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.     

A Way To Discover Maxwell's Equations Theoretically

The Coulomb force, established in the rest frame of a source-charge Q, when transformed to a new frame moving with a velocity V has a form F = q E + q v × B, where E = E′_∥ + γE′_⊥ and B = (1/c ²)v × E and E′ is the electric field in the rest frame of the source. The quantities E and B are then manifestly interdependent. We prove that they are determined by Maxwell's equations, so they represent the electric and magnetic fields in the new frame and the force F is the well known from experiments Lorentz force. In this way Maxwell's equations may be discovered theoretically for this particular situation of uniformly moving sources. The general solutions of the discovered Maxwell's equations lead us to fields produced by accelerating sources.     

CONSERVATION LAWS FOR THE SCHRÖDINGER–NEWTON EQUATIONS

In this Letter a first-order Lagrangian for the Schrödinger–Newton equations is derived by modifying a second-order Lagrangian proposed by Christian [Exactly soluble sector of quantum gravity, Phys. Rev. D 56(8) (1997) 4844–4877]. Then Noether's theorem is applied to the Lie point symmetries determined by Robertshaw and Tod [Lie point symmetries and an approximate solution for the Schrödinger–Newton equations, Nonlinearity 19(7) (2006) 1507–1514] in order to find conservation laws of the Schrödinger–Newton equations.     

A simple model of the classicalZitterbewegung: Photon wave function

We propose a simple classical model of the zitterbewegung. In this model spin is proportional to the velocity of the particle, the component parallel top is constant and the orthogonal components are oscillating with2p frequency. The quantization of the system gives wave equations for spin,0, 1/2, 1, 3/2,…, etc. respectively. These equations are convenient for massless particles. The wave equation of the spin-1, massless free particle is equivalent to the Maxwell equations and the state functions have a probability interpretation and exhibit conserved current densities. The ground state has zero energy.     

Contiguity relations for discrete and ultradiscrete Painlevé equations

Abstract

We show that the solutions of ultradiscrete Painlevé equations satisfy contiguity relations just as their continuous and discrete counterparts. Our starting point are the relations for q-discrete Painlevé equations which we then proceed to ultradiscretise. In this paper we obtain results for the one-parameter q-P_III, the symmetric q-P_IV and the q-P_IV. These results show that there exists a perfect parallel between the properties of continuous, discrete and ultradiscrete Painlevé equations.     

Triangular Newton Equations with Maximal Number of Integrals of Motion

Abstract

We study two-dimensional triangular systems of Newton equations (acceleration = velocity-independent force) admitting three functionally independent quadratic integrals of motion. The main idea is to exploit the fact that the first component M ₁(q ₁) of a triangular force depends on one variable only. By using the existence of extra integrals of motion we reduce the problem to solving a simultaneous system of three linear ordinary differential equations with nonconstant coefficients for M ₁(q ₁). With the help of computer algebra we have found and solved these ordinary differential equations in all cases. A complete list of superintegrable triangular equations in two dimensions is been given. Most of these equations were not known before.     

Calculating coefficients for a system of moment equations used to describe the magnetization kinetics of a superparamagnetic particle in a fluctuating field

A system of equations is derived for moments [averages of spherical harmonics 〈Y _l,m 〉(t)] that determine the dynamics of the magnetization M of a superparamagnetic particle in a fluctuating field. The system is derived by representing the Gilbert equation in a fluctuating field, and the corresponding Fokker-Planck equation for the distribution function of M, in terms of angular momentum operators, which in turn makes it possible to express the coefficients of the system of moment equations in terms of Clebsch-Gordan coefficients. Fiz. Tverd. Tela (St. Petersburg) 41, 2020–2027 (November 1999)     

Graded Symmetry Algebras of Time-Dependent Evolution Equations and Application to the Modified KP Equations

Abstract

By starting from known graded Lie algebras, including Virasoro algebras, new kinds of time-dependent evolution equations are found possessing graded symmetry algebras. The modified KP equations are taken as an illustrative example: new modified KP equations with m arbitrary time-dependent coefficients are obtained possessing symmetries involving m arbitrary functions of time. A particular graded symmetry algebra for the modified KP equations is derived in this connection homomorphic to the Virasoro algebras.

Dedicated to Prof. W.I. Fushchych on the occasion of his 60th birthday     

Seiberg-Witten-like Equations on 7-Manifolds with G 2-Structure

Abstract

The Seiberg-Witten equations are of great importance in the study of topology of smooth four-dimensional manifolds. In this work, we propose similar equations for 7-dimensional compact manifolds with G ₂-structure.     

On <Emphasis Type="Italic">osp</Emphasis>(<Emphasis Type="Italic">M</Emphasis>∣2<Emphasis Type="Italic">n</Emphasis>) Integrable Open Spin Chains

We consider open spin chains based on osp(M2n) Yangians and solve the reflection equations for some classes of reflection matrices, including the diagonal ones. Having then integrable open spin chains, we write the analytical Bethe Ansatz equations. More details and references can be found in D. Arnaudon et al.: Nucl. Phys B 668 (2003) 469 and 687 (2004) 257.     

The Riccati and Ermakov-Pinney hierarchies

Abstract

The concept and use of recursion operators is well-established in the study of evolution, in particular nonlinear, equations. We demonstrate the application of the idea of recursion operators to ordinary differential equations. For the purposes of our demonstration we use two equations, one chosen from the class of linearisable hierarchies of evolution equations studied by Euler et al (Stud Appl Math 111 (2003) 315–337) and the other from the class of integrable but nonlinearisible equations studied by Petersson et al (Stud Appl Math 112 (2004) 201–225). We construct the hierarchies for each equation. The symmetry properties of the first hierarchy are considered in some detail. For both hierarchies we apply the singularity analysis. For both we observe intersting behaviour of the resonances for the different possible leading order behaviours. In particular we note the proliferation of subsidiary solutions as one ascends the hierarchy.     

Discrete Symmetries and Generalized Fields of Dyons

We have studied the different symmetric properties of the generalized Maxwell’s–Dirac equation along with their quantum properties. Applying the parity (℘), time reversal ( T\mathcal{T} ), charge conjugation (C\mathcal{C}) and their combined effect like parity time reversal (PT\mathcal{PT}), charge conjugation and parity (CP\mathcal{CP}) and CPT\mathcal{CP}T transformations to various equations of generalized fields of dyons, it is shown that the corresponding dynamical quantities and equations of dyons are invariant under these discrete symmetries.     

Asymptotic Scaling in a Model Class of Anomalous Reaction-Diffusion Equations

Abstract

We analyze asymptotic scaling properties of a model class of anomalous reaction-diffusion (ARD) equations. Numerical experiments show that solutions to these have, for large t, well defined scaling properties. We suggest a general framework to analyze asymptotic symmetry properties; this provides an analytical explanation of the observed asymptotic scaling properties for the considered ARD equations.     

A normal coordinate treatment of the plane symmetrical XY4 molecule

The secular equations for the various irreducible representations of the plane XY₄ molecule (symmetry point group D _4h) are deduced by means of Wilson's F-G matrix method, using the most general harmonic force field. The equations for the case of the SVFF are also given.     

LAGRANGIANS FOR BIOLOGICAL MODELS

We show that a method presented in [S. L. Trubatch and A. Franco, Canonical Procedures for Population Dynamics, J. Theor. Biol. 48 (1974) 299–324] and later in [G. H. Paine, The development of Lagrangians for biological models, Bull. Math. Biol. 44 (1982) 749–760] for finding Lagrangians of classic models in biology, is actually based on finding the Jacobi Last Multiplier of such models. Using known properties of Jacobi Last Multiplier we show how to obtain linear Lagrangians of systems of two first-order ordinary differential equations and nonlinear Lagrangian of the corresponding single second-order equation that can be derived from them, even in the case where those authors failed such as the host-parasite model. Also we show that the Lagrangians of certain second-order ordinary differential equations derived by Volterra in [V. Volterra, Calculus of variations and the logistic curve, Hum. Biol. 11 (1939) 173–178] are particular cases of the Lagrangians that can be obtained by means of the Jacobi Last Multiplier. Actually we provide more than one Lagrangian for those Volterra's equations.     

Integrability and Explicit Solutions in Some Bianchi Cosmological Dynamical Systems

Abstract

The Einstein field equations for several cosmological models reduce to polynomial systems of ordinary differential equations. In this paper we shall concentrate our attention to the spatially homogeneous diagonal G ₂ cosmologies. By using Darboux’s theory in order to study ordinary differential equations in the complex projective plane ??² we solve the Bianchi V models totally. Moreover, we carry out a study of Bianchi VI models and first integrals are given in particular cases.     

Derivation of asymptotical formulas for resolution of systems of differential equations with symmetrical matrices

Abstract

Asymptotic formulae for resolution of L-diagonal systems of ordinary differential equations with symmetrical matrices are derived.