电磁场在超晶中传播的孤子解

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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.     

Numerical solution of sine-Gordon equation by variational iteration method

In this Letter, variational iteration method (VIM) is applied to obtain approximate analytical solution of the sine-Gordon equation without any discretization. Comparisons with the exact solutions reveal that VIM is very effective and convenient.     

Electromagnetic solitons in a system of quantum dots taking into account the Hubbard interaction

We consider the Maxwell equations for an electromagnetic field propagating in a solid with a three-dimensional superlattice of quantum dots linked by strong tunneling along one axis, where electrons with different spin projections are affected by the strong Coulomb repulsion at a single site. We obtain a phenomenological equation in the form of the classical 1+1-dimensional sine-Gordon equation. Electrons are considered within the framework of quantum formalism taking into account the changes in the dispersion law provided by the presence of Coulomb interactions. The phenomenological equation is solved numerically, and the influence of Coulomb repulsion and the degree of band population on the propagation of ultra-short optical pulses is analyzed.     

Separation Transformation and New Exact Solutions of the (N+1)-dimensional Dispersive Double sine-Gordon Equation

In this paper, the separation transformation approach is extended to the (N+1)-dimensional dispersive double sine-Gordon equation arising in many physical systems such as the spin dynamics in the B phase of ³He superfluid. This equation is first reduced to a set of partial differential equations and a nonlinear ordinary differential equation. Then the general solutions of the set of partial differential equations are obtained and the nonlinear ordinary differential equation is solved by F-expansion method. Finally, many new exact solutions of the (N+1)-dimensional dispersive double sine-Gordon equation are constructed explicitly via the separation transformation. For the case of N>2, there is an arbitrary function in the exact solutions, which may reveal more novel nonlinear structures in the high-dimensional dispersive double sine-Gordon equation.     

Separation Transformation and New Exact Solutions of the (N+1)-dimensional Dispersive Double sine-Gordon Equation

In this paper,the separation transformation approach is extended to the(N + 1)-dimensional dispersive double sine-Gordon equation arising in many physical systems such as the spin dynamics in the B phase of 3 He superfluid.This equation is first reduced to a set of partial differential equations and a nonlinear ordinary differential equation.Then the general solutions of the set of partial differential equations are obtained and the nonlinear ordinary differential equation is solved by F-expansion method.Finally,many new exact solutions of the(N + 1)-dimensional dispersive double sine-Gordon equation are constructed explicitly via the separation transformation.For the case of N 2,there is an arbitrary function in the exact solutions,which may reveal more novel nonlinear structures in the high-dimensional dispersive double sine-Gordon equation.     

A new auxiliary equation and exact travelling wave solutions of nonlinear equations

A new auxiliary ordinary differential equation and its solutions are used for constructing exact travelling wave solutions of nonlinear partial differential equations in a unified way. The main idea of this method is to take full advantage of the auxiliary equation which has more new exact solutions. More new exact travelling wave solutions are obtained for the quadratic nonlinear Klein–Gordon equation, the combined KdV and mKdV equation, the sine-Gordon equation and the Whitham–Broer–Kaup equations.     

sine-Gordon型方程的无穷序列新精确解

为了获得sine-Gordon型方程的无穷序列精确解,给出三角函数型辅助方程和双曲函数型辅助方程及其Bäcklund变换和解的非线性叠加公式,借助符号计算系统Mathematica,构造了sine-Gordon方程、mKdV-sine-Gordon方程、(n+1)维双sine-Gordon方程和sinh-Gordon方程的无穷序列新精确解.其中包括无穷序列三角函数解、无穷序列双曲函数解、无穷序列Jacobi椭圆函数解和无穷序列复合型解. 关键词： sine-Gordon型方程 解的非线性叠加公式 辅助方程 无穷序列精确解     

Degasperis-Procesi 方程的无穷序列尖峰孤立波解

本文为了构造非线性发展方程的无穷序列尖峰精确解,给出了Riccati方程的Bäcklund 变换和解的非线性叠加公式,并借助符号计算系统Mathematica,用Degasperis-Procesi方程为应用实例,构造了无穷序列尖峰孤立波解和无穷序列尖峰周期解. 关键词： Riccati方程 解的非线性叠加公式 尖峰孤立波解 Degasperis-Procesi 方程     

New Doubly Periodic Waves of the （2＋1）-Dimensional Double Sine-Gordon Equation

New exact solutions of the （2 ＋1）-dimensional double sine-Gordon equation are studied by introducing the modified mapping relations between the cubic nonlinear Klein-Gordon system and double sine-Gordon equation. Two arbitrary functions are included into the Jacobi elliptic function solutions. New doubly periodic wave solutions are obtained and displayed graphically by proper selections of the arbitrary functions.     

Dynamics of electromagnetic pulses with wide spectra in semiconductor superlattices

We considered the Maxwell equations for electromagnetic-field propagation in a solid with a one-dimensional semiconductor superlattice of quantum dots in the case where the spectral width of the electromagnetic pulse is sufficient to excite transitions between different minizones. A phenomenological equation was obtained in the form of the classical one-dimensional sine-Gordon equation with the perturbation caused by quantum transitions between the minizones. Quantum behavior of electrons was considered using the microscopic Hamiltonian, in the assumption that the pulse duration is small enough for the phonon effects to be neglected. The equation obtained was analyzed numerically, and cases where the adiabatic perturbation theory for the sine-Gordon equation can be used were found. Numerical solutions were obtained, and the domain where transitions between the minizones play a significant role in the electromagnetic-pulse dynamics was found. Talk presented at the oral issue of J. Russ. Laser Res. dedicated to the memory of Professor Vladimir A. Isakov, Professor Alexander S. Shumovsky, and Professor Andrei V. Vinogradov held in Moscow February 21–22, 2008.     

An irrational trial equation method and its applications

An irrational trial equation method was proposed to solve nonlinear differential equations. By this method, a number of exact travelling wave solutions to the Burgers-KdV equation and the dissipative double sine-Gordon equation were obtained. A more general irrational trial equation method was discussed, and many exact solutions to the Fujimoto-Watanabe equation were given.     

Nonlinear theory of waves in solid state with cardinally changing crystalline structure

A nonlinear theory of propagating periodic and nonlinear solitary waves (like kinks and solitons) related to the motion of defects in crystals and of specific periodic waves into which the former ones transform in the field of the compression stress was developed. The role of intense tension stress leading to the heavy structural rearrangement of the crystal as a result of the effect of the external stress on the interatomic potential barriers was taken into account as well. Crystals with a complex lattice consisting of two sublattices were considered. Arbitrarily large displacements of sublattices were analyzed. The nonlinear theory is based on an additional element of the translational symmetry typical for complex lattices but not introduced earlier in solid-state physics. The variational equations of macroscopic and microscopic displacements turn out to be a nonlinear generalization of the linear equations of acoustic and optical modes obtained by Carman, Born, and Huang Kun. The microscopic displacement fields are described by the nonlinear sine-Gordon equation. In the one-dimensional case, exact solutions of the nonlinear equations were found and their features were revealed. In the case of two-dimensional (2+1) fields, new methods of the exact solutions of the sine-Gordon equation were developed. They describe the interaction of the nonlinear waves with the structural inhomogeneities of solid state due to the external fields of stress and deformations.     

The tanh method and a variable separated ODE method for solving double sine-Gordon equation

We use the tanh method and a variable separated ODE method for solving the double sine-Gordon equation and a generalized form of this equation. Several exact travelling wave solutions are formally derived. The two methods provide distinct solutions of different physical structures.     

(n+1)维双Sine-Gordon方程的新精确解

给出包含第一种椭圆方程的三角函数型辅助方程及其解的叠加公式.在一般函数变换下,借助符号计算系统Mathematica,构造了(n+1)维双sine-Gordon方程新的Jacobi椭圆函数精确解.这些解包括了行波变换下的Jacobi椭圆函数精确解、精确孤立波解和三角函数解.     

Analytical solution of fluxons in a non-homogeneous Josephson junction

In order to study the fluxon behavior in a Josephson junction, a dielectric with variable thickness was inserted between two superconductors, which can be position-dependent. An analytical solution of the sine-Gordon equation is obtained. Subsequently, the relation between the kinetic and potential energy of a fluxon is obtained along with the analytic relationships between the velocity of the fluxon and the dielectric thickness. Finally, an exact topological soliton solution is obtained for the sine-Gordon equation with and without strong perturbations through the use of the ansatz method.     

强阻尼广义sine-Gordon方程特征问题的变分迭代法

研究了在数学、力学中广泛出现的一类非线性强阻尼广义sine-Gordon扰动微分方程问题. 首先, 引入行波变换, 求出退化方程的精确解. 再构造一个泛函, 创建了一个变分迭代算法, 最后, 求出原非线性强阻尼广义sine-Gordon扰动微分方程问题的近似行波解析解. 用变分迭代法可得到的各次近似解, 具有便于求解、精度高等特点. 求得的近似解析解弥补了单纯用数值方法的模拟解的不足.     

An exact (2 + 1)-dimensional optical soliton with spatially modulated nonlinearity and an external potential

An exact (2 + 1)-dimensional spatial optical soliton of the nonlinear Schrödinger equation with a spatially modulated nonlinearity and a special external potential is discovered in an inhomogeneous nonlinear medium, by utilizing the similarity transformation. Exact analytical solutions are constructed by the products of Whittaker functions and the bright and dark soliton solutions of the standard stationary nonlinear Schrödinger equation. Some examples of such composed solutions are given, in which these spatial solitons display different localized structures. Numerical calculation shows that the soliton is stable in propagating over long distances, thus also confirming the validity of the exact solution.