Bifurcation analysis and the travelling wave solutions of the Klein–Gordon–Zakharov equations

In this paper, we investigate the bifurcations and dynamic behaviour of travelling wave solutions of the Klein–Gordon–Zakharov equations given in Shang et al, Comput. Math. Appl. 56, 1441 (2008). Under different parameter conditions, we obtain some exact explicit parametric representations of travelling wave solutions by using the bifurcation method (Feng et al, Appl. Math. Comput. 189, 271 (2007); Li et al, Appl. Math. Comput. 175, 61 (2006)).     

Bright and singular soliton solutions of the conformable time-fractional Klein–Gordon equations with different nonlinearities

Finding the exact solutions of nonlinear fractional differential equations has gained considerable attention, during the past two decades. In this paper, the conformable time-fractional Klein–Gordon equations with quadratic and cubic nonlinearities are studied. Several exact soliton solutions, including the bright (non-topological) and singular soliton solutions are formally extracted by making use of the ansatz method. Results demonstrate that the method can efficiently handle the time-fractional Klein–Gordon equations with different nonlinearities.     

Solitons and cnoidal waves of the Klein–Gordon–Zakharov equation in plasmas

This paper studies the Klein?CGordon?CZakharov equation with power-law nonlinearity. This is a coupled nonlinear evolution equation. The solutions for this equation are obtained by the travelling wave hypothesis method, (G??/G) method and the mapping method.     

Moving potential for Dirac and Klein–Gordon equations

Using the Lorentz transformation, the Klein–Gordon and Dirac equations with moving potentials are reduced to one standard where the potential is time-independent. As application, the reflection and transmission coefficients are determined by considering the moving step with a constant velocity v. It has been found that R ± T = 1 only at x = vt. The problem of massless (2 + 1) Dirac particle is also considerered.     

Analytic solutions of the space–time conformable fractional Klein–Gordon equation in general form

ABSTRACT

The Klein–Gordon equation plays an important role in mathematical physics. In this paper, a direct method which is very effective, simple, and convenient, is presented for solving the conformable fractional Klein–Gordon equation. Using this analytic method, the exact solutions of this equation are found in terms of the Jacobi elliptic functions. This method is applied to both time and space fractional equations. Some solutions are also illustrated by the graphics.     

Coupled fractional nonlinear differential equations and exact Jacobian elliptic solutions for exciton–phonon dynamics

An improved quantum model for exciton–phonon dynamics in an α-helix is investigated taking into account the interspine coupling and the influence of power-law long-range exciton–exciton interactions. Having constructed the model Hamiltonian, we derive the lattice equations and employ the Fourier transforms to go in continuum space showing that the long-range interactions (LRI) lead to a nonlocal integral term in the equations of motion. Indeed, the non-locality originating from the LRI results in the dynamic equations with space derivatives of fractional order. New theoretical frameworks are derived, such that: fractional generalization of coupled Zakharov equations, coupled nonlinear fractional Schrödinger equations, coupled fractional Ginzburg–Landau equations, coupled Hilbert–Zakharov equations, coupled nonlinear Hilbert–Ginzburg–Landau equations, coupled nonlinear Schrödinger equations and coupled nonlinear Hilbert–Schrödinger equations. Through the F-expansion method, we derive a set of exact Jacobian solutions of coupled nonlinear Schrödinger equations. These solutions include Jacobian periodic solutions as well as bright and dark soliton which are important in the process of energy transport in the molecule. We also discuss of the impact of LRI on the energy transport in the molecule.     

Soliton solutions of the conformable fractional Zakharov–Kuznetsov equation with dual-power law nonlinearity

In this article, the Riccati sub equation method is employed to solve fractional Zakharov–Kuznetsov equation with dual-power law nonlinearity in the sense of the conformable derivative. By using this method, new exact solutions involving parameters, expressed by generalized hyperbolic functions are obtained. This method presents a wider applicability for handling nonlinear fractional wave equations.     

Klein–Gordon and Dirac Equations with Thermodynamic Quantities

We study the thermodynamic quantities such as the Helmholtz free energy, the mean energy and the specific heat for both the Klein–Gordon, and Dirac equations. Our analyze includes two main subsections: (1) statistical functions for the Klein–Gordon equation with a linear potential having Lorentz vector, and Lorentz scalar parts (2) thermodynamic functions for the Dirac equation with a Lorentz scalar, inverse-linear potential by assuming that the scalar potential field is strong (A ? 1). We restrict ourselves to the case where only the positive part of the spectrum gives a contribution to the sum in partition function. We give the analytical results for high temperatures.     

New exact solutions of the generalized Zakharov–Kuznetsov modified equal-width equation

In this paper, new exact solutions, including soliton, rational and elliptic integral function solutions, for the generalized Zakharov–Kuznetsov modified equal-width equation are obtained using a new approach called the extended trial equation method. In this discussion, a new version of the trial equation method for the generalized nonlinear partial differential equations is offered.     

q-nonlinear Schrodinger and q-nonlinear Klein–Gordon equations in the frame work of GUP

In this work, we first introduce a nonlinear Schrodinger and a nonlinear Klein–Gordon equations. Then we deform these equations to a the q-nonlinear Schrodinger and q-nonlinear Klein–Gordon equations. This is done using the formalism of generalized uncertainty principle (GUP). We also study the deformed nonlinear solutions.     

Auto-Bäcklund transformation and exact solutions for compound KdV-type and compound KdV–Burgers-type equations with nonlinear terms of any order

In this Letter, based on the idea of homogeneous balance (HB) method and with help of Mathematica, we obtain a new auto-Bäcklund transformation for compound KdV-type and compound KdV–Burgers-type equations with nonlinear terms of any order. Then based on the Bäcklund transformation, some solutions for these two equations are derived.     

Exact solutions of the Klein–Gordon equation in the Kerr–Newman background and Hawking radiation

This work considers the influence of the gravitational field produced by a charged and rotating black hole (Kerr–Newman spacetime) on a charged massive scalar field. We obtain exact solutions of both angular and radial parts of the Klein–Gordon equation in this spacetime, which are given in terms of the confluent Heun functions. From the radial solution, we obtain the exact wave solutions near the exterior horizon of the black hole, and discuss the Hawking radiation of charged massive scalar particles.     

Generalized Klein–Gordon field equations with octonion space-time (OST) algebra

The concept of space-time representation is redefined using the octonion space-time (OST) algebra. In this study, describing the properties of octonions and their possible connection with Euclidean space-times, the internal and external space-time events are represented within the OST algebra. Keeping in mind the octonionic dual-Euclidean space-times, we express the homogeneous field equations which leads to the symmetrical nature of internal and external space-times. We derive the generalized Proca–Maxwell equations for massive-dyons in the case of the OST algebra. Accordingly, we have obtained a new set of octonionic Klein–Gordon potential (KGP) and Klein–Gordon field (KGF) equations for massive dyons from the generalized Proca–Maxwell equations. This formalism demonstrates that the octonionic KGP and KGF equations can be expressed in a single equation and it is equivalent to energy-momentum relation for dyons. As such, we have made an attempt to write the conservation of Noetherian current from the octonionic Klein–Gordon equations.     

Traveling wave solutions of conformable time-fractional Zakharov–Kuznetsov and Zoomeron equations

To model physical phenomena more accurately, fractional order differential equations have been widely used. Investigating exact solutions of the fractional differential equations have become more important because of the applications in applied mathematics, mathematical physics, and other areas. In this work, by means of the trial solution method and complete discrimination system, exact traveling wave solutions of the conformable time-fractional Zakharov–Kuznetsov equation and conformable time-fractional Zoomeron equation have been obtained and also solutions have been illustrated. Finding exact solutions of these equations that are encountered in plasma physics, nonlinear optics, fluid mechanics, and laser physics can help to understand nature of the complex phenomena.     

Exact Solutions of the Klein–Gordon Equation with Hylleraas Potential

We present the exact solution of the Klein–Gordon with Hylleraas Potential using the Nikiforov–Uvarov method. We obtain explicitly the bound state energy eigenvalues and the corresponding eigen function for s-wave. The wave functions obtained are expressed in terms of Jacobi polynomials.     

Darboux Transformations for Energy-Dependent Potentials and the Klein–Gordon Equation

We construct explicit Darboux transformations for a generalized Schrödinger-type equation with energy-dependent potential, a special case of which is the stationary Klein–Gordon equation. Our results complement and generalize former findings (Lin et al., Phys Lett A 362:212–214, 2007).     

The fractional coupled KdV equations： Exact solutions and white noise functional approach

Variable coefficients and Wick-type stochastic fractional coupled KdV equations are investigated. By using the mod- ified fractional sub-equation method, Hermite transform, and white noise theory the exact travelling wave solutions and white noise functional solutions are obtained, including the generalized exponential, hyperbolic, and trigonometric types.