Radial asymptotically periodic solutions of the Kuramoto-Sivashinsky equation

Radial steady solutions of the Kuramoto-Sivashinsky equation are studied. It is shown that there exist solutions that approach at infinity the one-dimensional periodic solutions. Both hyperbolic and elliptic periodic solutions are considered.     

Davey-Stewartson方程组的包络周期解和孤立波解

应用Jacobi椭圆函数展开法,求得了Davey-Stewartson方程组的包络周期解和孤立波解． 关键词： Davey-Stewartson方程 Jacobi椭圆函数 包络周期解 孤立波解     

Exact periodic wave solutions to the generalized Nizhnik-Novikov-Veselov equation

The extended mapping method with symbolic computation is developed to obtain exact periodic wave solutions to the generalized Nizhnik-Novikov-Veselov equation. Limit cases are studied and new solitary wave solutions and triangular periodic wave solutions are obtained. The method is applicable to a large variety of non-linear partial differential equations, as long as odd-and even-order derivative terms do not coexist in the equation under consideration.     

Compacton and periodic wave solutions of the non-linear dispersive Zakharov-Kuznetsov equation

In this paper, the nonlinear dispersive Zakharov-Kuznetsov equation is solved by using the sine-cosine method. As a result, compactons, periodic, and singular periodic wave solutions are found.      

Diffusive stability of spatial periodic solutions of the Swift-Hohenberg equation

We are interested in the nonlinear stability of the Eckhaus-stable equilibria of the Swift-Hohenberg equation on the infinite line with respect to small integrable perturbations. The difficulty in showing stability for these stationary solutions comes from the fact that their linearizations possess continuous spectrum up to zero. The nonlinear stability problem is treated with renormalization theory which allows us to show that the nonlinear terms are irrelevant, i.e. that the fully nonlinear problem behaves asymptotically as the linearized one which is under a diffusive regime.     

On the stability of periodic solutions of the generalized Benjamin-Bona-Mahony equation

We study the stability of a four-parameter family of spatially periodic traveling wave solutions of the generalized Benjamin-Bona-Mahony equation under two classes of perturbations: periodic perturbations with the same periodic structure as the underlying wave, and long wavelength localized perturbations. In particular, we derive necessary conditions for spectral instability under perturbation for both classes of perturbations by deriving appropriate asymptotic expansions of the periodic Evans function, and we outline a theory of nonlinear stability under periodic perturbations based on variational methods which effectively extends our periodic spectral stability results.     

New binary travelling-wave periodic solutions for the modified KdV equation

In this Letter, the modified Korteweg-de Vries (mKdV) equations with the focusing (+) and defocusing (−) branches are investigated, respectively. Many new types of binary travelling-wave periodic solutions are obtained for the mKdV equation in terms of Jacobi elliptic functions such as sn(ξ,m)cn(ξ,m)dn(ξ,m) and their extensions. Moreover, we analyze asymptotic properties of some solutions. In addition, with the aid of the Miura transformation, we also give the corresponding binary travelling-wave periodic solutions of KdV equation.     

On some single-hump solutions of the short-pulse equation and their periodic generalizations

In the present work, we consider both localized (e.g. peakon and breather) and extended waveforms (peakon-lattice and breather-lattice, as well as some periodic ones) that arise in the context of the short-pulse equation, as emanating from their sine-Gordon equation analogs. Through direct numerical simulations, we find that the most robust solution is the breather, although some of the single-hump variants of the periodic solutions may be preserved upon the time dynamics as well. Multi-peakon, as well as multi-breather and multi-hump profiles more generally are found to be subject to symmetry-breaking instabilities and are, thus, less robust.     

Physical solutions and Mathematical properties of the Lorentz-Dirac equation

The rectilinear motion of a charge obeying the Lorentz-Dirac equation and moving in an external electrostatic field is analyzed. Imposing no additional constraints, it is shown that the asymptotic motion of this charge is inertial. Energy conservation by the entire process is proved. Mathematical arguments concerning the irrelevance of runaway solutions are discussed.     

Riemann theta function periodic wave solutions for the variable-coefficient mKdV equation

<正>In this paper,a variable-coefficient modified Korteweg-de Vries(vc-mKdV) equation is considered.Bilinear forms are presented to explicitly construct periodic wave solutions based on a multidimensional Riemann theta function,then the one and two periodic wave solutions are presented,and it is also shown that the soliton solutions can be reduced from the periodic wave solutions.     

Solitons and periodic solutions for the fifth-order KdV equation with the Exp-function method

In this Letter the Exp-function method is applied to obtain new generalized solitonary solutions and periodic solutions of the fifth-order KdV equation. It is shown that the Exp-function method, with the help of symbolic computation, provides a powerful mathematical tool for solving nonlinear equations arising in mathematical physics.     

Exact periodic and explicit solutions of the conformable time fractional Ginzburg Landau equation

The Ginzburg–Landau (GL) equation is one of the most important nonlinear equation in physics. It is used to model a vast variety of phenomena in physics like nonlinear waves, second order phase transitions, Bose–Einstein condensation, superfluidity, superconductivity, liquid crystals and strings in field theory. In this work, new exact, periodic and explicit solutions of a time fractional GL equation involving conformable fractional derivatives with Kerr law nonlinearity have been found. The Kerr law nonlinearity is due to the non-harmonic motion of electrons under the influence of an applied field. To determine the solution of the model, we have employed a couple of integration algorithms, solitary wave ansatz and \(\exp (-\varphi ({\chi }\))) methods. New periodic and hyperbolic soliton solutions are found as well as the constraint condition for the existence of the solution.     

Analytic properties of the solutions of Teukolsky's equation

A method is proposed for studying the low-frequency solutions of Teukolsky's equation, describing the behavior of massless fields in the vicinity of a Kerr black hole. Approximate basis solutions and an integral equation for exact solutions are constructed. The analytic properties of the exact solutions and the asymptotic behavior of the leading nonanalytic parts are investigated and the approximate solutions are estimated.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 47–51, July, 1981.The author is grateful to Professor K. A. Piragas and the participants of the seminar directed by him for their attention to the present work and for useful discussions.     

The linear Boltzmann equation properties and solutions

This work presents results on the transport equation obtained recently. Properties and distribution functions obeying the Boltzmann equation in one and many-dimensional spaces are derived and discussed. New polynomials have been found that under certain conditions represent solutions of the linear transport equation. A number of structural and spectral theorems have been demonstrated permitting a better understanding of the equation. It is shown that the streaming operator, $\text{Ω·Δ + σ(v)Σ}{}_{\mathrm{tot}}\text{(v)}$, maps a class of functions {$\text{;ψ(}\text{x, v}\text{)}$} of two variables onto another class of functions, {$\text{ψ(}\text{x}\text{)}$}, depending only on one variable. The distribution functions obtained here satisfy rigorously homogeneous or inhomogeneous Dirichlet conditions on the boundary surface of the convex system for any order of the polynomial representation. This is obtained by using the new polynomials which are characterised by particular structural properties. In many-dimensional cases the polynomials become operators with tempered distributional character. Numerical evaluations are given in the one dimensional case for both isotropic and anisotropic scattering. An application of the theory is also given for the heterogeneous system of plane geometry. This paper is organized in three parts. Part A gives an introduction to the present method and indicates the way leading from the original to the linearised Boltzmann equation. Other solution methods are comparatively discussed. Part B is dealing with the one-dimensional equation and the eigenvalues and eigenfunctions are found for various physical conditions. Degenerate kernels have been used throughout. Both the critical and non-critical problems have been solved and results are presented in form of graphs and tables. Part C proceeds to the examination of some problems in spaces of dimension p > 1. The main advantages of the structural approach are that (i) the solution is found in an elementary way completely analytically. (ii) The boundary conditions are rigorously satisfied. (iii) The heterogeneous system is very easily solved. (iv) The eigenvalues are found algebraically.     

圆杆波导中的一个非线性波动方程及准确周期解

在小变形条件下，采用Cox的非线性应力应变关系，计及横向Possion效应，借助Hamilton变分原理导出了非线性弹性圆杆波导中的纵向波动方程. 利用Jacobi椭圆余弦函数展开法，对该方程与截断的非线性波动方程进行求解，得到了两类非线性波动方程的准确周期解，它们可以进一步退化为孤波解. 关键词： 非线性波 Possion效应 Jacobi椭圆余弦函数     

Exact periodic and blow up solutions for 2D Ginzburg-Landau equation

New exact periodic wave solutions for the 2D Ginzburg-Landau equation are obtained using the homogeneous balance principle and general Jacobi elliptic-function method. Furthermore, a blow up solution is provided. At the end, some properties about these solutions are showed by the graphs.     

(3+1)维Nizhnik-Novikov-Veselov方程的孤子解和周期解

采用行波法约化方程,建立一种变换关系,把求解(3+1)维NizhnikNovikovVeselov(NNV)方程的解转化为求解一维非线性KleinGordon方程的解,从而得到了(3+1)维NNV方程的孤子解和周期解. 关键词： (3+1)维Nizhnik-Novikov-Veselov方程 非线性Klein-Gordon方程 孤子解 周期解