Related papers: On the recursion operators for integrable equation…

In this chapter of the book entitled, "Extending the Theory of Composites to Other Areas of Science" [edited by Graeme W. Milton, 2016] we give a rigorous derivation of the field equation recursion method in the abstract theory of…

We perform a classification of third order integrable systems of evolution equations with respect to higher symmetries. Applying it, we consider polynomial systems that are 0-homogeneous under a suitable weighting of variables with main…

A new recursion formula is presented for the correlation functions of the integrable spin 1/2 XXX chain with inhomogeneity. It relates the correlators involving n consecutive lattice sites to those with n-1 and n-2 sites. In a series of…

When applying Hamiltonian operator splitting methods for the time integration of multi-species Vlasov-Maxwell-Landau systems, the reliable and efficient numerical approximation of the Landau equation represents a fundamental component of…

We have developed a McMurchie-Davidson-like recursion formula for efficient evaluation of the Coulomb attraction and interaction matrix elements between two-dimensional primitive Cartesian Gaussian type orbitals. We also present recurrence…

This review is an extended version of the Seoul ICM 2014 proceedings.It is a short overview of the "topological recursion", a relation appearing in the asymptotic expansion of many integrable systems and in enumerative problems. We recall…

In this paper we show how to find a closed form solution for third order difference operators in terms of solutions of second order operators. This work is an extension of previous results on finding closed form solutions of recurrence…

In this paper we found a Lagrangian representation and corresponding Hamiltonian structure for the constant astigmatism equation. Utilizing this Hamiltonian structure and extra conservation law densities we construct a first evolution…

We suggest a numerical integration procedure for solving the equations of motion of certain classical spin systems which preserves the underlying symplectic structure of the phase space. Such symplectic integrators have been successfully…

We present a recursive algorithm for multi-coefficient inversion in nonlinear Helmholtz equations with polynomial-type nonlinearities, utilizing the linearized Dirichlet-to-Neumann map as measurement data. To achieve effective recursive…

We propose a linearly implicit structure-preserving numerical method for semilinear Hamiltonian systems with polynomial nonlinearities, combining Kahan's method and exponential integrator. This approach efficiently balances computational…

We apply Cartan's method of equivalence to find a B\"acklund autotransformation for the tangent covering of the universal hierarchy equation. The transformation provides a recursion operator for symmetries of this equation.

The problem of inverting the total divergence operator is central to finding components of a given conservation law. This might not be taxing for a low-order conservation law of a scalar partial differential equation, but integrable systems…

In this Chapter, using Riccati equation as our main example, we tried to demonstrate at least some of the ideas and notions introduced in Chapter 1 - integrability in quadratures, conservation laws, etc. Regarding transformation group and…

A notion of a particular integrability is introduced when two operators commute on a subspace of the space where they act. Particular integrals for one-dimensional (quasi)-exactly-solvable Schroedinger operators and Calogero-Sutherland…

The article deals with the problem of the integrable discretization of the well-known Drinfeld-Sokolov hierarchies related to the Kac-Moody algebras. A class of discrete exponential systems connected with the Cartan matrices has been…

We show that if a numerical method is posed as a sequence of operators acting on data and depending on a parameter, typically a measure of the size of discretization, then consistency, convergence and stability can be related by a…

Using standard calculus, explicit formulas for one-, two- and three-dimensional homotopy operators are presented. A derivation of the one-dimensional homotopy operator is given. A similar methodology can be used to derive the…

We establish a polynomial recursion formula for linear Hodge integrals. It is obtained as the Laplace transform of the cut-and-join equation for the simple Hurwitz numbers. We show that the recursion recovers the Witten-Kontsevich theorem…

From the literature it is known that orthogonal polynomials as the Jacobi polynomials can be expressed by hypergeometric series. In this paper, the authors derive several contiguous relations for terminating multivariate hypergeometric…