Related papers: $w$-function of the KdV hierarchy
In a 2004 paper by V. M. Buchstaber and D. V. Leykin, published in "Functional Analysis and Its Applications," for each $g > 0$, a system of $2g$ multidimensional heat equations in a nonholonomic frame was constructed. The sigma function of…
In investigation of boundary-value problems for certain partial differential equations arising in applied mathematics, we often need to study the solution of system of partial differential equations satisfied by hypergeometric functions and…
We introduce the notion of formally self-adjoint conformally covariant polydifferential operators and give some constructions of families of such operators. In one direction, we show that any homogeneous conformally variational scalar…
The nature of so-called differential-algebraic operators and their approximations is constitutive for the direct treatment of higher-index differential-algebraic equations. We treat first-order differential-algebraic operators in detail and…
On the basis of the generalization of the theorem about K-odd operators (K is the Dirac's operator), certain linear combination is constructed, which appears to commute with the Dirac Hamiltonian for Coulomb field. This operator coincides…
We consider representations of quadratic $R$-matrix algebras by means of certain first order ordinary differential operators. These operators turn out to act as parameter shifting operators on the Gauss hypergeometric function and its limit…
Using the Lax operator formalism, we construct a family of pairwise commuting operators such that the Macdonald symmetric functions of infinitely many variables and of two parameters $q,t$ are their eigenfunctions. We express our operators…
A generic differential operator on the vectorial space of polynomial functions was presented in a recent work and applied in the study of differential relations fulfilled by polynomial sequences either orthogonal or 2-orthogonal. Using the…
Let KdV stand for the Nth Gelfand-Dickey reduction of the KP hierarchy. The purpose of this paper is to show that any KdV solution leads effectively to a solution of the q-approximation of KdV. Two different q-KdV approximations were…
We introduce and systematically develop two classes of discrete integrable operators: those with $2\times 2$ matrix kernels and those possessing general differential kernels, thereby generalizing the discrete analogue previously studied. A…
By applying the derivative operator to the corresponding hypergeometric form of a $q$-series transformation due to Andrews [1,Theorem 4], we establish a general harmonic number identity. As the special cases of it, several interesting…
The Schrodinger equations which are exactly solvable in terms of associated special functions are directly related to some self-adjoint operators defined in the theory of hypergeometric type equations. The fundamental formulae occurring in…
In this thesis, we construct an approximate series solution of the Wigner equation in terms of Airy functions, which are semiclassically concentrated on certain Lagrangian curves in two-dimensional phase space. These curves are defined by…
We construct a large family of commutative algebras of partial differential operators invariant under rotations. These algebras are isomorphic extensions of the algebras of ordinary differential operators introduced by Grunbaum and Yakimov…
By using pseudo-differential operators containing two derivations, we extend the Kadomtsev-Petviashvili (KP) hierarchy to a certain KP-mKP hierarchy. For the KP-mKP hierarchy, we obtain its B\"{a}cklund transformations, bilinear equations…
We study relations of the Weierstrass's hyperelliptic al-functions over a non-degenerated hyperelliptic curve $y^2 = f(x)$ of arbitrary genus $g$ as solutions of sine-Gordon equation using Weierstrass's local parameters, which are…
We investigate the Schr\"{o}dinger operators $H_\varepsilon=-\Delta +W+V_\varepsilon$ in $\mathbb{R}^2$ with the short-range potentials $V_\varepsilon$ which are localized around a smooth closed curve $\gamma$. The operators $H_\varepsilon$…
In this article we introduce Hecke operators on the differential algebra of geometric quasi-modular forms. As an application for each natural number $d$ we construct a vector field in six dimensions which determines uniquely the polynomial…
We develop the theory of Abelian functions defined using a tetragonal curve of genus six, discussing in detail the cyclic curve $y^4 = x^5 + \lambda_4x^4 + \lambda_3x^3 + \lambda_2x^2 + \lambda_1x + \lambda_0$. We construct Abelian…
We solve some forms of non homogeneous differential equations in one and two dimensions. By expanding the solution into whell-posed closed form-Eisenstein series the solution itself is quite simple and elementary. Also we consider Fourier…