A family of integrable evolution equations of third order

We construct a family of integrable equations of the form v_t = f(v; v_x; v_xx; v_xxx) such that f is a transcendental function in v; v_x; v_xx. This family is related to the Krichever-Novikov equation by a differential substitution. Our construction of integrable equations and the corresponding differential substitutions involves geometry of a family of genus two curves and their Jacobians.     

String branchings on complex tori and algebraic representations of generalized Krichever-Novikov algebras

The propagation differential for bosonic strings on a complex torus with three symmetric punctures is investigated. We study deformation aspects between two-and three-point differentials as well as the behaviour of the corresponding Krichever-Novikov algebras. The structure constants are calculated and from this we derive a central extension of the Krichever-Novikov algebras by means of b–c systems. The defining cocycle for this central extension deforms to the well-known Virasoro cocycle for certain kinds of degenerations of the torus.     

First Integrals and Parametric Solutions for Equations Integrable Through Lie Symmetries

Abstract

We present here the explicit parametric solutions of second order differential equations invariant under time translation and rescaling and third order differential equations invariant under time translation and the two homogeneity symmetries. The computation of first integrals gives in the most general case, the parametric form of the general solution. For some polynomial functions we obtain a time parametrisation quadrature which can be solved in terms of “known” functions.     

Dynamical coupling in the differential equations approach to atom-diatom exchange reactions

The atomic exchange reaction A + BC → AB + C is investigated quantum mechanically employing a coupled differential equations approach. The relative motion in reactant and product channels is described in the common coordinate R ₃ (the AC nuclear separation) and is developed in three-dimensional space. The total wave functions of the system are expressed as a superposition of valence bond electronic states of the initial (A, BC) and final (AB, C) configurations, with the coefficients describing the relative and internal (vibrational, rotational) nuclear motions. Choosing convenient trial functions with the appropriate boundary conditions and using the Kohn variational principle, a set of differential (rather than the usual integro-differential) equations is obtained for the relative motion wave functions in R ₃. The potential matrix elements turn out to be dynamical in that they depend on the initial k ₁ and final k ₂ wave vectors. Two-state coupled channel calculations of the differential and integral cross sections for the isotopic species D + H₂, H + H₂ and D + D₂ are presented for collision energies up to 0·8 eV.     

On the Bilinear Equations for Fredholm Determinants Appearing in Random Matrices

Abstract

It is shown how the bilinear differential equations satisfied by Fredholm determinants of integral operators appearing as spectral distribution functions for random matrices may be deduced from the associated systems of nonautonomous Hamiltonian equations satisfied by auxiliary canonical phase space variables introduced by Tracy and Widom. The essential step is to recast the latter as isomonodromic deformation equations for families of rational covariant derivative operators on the Riemann sphere and interpret the Fredholm determinants as isomonodromic τ -functions.     

On Lie Reduction of the Navier-Stokes Equations

Abstract

Lie reduction of the Navier-Stokes equations to systems of partial differential equations in three and two independent variables and to ordinary differential equations is described.     

Bound-State Solutions of the Klein-Gordon Equation for the Generalized <Emphasis Type="Italic">PT</Emphasis>-Symmetric Hulthén Potential

The one-dimensional Klein-Gordon equation is solved for the PT-symmetric generalized Hulthén potential in the scalar coupling scheme. The relativistic bound-state energy spectrum and the corresponding wave functions are obtained by using the Nikiforov-Uvarov method which is based on solving the second-order linear differential equations by reduction to a generalized equation of hypergeometric type. PACS numbers: 03.65.Fd, 03.65.Ge     

Linearization of one-dimensional nonautonomous jump-diffusion stochastic differential equations

Abstract

Necessary and sufficient conditions for the linearization of one-dimensional nonautonomous jump-diffusion stochastic differential equations are given. Stochastic integrating factor is introduced to solve the linear jump- diffusion stochastic differential equations. Closed form solutions to certain linearizable jump-diffusion stochastic differential equations are obtained.     

Differential Equations Compatible with KZ Equations

We define a system of dynamical differential equations compatible with the KZ differential equations. The KZ differential equations are associated to a complex simple Lie algebra g. These are equations on a function of n complex variables z _i taking values in the tensor product of n finite dimensional g-modules. The KZ equations depend on the dual variable in the Cartan subalgebra of g. The dynamical differential equations are differential equations with respect to the dual variable. We prove that the standard hypergeometric solutions of the KZ equations also satisfy the dynamical equations. As an application we give a new determinant formula for the coordinates of a basis of hypergeometric solutions.     

Nonstationary two-flux model of radiation transport for optical tomography of scattering media

A nonstationary two-flux model is formulated for the transport of radiation in an inhomogeneous scattering medium and is applied to the situation where such a medium is irradiated by the narrow beam of a pulsed laser. It is shown that when the time distribution of the transmitted photons is measured it is possible simultaneously to reconstruct the two spatial functions (the coefficients of absorption coefficient and of scattering of the radiation by the medium) by means of an inverse Radon transformation and the solution of a system of nonlinear differential equations on the projection lines. An analytic solution is obtained in quadratures for these differential equations. The results constitute a method of solving problems in optical tomography in an inhomogeneous scattering medium Zh. Tekh. Fiz. 67, 61–65 (May 1997)     

Massive Field Equations of Arbitrary Spin in Schwarzschild Geometry: Separation Induced by Spin-{{3\over2}} Case

The separation of variables of the spin- field equation is performed in detail in the Schwarzschild geometry by means of the Newman Penrose formalism. The separated angular equations coincide with those relative to the Robertson-Walker space-time. The separated radial equations, that are much more entangled, can be reduced to four ordinary differential equations, each in one only radial function. As a consequence of the particular nature of the spin coefficients it is shown, by induction, that the massive field equations can be separated for arbitrary spin. baselineskip=12 pt PACS 04.20.Cv- Fundamental problems and general formalism. PACS 03.65.Pm- Relativistic wave equations. PACS 02.30.Jr- Partial differential equations. PACS 04.20.Jb- Exact solutions.     

Graph Expansions and Graphical Enumeration Applied to Semiclassical Propagator Expansions

Abstract

Reduction of multidimensional Poincaré-invariant equations to ordinary differential equations and 2-dimensional equations is considered.     

Transient Optical Regime of Two Level Atom in Laser Light

An extended analytic approach is considered for optical Bloch equations in the two level atom interacting with laser light. The separation approach of coupled differential equations is always possible with a sequence of special transformation into the Riccati nonlinear differential equation. The conditions that permit an exact solutions of three coupled system are extracted in a natural manner. The case of sodium atom moving along the axis of a monochromatic wave is treated in some details including a discussion on the radiation pressure forces exerted by laser light in the transient regime. PACS numbers: 32.80.Pj, 42.50.Vk, 42.50.Hz, 42.50.Lc.     

Differential Constraints Compatible with Linearized Equations

Abstract

Differential constraints compatible with the linearized equations of partial differential equations are examined. Recursion operators are obtained by integrating the differential constraints.     

On Integrability of Differential Constraints Arising from the Singularity Analysis

Abstract

Integrability of differential constraints arising from the singularity analysis of two (1+1)-dimensional second-order evolution equations is studied. Two nonlinear ordinary differential equations are obtained in this way, which are integrable by quadratures in spite of very complicated branching of their solutions.     

The Method of an Exact Linearization of n-order Ordinary Differential Equations

Abstract

Necessary and sufficient conditions are found that the n-order nonlinear and nonautonomous ordinary differential equation could be transformed into a linear equation with constant coefficients with the help, generally speaking, nonlocal transformation of dependent and independent variables. These conditions are expressed in termes of factorization through first order nonlinear differential operators. Examples are considered also.

“Two subjects that are theoretical physics and integration of differential equations, are quitely impossible one without another, were always developing together, and the success of one of them influenced another”

(V.P. Ermakov)     

Generalised Symmetries and the Ermakov-Lewis Invariant

Abstract

Generalised symmetries and point symmetries coincide for systems of first-order ordinary differential equations and are infinite in number. Systems of linear first-order ordinary differential equations possess a generalised rescaling symmetry. For the system of first-order ordinary differential equations corresponding to the time-dependent linear oscillator the invariant of this symmetry has the form of the famous Ermakov-Lewis invariant, but in fact reveals a richer structure.     

On Reduction of the Euler Equations by Means of Two-Dimensional Algebras

Abstract

A complete set of inequivalent two-dimensional subalgebras of the maximal Lie invariance algebra of the Euler equations is constructed. Using some of them, the Euler equations are reduced to systems of partial differential equations in two independent variables which are integrated in quadratures.