Differential Spectral Problem Research Articles

Eigenvibrations of a beam with two mechanical resonators attached to the ends

The fourth-order ordinary differential spectral problem describing vertical eigenvibrations of a beam with two mechanical resonators attached to the ends is studied. This problem has positive simple eigenvalues and corresponding eigenfunctions. We define limit differential spectral problem and establish the convergence of the eigenvalues and eigenfunctions of the original spectral problem to the eigenvalues and eigenfunctions of the limit spectral problem as parameters of the attached resonators tending to infinity. The initial fourth-order ordinary differential spectral problem is approximated by the finite difference method. Theoretical error estimates for approximate eigenvalues and eigenfunctions are derived. Obtained theoretical results are illustrated by computations for model problem with constant coefficients. Theoretical and experimental results of this paper can be developed for the problems on eigenvibrations of complex mechanical constructions with systems of resonators.

Finite difference approximation of eigenvibrations of a bar with oscillator

The second-order ordinary differential spectral problem governing eigenvibrations of a bar with attached harmonic oscillator is investigated. We study existence and properties of eigensolutions of formulated bar-oscillator spectral problem. The original second-order ordinary differential spectral problem is approximated by the finite difference mesh scheme. Theoretical error estimates for approximate eigenvalues and eigenfunctions of this mesh scheme are established. Obtained theoretical results are illustrated by computations for a model problem with constant coefficients. Theoretical and experimental results of this paper can be developed and generalized for the problems on eigenvibrations of complex mechanical constructions with systems of harmonic oscillators.

Linear integral equations, infinite matrices, and soliton hierarchies

A systematic framework is presented for the construction of hierarchies of soliton equations. This is realised by considering scalar linear integral equations and their representations in terms of infinite matrices, which give rise to all (2 + 1)- and (1 + 1)-dimensional soliton hierarchies associated with scalar differential spectral problems. The integrability characteristics for the obtained soliton hierarchies, including Miura-type transforms, τ-functions, Lax pairs, and soliton solutions, are also derived within this framework.

On the eigenvectors and eigenvalues of an operator pair

We establish conditions for orthogonality of eigenvectors and the type of eigenvalues (real or purely imaginary) of an operator pair A, B in an equation Au + λBu = 0which permute (or skew-permute) with a given third operator. We consider equations with differential operators A and B (the differential spectral problem), with a differential operator A and an integro-differential operator B, and also with algebraic operators A and B.

The two-dimensional second-order differential spectral problem: compatibility conditions, general BTs and integrable equations

Nonlinear evolution systems in two spatial dimensions integrable by the spectral problem ( delta x2- sigma 2 delta y2+ phi 1 delta x+ phi 2 delta y+U(x, y)) psi =0 are considered. It is shown that such systems possess the matrix commutativity representation (T1M,T2M)=0 which is equivalent to the usual 'L-A-B triad' representation of the compatibility condition. General Backlund transformations (BTS) and the general form of integrable equations are found by the recursion operator method.

The hierarchy of the Poisson brackets between the elements of the scattering matrix for the general differential spectral problem

The infinite family of Poisson brackets (S(i1k1)( lambda 1), S(i2k2)( lambda 2))(n(1),n(2)) (n1, n2=0, 1,.) between the elements of the scattering matrix S( lambda ) is calculated for the differential spectral problem of an arbitrary order N with respect to the family of brackets of the form (F, H)(n(1),n(2))=1/2N integral - infinity + infinity dx tr ( delta F/ delta VT. (L++)n1I(L-)n2 delta H/ delta V)-(F to or from H) where I is Gelfand-Dikij Hamiltonian operator and L+, L- are recursion operators. It is shown that the highest Poisson brackets (S,S)(n(1),n(2)) (n1+n2>or=1) in general contain new t-type terms in addition to the classical r-matrices terms. The classical r matrices depend only on n1+n2. The t terms depend separately on n1 and n2. The t-type terms are absent for the two brackets (S, S)(0,0) and (S, S)(0,1). It is demonstrated that both the classical r matrices and t terms are bounded only for special values of arg lambda 1 and arg lambda 2. The cases N=2 and N=3 are considered in detail. For N=2, 3 the quadratic algebras of the Poisson brackets (,)(n(1),n(2)) between scattering data are also calculated. The t-type terms do not contribute to these Poisson brackets and the algebras of the Poisson brackets of the scattering data are the same for different n1 and n2 with a fixed value of n1+n2.

Backlund-Calogero group for the general differential spectral problem of an arbitrary order

The infinite-dimensional Abelian group of general Backlund-Calogero transformations is constructed for the evolution equations which can be integrated by a Gelfand-Dikij-Zakharov-Shabat spectral problem of an arbitrary order. The structure of the recursion operator and transformation properties of the Backlund-Calogero transformations under the gauge group are considered.

General Nth-order differential spectral problem: General structure of the integrable equations, nonuniqueness of recursion operator and gauge invariance

The general scalar Gelfand-Dikij-Zakharov-Shabat spectral problem of arbitrary order is considered within the framework of the AKNS method. The general form of the integrable equations is found. Uncertainties which appear in the construction of recursion operator and transformation properties of the integrable equations under the gauge transformations are considered. The manifestly gauge-invariant formulation of the integrable equations is given. It is shown that the intergrable equations under consideration are Hamiltonian ones with respect to the infinite family of Hamiltonian structures.

General nth order differential spectral problem: General structure of the integrable equations, nonuniqueness of recursion operator, and gauge invariance: B. G. Konopelchenko and V. G. Dubrovsky. Institute of Nuclear Physics, 630090, Novosibirsk 90, U.S.S.R.

General nth order differential spectral problem: General structure of the integrable equations, nonuniqueness of recursion operator, and gauge invariance: B. G. Konopelchenko and V. G. Dubrovsky. Institute of Nuclear Physics, 630090, Novosibirsk 90, U.S.S.R.