Soliton Stability in a Generalized Sine-Gordon Potential

We study stability of a generalized sine-Gordon model with two coupled scalar fields in two dimensions. Topological soliton solutions are found from the first-order equations that solve the equations of motion. The perturbation equations can be cast in terms of a Schrödinger-like operators for fluctuations and their spectra are calculated.     

Pure soliton solutions of some nonlinear partial differential equations

A general approach is given to obtain the system of ordinary differential equations which determines the pure soliton solutions for the class of generalized Korteweg-de Vries equations (cf. [6]). This approach also leads to a system of ordinary differential equations for the pure soliton solutions of the sine-Gordon equation.     

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.     

Solutions of local fractional sine-Gordon equations

In this paper, we study sine-Gordon equation in order to obtain exact solitary wave solutions in the domain of fractional calculus. By using the definition of conformable fractional derivative, we obtain analytical solutions of time, space and time-space fractional sine-Gordon equations. We analyze graphically the effect of fractional order on evolution of the kink and antikink type solitons.     

Soliton equations and pseudospherical surfaces

All the soliton equations in 1 + 1 dimensions that can be solved by the AKNS 2 × 2 inverse scattering method (for example, the sine-Gordon, KdV or modified KdV equations) are shown to describe pseudospherical surfaces, i.e., surfaces of constant negative Gaussian curvature. This result provides a unified picture of all these soliton equations. Geometrical interpretations of characteristic properties like infinite numbers of conservation laws and Bäcklund transformations and of the soliton solutions themselves are presented. The important role of scale transformations as generating one parameter families of pseudospherical surfaces is pointed out.     

Scattering of solitons with fermion number ±12

In a model of a Dirac field coupled to a sine-Gordon system, we have soliton solutions with half-integral fermion number. Scattering problems of such solitons are investigated. For soliton-soliton and soliton-antisoliton solutions, we find explicit solutions to the Dirac equations which are localized at the position of the (anti-)solitons. It is shown that the fermion number is generally transferred between the soliton and the (anti-)soliton in the scattering process.     

Soliton Asymptotics of Solutions of the Sine-Gordon Equation

An asymptotic analysis of the Marchenko integral equation for the sine-Gordon equation is presented. The results are used for a construction of soliton asymptotics of decreasing and some non-decreasing solutions of the sine-Gordon equation. The soliton phases are shown to have an additional shift with respect to the reflectionless case caused by the non-zero reflection coefficient of the corresponding Dirac operator. Explicit formulas for the phases are also obtained. The results demonstrate an interesting phenomenon of splitting of non-decreasing solutions into an infinite series of asymptotic solitons.     

Exact solutions of stochastic fractional Korteweg de–Vries equation with conformable derivatives

We deal with the Wick-type stochastic fractional Korteweg de–Vries(KdV) equation with conformable derivatives.With the aid of the Exp-function method, white noise theory, and Hermite transform, we produce a novel set of exact soliton and periodic wave solutions to the fractional KdV equation with conformable derivatives. With the help of inverse Hermite transform, we get stochastic soliton and periodic wave solutions of the Wick-type stochastic fractional KdV equation with conformable derivatives. Eventually, by an application example, we show how the stochastic solutions can be given as Brownian motion functional solutions.     

Analytical treatments of the space–time fractional coupled nonlinear Schrödinger equations

Under investigation in this work is a (\(2+1\))-dimensional the space–time fractional coupled nonlinear Schrödinger equations, which describes the amplitudes of circularly-polarized waves in a nonlinear optical fiber. With the aid of conformable fractional derivative and the fractional wave transformation, we derive the analytical soliton solutions in the form of rational soliton, periodic soliton, hyperbolic soliton solutions by four integration method, namely, the extended trial equation method, the \(\exp (-\,\Omega (\eta ))\)-expansion method and the improved \(\tan (\phi (\eta )/2)\)-expansion method and semi-inverse variational principle method. Based on the the extended trial equation method, we derive the several types of solutions including singular, kink-singular, bright, solitary wave, compacton and elliptic function solutions. Under certain condition, the 1-soliton, bright, singular solutions are driven by semi-inverse variational principle method. Based on the analytical methods, we find that the solutions give birth to the dark solitons, the bright solitons, combine dark-singular, kink, kink-singular solutions with fractional order for nonlinear fractional partial differential equations arise in nonlinear optics.     

Connections defining representations of zero curvature and the solitons of sine-Gordon and Korteweg-de Vries equations

It is claimed that solutions of travelling-wave type (and, in particular, soliton solutions) of partial differential equations can be created by using connections defining representations of zero curvature. In this paper, we construct solitons of the sine-Gordon and Korteweg-de Vries equations. By previous results of the author, the connections defining representations of zero curvature for a given differential equation generate Bäcklund transformations for this equation. It can be shown that the well-known Lax system (the so-called Lax pair) for the Korteweg-de Vries equation is a special case of a Bäcklund system (i.e., the system of partial differential equations defining a Bäcklund transformation). Note that the creation of solitons by means of the inverse scattering method is in fact a creation of solitons by means of the Lax system (without using connections defining the representations of zero curvature from the very beginning). Moreover, the inverse scattering method is essentially more labor-consuming than the method suggested in the present paper. Further, it is not required to involve any physical notions when using the suggested method. In the final section of the paper, we consider the so-called 2-soliton solutions of sine-Gordon and Korteweg — de Vries equations. Here we systematically use the invariant analytic method developed by G. F. Laptev, which is well-known in differential geometry under the title of Cartan-Laptev method.     

Dynamics of generalized sine-Gordon soliton in inhomogeneous media

In this paper we introduce a few novel generalized sine-Gordon equations and study the dynamics of its solitons in inhomogeneous media. We consider length, mass, gravitational acceleration and spring stiffness of a coupled pendulums chain as a function of position x. Then in the continuum limit we derive semi-analytical and numerical soliton solutions of the modified sine-Gordon equation in the inhomogeneous media. The obtained results confirm that the behavior of solitons in these media is similar to that of a classical point particle moved in an external potential.     

Study of a soliton in a medium

The solution of the sine-Gordon equation for a medium is studied in a simplified “quasi-linear” approximation. Several aspects of this approximation are explored and compared with other approximate solutions for the sine-Gordon soliton.     

奇异物质与暗能量作用的sine-Gordon孤子星模型

研究了具有奇异物质和暗能量作用的sine-Gordon孤子星模型,根据场方程计算了物态方程的解和星体质量,发现物质密度和压强与孤子态和星体质量有关.另外,还对星体平衡和暗能量的稳定性质进行了分析和讨论,结果表明孤子星内部以奇异物质与暗能量的混合态形式存在. 关键词： 奇异物质 暗能量 sine-Gordon孤子星     

Solitons in Stimulated Raman Scattering

The semiclassical equations of SRS are derived by the following method: The many-level molecule is described by the Schrödinger equation and subsequently all amplitudes but those of the essential two levels are eliminated by approximate integration. Especially the case of equal group velocities for pump and Stokes wave is considered. Terms describing a frequency shift due to dynamical Stark effect and a change of refraction index by vibrational excitation can be removed by a suitable transformation. Under various, physically different, assumptions the system of molecular and field equations is reduced to the equations describing a quasimonochromatic pulse propagating in a resonant two-level system, and in this way we can find a rich manifold of solutions. E.g., the 2π-pulse of self-induced transparency can be transferred to SRS. By other assumptions we are led to equations defining the Bäcklund transformation for the sine-Gordon equation. For this case also soliton type solutions can be constructed.     

修正cKdV方程组的孤立波结构及其稳定性

利用函数展开法求解修正耦合KdV(Coupled KdV,cKdV)方程组,得到几类孤立波解,包括扭结型-钟型、双扭结型、双钟型以及双扭结-双钟型结构的单孤子解.在不同的极限情况下,这些解分别退化为修正cKdV方程的扭结状或钟状孤波解.对孤立波的稳定性进行了数值研究,结果表明:修正cKdV方程既存在稳定的孤立波解,也存在不稳定的孤立波解.     

Some Generalized Scalar Field Models with Abundant Soliton-Like configurations

Some generahed scalar field models are presented. It is shown that there exist generally three types of soliton solutions for the extended models. The form of the first type of the soliton solutions is model-independent. An arbitrary constant can be included in the second type of the soliton excitations which is model-dependent. The third type of the soliton solutions is also modeldependent but with no arbitrary constant. The extended φ⁴ model and sine-Gordon model are studied in detail.