Integrable lattice equations and their growth properties

In this Letter we investigate the integrability of two-dimensional partial difference equations using the newly developed techniques of study of the degree of the iterates. We show that while for generic, nonintegrable equations, the degree grows exponentially fast, for integrable lattice equations the degree growth is polynomial. The growth criterion is used in order to obtain the integrable deautonomisations of the equations examined. In the case of linearisable lattice equations we show that the degree growth is slower than in the case of equations integrable through inverse scattering transform techniques.     

Plasmon-exciton self-induced transparency

The possibility of forming stable bound plasmon-polariton states in an extended metallic cylinder surrounded by a two-level medium has been investigated. The dynamics of plasmons is described in the hydrodynamic approximation. It has been shown that the equations of motion of charge-density bunches and the Bloch equations for the two-level medium are reduced in certain approximations to integrable equations for both transverse and longitudinal plasmons. In the former case, the initial system of equations after the application of the slow-envelope approximation is reduced to equations equivalent to the Maxwell-Bloch equations. In the latter case, the equations describe wave dynamics beyond the slow-envelope approximation. In the approximation of unidirectional wave propagation, the initial system of equations is reduced to equations related to the reduced Maxwell-Bloch equations. Soliton and breather-like solutions of the derived equations describe plasmon-exciton self-induced transparency.     

Flat bianchi I Friedmann-Robertson-Walker space-times

In a recent paper Hajj-Boutros and Sfeila obtained the field equations in general relativity for the case of Bianchi I space-time filled with a perfect fluid and solved the equations for a particular case. In the present note the complete set of solutions of these equations is obtained.     

On surface elasticity theory for plane interfaces

A closed system of surface elasticity equations was derived in terms of surface quantities defined as integrals of respective excess bulk quantities normal to the interface. The equations were consistently linearized for the case of small strains. It is shown that these equations are more general than the Shuttleworth equations. Equations of this type were also derived for the particular case of an interface formed by a thin layer with constant properties. The derived equations were used to consider bending of a plate under pressure applied to both sides.     

Coupled KdV Equations and Their Explicit Solutions Through Two-Dimensional Hamiltonian System with a Quartic Potential

Based on the second integrable ease of known two-dimensional Hamiltonian system with a quartie potentiM, we propose a 4 × 4 matrix speetrM problem and derive a hierarchy of coupled KdV equations and their Hamiltonian structures. It is shown that solutions of the coupled KdV equations in the hierarchy are reduced to solving two compatible systems of ordinary differentiM equations. As an application, quite a few explicit solutions of the coupled KdV equations are obtained via using separability for the second integrable ease of the two-dimensional Hamiltonian system.     

Kink dynamics in one-dimensional nonlinear systems

A certain class of nonlinear evolution equations of one space dimension which permits kink type solutions and includes one-dimensional time-dependent Ginzburg-Landau (TDGL) equations and certain nonlinear wave equations is studied in some strong coupling approximation where the problem can be reduced to the study of kink dynamics. A detailed study is presented for the case of TDGL equation with possible applications to the late stage kinetics of order-disorer phase transitions and spinodal decompositions. A special case of kink dynamics of nonlinear wave equations is found to reduce to the Toda lattice dynamics. A new conservation law for dissipative systems is found which corresponds to the momentum conservation law for wave equations.     

A cumulant expansion for the two-time correlation functions for linear stochastic differential equations

A cumulant expansion for the two-time correlation functions for linear stochastic differential equations is presented. This extends the work of Kubo and Van Kampen on one-time averages to the case of two-time averages. The resulting equations become exact in the second order approximation in the scalar gaussian case.     

Variational Equations and Symmetries in the Lagrangian Formalism; Arbitrary Vector Fields

We continue the study of symmetries in the Lagrangian formalism of arbitrary order with the help of the generalized Helmholtz equations (sometimes called the Anderson-Duchamp-Krupka equations). For the case of second-order equations and arbitrary vector fields we are able to establish a polynomial structure in the second-order derivatives. This structure is based on the some linear combinations of Olver hyper-Jacobians. We use as the main tools Fock space techniques and induction. This structure can be used to analyze Lagrangian systems with groups of Noetherian symmetries. As an illustration we analyze the case of Lagrangian equations with Abelian gauge invariance.     

On the relativistic classical motion of a radiating spinning particle in a magnetic field

We propose classical equations of motion for a charged particle with magnetic moment, taking radiation reaction into account. This generalizes the Landau–Lifshitz equations for the spinless case. In the special case of spin-polarized motion in a constant magnetic field (synchrotron motion) we verify that the particle does lose energy. Previous proposals did not predict dissipation of energy and also suffered from runaway solutions analogous to those of the Lorentz–Dirac equations of motion.     

Solution of the five-dimensional vacuum Einstein equations in Kaluza-Klein theory

In a recent paper Ross obtained the five-dimensional vacuum Einstein equations in Kaluza-Klein theory with energy-momentum tensor equal to zero and solved the equations for a particular case. Here we obtain the complete set of solutions of these equations.     

Brans-Dicke cosmological exact solution in a radiation-filled Robertson-Walker universe

In a recent paper Singh and Deo obtained the field equations in Brans-Dicke theory for a radiation-filled universe with Robertson-Walker metric and solved the equations for a particular case. Here we obtain the complete set of solutions of these equations.     

Nonlinear dynamics of magnetohydrodynamic flows of a heavy fluid on slope in the shallow water approximation

Magnetohydrodynamic equations for a heavy fluid over an arbitrary surface are studied in the shallow water approximation. While solutions to the shallow water equations for a neutral fluid are well known, shallow water magnetohydrodynamic (SMHD) equations over a nonflat boundary have an additional dependence on the magnetic field, and the number of equations in the magnetic case exceeds that in the neutral case. As a consequence, the number of Riemann invariants defining SMHD equations is also greater. The classical simple wave solutions do not exist for hyperbolic SMHD equations over an arbitrary surface due to the appearance of a source term. In this paper, we suggest a more general definition of simple wave solutions that reduce to the classical ones in the case of zero source term. We show that simple wave solutions exist only for underlying surfaces that are slopes of constant inclination. All self-similar discontinuous and continuous solutions are found. Exact explicit solutions of the initial discontinuity decay problem over a slope are found. It is shown that the initial discontinuity decay solution is represented by one of four possible wave configurations. For each configuration, the necessary and sufficient conditions for its realization are found. The change of dependent and independent variables transforming the initial equations over a slope to those over a flat plane is found.     

Separation of the massless spin-1 equation in Robertson-Walker space-time

The massless spin-1 free field equation is studied via the Newman-Penrose formalism and separated by the Chandrasekhar-Teukolski method. The temporal and angular equations are explicitly integrated. The radial equations are solved in the flat-universe case. The closed-universe case shows, in principle, the existence of a discrete spectrum of the energy of the massless particles.     

Two-level model of recording and reading of holographic gratings in photorefractive crystals: nonstationary case

A two-level model of recording and readout of holographic gratings in photorefractive crystals proposed by Rustamov has been developed for the nonstationary case. Based on the joint solution of kinetic equations and Maxwell's equations in the quasistationary approximation, a system of coupled partial differential equations for slowly varying optical wave amplitudes has been obtained. The use of a truncated power series expansion has allowed an expression to be obtained for diffraction efficiency in the nonstationary case. The dynamics of diffraction efficiency development is in satisfactory agreement with the available experimental data.     

Hurst exponents for interacting random walkers obeying nonlinear Fokker-Planck equations

Anomalous diffusion of random walks has been extensively studied for the case of non-interacting particles. Here we study the evolution of nonlinear partial differential equations by interpreting them as Fokker-Planck equations arising from interactions among random walkers. We extend the formalism of generalized Hurst exponents to the study of nonlinear evolution equations and apply it to several illustrative examples. They include an analytically solvable case of a nonlinear diffusion constant and three nonlinear equations which are not analytically solvable: the usual Fisher equation which contains a quadratic nonlinearity, a generalization of the Fisher equation with density-dependent diffusion constant, and the Nagumo equation which incorporates a cubic rather than a quadratic nonlinearity. We estimate the generalized Hurst exponents.     

Exponential discretization of the Perfectly Matched Layer (PML) absorbing boundary condition simulation in FD-TD 3D

We present the discretized equations of the 12 PML (perfectly matched layer) in the three-dimensional case using the Cartesian geometry. These equations can be used in different fields where Maxwell equations need to be solved.     

Transport equations for chiral fermions to order ? and electroweak baryogenesis: Part I

This is the first in a series of two papers. In this first part, we use the Schwinger-Keldysh formalism to derive semiclassical Boltzmann transport equations, accurate to order ?, for massive chiral fermions, scalar particles, and for the corresponding CP-conjugate states. Our considerations include complex mass terms and mixing fermion and scalar fields, such that CP-violation is naturally included, rendering the equations particularly suitable for studies of baryogenesis at a first order electroweak phase transition. We provide a quantitative criterion in which case the reduction to the diagonal kinetic equations in the mass eigenbasis is justified, leading to a quasiparticle picture even in the case of mixing scalar or fermionic particles. Within the approximations we make, it is possible to first study the Boltzmann equations without the collision term. In a second paper [Ann. Phys. xxx (2004) xxx] we discuss the collision terms and reduce the Boltzmann equations to fluid equations.     

Subdiffusion with the disappearance of particles at the time of a jump

The reaction-subdiffusion equations corresponding to a monomolecular chemical reaction at the time of a diffusion jump are derived. It is shown that the approach to deriving such equations suggested previously in [8] gives the correct result in the case of asymptotic subdiffusion but is inapplicable in the practically important case of transient subdiffusion.     

Solutions of Nonlinear Differential and Difference Equations with Superposition Formulas

Matrix Riccati equations and other nonlinear ordinary differential equations with superposition formulas are, in the case of constant coefficients, shown to have the same exact solutions as their group theoretical discretizations. Explicit solutions of certain classes of scalar and matrix Riccati equations are presented as an illustration of the general results.