Lie-algebraic structure of integrable nonlinear dynamical systems on extended functional manifolds

We consider the general Lie-algebraic scheme of construction of integrable nonlinear dynamical systems on extended functional manifolds. We obtain an explicit expression for consistent Poisson structures and write explicitly nonlinear equations generated by the spectrum of a periodic problem for an operator of Lax-type representation.     

Structure of integrable supersymmetric nonlinear dynamical systems on reduced invariant submanifolds

Based on an analysis of a supersymmetric extension of the algebra of pseudodifferential operators on ¹ an infinite hierarchy of supersymmetric Lax-integrable nonlinear dynamical systems is constructed by means of the Yang-Baxter -equation method. The structure of these systems on reduced invariant submanifolds specified by a natural invariant Lax-type spectral problem is investigated.Translated from Ukrayins'kyy Matematychnyy Zhurnal, Vol. 44, No. 9, pp. 1292–1295, September, 1992.     

Finite-Dimensional Nonlocal Reductions of the Inverse Korteweg–de Vries Dynamical System

We study finite-dimensional Moser-type reductions for the inverse nonlinear Korteweg–de Vries dynamical system and the Liouville integrability of these reductions in quadratures.     

On the complete integrability and linearization of a Burgers– Korteweg–de Vries-type nonlinear equation

In this work, on the basis of the Bogolyubov–Prykarpats’kyi gradient–holonomic algorithm for the investigation of the integrability of nonlinear dynamical systems on functional manifolds, the exact linearization of a Burgers–Korteweg–de Vries-type nonlinear dynamical system is established. As a result, we describe the linear structure of the space of solutions and show its relation to the convexity of certain functional subsets. The bi-Hamiltonian property of the Burgers–Korteweg–de Vries dynamical system is also established, and the infinite hierarchy of functionally independent invariants is constructed.     

Modification of the lie‐algebraic scheme and approximation error estimations

A new modification of the the Lie‐algebraic scheme for solving partial differential equations with initial and boundary conditions based on constructing quasirepresentations of the Heisenberg‐Weyl algebra operators involving boundary conditions is proposed. Approximation errors for the modified scheme are evaluated. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonlinear integrable systems related to the elliptic lie—baxter algebra

We construct a hierarchy of Poisson Hamiltonian structures related to an “elliptic” spectral problem and determine the generating operators for the equation of asymmetric chiral 0 (3) — field.     

Collapse in coupled nonlinear Schrödinger equations: Sufficient conditions and applications

In this paper we study blow-up phenomena in general coupled nonlinear Schrödinger equations with different dispersion coefficients. We find sufficient conditions for blow-up and for the existence of global solutions. We discuss several applications of our results to heteronuclear multispecies Bose-Einstein condensates and to degenerate boson-fermion mixtures.     

Synthesis,Crystal Structure and Hirshfeld Surfaces of the N-(bis(benzylamino)phosphoryl)-4-Methylbenzenesulfonamide Polymeric tetrakis-Complex {Na[Yb(L)4(iPrOH)]}n

Lanthanide containing polymer chain of title compound {Na[Yb(L)₄(ⁱPrOH)]}_n is formed by alternating complex anions and sodium cations. The anion is composed by four deprotonated bidentate chelate ligands of sulfonylamidophosphate type N-(bis(benzylamino)phosphoryl)-4-methylbenzenesulfonamide (L^?) coordinated to the Yb(III) ion via O atoms belonging to the sulfonyl and phosphoryl groups. The coordination polyhedron of the ytterbium has a slightly distorted triangular dodecahedron conformation, TDD-8, D_2d. The Na⁺ cation is coordinated by four sulfonyl O atoms, the NaO₄ polyhedra shape is intermediate between axially vacant trigonal bipyramid C_3v and tetrahedron T_d. Single-crystal X-ray diffraction studies show that one 2-propanol molecule is hydrogen-bonded to the sulphonyl oxygen of one of the chelate ligands. Also the extensive network of hydrogen bonding interactions between the aromatic rings both in the structural unit and between the polymer chains were detected. They were investigated by the analysis of Hirshfeld surfaces and fingerprint plots. The correlations between structure parameters obtained and the physico-chemical properties of the {Na[Yb(L)₄(ⁱPrOH)]}_n were investigated.

Graphic Abstract

New tetrakis-complex {Na[Yb(L)₄(ⁱPrOH)]}_n was synthesized and characterized by X-ray crystallography. Structural packaging characteristics were investigated by the analysis of Hirshfeld surfaces.