A q-analog of the sixth Painlevé equation

A q-difference analog of the sixth Painlevé equation is presented. It arises as the condition for preserving the connection matrix of linear q-difference equations, in close analogy with the monodromy-preserving deformation of linear differential equations. The continuous limit and special solutions in terms of q-hypergeometric functions are also discussed.     

Exact linear hydrodynamics from the Boltzmann equation

Exact (to all orders in Knudsen number) equations of linear hydrodynamics are derived from the Boltzmann kinetic equation with the Bhatnagar-Gross-Krook collision integral. The exact hydrodynamic equations are cast in a form which allows us to immediately prove their hyperbolicity, stability, and existence of an H theorem.     

On the canonical character of the Sawada-Kotera equation

Recursion operators for solutions of two linear equations, associated with the SK equation, are given. A method to construct constants on the motion is described. The hamiltonian structure and the relation with the inverse scattering problem is indicated.     

Higher-order quasilinear approximations for the propagator of the Kramers equation

We present a systematic procedure for constructing higher-order quasilinear approximations for the propagator of the Klein-Kramers equation describing the motion of a Brownian particle in a general force field. Its key points are splitting the full force field into a linear contribution and an anharmonic correction, replacing the underlying Langevin equations by difference equations and solving these equations iteratively. An accurate single step propagator is then derived in terms of known statistical properties of the noise terms. Its use in a path integral shows this approach to be advantageous over a Taylor series expansion for the propagator recently derived employing standard techniques.     

New formulation of the equation for synchrotron oscillations

We develop an orbit theory for synchrotron oscillations using the orbit length, , as an independent variable. This is commonplace for static magnetic fields (storage rings). We extend this to the case of adiabatically varying magnetic fields (synchrotrons). Contrary to conventional treatments, betatron acceleration terms appear in both the energy and phase equations. We derive one-turn difference equations in the linear and adiabatic approximations. By a smooth approximation instead of the traveling-wave approximation, and by combining the two equations, we obtain a differential equation where the betatron acceleration terms are canceled. This equation is an extension of McMillan's equation to the case of strong-focusing synchrotrons.     

Lax pairs for certain generalizations of the sine-Gordon equation

We derive the one-parameter family of isospectral linear eigenvalue problems which is the basic tool for treating certain generalized sine-Gordon equations by the inverse scattering method.     

The complexiton solutions to the nonisospectral Korteweg-de Vries equation

Starting with the solution classification for a linear differential equations, the complexiton solutions to nonisospectral Korteweg-de Vries equation are presented. The basic technique adopted is the Wronskian technique for Hirota's bilinear equation.     

Approximate direct reduction method: infinite series reductions to the perturbed mKdV equation

The approximate direct reduction method is applied to the perturbed mKdV equation with weak fourth order dispersion and weak dissipation. The similarity reduction solutions of different orders conform to formal coherence, accounting for infinite series reduction solutions to the original equation and general formulas of similarity reduction equations. Painlevé II type equations, hyperbolic secant and Jacobi elliptic function solutions are obtained for zero-order similarity reduction equations. Higher order similarity reduction equations are linear variable coefficient ordinary differential equations.     

Evidence for the anomalous scaling behaviour of the molecular-beam epitaxy growth equation

According to the scaling idea of local slope, we investigate numerically and analytically anomalous dynamic scaling behaviour of (1+1)-dimensional growth equation for molecular-beam epitaxy. The growth model includes the linear molecular-beam epitaxy (LMBE) and the nonlinear Lai--Das Sarma--Villain (LDV) equations. The anomalous scaling exponents in both the LMBE and the LDV equations are obtained, respectively. Numerical results are consistent with the corresponding analytical predictions.     

High order accurate, explicit, difference formulas for the classical wave equation

This article presents a simple and yet very novel approach to developing difference schemes for wave equations. The schemes that are developed are explicit in nature. The schemes are of such generality that one can transform from one difference scheme to another with only the slightest of computational effort. The schemes exhibit dispersive errors. The errors can be minimized, however, by increasing the order of truncation error. Numerical results are presented for two linear model equations with truncation error ranging up to O(h⁵). Numerical results are also presented for a system of shallow water equations. By choosing the appropriate a for a first order linear equation (a defines the geometry of an element) we may generate stable schemes for an arbitrary Courant number.     

远场模型方程的同伦近似对称约化

以同伦近似对称法为理论依据研究了远场模型方程, 通过归纳各阶相似约化解和各阶相似约化方程的通式构造相应的同伦级数解. 各阶相似约化方程均为线性变系数常微分方程, 并且可以从零阶开始依次求解. 同伦模型中的辅助参数影响同伦级数解的收敛性. 关键词： 同伦近似对称法 远场模型方程 同伦级数解     

An integral equation analysis of the harmonic response of three-layer beams

The integral equations of harmonic motion have been derived and solved for three-layer sandwich beams with a constrained linear viscoelastic core. The method of solution required first the construction of the Green's vector for a beam in analytical form. Following this, the integral equations were derived and readily approximated by matrix equations which were finally solved numerically. In addition to this analysis, the corresponding eigenvalue problem has been solved so that the modal frequencies and the beam loss factor could be calculated directly. The integral equation analysis offers a fast and efficient alternative to the traditional methods based on the solution of the differential equations of motion. The method has been verified by comparison with experimental results for three-layer cantilevers and simply supported beams.     

mKdV-sine-Gordon方程丰富的相互作用解

基于mKdV-sine-Gordon方程的Wronsk解的形式和结构,提出了Wronsk形式展开法,通过这一方法求得了该方程的丰富的相互作用解,该方法的主要特征是不要求Wronsk行列式元素满足线性偏微分方程组。     

On a transformed stochastic equation

A linear stochastic equation is considered. As a result of the transformation used in the theory of integral equations for improving the convergence of successive approximations, transformed stochastic equations are obtained. The latter are exact and are equivalent to the original equation. By solving the transformed stochastic equations by the method of small perturbations the conditions are derived for the applicability of the approximate Keller equations for a value of the field averaged over the ensemble, which satisfies the original stochastic equation. As an application, the applicability boundaries of the Dyson equations are estimated in the Foldy and Burre approximation. In the first case it is assumed that the medium consists of Rayleigh scatters, while in the second case it is assumed that the fluctuations of the permeability of the medium are small-scale ones. If the medium is bounded and has the form of a sphere, the applicability condition of the Dyson equations impose an upper constraint on the radius of the sphere which nevertheless may take values that exceed the extinction length.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 15, No. 1, pp. 66–72, January, 1972.     

New interaction solutions to the KdV equation

In this Letter, a few new types of interaction solutions to the KdV equation are obtained through a constructed Wronskian form expansion method. The method takes advantage of the forms and structures of Wronskian solutions to the KdV equation, and the functions used in the Wronskian determinants don't satisfy the systems of linear partial differential equations.     

Noncommutative solutions of the d'Alembert equation

All the subalgebras of first-order symmetry operators for the d'Alembert equation, generating the bases of solutions in the method of noncommutative integration of linear differential equations, which cannot be constructed in the method of separation of variables, are found. These bases themselves are then given in explicit form. The complete systems of solutions of the d'Alembert equation, determined by noncommutative sets of first-order symmetry operators, are thereby classified. Tomsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 25–30, June, 1998.     

A simple derivation of the three magnon bound state equation

A simple derivation of the equation for determining the bound states of three magnons in the Heisenberg linear chain with longitudinal anisotropy is given. The present method utilizes nothing more than the Schrödinger equation and Faddeev’s three body equations, and avoids the introduction of the ideal spin wave Hilbert space.     

Backlund transformation and variable separation solutions for the generalized Nozhnik—Novikov—Veselov equation

Using the extended homogeneous balance method, the B?cklund transformation for a (2+1)-dimensional integrable model, the generalized Nizhnik－Novikov－Veselov (GNNV) equation, is first obtained. Also, making use of the B?cklund transformation, the GNNV equation is changed into three equations: linear, bilinear and trilinear form equations. Starting from these three equations, a rather general variable separation solution of the model is constructed. The abundant localized coherent structures of the model can be induced by the entrance of two variable-separated arbitrary functions.     

Noncommutative integration of the Dirac equation in Riemann spaces possessing a group of automorphisms

Applying the method of noncommutative integration for linear differential equations, we build exact solutions for the Dirac equation in 4-dimensional Riemann spaces, which have a 5-parameter group of automorphisms and where the Klein-Gordon and the Dirac equations are nonintegrable using the technique of complete separation of variables.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 43–46, September, 1991.