Related papers: On some class of differential-difference equations…
We classify generalised Camassa-Holm type equations which possess infinite hierarchies of higher symmetries. We show that the obtained equations can be treated as negative flows of integrable quasi-linear scalar evolution equations of…
A general form of the fifth-order nonlinear evolution equation is considered. Helmholtz solution of the inverse variational problem is used to derive conditions under which this equation admits an analytic representation. A Lennard type…
The theory of evolution equations has been applied in various ways in general relativity. Following some general considerations about this, some illustrative examples of the use of ordinary differential equations in general relativity are…
The purpose of this note is to provide an alternative proof of two quadratic transformation formulas contiguous to that of Gauss using a differential equation approach.
We introduce a class of ${\mathbb{Z}}_N$ graded discrete Lax pairs, with $N\times N$ matrices, linear in the spectral parameter. We give a classification scheme for such Lax pairs and the associated discrete integrable systems. We present…
We introduce a notion of elliptic differential graded Lie algebra. The class of elliptic algebras contains such examples as the algebra of differential forms with values in endomorphisms of a flat vector bundle over a compact manifold, etc.…
In this paper, we introduce a generalization of Balancing and Balancing-Lucas numbers. We describe some of their properties also we give the related matrix representation and divisibility properties.
We show that the integrable equations of hydrodynamic type admit nonlocal reductions. We first construct such reductions for a general Lax equation and then give several examples. The reduced nonlocal equations are of hydrodynamic type and…
The equivalence transformation algebra $L_{\cal E}$ for the class of equations $u_t -u_{xx}=f(u, u_x) $ is obtained. After getting the differential invariants with respect to $L_{\cal E}$, some results which allow to linearize a subclass of…
We present examples of Lax-integrable multi-dimensional systems of partial differential equations with higher local symmetries. We also consider Lagrangian deformations of these equations and construct variational bivectors on them.
Several classes of systems of evolution equations with one or two vector unknowns are considered. We investigate also systems with one vector and one scalar unknown. For these classes all equations having the simplest higher symmetry are…
The linearization of complex ordinary differential equations is studied by extending Lie's criteria for linearizability to complex functions of complex variables. It is shown that the linearization of complex ordinary differential equations…
We determine the most general group of equivalence transformations for a family of differential equations defined by an arbitrary vector field on a manifold. We also find all invariants and differential invariants for this group up to the…
An evolution-type differential equation encodes the intersection theory of tautological classes on the Hilbert scheme of a family of nodal curves.
In this article we introduce theory and algorithms for learning discrete representations that take on a lattice that is embedded in an Euclidean space. Lattice representations possess an interesting combination of properties: a) they can be…
Fundamentals on Lie group methods and applications to differential equations are surveyed. Many examples are included to elucidate their extensive applicability for analytically solving both ordinary and partial differential equations.
The generalized divided differences are introduced. They are applied to investigate some properties characterizing generalized higher-order convexity. Among others some support-type property is proved.
The aim of this note is to characterize those doubly ordered frames $\langle X, \leq_1, \leq_2 \rangle$ which are embeddable into the canonical frame of its Urquhart complex algebra.
Admissible point transformations of classes of $r$th order linear ordinary differential equations (in particular, the whole class of such equations and its subclasses of equations in the rational form, the Laguerre-Forsyth form, the first…
The problem of computing differential constraints for a family of evolution PDEs is discussed from a constructive point of view. A new method, based on the existence of generalized characteristics for evolution vector fields, is proposed in…