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The description of many dynamical problems like the particle motion in higher dimensional spherically and axially symmetric space-times is reduced to the inversion of a holomorphic hyperelliptic integral. The result of the inversion is…
Integrable N-particle systems have an important property that the associated Seiberg-Witten prepotentials satisfy the WDVV equations. However, this does not apply to the most interesting class of elliptic and double-elliptic systems.…
We introduce two ordinary second-order linear differential equations of the Laguerre- and Jacobi-type. Solutions are written as infinite series of square integrable functions in terms of the Laguerre and Jacobi polynomials, respectively.…
The paper discusses relations between the structure of the complex Fermi surface below the spectrum of a second order periodic elliptic equation and integral representations of certain classes of its solutions. These integral…
This paper introduces in a natural way a notion of horizontal convex envelopes of continuous functions in the Heisenberg group. We provide a convexification process to find the envelope in a constructive manner. We also apply the…
We derive a priori estimates for solutions of a general class of fully non-linear equations on compact Hermitian manifolds. Our method is based on ideas that have been used for different specific equations, such as the complex…
Properties of solutions of generic hyperbolic systems with multiple characteristics with diagonalizable principal part are investigated. Solutions are represented as a Picard series with terms in the form of iterated Fourier integral…
We present new addition formulae for the Weierstrass functions associated with a general elliptic curve. We prove the structure of the formulae in n-variables and give the explicit addition formulae for the 2- and 3-variable cases. These…
In this paper, we introduce a new class of confluent hypergeometric functions of many variables, study their properties, and determine a system of partial differential equations that this function satisfies. It turns out that all the…
We discuss the application of a variant of the method of simplest equation for obtaining exact traveling wave solutions of a class of nonlinear partial differential equations containing polynomial nonlinearities. As simplest equation we use…
Using a mixture of classical and probabilistic techniques we investigate the convexity of solutions to the elliptic pde associated with a certain generalized Ornstein-Uhlenbeck process.
In the framework of differential Galois theory we treat the classical spectral problem $\Psi"-u(x)\Psi=\lambda\Psi$ and its finite-gap potentials as exactly solvable in quadratures by Picard--Vessiot without involving special functions; the…
In the present paper, a hierarchy of the mKdV equation is integrated by the methods of algebraic geometry. The mKdV hierarchy in question arises on coadjoint orbits in the loop algebra of $\mathfrak{sl}(2)$, and employs a family of…
The article presents various Witt type vector field realizations of 2-D Cayley-Klein algebras with non-vanishing curvatures. The expressions of the vector fields involve Jacobi elliptic functions whose moduli are directly related to the…
In this work an extended elliptic function method is proposed and applied to the generalized shallow water wave equation. We systematically investigate to classify new exact travelling wave solutions expressible in terms of quasi-periodic…
The periodic and quasi-periodic solutions of the integrable system have been studied for four decades based on the Riemann theta functions. However, there is a fundamental difficulty in representing the solutions graphically because the…
The classical phase retrieval problem involves estimating a signal from its Fourier magnitudes (power spectrum) by leveraging prior information about the desired signal. This paper extends the problem to compact groups, addressing the…
We consider complementary dynamical systems related to stationary Korteweg-de Vries hierarchy of equations. A general approach for finding elliptic solutions is given. The solutions are expressed in terms of Novikov polynomials in general…
We give combinatorial models for the homotopy type of complements of elliptic arrangements (i.e., certain sets of abelian subvarieties in a product of elliptic curves). We give a presentation of the fundamental group of such spaces and, as…
We consider the solution of spectral problems with elliptic coefficients in the framework of the Hermite ansatz. We show that the search for exactly solvable potentials and their spectral characteristics is reduced to a system of polynomial…