非线性孤子方程的齐次平衡法

齐次平衡法是求非线性孤子方程孤波解的一种十分有效的方法.对齐次平衡法的一些关键步骤进行拓广使用，获得了非线性孤子方程一批新的具有更为丰富形式的精确解，孤波解仅是其中的一种特殊情形，使得结果更加完美. 关键词：     

The Electromagnetic Analogue of Some Gravitational Perturbations in Cosmology

Recently attention has been drawn to the fact that perfect fluid tensor perturbations (with perturbed vorticity and acceleration vanishing) of isotropic cosmological models have a perturbed Weyl tensor with electric part satisfying a linear, homogeneous, third-order wave equation while the magnetic part satisfies a linear, homogeneous, second-order wave equation. We construct an analogous class of electromagnetic test fields in the isotropic cosmological models for which the electric vector satisfies a third-order, linear and homogeneous wave equation while the magnetic vector satisfies a second-order, linear and homogeneous wave equation. If the perfect fluid has an equation of state we give a simplified derivation of the authors' previous perturbation analysis describing gravitational waves carrying arbitrary information. We also present the analogous solutions of Maxwell's equations which contain electromagnetic waves conveying arbitrary information.     

Similarity transformations and exact solutions for a family of higher-dimensional generalized Boussinesq equations

In this Letter, we investigate explicitly exact solutions of the higher-dimensional generalized Boussinesq equation. We firstly reduce this equation to one nonlinear ODE and a set of two nonlinear homogeneous PDEs via semi-traveling wave similarity transformation. And then we study solutions of the obtained nonlinear ODE and the set of two nonlinear homogeneous PDEs, respectively. Finally, we can obtain many types of exact solutions of higher-dimensional generalized Boussinesq equation via the semi-traveling wave similarity transformations. These solutions contain an arbitrary function which leads to abundant structures.     

New multi—soliton solutions and travelling wave solutions of the dispersive long—wave equations

Using the extended homogeneous balance method,the (1 1)-dimensional dispersive long-wave equations have been solved.Starting from the homogeneous balance method,we have obtained a nonlinear transformation for simplifying a dispersive long-wave equation into a linear partial differential equation.Usually,we can obtain only a type of soliton-like solution.In this paper,we have further found some new multi-soliton solutions and exact travelling solutions of the dispersive long-wave equations from the linear partial equation.     

Multisoliton Solutions of the (2 1)-Dimensional KdV Equation

Using the extension homogeneous balance method,we have obtained some new special types of soliton solutions of the (2 1)-dimensional KdV equation.Starting from the homogeneous balance method,one can obtain a nonlinear transformation to simple (2 1)-dimensional KdV equation into a linear partial differential equation and two bilinear partial differential equations.Usually,one can obtain only a kind of soliton-like solutions.In this letter,we find further some special types of the multisoliton solutions from the linear and bilinear partial differential equations.``     

Multisoliton Solutions of the (2+1)-Dimensional KdV Equation

Using the extension homogeneous balance method,we have obtained some new special types of soliton solutions of the (2+1)-dimensional KdV equation.Starting from the homogeneous balance method,one can obtain a nonlinear transformation to simple (2+1)-dimensional KdV equation into a linear partial differential equation and two bilinear partial differential equations.Usually,one can obtain only a kind of soliton-like solutions.In this letter,we find further some special types of the multisoliton solutions from the linear and bilinear partial differential equations.     

Analytic solutions of the susceptibility for Doppler-broadened two-level atoms

We derive the rigorous analytic solutions for the susceptibility of Doppler-broadened two-level atoms, valid for arbitrary laser intensities. As is usually the case with real atoms, we observe that, when the homogeneous linewidth is much narrower than the Doppler width, the real part of the susceptibility is given by the product of an imaginary error function and a Gaussian, whereas the imaginary part consists simply of a Gaussian function. By employing the analytic results, in particular, we study the effects of optical pumping and saturation on the atomic susceptibility. We also find that the real (imaginary) part of the susceptibility is weakly (strongly) dependent on such parameters as the laser intensity or the optical pumping rate. The availability of analytic solutions for the simple laser-atom interaction may provide convenient tools for its understanding as well as application to more general and realistic situations.     

Affine symmetry,geodesics, and homogeneous spacetimes

We show that the conservation laws for the geodesic equation which are associated to affine symmetries can be obtained from symmetries of the Lagrangian for affinely parametrized geodesics according to Noether’s theorem, in contrast to claims found in the literature. In particular, using Aminova’s classification of affine motions of Lorentzian manifolds, we show in detail how affine motions define generalized symmetries of the geodesic Lagrangian. We compute all infinitesimal proper affine symmetries and the corresponding geodesic conservation laws for all homogeneous solutions to the Einstein field equations in four spacetime dimensions with each of the following energy–momentum contents: vacuum, cosmological constant, perfect fluid, pure radiation, and homogeneous electromagnetic fields.     

Dynamics of Robertson–Walker spacetimes with diffusion

We study the dynamics of spatially homogeneous and isotropic spacetimes containing a fluid undergoing microscopic velocity diffusion in a cosmological scalar field. After deriving a few exact solutions of the equations, we continue by analyzing the qualitative behavior of general solutions. To this purpose we recast the equations in the form of a two dimensional dynamical system and perform a global analysis of the flow. Among the admissible behaviors, we find solutions that are asymptotically de-Sitter both in the past and future time directions and which undergo accelerated expansion at all times.     

New Explicit Solitary Wave Solutions and Periodic Wave Solutions for the Generalized Coupled Hirota-Satsuma KdV System

In this paper, we study the generalized coupled Hirota Satsuma KdV system by using the new generalizedtransformation in homogeneous balance method. As a result, many explicit exact solutions, which contain new solitarywave solutions, periodic wave solutions, and the combined formal solitary wave solutions, and periodic wave solutions,are obtained.     

Propagation of gravitational waves in Robertson-Walker backgrounds

The electric and magnetic parts of the linearized Weyl tensor, when the stress-energy tensor is that of a perfect fluid and the background is of Robertson-Walker type, are known to satisfy wave equations that differ by the presence of a source term for the electric part. It is shown here that all of the allowed solutions of the inhomogeneous equation can be obtained by applying a differential operator to the solutions of the homogeneous equation; consequently, electric-type and magnetic-type gravitational waves have the same propagation properties. The results of a complete integration of the appropriately linearized Newman-Penrose equations are given.     

Auto-B(a)cklund Transformation and Exact Solutions to the Generalized Kadomtsev-Petviashvili Equation with Variable Coefficients

By using the extended homogeneous balance method, a new auto-Backlund transformation(BT) to thegeneralized Kadomtsev-Petviashvili equation with variable coefficients (VCGKP) are obtained. And making use of theauto-BT and choosing a special seed solution, we get many families of new exact solutions of the VCGKP equations,which include single soliton-like solutions, multi-soliton-like solutions, and special-soliton-like solutions. Since the KPequation and cylindrical KP equation are all special cases of the VCGKP equation, and the corresponding results ofthese equations are also given respectively.     

Homogeneous Statistical Solutions and Local Energy Inequality for 3D Navier-Stokes Equations

We are interested in space-time spatially homogeneous statistical solutions of Navier-Stokes equations in space dimension three. We first review the construction of such solutions, and introduce convenient tools to study the pressure gradient. Then we show that given a spatially homogeneous initial measure with finite energy density, one can construct a homogeneous statistical solution concentrated on weak solutions which satisfy the local energy inequality.     

Singularities in Bianchi cosmologies

We discuss how infinite density singularities may be shown to occur in Friedmann-Robertson-Walker universes and orthogonal spatially homogeneous universes, but how very different behaviours are possible in tilted homogeneous cosmologies. After considering various possibilities that arise in this case, we illustrate them by examining the behaviour of exact solutions of Einstein's equations for a homogeneous cosmology which is a locally rotationally symmetric tilted Bianchi type V universe. These universes - which can be arbitrarily similar to a Robertson-Walker universe at late times - show a variety of singular behaviours quite different from those in the ‘orthogonal’ case. In particular, there exist such universes in which two singularities occur at the early stages of the universe, but in which the density of matter is finite at all times.     

The Klein-Gordon Operator on Möbius Strip Domains and the Klein Bottle in ℝ n

In this paper we present explicit formulas for the fundamental solution to the Klein-Gordon operator on some higher dimensional generalizations of the Möbius strip and the Klein bottle with values in distinct pinor bundles. The fundamental solution is described in terms of generalizations of the Weierstraß $\wp $ -function that are adapted to the context of these geometries. The explicit formulas for the kernel then allow us to express all solutions to the homogeneous and inhomogeneous Klein-Gordon problem with given boundary data in the context of these manifolds. In the case of the Klein bottle we are able to describe all null solutions of the Klein-Gordon equation in terms of finite linear combinations of the fundamental solution and its partial derivatives.     

A standard and direct method to obtain multiple soliton solutions of the nonlinear partial differential equation

In this paper, we improve some key steps in the homogeneous balance method (HBM), and propose a modified homogeneous balance method (MHBM) for constructing multiple soliton solutions of the nonlinear partial differential equation (PDE) in a unified way. The method is very direct and primary; furthermore, many steps of this method can be performed by computer. Some illustrative equations are investigated by this method and multiple soliton solutions are found.     

Solutions of Klein-Gordon and Dirac equations on quantum Minkowski spaces

Covariant differential calculi and exterior algebras on quantum homogeneous spaces endowed with the action of inhomogeneous quantum groups are classified. In the case of quantum Minkowski spaces they have the same dimensions as in the classical case. Formal solutions of the corresponding Klein-Gordon and Dirac equations are found. The Fock space construction is sketched.This research was supported in part by NSF grant DMS-9508597 and in part by Polish KBN grant No. 2 P301 02007.     

Auto-Backlund Transformation and Exact Solutions to the Generalized Kadomtsev-Petviashvili Equation with Variable Coefficients

By using the extended homogeneous balance method, a new auto-Ba^ecklund transformation(BT) to the generalized Kadomtsew-Petviashvili equation with variable coefficients (VCGKP) are obtained. And making use of the auto-BT and choosing a special seed solution, we get many families of new exact solutions of the VCGKP equations, which include single soliton-like solutions, multi-soliton-like solutions, and special-soliton-like solutions. Since the KP equation and cylindrical KP equation are all special cases of the VCGKP equation, and the corresponding results of these equations are also given respectively.     

Modulation of localized solutions in an inhomogeneous saturable nonlinear Schrödinger equation

In this paper we study the modulation of localized solutions by an inhomogeneous saturable nonlinear medium. Throughout an appropriate ansatz we convert the inhomogeneous saturable nonlinear Schrödinger equation in a homogeneous one. Then, via a variational approach we construct localized solutions of the autonomous equation and we present some modulation patterns of this localized structures. We have checked the stability of such solutions through numerically simulations.     

New explicit exact solutions for a generalized Hirota—Satsuma coupled KdV system and a coupled MKdV equation

In this paper, we make use of a new generalized ansatz in the homogeneous balance method, the well-known Riccati equation and the symbolic computation to study a generalized Hirota--Satsuma coupled KdV system and a coupled MKdV equation, respectively. As a result, numerous explicit exact solutions, comprising new solitary wave solutions, periodic wave solutions and the combined formal solitary wave solutions and periodic wave solutions, are obtained.