Inverse Hyperbolic Problems and Optical Black Holes

In this paper we state a uniqueness theorem for the inverse hyperbolic problem in the case of a finite time interval. We apply this theorem to the inverse problem for the equation of the propagation of light in a moving medium (the Gordon equation). Then we study the existence of black and white holes for the general second order hyperbolic equation and for the Gordon equation and we discuss the impact of this phenomenon on the inverse problems.     

Exact solutions of nonlinear Schrodinger’s equation with dual power-law nonlinearity by extended trial equation method

In this paper, we acquire the soliton solutions of the nonlinear Schrodinger’s equation with dual power-law nonlinearity. Primarily, we use the extended trial equation method to find exact solutions of this equation. Then, we attain some exact solutions including soliton solutions, rational and elliptic function solutions of this equation using the extended trial equation method.     

Remarks concerning the derivation and the expansion of the master equation

The present work consist of two parts: In the first part we apply the method of quasilinearization to the differential equation describing the time development of the quantum-mechanical probability density. In this way we derive the master equation without resorting to perturbation theory. In the second part of the paper, for a general form of the master equation which is an integro-differential equation, we test the accuracy of the Fokker-Planck approximation with the help of a solvable model. Then we study an alternative way of reducing the integro-differential equation to a partial differential equation. By expanding the transition probability W(q, q′), and the distribution function in terms of a complete set of functions, we show that for certain forms of W(q, q′), the master equation can be transformed exactly to partial differential equations of finite order.     

Simple Soliton Solution Method for the Combined KdV and MKdV Equation

Malfliet first proposed a simple solution method for the multisoliton solutionofthe KdV equation. Abdel-Rahman used Malfliet's method in a slightly modifiedform, and gave the multisoliton solution of the mKdV equation, RLW equation,Boussinesq equation, and modified Boussinesq equation. In this paper, we solvethe soliton solution of the cKdV=nmKdV equation by using this method.     

A Super Generalization of KdV6 Equation

In this paper, we propose a fermionic generalization of KdV6 equation and study its integrability. Moreover, we show that this equation is a constraint Hamiltonian flow on the coadjoint orbit of Neveu-Schwarz superalgebra.     

Pulses in a complex Ginzburg-Landau system: Persistence under coupling with slow diffusion

The Ginzburg-Landau (GL) equation is essential for understanding the dynamics of patterns in a wide variety of physical contexts. It governs the evolutions of small amplitude instabilities near criticality. If the instabilities are, however, driven by two coupled instability mechanisms, of which one corresponds with a neutrally stable mode, their evolution is described by a GL equation coupled to a diffusion equation.In this paper, we study the influence of an additional diffusion equation on the existence of pulse solutions in the complex GL equation. In light of recently developed insights into the effect of slow diffusion on the stability of pulses, we consider the case of slow diffusion, i.e., in which the additional diffusion equation acts on a long spatial scale.In previous work [A. Doelman, G. Hek, N. Valkhoff, Stabilization by slow diffusion in a real Ginzburg-Landau system, J. Nonlinear Sci. 14 (2004) 237-278; A. Doelman, G. Hek, N.J.M. Valkhoff, Algebraically decaying pulses in a Ginzburg-Landau system with a neutrally stable mode, Nonlinearity 20 (2007) 357-389], we restricted ourselves to a model with both real coefficients and, more importantly, a real amplitude A rather than the complex-valued A that is needed to completely describe the pattern formation near criticality. In this simpler setting, we proved that pulse solutions of the GL equation can both persist and be stabilized under coupling with a slow diffusion equation. In the current work, we no longer make these restrictions, so that the problem is higher-dimensional and intrinsically harder. By a combination of a geometrical approach and explicit perturbation analysis, we consider the persistence of the solitary pulse solution of the GL equation under coupling with the additional diffusion equation. In the two limiting situations of the nearly real GL equation and the near nonlinear Schrödinger equation, we show that the pulse solutions can indeed persist under this coupling.     

Euler-Poincaré Formalism of (Two Component) Degasperis-Procesi and Holm-Staley type Systems

Abstract

In this paper we propose an Euler-Poincaré formalism of the Degasperis and Procesi (DP) equation. This is a second member of a one-parameter family of partial differential equations, known as b-field equations. This one-parameter family of pdes includes the integrable Camassa-Holm equation as a first member. We show that our Euler-Poincaré formalism exactly coincides with the Degasperis-Holm-Hone (DHH) Hamiltonian framework. We obtain the DHH Hamiltonian structues of the DP equation from our method. Recently this new equation has been generalized by Holm and Staley by adding viscosity term. We also discuss Euler-Poincaré formalism of the Holm-Staley equation. In the second half of the paper we consider a generalization of the Degasperis and Procesi (DP) equation with two dependent variables. we study the Euler-Poincaré framework of the 2-component Degasperis-Procesi equation. We also mention about the b-family equation.     

On the dynamics of a continuum spin system

For a one-dimensional system of classical spins with nearest neighbour Heisenberg interaction we derive the equation of motion for each three-dimensional spin vector. In the continuum limit where the spins lie dense on a line this set of equations reduces to a nonlinear partial differential equation. In addition to spin-wave solutions we obtain some other special solutions of this equation. In particular we find solitary waves having total energy localised in a finite region, with velocity of propagation inversely proportional to the width of this region. Solutions of still another type are shown to have a diffusive character. The stability of such solutions and the possibility of interaction of two or more solitary waves have not yet been studied.     

Painlevé Property and Allowed Transformations of the Variable-Coefficient Zakharov-Kuznetsov Equation

In this letter, we discuss the Painlevé property and allowed transformation for the variablecoefficient Zakharov-Kuznetsov equation which governs nonlinear ion-acoustic wqves in a magnetized plasma. The general solution of the singular manifold equation and stability solution of the VCZK equation are obtained. To prove some information of the integrability of the ZK equation, we prove the constraints that the variable-coefficient functions for the equation to possess the Painlevé property are not equivalent in order that the equations may be transformed into the constant coefficient equation. So we confirm that the ZK equation is not integrable.     

Evolution equations of the expected phase space ion densities and correlation functions in a D+T plasma

Summary In this paper we formualte a master equation approach describing a D+T thermonuclear plasma in a lumped phase space. From the first moments of this master equation and performing the pass to the continuous limit the evolution equations for the expected phase space ion densities emerge. Also we have obtained the evolution equations of the equal time correlation and covariance functions. Finally we have deduced the hydrodynamic equations that arise from a master equation approach.     

一种产生Einstein约化场方程解的方法

本文给出了在Ernst场方程的解空间上的一种新变换,并且研究了这种变换与Ernst场方程解之间的关系。证明了在这种变换下,Ernst场方程是不变的,即由我们的这种变换可产生Ernst场方程的新解。最后还讨论了这种变换与Virasoro代数的关系。 关键词：     

A Note on the Relationship Between Solutions of Einstein, Ramanujan and Chazy Equations

The Einstein equation for the Friedmann-Robertson-Walker metric plays a fundamental role in cosmology. The direct search of the exact solutions of the Einstein equation even in this simple metric case is sometime a hard job. Therefore, it is useful to construct solutions of the Einstein equation using a known solutions of some other equations which are equivalent or related to the Einstein equation. In this work, we establish the relationship the Einstein equation with two other famous equations namely the Ramanujan equation and the Chazy equation. Both these two equations play an important role in the number theory. Using the known solutions of the Ramanujan and Chazy equations, we find the corresponding solutions of the Einstein equation.     

An Algebraic Method for Constructing Exact Solutions to Difference-Differential Equations

In this paper, we present a method to solve difference differential equation(s). As an example, we apply this method to discrete KdV equation and Ablowitz-Ladik lattice equation. As a result, many exact solutions are obtained with the help of Maple including soliton solutions presented by hyperbolic functions sinh and cosh, periodic solutions presented by sin and cos and rational solutions. This method can also be used to other nonlinear difference-differential equation(s).     

Symmetry properties,conservation laws and exact solutions of time-fractional irrigation equation

In this paper, we deal with the complete algebra of Lie point symmetries for the generalized model of an irrigation system of fractional order. By means of Lie symmetry method, the vector fields has been investigated which are utilized for obtaining the conservation laws of equation. In addition, through the sub-equation method, we construct some exact solutions for the considered equation by reducing the fractional partial differential equation to a ordinary fractional differential equation.     

A note on the prolongation structure of the cubically nonlinear integrable Camassa-Holm type equation

In this Letter, we formulate an exterior differential system for the newly discovered cubically nonlinear integrable Camassa-Holm type equation. From the exterior differential system we establish the integrability of this equation. We then study Cartan prolongation structure of this equation. We also discuss the method of identifying conservation laws and Bäcklund transformation for this equation from the identified exterior differential system.     

Symmetry Groups and Exact Solutions of New （4＋1）-Dimensional Fokas Equation

In this paper, we investigate symmetries of the new （4＋1）-dimensional Fokas equation, including point symmetries and the potential symmetries. We firstly employ the algorithmic procedure of computing the point symmetries. And then we transform the Fokas equation into a potential system and gain the potential symmetries of Fokas equation. Finally, we use the obtained point symmetries wave solutions and other solutions of the Fokas equation. and some constructive methods to get some doubly periodic In particular, some solitary wave solutions are also given.     

Perturbation,symmetry analysis,Bäcklund and reciprocal transformation for the extended Boussinesq equation in fluid mechanics

In this work, we study a generalized double dispersion Boussinesq equation that plays a significant role in fluid mechanics, scientific fields, and ocean engineering. This equation will be reduced to the Korteweg–de Vries equation via using the perturbation analysis. We derive the corresponding vectors, symmetry reduction and explicit solutions for this equation. We readily obtain B?cklund transformation associated with truncated Painlevéexpansion. We also examine the related conservation laws of this equation via using the multiplier method. Moreover, we investigate the reciprocal B?cklund transformations of the derived conservation laws for the first time.     

Direct linearization of nonlinear difference-difference equations

Starting from the linear integral equation for the solutions of the Korteweg-de Vries (KdV) equation, we obtain the direct linearization of a general nonlinear difference-difference equation. In a continuum limit this equation reduces to a general integrable differential-difference equation which contains e.g. the Toda equation and the discrete KdV and MKdV as special cases.     

New exact travelling wave solutions for the Ostrovsky equation

In this Letter, auxiliary equation method is proposed for constructing more general exact solutions of nonlinear partial differential equation with the aid of symbolic computation. In order to illustrate the validity and the advantages of the method we choose the Ostrovsky equation. As a result, many new and more general exact solutions have been obtained for the equation.     

An Invariance for (2+1)-Extension of Burgers Equation and Formulae to Obtain Solutions of KP Equation

In this article, we study the (2+1)-extension of Burgers equation and the KP equation. At first, based on a known Bäcklund transformation and corresponding Lax pair, an invariance which depends on two arbitrary functions for (2+1)-extension of Burgers equation is worked out. Given a known solution and using the invariance, we can find solutions of the (2+1)-extension of Burgers equation repeatedly. Secondly, we put forward an invariance of Burgers equation which cannot be directly obtained by constraining the invariance of the (2+1)-extension of Burgers equation. Furthermore, we reveal that the invariance for finding the solutions of Burgers equation can help us find the solutions of KP equation. At last, based on the invariance of Burgers equation, the corresponding recursion formulae for finding solutions of KP equation are digged out. As the application of our theory, some examples have been put forward in this article and some solutions of the (2+1)-extension of Burgers equation, Burgers equation and KP equation are obtained.