Orbital stability of the smooth solitary wave with nonzero asymptotic value for the mCH equation

This paper is concerned with orbital stability of the smooth solitary wave with nonzero asymptotic value for the mCH equation

Under the parametric conditions a > 0 and , an interesting phenomenon is discovered, that is, for the stability there exist three bifurcation wave speeds

such that the following conclusions hold.

1. When wave speed belongs to the interval (c₁, c₂) for , the smooth solitary wave is orbitally stable.

2. When wave speed belongs to the interval (c₂, c₃) for , the smooth solitary wave is orbitally unstable.

3. When wave speed belongs to the interval (c₁, c₃) for , the smooth solitary wave is orbitally unstable.

     

Interrelations of discrete Painlevé equations through limiting procedures

We study the discrete Painlevé equations associated to the affine Weyl group which can be obtained by the implementation of a special limits of -associated equations. This study is motivated by the existence of two -associated discrete both having a double ternary dependence in their coefficients and which have not been related before. We show here that two equations correspond to two different limits of a -associated discrete Painlevé equation. Applying the same limiting procedures to other -associated equations we obtained several -related equations most of which have not been previously derived.     

Products in the category of -manifolds

We prove that the category of -manifolds has all finite products. Further, we show that a -manifold (resp., a -morphism) can be reconstructed from its algebra of global -functions (resp., from its algebra morphism between global -function algebras). These results are of importance in the study of Lie groups. The investigation is all the more challenging, since the completed tensor product of the structure sheafs of two -manifolds is not a sheaf. We rely on a number of results on (pre)sheaves of topological algebras, which we establish in the appendix.     

Connection between the ideals generated by traces and by supertraces in the superalgebras of observables of Calogero models

If G is a finite Coxeter group, then symplectic reflection algebra H := H_1,η (G) has Lie algebra of inner derivations and can be decomposed under spin: H = H₀ ⊕ H_1/2 ⊕ H₁ ⊕ H_3/2 ⊕ …?We show that if the ideals of all the vectors from the kernel of degenerate bilinear forms B_i(x, y) := sp_i (x · y), where sp_i are (super)traces on H, do exist, then if and only if .     

On (2)-relative cohomology of the Lie algebra of vector fields and differential operators

Abstract

Let Vect(?) be the Lie algebra of smooth vector fields on ?. The space of symbols Pol(T*?) admits a non-trivial deformation (given by differential operators on weighted densities) as a Vect(?)-module that becomes trivial once the action is restricted to (2) ? Vect(?). The deformations of Pol(T*?), which become trivial once the action is restricted to (2) and such that the Vect(?)-action on them is expressed in terms of differential operators, are classified by the elements of the weight basis of , where denotes the differential cohomology (i.e., we consider only cochains that are given by differential operators) and where D _λ,μ = Homdiff(F _λ, F _μ) is the space of differential operators acting on weighted densities. The main result of this paper is computation of this cohomology. In addition to relative cohomology, we exhibit 2-cocycles spanning and (2).     

SO(4)-symmetry of mechanical systems with 3 degrees of freedom

We answered an old question: does there exist a mechanical system with 3 degrees of freedom, except for the Coulomb system, which has 6 first integrals generating the Lie algebra (4) by means of the Poisson brackets? A system which is not centrally symmetric, but has 6 first integrals generating Lie algebra (4), is presented. It is shown also that not every mechanical system with 3 degrees of freedom has first integrals generating (4).     

Differential Operators,Symmetries and the Inverse Problem for Second-Order Differential Equations

Abstract

With each second-order differential equation Z in the evolution space J ¹(M _n+1) we associate, using the natural f(3, ?1)-structure and the f(3, 1)-structure K, a group of automorphisms of the tangent bundle T (J ¹(M _n+1)), with isomorphic to a dihedral group of order 8. Using the elements of and the Lie derivative, we introduce new differential operators on J ¹(M _n+1) and new types of symmetries of Z. We analyze the relations between the operators and the “dynamical” connection induced by Z. Moreover, we analyze the relations between the various symmetries, also in connection with the inverse problem for Z. Both the approach based on the Poincaré–Cartan two forms and the one relying on the introduction of the so-called metrics compatible with Z are explicitly worked out.     

Note on algebro-geometric solutions to triangular Schlesinger systems

We construct algebro-geometric upper triangular solutions of rank two Schlesinger systems. Using these solutions we derive two families of solutions to the sixth Painlevé equation with parameters (1/8, ?1/8, 1/8, 3=8) expressed in simple forms using periods of differentials on elliptic curves. Similarly for every integer n different from 0 and ?1 we obtain one family of solutions to the sixth Painlevé equation with parameters .     

The Shapovalov Determinant for the Poisson Superalgebras

Abstract

Among simple ?-graded Lie superalgebras of polynomial growth, there are several which have no Cartan matrix but, nevertheless, have a quadratic even Casimir element C ₂: these are the Lie superalgebra of vector fields on the (1|6)-dimensional supercircle preserving the contact form, and the series: the finite dimensional Lie superalgebra of special Hamiltonian fields in 2k odd indeterminates, and the Kac–Moody version of . Using C ₂ we compute N. Shapovalov determinant for and , and for the Poisson superalgebras associated with . A. Shapovalov described irreducible finite dimensional representations of and ; we generalize his result for Verma modules: give criteria for irreducibility of the Verma modules over and      

Generalized Solvable Structures and First Integrals for ODEs Admitting an Symmetry Algebra

The notion of solvable structure is generalized in order to exploit the presence of an algebra of symmetries for a kth-order ordinary differential equation with k > 3. In this setting, the knowledge of a generalized solvable structure for allows us to reduce to a family of second-order linear ordinary differential equations depending on k ? 3 parameters. Examples of explicit integration of fourth and fifth order equations are provided in order to illustrate the procedure.