Matrix Differential Operators Research Articles

Local regularity of weak solutions of semilinear parabolic systems with critical growth

We show that, under so called controllable growth conditions, any weak solution in the energy class of the semilinear parabolic system $$ u_{t} {\left( {t,x} \right)} + Au{\left( {t,x} \right)} = f{\left( {t,x,u, \ldots \nabla ^{m} u} \right)},\quad {\left( {t,x} \right)} \in {\left( {0,T} \right)} \times \Omega . $$ is locally regular. Here, A is an elliptic matrix differential operator of order 2m. The result is proved by writing the system as a system with linear growth in u,... , ∇ m u but with “bad” coefficients and by means of a continuity method, where the time serves as parameter of continuity.

Classification of quasifinite modules over Lie algebras of matrix differential operators on the circle

We prove that an irreducible quasifinite module over the central extension of the Lie algebra of N × N N\times N -matrix differential operators on the circle is either a highest or lowest weight module or else a module of the intermediate series. Furthermore, we give a complete classification of indecomposable uniformly bounded modules.

Variable coefficient Schrödinger flows for ultrahyperbolic operators

We consider Schrödinger flows which are given by a real symmetric non-degenerate matrix of variable coefficient second order differential operators. After establishing the local smoothing effect we treat non-linear perturbations for first and zero order terms. A fundamental step is the construction of an integrating factor using some non-standard symbols.

Matrix Representations for Vector Differential Operators in General Orthogonal Coordinates

The restrictions in arbitrary orthogonal coordinates to a unified matrix presentation of vector operations using generalized differential matrix operators [Y. Chen et al.: IEEE Trans. on Educ., Vol. 41, (1998), p. 61] are pointed out. The corrected matrix representation for the operations ∇⋅(∇A) and B⋅(∇A) valid in arbitrary orthogonal curvilinear coordinates are obtained and shown to be consistent with the calculation using the dyadic technique. The presentation is accessible to undergraduate students.

On One Class of Matrix Differential Operators

We consider one class of matrix differential operators in the whole space. For this class of operators we establish the isomorphic properties in some special scales of weighted Sobolev spaces and study the regularity properties for solutions to the system of differential equations defined by these operators. The class of operators under consideration contains the stationary Navier–Stokes operator.

Factorization of second-order matrix differential operators and a matrix Darboux transformation

It is shown that a class of matrix Schrödinger operators can be factored into a product of two first-order matrix operators. The equations that relate the elements in these first-order operators to the elements of the potential matrix of the Schrödinger operator are obtained. They are found to be coupled first-order differential equations in the variables of the first-order matrix operators. Finally, an example of a factorization of a matrix operator is obtained, and a general solution associated to a value of the spectral parameter is given. PACS Nos.: 02.30.Mq, 12.39.Pn

Opers and theta functions

We construct natural maps (the Klein and Wirtinger maps) from moduli spaces of semistable vector bundles over an algebraic curve X to affine spaces, as quotients of the nonabelian theta linear series. We prove a finiteness result for these maps over generalized Kummer varieties (moduli space of torus bundles), leading us to conjecture that the maps are finite in general. The conjecture provides canonical explicit coordinates on the moduli space. The finiteness results give low-dimensional parametrizations of Jacobians (in P 3g−3 for generic curves), described by 2 Θ functions or second logarithmic derivatives of theta. We interpret the Klein and Wirtinger maps in terms of opers on X. Opers are generalizations of projective structures, and can be considered as differential operators, kernel functions or special bundles with connection. The matrix opers (analogues of opers for matrix differential operators) combine the structures of flat vector bundle and projective connection, and map to opers via generalized Hitchin maps. For vector bundles off the theta divisor, the Szegö kernel gives a natural construction of matrix oper. The Wirtinger map from bundles off the theta divisor to the affine space of opers is then defined as the determinant of the Szegö kernel. This generalizes the Wirtinger projective connections associated to theta characteristics, and the associated Klein bidifferentials.

Uniform Asymptotic Expansion of the Square-Root Helmholtz Operator and the One-Way Wave Propagator

The Bremmer coupling series solution of the wave equation, in generally inhomogeneous media, requires the introduction of pseudodifferential operators. Such operators appear in the diagonalization process of the acoustic system's matrix of partial differential operators upon extracting a principal direction of (one-way) propagation. In this paper, in three dimensions, uniform asymptotic expansions of the Schwartz kernels of these operators are derived. Also, we derive a uniform asymptotic expansion of the one-way propagator appearing in the series. We focus on designing closed-form representations, valid in the high-frequency limit, taking into account critical scattering-angle phenomena. The latter phenomena are not dealt with in the standard calculus of pseudodifferential operators. Our expansion is not limited by propagation angle. In principle, the uniform asymptotic expansion of a kernel follows by matching its asymptotic behaviors away and near its diagonal.

Preconditioning Techniques for Newton's Method for the Incompressible Navier–Stokes Equations

Newton's method for the incompressible Navier—Stokes equations gives rise to large sparse non-symmetric indefinite matrices with a so-called saddle-point structure for which Schur complement preconditioners have proven to be effective when coupled with iterative methods of Krylov type. In this work we investigate the performance of two preconditioning techniques introduced originally for the Picard method for which both proved significantly superior to other approaches such as the Uzawa method. The first is a block preconditioner which is based on the algebraic structure of the system matrix. The other approach uses also a block preconditioner which is derived by considering the underlying partial differential operator matrix. Analysis and numerical comparison of the methods are presented.

Geometry of the Kaup-Newell equation

It is known that the KdV equation appears naturally in the geometry of the orientation preserving diffeomorphic group Diff( S 1). It is a geodesic flow of a L 2 metric on the Bott-Virasoro group. The nonlinear Schrödinger equation (NLSE) is an evolution equation analogous to the KdV equation which describes an isospectral deformation of the first order 2 × 2 matrix differential operator, yet the family of NSLE has not been studied via diffeomorphic groups. In this paper we derive the Kaup-Newell equation and its generalizations are the Euler-Poincaré flows on the space of first-order scalar (or matrix) differential operators. We show that the operators involved in the flow generated by the action of Vect( S 1) are neither Poisson nor skew symmetric. We also discuss the relation between the KdV flow and the Kaup-Newell flow.

On the essential spectrum of a class of singular matrix differential operators. I: Quasiregularity conditions and essential self-adjointness

The essential spectrum of singular matrix differential operator determined by the operator matrix (-d/dx rho(x)d/dx + q(x) d/dx . beta/x - beta/x . d/dx m(x)/x(2))) is studied. It is proven that the essential spectrum of any self-adjoint operator associated with this expression consists of two branches. One of these branches (called regularity spectrum) can be obtained by approximating the operator by regular operators (with coefficients which are bounded near the origin), the second branch (called singularity spectrum) appears due to singularity of the coefficients.

On the Essential Spectrum of a Class of Matrix Differential Operators

We study the essential spectrum of a class of nonelliptic matrix partial differential operators related to a linear magnetohydrodynamic model.

Commutative Rings of Differential Operators Corresponding to Multidimensional Algebraic Varieties

We construct new examples of multidimensional commuting matrix differential operators and a multidimensional analog of the Kadomtsev-Petviashvili hierarchy.

Essential spectrum of a system of singular differential operators and the asymptotic Hain–Lüst operator

We consider a matrix differential operator with singular entries which arises in magnetohydrodynamics. By means of the asymptotic Hain-Lüst operator and some pseudo-differential operator techniques, we determine the essential spectrum of this operator. Whereas in the regular case, the essential spectrum consists of two intervals, it turns out that in the singular case two additional intervals due to the singularity may arise. In addition, we establish criteria for the essential spectrum to lie in the left half-plane.

Dynamic Green's function for homogeneous and isotropic porous media

SUMMARY The source terms that are meaningful in dynamic poroelasticity are those exciting the centre-of-mass field and the internal field. These fields are the sum of the mass weighted motion and the difference motion of the solid and fluid constituents, respectively. The corresponding homogeneous and isotropic Green's function valid for a uniform whole-space is obtained using Kupradze's method after the vector differential equations for these two fields are combined and expressed as a 6×6 matrix differential operator. The solution is quite amenable to numerical calculations and the results for a saturated Berea sandstone show that the fast P and S waves correspond to those usually detected by geophones at large distances from the source. The slow P wave, which is associated with fluid flow, is rapidly attenuated with distance from the source while the slow S wave, which is part of the solution, dies off rapidly in the near-neighbourhood of the source.

Estimates for the Scattering Map Associated with a Two-Dimensional First-Order System

We consider the scattering transform for the first order system in the plane, (D-Q) \psi =0 where D is the 2x2 diagonal matrix differential operator whose diagonal entries are d-bar and d and Q is a 2x2 off-diagonal matrix. We show that the scattering map is Lipschitz continuous on a neighborhood of zero in L^2.

The trace formula of a matrix differential operator with application of integrable system

The formula of a matrix differential operator is obtained by an approach that is developed of Zakharov–Faddeev method. A new application for integrable system is presented.

Classical Lie subalgebras of the Lie algebra of matrix differential operators on the circle

We give a complete description of the anti-involutions of the algebra DN of N×N-matrix differential operators on the circle, preserving the principal ℤ gradation. We obtain, up to conjugation, two families σ±,m with 1⩽m⩽N, getting two families D±,mN of simple Lie subalgebras fixed by −σ±,m. We also give a geometric realization of σ±,m, concluding that D+,mN is a subalgebra of DN of type o(m,n) and D−,mN is a subalgebra of DN of type osp(m,n) (ortho-symplectic). Finally, we study the conformal algebras associated with D+,mN and D−,mN.

Commuting matrix differential operators and loop algebras

We consider for fixed positive integers p and q which are coprime the space of all pairs ( P, Q) of commuting matrix differential operators (of a fixed size n), where P is monic of order p and Q is normalized of order q. We use the vector valued Sato Grassmannian to construct a natural bijection to an affine subspace of the loop algebra gl(nq)((λ −1)) . In the scalar case ( n=1) the KP flows on the Grassmannian, which are known to trace out Jacobians, lead to commuting flows on this affine space. These flows are Hamiltonian with respect to a family of Poisson structures which are obtained from a family of Lie brackets on the loop algebra.

Colored brackets and 2-manifolds

Frequently, although a set of matrix differential operators will not be closed under, say, a Lie bracket, it might be closed under more general Lie Γ-graded brackets. Some of these operator products are defined using commutation factors for Γ. The classification of commutation factors, and hence Lie Γ-graded brackets, is one of the stepping stones in an attempt to classify algebras of matrix differential operators, generalizing a question outline in [Am. J. Math. 114 (6) (1992) 1163]. Many examples of Lie superalgebras of matrix differential operators exist, but there is still a question of what other possible algebra products may close a space of matrix differential operators. Studying Lie color algebras and Lie color superalgebras represents an attempt in this direction. It is a curious fact that commutation factors for groups of the form ( Z/2 Z) n coincide with homeomorphism-equivalence classes of connected compact 2-manifolds, so this coincidence is also given in Section 4 of this paper.