New finite-gap solutions for the coupled Burgers equations engendered by the Neumann systems

On the tangent bundle TS^N-1 of the unit sphere S^N-1, this paper reduces the coupled Burgers equations to two Neumann systems by using the nonlinearization of the Lax pair, whose Liouville integrability is displayed in the scheme of the r-matrix technique. Based on the Lax matrix of the Neumann systems, the Abel--Jacobi coordinates are appropriately chosen to straighten out the restricted Neumann flows on the complex torus, from which the new finite-gap solutions expressed by Riemann theta functions for the coupled Burgers equations are given in view of the Jacobi inversion.     

Properties of certain matrices related to the equilibrium configuration of the one-dimensional many-body problems with the pair potentialsV 1(x)=−log ∣sinx∣ andV 2(x)=1/sin2 x

It is shown that at equilibrium certain matrices associated to the one-dimensional many-body problems with the pair potentialsV ₁(x)=–logsinx andV ₂(x)=1/sin² x have a very simple structure. These matrices are those that characterize the small oscillations of these systems around their equilibrium configurations, and, for the second system, the Lax matrices that demonstrate its integrability.     

Initial-boundary value problem for the two-component Gerdjikov-Ivanov equation on the interval

In this paper, we apply Fokas unified method to study initial-boundary value problems for the two-component Gerdjikov-Ivanov equation formulated on the finite interval with 3×3 Lax pairs. The solution can be expressed in terms of the solution of a 3×3 Riemann-Hilbert problem. The relevant jump matrices are explicitly given in terms of three matrix-value spectral functions s (λ), S (λ) and S_L(λ), which arising from the initial values at t = 0, boundary values at x = 0 and boundary values at x = L, respectively. Moreover, The associated Dirichlet to Neumann map is analyzed via the global relation. The relevant formulae for boundary value problems on the finite interval can reduce to ones on the half-line as the length of the interval tends to infinity.     

Real Forms of Quantum Orthogonal Groups,q-Lorentz Groups in any Dimension

We review known real forms of the quantum orthogonal groups SO _q (N). New *-conjugations are then introduced and we contruct all real forms of quantum orthogonal groups. We thus give an RTT formulation of the *-conjugations on SO _q (N) that is complementary to the U _q (g) *-structure classification of Twietmeyer. In particular, we easily find and describe the real forms SO _q (N-1,1) for any value of N. Quantum subspaces of the q-Minkowski space are analyzed.     

Solvable Systems Featuring 2 Dependent Variables Evolving in Discrete-Time via 2 Nonlinearly-Coupled First-Order Recursion Relations with Polynomial Right-Hand Sides

The evolution equations mentioned in the title of this paper read as follows:

where ? is the “discrete-time” independent variable taking integer values (? = 0, 1, 2,?…?), xn ≡ xn(?) are the 2 dependent variables, , and the 2 functions P⁽ⁿ⁾(x₁, x₂), n = 1, 2, are 2 polynomials in the 2 dependent variables x₁(?) and x₂(?). The results reported in this paper have been obtained by an appropriate modification of a recently introduced technique to obtain analogous results in continuous-time t—in which case x_n ≡ x_n(t) and the above recursion relations are replaced by first-order ODEs. Their potential interest is due to the relevance of this kind of evolution equations in various applicative contexts.     

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.     

∗-Representations of the Quantum Algebra U q(sl(3))

Abstract

Studied in this paper are real forms of the quantum algebra U _q(sl(3)). Integrable operator representations of ?-algebras are defined. Irreducible representations are classified up to a unitary equivalence.     

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).     

Poisson-Lie Interpretation of Trigonometric Ruijsenaars Duality

A geometric interpretation of the duality between two real forms of the complex trigonometric Ruijsenaars-Schneider system is presented. The phase spaces of the systems in duality are viewed as two different models of the same reduced phase space arising from a suitable symplectic reduction of the standard Heisenberg double of U(n). The collections of commuting Hamiltonians of the systems in duality are shown to descend from two families of ‘free’ Hamiltonians on the double which are dual to each other in a Poisson-Lie sense. Our results give rise to a major simplification of Ruijsenaars’ proof of the crucial symplectomorphism property of the duality map.     

Spectral and scattering theory for a class of strongly oscillating potentials

Following a recent investigation by Pearson [23] on scattering theory for some class of oscillating potentials, we consider the Schrödinger operator inL ²(IR ⁿ ) given by:H =-e ^?U?·e ^2U ?e ^?U +e ^?2U (F + (?·Q)). HereU andF are real functions ofx, andQ is a IR ⁿ -valued function ofx, such that:

1. U is bounded, and the local singularities ofF andQ ² are controlled in a suitable sense by the kinetic energy,
2. U, Q, andF tend to zero at infinity faster than ‖x‖^?1. We defineH by a method of quadratic forms and derive the usual results of scattering theory, namely: the absolutely continuous spectrum is [0, ∞) and the singular continuous spectrum is empty, the wave operators exist and are asymptotically complete. This enlarges the class of already studied strongly oscillating potentials that give rise to the scattering and spectral properties mentioned above.

     

Hamiltonian Structures and Integrability of Frobenius Algebra-Valued (n,m)th KdV Hierarchy

We introduce Frobenius algebra ?-valued (n, m)^th KdV hierarchy and construct its bi-Hamiltonian structures by employing ?-valued pseudo-differential operators. As an illustrative example, the (1, 1)^th -valued case is analyzed in detail. Its Hamiltonian structures and recursion operator are derived. Infinitely many symmetries, conservation laws and explicit flow equations are also obtained.     

r-Matrix for the Restricted KdV Flows with the Neumann Constraints

Abstract

Under the Neumann constraints, each equation of the KdV hierarchy is decomposed into two finite dimensional systems, including the well-known Neumann model. Like in the case of the Bargmann constraint, the explicit Lax representations are deduced from the adjoint representation of the auxiliary spectral problem. It is shown that the Lax operator satisfies the r-matrix relation in the Dirac bracket. Thus, the integrabilities of these resulting systems with the Neumann constraints are obtained.