Two results concerning asymptotic behavior of solutions of the Burgers equation with force

We consider the Burgers equation with an external force. For the case of the force periodic in space and time we prove the existence of a solution periodic in space and time which is the limit of a wide class of solutions ast . If the force is the product of a periodic function ofx and white noise in time, we prove the existence of an invariant distribution concentrated on the space of space-periodic functions which is the limit of a wide class of distributions ast .     

Some properties of the asymptotic solutions of the Montroll-Weiss equation

Curves of asymptotic probability densities appropriate to the continuous time random walk model of Montroll and Weiss are presented and are calculated numerically using the fast Fourier transform. The behavior of the moments is briefly discussed and it is shown that the Einstein formula relating the diffusion and mobility coefficients can be generalized to include the case where the mean waiting time between hops is infinite.     

Quasi-periodic solutions and asymptotic properties for the nonlocal Boussinesq equation

We construct the Hirota bilinear form of the nonlocal Boussinesq(nlBq) equation with four arbitrary constants for the first time. It is special because one arbitrary constant appears with a bilinear operator together in a product form. A straightforward method is presented to construct quasiperiodic wave solutions of the nl Bq equation in terms of Riemann theta functions. Due to the specific dispersion relation of the nl Bq equation, relations among the characteristic parameters are nonlinear, then the linear method does not work for them. We adopt the perturbation method to solve the nonlinear relations among parameters in the form of series. In fact, the coefficients of the governing equations are also in series form.The quasiperiodic wave solutions and soliton solutions are given. The relations between the periodic wave solutions and the soliton solutions have also been established and the asymptotic behaviors of the quasiperiodic waves are analyzed by a limiting procedure.     

On the asymptotic behavior of an earthquake model

We present evidence showing that there are important differences in the relaxation times associated with different aspects of the model of Olami, Feder and Christensen. The asymptotic behavior is characterized by a critical exponent that is independent of α, the degree of conservation of the model. The analysis of temporal series of avalanches gives helpful information to study the approach to self-organized criticality, and to put in evidence any quasi-periodic evolution.     

Large-time behavior of solutions of the discrete Boltzmann equation

Large-time behavior of solutions of the one-dimensional discrete Boltzmann equation is studied. Under suitable assumptions it is proved that as time tends to infinity, the solution approaches a function which is constructed explicitly in terms of the self-similar solutions of the Burgers equation and the linear heat equation.     

Localized asymptotic solutions of the wave equation with variable velocity on the simplest graphs

The asymptotic behavior of the Cauchy problem for the wave equation with variable velocity and localized initial conditions on the line, semi-axis, and an infinite starlike graph is described. The solution consists of a short-wave and long-wave parts; the shortwave part moves along the characteristics, while the long-wave part satisfies the Goursat or Darboux problem. In the case of a star-like graph, the distribution of energy with respect to the edges is discussed; this distribution depends on the arrangement of the eigensubspaces of the unitary matrix that defines the boundary condition at the vertex of the star.     

On exact solutions of the Bogoyavlenskii equation

Exact solutions for the Bogoyavlenskii equation are studied by the travelling wave method and the singular manifold method. It is found that the linear superposition of the shock wave solution and the complex solitary wave solution for the physical field is still a solution of the equation of interest, except for a phase-shift. The dromion-like structures with elastic and nonelastic interactions are found.     

On multimeron solutions of the Yang-Mills equation

We study a singular boundary value problem introduced by Glimm and Jaffe for the purpose of obtaining solutions of the Euclidean Yang-Mills equations with isolated singularities along an axis. Using comparison techniques, we prove existence, asymptotic behavior and also uniqueness in some special cases.     

On similarity solutions of the Boussinesq equation

It is shown that the similarity solutions of the Boussinesq equation satisfy the first or second Painlevéequation. We also discuss properties of the soliton solution.     

On twisting solutions to the Yang-Baxter equation

Sufficient conditions for an invertible two-tensor F to relate two solutions to the Yang-Baxter equation via the transformation R F ₂₁ ^-1 RF are formulated. Those conditions are equivalent to the problem of twisting for certain quasitriangular bialgebras.     

On particular solutions of the equation of transfer

The particular solutions corresponding to two very special inhomogeneous terms of the equation of transfer with linearly anisotropic scattering are reported.     

Adiabatic invariants and asymptotic behavior of Lyapunov exponents of the Schrödinger equation

We give an upper bound for the high-energy behavior of the Lyapunov exponent of the one-dimensional Schrödinger equation. We relate this behavior to the diffrentiability properties of the potential. As an application, this result provides an upper bound for the asymptotic length of the gaps of the Schrödinger equation.     

Asymptotic behavior of the solutions of the Kadomtsev-Pyatviashvili equation (two-dimensional Korteweg-De Vries equation)

The asymptotic form of the solutions of the Kadomtsev-Pyatviashvili equation for t → ± ∞ is presented. The reverse problem of reconstructing the solution from its asymptotic form is also solved.     

On multimeron solutions of the classical Yang-Mills equation

The double-meron solution and the solution indicated by Protogenov [1] of the SU(2) Yang-Mills equation in four-dimensional euclidean space are generalised. It is noted that the double-meron solution has an analogous solution in Minkowski space which has a finite energy integral.     

On Lipschitz solutions of the constant astigmatism equation

We show that the solutions of the constant astigmatism equation that correspond to a class of surfaces found by Lipschitz in 1887, exactly match the Lie symmetry invariant solutions and constitute a four-dimensional manifold. The two-dimensional orbit space with respect to the Lie symmetry group is described. Our approach relies on the link between constant astigmatism surfaces and orthogonal equiareal patterns. The counterpart sine–Gordon solutions are shown to be Lie symmetry invariant as well.     

On the asymptotic behavior of effective QED at finite temperature

The aim of this paper is to study finite temperature effects in effective quantum electrodynamics using Weisskopfs zero-point energy method in the context of thermo field dynamics. After a general calculation for a weak magnetic field at fixed T, the asymptotic behavior of the Euler-Kockel-Heisenberg Lagrangian density is investigated focusing on the regularization requirements in the high temperature limit. In scalar QED the same problem is also discussed.Received: 14 June 2003, Revised: 14 August 2003, Published online: 2 October 2003This revised version was published online in Oktober 2003 with corrections to the citation of equations in the text after equation (17).     

On the asymptotic behavior of Wightman functions in space-like directions

The asymptotic behavior of the truncated vacuum expectation value of a product ofN (unbounded) quasilocal operators,F(x)=Q ₁(x ₁) ...Q _N (x _N ) _T , is investigated for some of the separations space-like. It is shown that unless all clusters {x _i1, ...,x _ij } are partially time-like (or light-like) separated from their complements ,F(x) decreases faster than any inverse power of the diameter of the set {x ₁, ...,x _N }.This work was supported in part by the U.S. Atomic Energy Commission.Research supported in part by the National Science Foundation.     

On the path integral solutions of the master equation

The lagrangian in the path integral solution of the master equation of a stationary Markov process is derived by application of the Ehrenfest-type theorem of quantum mechanics and the Cauchy method of finding inverse functions. Applied to the non-linear Fokker-Planck equation we reproduce the result obtained by integrating over Fourier series coefficients and by other methods.