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.     

The appearance and development of turbulence in a flow past a sphere: Problems and the existing approaches to their solution

The results of the direct numerical integration of the Navier-Stokes equations and integration of the pair functions theory equations are evaluated against experimental data for the problem of a flow past a hard sphere at rest in an unstable regime. Calculations based on the Navier-Stokes equations satisfactorily reproduced three stable medium states observed for a flow past a sphere. In agreement with experiment, each of these three states begins to develop after stability loss in its own direction different from those characteristic of the other states. Calculations were, however, incapable of reproducing any of the three directions of turbulence development recorded experimentally. Most likely, the reason for this is the Navier-Stokes equations themselves. The possibility is discussed that the assumption made in the derivation of the Boltzmann equation, namely, the molecular chaos hypothesis (Stosszahlansatz), may be responsible for the failure of classic hydrodynamics. This assumption is a closure to the Boltzmann equation that allows hydrodynamics to be constructed on the lower three hydrodynamic values. The inaccuracy mentioned introduced no substantial error into stable flow calculations. The error, however, increased rapidly after stability loss. We suggest the use of hydrodynamic equations based on pair functions theory as an alternative to the Navier-Stokes equations for unstable modes. These equations are derived without invoking any additional assumptions such as the Stosszahlansatz hypothesis. As distinct from classic hydrodynamics equations, pair functions theory equations predict the direction of turbulence development close to that observed experimentally.     

Dynamical behaviour of transverse plasmons in pair plasmas

This paper analytically investigates the nonlinear behaviour of transverse plasmons in pair plasmas on the basis of the nonlinear governing equations obtained from Vlasov--Maxwell equations. It shows that high frequency transverse plasmons are modulationally unstable with respect to the uniform state of the pair plasma. Such an instability would cause wave field collapse into a localized region. During the collapse process, ponderomotive expulsion is greatly enhanced for the increase of wave field strength, leading to the formation of localized density cavitons which are significant for the future experimental research in the interaction between high frequency electromagnetic waves and pair plasmas.     

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.     

Constraints and Soliton Solutions for KdV Hierarchy and AKNS Hierarchy

It is well-known that the finite-gap solutions of the KdV equationcan be generated by its recursion operator.We generalize the result to a special form of Lax pair,from which a method to constrain the integrable system to alower-dimensional or fewer variable integrable system is proposed.A direct result is that the n-soliton solutions of the KdV hierarchy can be completely depictedby a series of ordinary differential equations (ODEs), which may be gotten by a simple but unfamiliar Lax pair. Furthermore the AKNS hierarchy is constrained to a series of univariate integrable hierarchies. The key is a special form of Lax pair for the AKNS hierarchy. It is proved that under the constraints all equations of the AKNS hierarchy are linearizable.     

Effect of an external action on a pair distribution function in a steady state

An external action that reduces a two-component equilibrium thermodynamic system to a nonequilibrium steady state with scalar fluxes has been studied. A system of integrodifferential equations for pair correlation functions has been obtained. These equations coincide with the second equations of the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy, but with different effective temperatures. Thus, ordinary integrodifferential equations for a pair correlation function with effective temperatures expressed in terms of the perturbed (nonequilibrium) Maxwellian momentum distribution function can be used to calculate the structural and thermodynamic properties of such a system.     

Invasion and Extinction in the Mean Field Approximation for a Spatial Host-Pathogen Model

We derive the mean field equations of a simple spatial host-pathogen, or predator-prey, model that has been shown to display interesting evolutionary properties. We compare these equations, and the equations including pair-correlations, with the low-density approximations derived by other authors. We study the process of invasion by a mutant pathogen, both in the mean field and in the pair approximation, and discuss our results with respect to the spatial model. Both the mean field and pair correlation approximations do not capture the key spatial behaviors—the moderation of exploitation due to local extinctions, preventing the pathogen from causing its own extinction. However, the results provide important hints about the mechanism by which the local extinctions occur.     

On perturbations of solutions of the Einstein-Maxwell equations

By using the expressions for the solutions of the Einstein-Maxwell equations in terms of potentials, valid in the case where the spacetime admits a shear-free geodesic null congruence and the electromagnetic field is aligned to it, we show that a pair of complex potentials generates simultaneous perturbations of the gravitational and the electromagnetic fields. We also show that if the background electromagnetic field is null, then the pair of complex potentials is determined by a pair of coupled, linear, second-order differential equations.     

Integrable systems of partial differential equations determined by structure equations and Lax pair

It is shown how a system of evolution equations can be developed both from the structure equations of a submanifold embedded in three-space as well as from a matrix SO(6) Lax pair. The two systems obtained this way correspond exactly when a constraint equation is selected and imposed on the system of equations. This allows for the possibility of selecting the coefficients in the second fundamental form in a general way.     

Relaxation of Excited States in C60

The static study has shown that the excited state with electron-hole pair in C₆₀ can cause the distortion of the bond structure to form a polaron-like exciton with symmetry D_5d. This paper further reveals the relaxation process from the initial electron-hole pair state to the polaron-like exciton by solving the dynamical equations. The relaxation time of this dynamical process can be determined from the time-dependent bond distortion with time step 4 femtoseconds.     

A reformulation of the one-dimensional surface field integral equations

By starting from the matrix forms of the two coupled, inhomogeneous integral equations for the values of the magnetic field and its normal derivative on a one-dimensional, rough metal surface, or for the values of the electric field and its normal derivative on such a surface, we have obtained an equivalent pair of equations for these quantities in which the inhomogeneous terms are just the Kirchhoff approximations to them. The new pair of equations for the surface values of the magnetic field and its normal derivative is solved iteratively to generate a multiple-scattering expansion for the scattering amplitude when p-polarized light is scattered from a large RMS height, large RMS slope, one-dimensional, random silver surface, with the plane of incidence perpendicular to the generators of the surface. It is shown that the Kirchhoff approximation to the contribution to the mean differential reflection coefficient from the incoherent component of the scattered light displays no evidence of enhanced backscattering. However, the pure double-scattering contribution already displays this effect, stamping it as a multiple-scattering phenomenon.     

Direct calculation of ionization energies

The advantages of the propagator formalism, as a direct method of calculating ionization energies, are stressed. The propagator equations are derived for closed-shell systems using an operator method instead of the usual diagrammatic derivations. The equations enable the development of an interpretation of the ionization energies in terms of conceptually simple quantities, such as pair correlation energies and associated relaxation effects, and retain the idea of orbital ionization. Infinite summations appearing in the self-energy terms are replaced by finite expressions involving functions satisfying uncoupled inhomogeneous differential equations. Certain high-order propagator equations are derived, and a connection with the Bethe-Goldstone formulation of pair correlation is made. Several computational procedures are advocated as forming the basis for balanced calculations of atomic and molecular ionization energies.     

Modulational instability of nonlinearly interacting incoherent sea states

The modulational instability of nonlinearly interacting spatially incoherent Stokes waves is analyzed. Starting from a pair of nonlinear Schrödinger equations, we derive a coupled set of wave-kinetic equations by using the Wigner transform technique. It is shown that the partial coherence of the interacting waves induces novel effects on the dynamics of crossing sea states.     

A reformulation of the one-dimensional surface field integral equations

Abstract

By starting from the matrix forms of the two coupled, inhomogeneous integral equations for the values of the magnetic field and its normal derivative on a one-dimensional, rough metal surface, or for the values of the electric field and its normal derivative on such a surface, we have obtained an equivalent pair of equations for these quantities in which the inhomogeneous terms are just the Kirchhoff approximations to them. The new pair of equations for the surface values of the magnetic field and its normal derivative is solved iteratively to generate a multiple-scattering expansion for the scattering amplitude when p-polarized light is scattered from a large RMS height, large RMS slope, one-dimensional, random silver surface, with the plane of incidence perpendicular to the generators of the surface. It is shown that the Kirchhoff approximation to the contribution to the mean differential reflection coefficient from the incoherent component of the scattered light displays no evidence of enhanced backscattering. However, the pure double-scattering contribution already displays this effect, stamping it as a multiple-scattering phenomenon.     

Pair creation in QED-strong pulsed laser fields interacting with electron beams

QED effects are known to occur in a strong laser pulse interaction with a counterpropagating electron beam, among these effects being electron-positron pair creation. We discuss the range of laser pulse intensities of J≥5×10(22) W/cm2 combined with electron beam energies of tens of GeV. In this regime multiple pairs may be generated from a single beam electron, some of the newborn particles being capable of further pair production. Radiation backreaction prevents avalanche development and limits pair creation. The system of integro-differential kinetic equations for electrons, positrons and γ photons is derived and solved numerically.     

The Matrix Kadomtsev–Petviashvili Equation as a Source of Integrable Nonlinear Equations

Abstract

A new integrable class of Davey–Stewartson type systems of nonlinear partial differential equations (NPDEs) in 2+1 dimensions is derived from the matrix Kadomtsev– Petviashvili equation by means of an asymptotically exact nonlinear reduction method based on Fourier expansion and spatio-temporal rescaling. The integrability by the inverse scattering method is explicitly demonstrated, by applying the reduction technique also to the Lax pair of the starting matrix equation and thereby obtaining the Lax pair for the new class of systems of equations. The characteristics of the reduction method suggest that the new systems are likely to be of applicative relevance. A reduction to a system of two interacting complex fields is briefly described.     

基于两对互相垂直平行线的自定标方法

提出了一种利用图像中的互相垂直的两对平行线来实现相机的自定标方法.利用模约束可以提供一个方程,图像中的两对互相垂直的平行线可以提供两个线性方程的特性,该方法通过两幅图像就可以实现相机的自定标.所提出的相机定标方法在求解过程中是准线性的,实验数据结果表明,该定标方法具有鲁棒性好、收敛性好等优点.     

Ehrenfest theorem for inhomogeneous superconductors and supercurrent force on Andreev reflected quasiparticles

Quasiclassical equations of motion are derived for quasiparticle wave packets in inhomogeneous superconductors with electromagnetic fields. Besides the Lorentz forces on electrons and holes there are two off-diagonal pair potential forces. The first one is due to gradients of the modulus of the pair potential and responsible for electron-hole scattering from inhomogeneities of the pair potential. The second one results from interactions with the ground state condensate and is proportional to the gauge invariant Cooper pair velocity of surface supercurrents in the interfaces between normal and superconducting regions. The most favorable experimental conditions for the observation of this new force may be found inN-S junctions composed of clean normal metals and type II superconductors with high critical fields, large London penetration depths, and large energy gaps.     

Nonlinear heat equations

The modified Smoluchowski equation, coupled to a temperature field, leads to a pair of nonlinear heat equations obeying the first and second laws of thermodynamics. We obtain a solution representing a particle under gravity, moving in a slab and maintained in stasis away from the Gibbs state by a temperature gradient. A two-state atom in a potential in isothermal conditions is described by coupled equations satisfying detailed balance. It is shown that the free energy is a monotonic decreasing function of time.