the rotational-vibrational energy spectrum for some diatomic molecules. These molecules are CO, HCl, H2, and LiH. We have also obtained the relativistic bound state solution of the Klein−Gordon equation with this potential. In the nonrelativistic limits, our result converges to that of the Schrödinger system.

Wave Systems with an Infinite Number of Localized Traveling Waves

In many wave systems, propagation of steadily traveling solitons or kinks is prohibited because of resonances with linear excitations. We show that wave systems with resonances may admit an infinite number of traveling solitons or kinks if the closest to the real axis singularities of a limiting asymptotic solution in the complex upper half plane are of the form z±=±α+iβ, α≠0. This quite general statement is illustrated by examples of the fifth-order Korteweg-de Vries equation, the discrete cubic-quintic Klein-Gordon equation, and the nonlocal double sine-Gordon equations.

Full time-dependent Hartree-Fock solution of largeNGross-Neveu models

We find the general solution to the time-dependent Hartree-Fock problem for scattering solutions of the Gross--Neveu models, with both discrete (${\mathrm{GN}}_{2}$) and continuous (${\mathrm{NJL}}_{2}$) chiral symmetry. We find new multibreather solutions both for the ${\mathrm{GN}}_{2}$ model, generalizing the Dashen--Hasslacher--Neveu breather solution, and also new twisted breathers for the ${\mathrm{NJL}}_{2}$ model. These solutions satisfy the full time-dependent Hartree-Fock consistency conditions, and only in the special cases of ${\mathrm{GN}}_{2}$ kink scattering do these conditions reduce to the integrable Sinh--Gordon equation. We also show that all baryons and breathers are composed of constituent twisted kinks of the ${\mathrm{NJL}}_{2}$ model. Our solution depends crucially on a general class of transparent, time-dependent Dirac potentials found recently by algebraic methods.

A sharp bilinear estimate for the Klein--Gordon equation in arbitrary space-time dimensions

We prove a sharp bilinear inequality for the Klein--Gordon equation on $\mathbb{R}^{d+1}$, for any $d \geq 2$. This extends work of Ozawa--Rogers and Quilodrán for the Klein--Gordon equation and generalizes work of Bez--Rogers for the wave equation. As a consequence, we obtain a sharp Strichartz estimate for the solution of the Klein--Gordon equation in five spatial dimensions for data belonging to $H^1$. We show that maximizers for this estimate do not exist and that any maximizing sequence of initial data concentrates at spatial infinity.

Analytical solution of the Klein--Gordon equation under the Coulomb-like scalar plus vector potential with the Laplace transforms approach

In this paper, analytical solution of the Klein-Gordon equation with Coulomb-like scalar plus vector potentials is obtained exactly. We considered the Laplace transform approach in our calculation. The exact bound state energy eigenvalues and the corresponding eigenfunctions are reported for various values of the quantum numbers n and l.

EXACT SOLUTIONS FOR NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS AND ITS APPLICATIONS

In this article, we use the improved general mapping deformation method based on the generalized Jacobi elliptic functions expansion method with computerized symbolic computation to construct some new exact solu- tions for some nonlinear partial differential equations via the cubic nonlinear Klein - Gordon equation and the modified Kawahara equation. As a result, new generalized Jacobi elliptic function-like solutions are obtained by using this method. This method is more powerful and will be used in further works to establish more entirely new solutions for other kinds of nonlinear partial differ- ential equations arising in mathematical physics when the balance is positive integer and the balance number is not positive integers.

Partitioning evapotranspiration – Testing the Craig and Gordon model with field measurements of oxygen isotope ratios of evaporative fluxes

Summary Stable oxygen isotopes of water provide a valuable tracer for water movements within ecosystems and are used to estimate the contribution of transpiration to total ecosystem evapotranspiration ( ft ). We tested the Craig and Gordon equation against continuous field measurements of isotopic composition of evaporation and assessed the impact for partitioning evapotranspiration. Therefore, evaporation ( E ) and its isotopic signature ( δ 18 O E ) on bare soil plots, as well as evapotranspiration ( ET ) and its corresponding isotopic composition of ( δ 18 O ET ) of an herbaceous layer was measured with a cavity ring-down spectrometer connected to a soil chamber on a field site in central Portugal. We quantified the variation in δ 18 O E arising from uncertainties in the determination of environmental input variables to the Craig and Gordon equation: the isotope signature ( δ 18 O e ) and the temperature at the evaporating site ( T e ), and the kinetic fractionation factor ( α k ). We could hence quantify ft based on measured δ 18 O ET , modeled δ 18 O E from observed soil water isotopic composition at the evaporating site ( δ 18 O e ), and modeled δ 18 O of transpiration ( δ 18 O T ) from observed total soil water isotopic composition. Our results demonstrate that predicting δ 18 O E using the Craig and Gordon equation leads to good agreement with measured δ 18 O E given that the temperature and 18 O isotope profiles of the soil are thoroughly characterized. However, modeled δ 18 O E is highly sensitive to changes in T e and δ 18 O e as well as α k . This markedly affected the partition results of transpiration and evaporation from the total ET flux: The fraction of transpiration ( ft ) varied strongly using different formulations for α k and assuming steady or non-steady state transpiration. These findings provide a first comparison of laser-based and modeled isotopic compositions of evaporation based on the Craig and Gordon equation under field conditions. This is of special interest for studies using stable isotopes to separate soil evaporation and plant transpiration fluxes and highlights the need for a thorough characterization of the micrometeorological and isotopic constitution of the upper soil layer to locate the evaporating front with a resolution of a few cm soil depths. We also call on a better characterization of the kinetic fractionation factor of soil evaporation.

Based on the Homotopy Analysis Method to Solve the Klein - Gordon Equation

This paper applies the homotopy analysis method to solve Klein-Gordon equation, Firstly, structure a zero-order deformation equation. Then get the formal approximation of the model from high-order deformation equation and prove the effectiveness of the solution in the end.

非线性离散的Klein-Gordon方程的对称约化

本文把离散的Lie点对称群分析方法应用于非线性离散的Klein-Gordon方程。由于该方程不易应用李点对称进行约化，所以本文首先引入一个相似变换将其转化为易被李点对称约化的新方程，然后用李点对称方法约化新方程得到其不变解，最后再通过相似变换得到原非线性离散的Klein-Gordon方程的解。 In this paper, the discrete Lie point symmetry group analysis method is applied on the discrete nonlinear Klein-Gordon equation. Since this equation is not easy to be reduced by Lie point symmetry method, firstly, this paper introduces a similarity transformation to change this equation into a new equation which can be reduced easily by Lie point symmetry method. Then the new equation is reduced by Lie point symmetry method and its invariant solutions are obtained. Finally, the solutions of the primal discrete nonlinear Klein- Gordon equation are acquired by the similarity transformation again.

The finite volume method for solving Buckmaster's equation, Fisher's equation and sine Gordon's equation for PDE's

The finite volume method for solving Buckmaster's equation, Fisher's equation and sine Gordon's equation for PDE's

Chiral Spin Waves in Fermi Liquids with Spin-Orbit Coupling

We predict the existence of chiral spin waves-collective modes in a two-dimensional Fermi liquid with the Rashba or Dresselhaus spin-orbit coupling. Starting from the phenomenological Landau theory, we show that the long-wavelength dynamics of magnetization is governed by the Klein-Gordon equations. The standing-wave solutions of these equations describe ''particles" with effective masses, whose magnitudes and signs depend on the strength of the electron-electron interaction. The spectrum of the spin-chiral modes for arbitrary wavelengths is determined from the Dyson equation for the interaction vertex. We propose to observe spin-chiral modes via microwave absorption by standing waves confined by an in-plane profile of the spin-orbit splitting.

Dual Klein–Gordon and Dirac equations

Olkhovsky [1], Pecam et al. [2], and Kholevo [3] (see also review [1]) have demonstrated and proved that, based on the Naimark theorems [4], the time operator can be introduced as a maximum Hermitian operator [5] for time as a quantum observable that is canonically conjugate to energy. Calculations of the observable average time moments ( ) t x with the help of measures depending on the probability flux density are used, in particular, for time analysis of

Quantum breathers in lithium tantalate ferroelectrics

Lithium tantalate is technologically one of the most important ferroelectric materials with a low poling field that has several applications in the field of photonics and memory switching devices. In a Hamiltonian system, such as dipolar system, the polarization behavior of such ferroelectrics can be well-modeled by Klein–Gordon (K-G) equation. Due to strong localization coupled with discreteness in a nonlinear K-G lattice, there is a formation of breathers and multi-breathers that manifest in the localization peaks across the domains in polarization–space–time plot. Due to the presence of nonlinearity and also impurities (as antisite tantalum defects) in the structure, dissipative effects are observed and hence dissipative breathers are studied here. To probe the quantum states related to discrete breathers, the same K-G lattice is quantized to give rise to quantum breathers (QBs) that are explained by a periodic boundary condition. The gap between the localized and delocalized phonon-band is a function of impurity content that is again related to the effect of pinning of domains due to antisite tantalum defects in the system, i.e., a point of easier switching within the limited amount of data on poling field, which is related to Landau coefficient (read, nonlinearity). Secondly, in a non-periodic boundary condition, the temporal evolution of quanta shows interesting behavior in terms of ‘critical’ time of redistribution of quanta that is proportional to QB’s lifetime in femtosecond having a possibility for THz applications. Hence, the importance of both the methods for characterizing quantum breathers is shown in these perspectives.

RELATIVISTIC PARTICLE IN ELECTROMAGNETIC FIELDS WITH A GENERALIZED UNCERTAINTY PRINCIPLE

In this paper, we propose to solve the relativistic Klein–Gordon and Dirac equations subjected to the action of a uniform electromagnetic field with a generalized uncertainty principle in the momentum space. In both cases, the energy eigenvalues and their corresponding eigenfunctions are obtained. The limit case is then deduced for a small parameter of deformation.

A scale-invariant Klein–Gordon model with time-dependent potential

The goal of the present paper is to derive statements about energy estimates as well as L p −L q decay estimates for a Klein–Gordon model with a particular time-dependent mass. The study of this special case of a scale-invariant model is an important step within a systematic investigation of Klein–Gordon models with time-dependent mass.

SELECTED TOPICS IN CLASSICAL INTEGRABILITY

Basic notions regarding classical integrable systems are reviewed. An algebraic description of the classical integrable models together with the zero curvature condition description is presented. The classical r-matrix approach for discrete and continuum classical integrable models is introduced. Using this framework the associated classical integrals of motion and the corresponding Lax pair are extracted based on algebraic considerations. Our attention is restricted to classical discrete and continuum integrable systems with periodic boundary conditions. Typical examples of discrete (Toda chain, discrete NLS model) and continuum integrable models (NLS, sine–Gordon models and affine Toda field theories) are also discussed.

Helicity and Wave Switching in a Nonlinear Model of DNA Dynamics

We have introduced helicity in the revised Hamiltonian in which the dipole-dipole interaction and dipoleinduced- dipole interaction have taken into account. The system has been shown to be govern by a set of coupled discrete sine- Gordon equations, which describe the bending and oscillations of the hydrogen bonds. The multiple scale expansion has been used and the stability of planar waves, solutions of the system, has been investigated. We have brought out the effect of discreteness and have shown that helicity brings about wave switching in the two-component DNA model under study. PACS number(s): 87.14.gk, 05.45.Yv, 87.15.H.

Breather continuation from infinity in nonlinear oscillator chains

Existence of large-amplitude time-periodic breathers localized neara single site is proved for the discrete Klein--Gordon equation, in the casewhen the derivative of the on-site potential has a compact support.Breathers are obtained at small couplingbetween oscillators and under nonresonance conditions.Our method is different from the classical anti-continuum limitdeveloped by MacKay and Aubry, and yields in generalbranches of breather solutions that cannotbe captured with this approach.When the coupling constant goes to zero,the amplitude and period of oscillationsat the excited site go to infinity.Our method is based on near-identity transformations, analysis ofsingular limits in nonlinear oscillator equations, andfixed-point arguments.

Lp - Lq decay estimates for Klein-Gordon models with effective mass

The goal of this paper is to derive Lp – Lq decay estimates for Klein– Gordon models with an effective mass. We explain first the WKB analysis which leads to a representation of the solution by Fourier multipliers. Then we use stationary phase method to get Lp – Lq decay estimates on the conjugate line. The effective mass allows to derive a Klein–Gordon type decay rate. Some comments to Lp – Lq decay rates for scale-invariant models and for Klein–Gordon models possessing scattering states to wave models complete the paper.

Classification of integrable discrete Klein–Gordon models

The Lie algebraic integrability test is applied to the problem of classification of integrable Klein–Gordon-type equations on quad graphs. The list of equations passing the test is presented, containing several well-known integrable models. A new integrable example is found; its higher symmetry is presented.