A Benney-Like Lattice

We propose a new Benney-like lattice and show that the new system of equations can be reduced to Chaplygin gas-like equations as well as the heavenly equation. We construct two infinite sets of conserved charges. The conserved densities are related to Legendre polynomials. We prove that the system is bi-Hamiltonian and that the conserved charges are in involution with respect to either of the Hamiltonian structures. We show that our Lax operator generates a new dispersionless Toda hierarchy.     

New exact solutions of Einstein's field equations: Gravitational force can also be repulsive!

This article has not been written for specialists of exact solutions of Einstein's field equations but for physicists who are interested in nontrivial information on this topic. We recall the history and some basic properties of exact solutions of Einstein's vacuum equations. We show that the field equations for stationary axisymmetric vacuum gravitational fields can be expressed by only one nonlinear differential equation for a complex function. This compact form of the field equations allows the generation of almost all stationary axisymmetric vacuum gravitational fields. We present a new stationary two-body solution of Einstein's equations as an application of this generation technique. This new solution proves the existence of a macroscopic, repulsive spin-spin interaction in general relativity. Some estimates that are related to this new two-body solution are given.     

Applications of Lie Symmetries to Higher Dimensional Gravitating Fluids

We consider a radiating shear-free spherically symmetric metric in higher dimensions. Several new solutions to the Einstein’s equations are found systematically using the method of Lie analysis of differential equations. Using the five Lie point symmetries of the fundamental field equation, we obtain either an implicit solution or we can reduce the governing equations to a Riccati equation. We show that known solutions of the Einstein equations can produce infinite families of new solutions. Earlier results in four dimensions are shown to be special cases of our generalised results.     

On the Geometry of Darboux Transformations for the KP Hierarchy and its Connection with the Discrete KP Hierarchy

We tackle the problem of interpreting the Darboux transformation for the KP hierarchy and its relations with the modified KP hierarchy from a geometric point of view. This is achieved by introducing the concept of a Darboux covering. We construct a Darboux covering of the KP equations and obtain a new hierarchy of equations, which we call the Darboux-KP hierarchy (DKP). We employ the DKP equations to discuss the relationships among the KP equations, the modified KP equations, and the discrete KP equations. Our approach also handles the various reductions of the KP hierarchy. We show that the KP hierarchy is a projection of the DKP, the mKP hierarchy is a DKP restriction to a suitable invariant submanifold, and that the discrete KP equations are obtained as iterations of the DKP ones. Received: 23 July 1996 / Accepted: 6 January 1997     

New Formal Solutions of Davey-Stewartson Equation via Combined tanh Function Method with Symmetry Method

We combine the tanh function method with the symmetry group method to construct new type of solutions of Davey-Stewartson equation and implemente it in a computer algebraic system. As a result, some new types of solutions are obtained. This method is also applied to other differential equations if the nonlinear evolution equations admit nontrivial one-parameter group of transformation.     

Painlevé Integrability of Nonlinear Schrödinger Equations with bothSpace- and Time-Dependent Coefficients

We investigate the Painlevé integrability of nonautonomous nonlinearSchrödinger (NLS) equations with both space- and time-dependent dispersion, nonlinearity, and external potentials. The Painlevé analysis is carried out without using the Kruskal's simplification, which results in more generalized form of inhomogeneous equations. The obtained equations are shown to be reducible to the standard NLS equation by using a point transformation. We also construct the corresponding Lax pair and carry out its Kundu-type reduction to the standard Lax pair. Special cases of equations from choosing limited form of coefficients coincide with the equations from the previous Painlevé analyses and/or become unknown new equations.     

New drift modes in a nonuniform quantum magnetoplasma

We report the existence of two new drift modes in a nonuniform quantum magnetoplasma. By using the electron and ion fluid velocities deduced from the quantum momentum equations, together with the continuity and Poisson equations, we derive the governing equations for the low-frequency drift modes. The equations are then Fourier transformed to obtain linear dispersion relations, which admit new drift modes. The results are relevant for identifying electrostatic fluctuations that can cause cross-field charged particle transport in an inhomogeneous, ultracold magnetized quantum plasma.     

Closed/open string diagrammatics

We introduce a combinatorial model based on measured foliations in surfaces which captures the phenomenology of open/closed string interactions. The predicted equations are derived in this model, and new equations can be discovered as well. In particular, several new equations together with known transformations generate the combinatorial version of open/closed duality. On the topological and chain levels, the algebraic structure discovered is new, but it specializes to a modular bi-operad on the level of homology.     

The evolution of longitudinal and transverse acoustic waves in a medium with paramagnetic impurities

We study the bipartial interaction of longitudinal and transverse acoustic pulses with a system of paramagnetic impurities with an effective spin S=1/2 in a crystalline layer or on a surface in the presence of an arbitrarily directed external constant magnetic field. We derive a new system of evolution equations that describes this interaction and show that, in the absence of losses, for equal phase velocities of these acoustic components, and under the condition of their unidirectional propagation, the original system reduces to a new integrable system of equations. The derived integrable system describes the pulse dynamics outside the scope of the slow-envelope approximation. For one of the reductions of the general model that corresponds to the new integrable model, we give the corresponding equations of the inverse scattering transform method and find soliton solutions. We investigate the dynamics and formation conditions of the phonon avalanche that arises when the initial completely or incompletely inverted state of the spin system decays. We discuss the application of our results to describing the interaction dynamics of spins and acoustic pulses in various systems with an external magnetic field.     

A simple and direct method for generating travelling wave solutions for nonlinear equations

We propose a simple and direct method for generating travelling wave solutions for nonlinear integrable equations. We illustrate how nontrivial solutions for the KdV, the mKdV and the Boussinesq equations can be obtained from simple solutions of linear equations. We describe how using this method, a soliton solution of the KdV equation can yield soliton solutions for the mKdV as well as the Boussinesq equations. Similarly, starting with cnoidal solutions of the KdV equation, we can obtain the corresponding solutions for the mKdV as well as the Boussinesq equations. Simple solutions of linear equations can also lead to cnoidal solutions of nonlinear systems. Finally, we propose and solve some new families of KdV equations and show how soliton solutions are also obtained for the higher order equations of the KdV hierarchy using this method.     

Prolongation structure of a new integrable system

We have obtained the inverse scattering equations associated with a new pair of coupled nonlinear evolution equations in two dimensions. The spectral parameter is introduced by invoking the invariance of the equation set, and imposing those on the Lax pair.     

Ballistic-diffusive heat-conduction equations

We present new heat-conduction equations, named ballistic-diffusive equations, which are derived from the Boltzmann equation. We show that the new equations are a better approximation than the Fourier law and the Cattaneo equation for heat conduction at the scales when the device characteristic length, such as film thickness, is comparable to the heat-carrier mean free path and/or the characteristic time, such as laser-pulse width, is comparable to the heat-carrier relaxation time.     

Averaged Lagrangians and the mean effects of fluctuations in ideal fluid dynamics

We begin by placing the generalized Lagrangian mean (GLM) equations for a compressible adiabatic fluid into the Euler–Poincaré (EP) variational framework of fluid dynamics, for an averaged Lagrangian. We then state the EP Averaging Result—that GLM equations arise from GLM Hamilton’s principles in the EP framework. Next, we derive a new set of approximate small-amplitude GLM equations (gℓm equations) at second order in the fluctuating displacement of a Lagrangian trajectory from its mean position. These equations express the linear and nonlinear back-reaction effects on the Eulerian mean fluid quantities by the fluctuating displacements of the Lagrangian trajectories in terms of their Eulerian second moments. The derivation of the gℓm equations uses the linearized relations between Eulerian and Lagrangian fluctuations, in the tradition of Lagrangian stability analysis for fluids. The gℓm derivation also uses the method of averaged Lagrangians, in the tradition of wave, mean flow interaction (WMFI). The gℓm EP motion equations for compressible and incompressible ideal fluids are compared with the Euler-alpha turbulence closure equations. An alpha model is a GLM (or gℓm) fluid theory with a Taylor hypothesis closure (THC). Such closures are based on the linearized fluctuation relations that determine the dynamics of the Lagrangian statistical quantities in the Euler-alpha closure equations. We use the EP Averaging Result to bridge between the GLM equations and the Euler-alpha closure equations. Hence, combining the small-amplitude approximation with THC yields in new turbulence closure equations for compressible fluids in the EP variational framework.     

CONSERVATION LAWS FOR SELF-ADJOINT FIRST-ORDER EVOLUTION EQUATION

We consider the problem on group classification and conservation laws for first-order evolution equations. Subclasses of these general equations which are quasi-self-adjoint and self-adjoint are obtained. By using the recent new conservation theorem due to Ibragimov, conservation laws for equations admiting self-adjoint equations are established. The results are illustrated applying them to the inviscid Burgers equation. In particular an infinite number of new symmetries of this equation are found.     

扩展的(G’/G)展开法和Zakharov方程组的新精确解

对(G’/G)展开法进行了扩展, 引入了新的辅助方程, 对(G’/G)展开式附加了负指数幂, 并利用扩展的(G’/G)展开法求出了Zakharov方程组的一些新精确解. 该方法还可被应用到其他非线性演化方程中去. 关键词： G’/G)展开法')" href="#">(G’/G)展开法 Zakharov方程组 精确解     

On the relation between the conditional moment closure and unsteady flamelets

We consider the relation between the conditional moment closure (CMC) and the unsteady flamelet model (FM). The CMC equations were originally constructed as global equations, while FM was derived asymptotically for a thin reaction zone. The recent tendency is to use FM-type equations as global equations. We investigate the possible consequences and suggest a new version of FM: coordinate-invariant FM (CIFM). Unlike FM, CIFM complies with conditional properties of the exact transport equations which are used effectively in CMC. We analyse the assumptions needed to obtain another global version of FM: representative interactive flamelets (RIF), from original FM and demonstrate that, in homogeneous turbulence, one of these assumptions is equivalent to the main CMC hypothesis.     

On non-Lie ansatzes and new exact solutions of the classical Yang-Mills equations

Abstract

We ssuggest an effective method for reducing Yang-Mills equations to systems of ordinary differential equations. With the use of this method, we construct wide families of new exact solutions of the Yang-Mills equations. Analysis of the solutions obtained shows that they correspond to conditional symmetry of the equations under study.     

New absorbing layers conditions for short water waves

We develop a new PML formulation for the linearized shallow-water equations including the Coriolis force. The construction process is based on the uncoupling of the velocity components with the depth of water. Then the damping effect is only applied to the propagative modes just as was formerly done by Nataf [1] to the linearized Euler equations to enforce the long-time stability. We assess numerically the performance of the new absorbing condition and we illustrate in particular that it is stable for long-time simulations.     

自由电子激光各类纵模的统一描述

给出一种研究自由电子激光各类纵模发展过程的新解析方法。建立了统一描述各类模的基本方程。并用这一方法研究了各次谐波的诱导辐射、超辐射和边带超辐射。