Matrix Pseudodifferential Operator Research Articles

Topological obstructions to the diagonalisation of pseudodifferential systems

Given a matrix pseudodifferential operator on a smooth manifold, one may be interested in diagonalising it by choosing eigenvectors of its principal symbol in a smooth manner. We show that diagonalisation is not always possible, on the whole cotangent bundle or even in a single fibre. We identify global and local topological obstructions to diagonalisation and examine physically meaningful examples demonstrating that all possible scenarios can occur.

A Geometric Construction of Solutions of the Strict h-Hierarchy

Let h be a complex commutative subalgebra of the n×n matrices Mn(ℂ). In the algebra MPsd of matrix pseudodifferential operators in the derivation ∂, we previously considered deformations of h[∂] and of its Lie subalgebra h[∂]>0 consisting of elements without a constant term. It turned out that the different evolution equations for the generators of these two deformed Lie algebras are compatible sets of Lax equations and determine the corresponding h-hierarchy and its strict version. Here, with each hierarchy, we associate an MPsd-module representing perturbations of a vector related to the trivial solution of each hierarchy. In each module, we describe so-called matrix wave functions, which lead directly to solutions of their Lax equations. We next present a connection between the matrix wave functions of the h-hierarchy and those of its strict version; this connection is used to construct solutions of the latter. The geometric data used to construct the wave functions of the strict h-hierarchy are a plane in the Grassmannian Gr(H), a set of n linearly independent vectors {wi} in W, and suitable invertible maps δ: S1 → h, where S1 is the unit circle in ℂ*. In particular, we show that the action of a corresponding flow group can be lifted from W to the other data and that this lift leaves the constructed solutions of the strict h-hierarchy invariant. For n > 1, it can happen that we have different solutions of the strict h-hierarchy for fixed W and {wi}. We show that they are related by conjugation with invertible matrix differential operators.

Integrable deformations in the matrix pseudo differential operators

Inside the algebra MPsd of matrix pseudo differential operators we consider deformations of the generators of a commutative subalgebra, built out of a commutative subalgebra of the n × n -matrices of maximal dimension. The evolution equations that we impose on these generators depend of the way in which one decomposes the algebra MPsd into the direct sum of two Lie subalgebras. We treat two cases that both lead to a compatible system of Lax equations inside MPsd. These systems generalize well-known systems as the AKNS-hierarchy and the multicomponent KP hierarchy. Finally, one shows that the Lax form of the systems is equivalent to a set of zero curvature relations.

A New Scheme of Integrability for (bi)Hamiltonian PDE

We develop a new method for constructing integrable Hamiltonian hierarchies of Lax type equations, which combines the fractional powers technique of Gelfand and Dickey, and the classical Hamiltonian reduction technique of Drinfeld and Sokolov. The method is based on the notion of an Adler type matrix pseudodifferential operator and the notion of a generalized quasideterminant. We also introduce the notion of a dispersionless Adler type series, which is applied to the study of dispersionless Hamiltonian equations. Non-commutative Hamiltonian equations are discussed in this framework as well.

Singular Degree of a Rational Matrix Pseudodifferential Operator

In our previous work we studied minimal fractional decompositions of a rational matrix pseudodifferential operator: H=A/B, where A and B are matrix differential operators, and B is non-degenerate of minimal possible degree deg(B). In the present paper we introduce the singular degree sdeg(H)=deg(B), and show that for an arbitrary rational expression H=sum_a (A^a_1)/(B^a_1)...(A^a_n)/(B^a_n), we have that sdeg(H) is less than or equal to sum_{a,i} deg(B^a_i). If the equality holds, we call such an expression minimal. We study the properties of the singular degree and of minimal rational expressions. These results are important for the computations involved in the Lenard-Magri scheme of integrability.

Rational matrix pseudodifferential operators

The skewfield K(d) of rational pseudodifferential operators over a differential field K is the skewfield of fractions of the algebra of differential operators K[d]. In our previous paper we showed that any H from K(d) has a minimal fractional decomposition H=AB^(-1), where A,B are elements of K[d], B is non-zero, and any common right divisor of A and B is a non-zero element of K. Moreover, any right fractional decomposition of H is obtained by multiplying A and B on the right by the same non-zero element of K[d]. In the present paper we study the ring M_n(K(d)) of nxn matrices over the skewfield K(d). We show that similarly, any H from M_n(K(d)) has a minimal fractional decomposition H=AB^(-1), where A,B are elements of M_n(K[d]), B is non-degenerate, and any common right divisor of A and B is an invertible element of the ring M_n(K[d]). Moreover, any right fractional decomposition of H is obtained by multiplying A and B on the right by the same non-degenerate element of M_n(K [d]). We give several equivalent definitions of the minimal fractional decomposition. These results are applied to the study of maximal isotropicity property, used in the theory of Dirac structures.

Some algebraic properties of differential operators

First, we study the subskewfield of rational pseudodifferential operators over a differential field \documentclass[12pt]{minimal}\begin{document}${\mathcal K}$\end{document}K generated in the skewfield \documentclass[12pt]{minimal}\begin{document}${\mathcal K}((\partial ^{-1}))$\end{document}K((∂−1)) of pseudodifferential operators over \documentclass[12pt]{minimal}\begin{document}${\mathcal K}$\end{document}K by the subalgebra \documentclass[12pt]{minimal}\begin{document}${\mathcal K}[\partial ]$\end{document}K[∂] of all differential operators. Second, we show that the Dieudonnè determinant of a matrix pseudodifferential operator with coefficients in a differential subring \documentclass[12pt]{minimal}\begin{document}${\mathcal A}$\end{document}A of \documentclass[12pt]{minimal}\begin{document}${\mathcal K}$\end{document}K lies in the integral closure of \documentclass[12pt]{minimal}\begin{document}${\mathcal A}$\end{document}A in \documentclass[12pt]{minimal}\begin{document}${\mathcal K}$\end{document}K, and then we give an example of a 2 × 2 matrix with entries in \documentclass[12pt]{minimal}\begin{document}${\mathcal A}[\partial ]$\end{document}A[∂] whose Dieudonnè determinant does not lie in \documentclass[12pt]{minimal}\begin{document}${\mathcal A}$\end{document}A.

Exponential Function of Pseudo-Differential Operators in Gevrey Spaces

The article is devoted to the construction of the exponential function of the matrix pseudo-differential operator, which does not satisfy conditions of any known theorem (see, e.g. Sec. 8 Ch. VIII and Sec. 2 Ch. XI of Treves in Introduction to the theory of pseudodifferential and Fourier integral operator, vols. 1 & 2, Plenum Press, New York, 1980). An application of the exponential function to the fundamental solution of the Cauchy problem for the hyperbolic operators with the characteristics of variable multiplicity is given.

The Potential Method for the Reactance Wave Diffraction Problem in a Scale of Spaces

Abstract This paper is concerned with a screen type boundary value problem arising from the wave diffraction problem with a reactance condition. We consider the problem in a weak formulation within Bessel potential spaces, and where both cases of a complex and a pure real wave number are analyzed. Using the potential method, the boundary value problem is converted into a system of integral equations. The invertibility of the corresponding matrix pseudodifferential operator is shown in appropriate function spaces which allows the conclusion about the existence and uniqueness of a weak solution to the original problem. Higher regularity properties of solutions are also proved to exist in some scale of Bessel potential spaces, upon the corresponding smoothness improvement of given data. In particular, the 𝐶 α -smoothness of solutions in a neighbourhood of the screen edge is established with arbitrary α < 1 in the two-dimensional case and α < 1/2 in the three-dimensional case.

Multi-component hierarchies of double bracket equations

A new integrable hierarchy, with equations defined by double brackets of two matrix pseudo-differential operators (Lax pairs), is constructed. Some algebraic properties are demonstrated. It is also shown that each equation is equivalent to a certain gradient flow. A new version of the Zakharov-Shabat type equations is proved. Formal solutions of this hierarchy are constructed using a matrix “double bracket bilinear identity”.

Generalization of the algebra of the matrix pseudodifferential operators

We introduce the new nonhomogeneous graded matrix algebra Matn(R) over the ring ℜ of the pseudodifferential operators of the arbitrary orders. This algebra forms the generalization of the algebra of the standard matrix pseudodifferential operators. We show that the Matn(R) algebra admits the nontrivial central extension. The case n=2 is considered in some detail. As the example we consider the Mat2(W1+∞), Mat2(Virasoro) and Mat2(sl(2,C)) algebras.

Interaction between thermoelastic and scalar oscillation fields

Three-dimensional mathematical problems of the interaction between thermoelastic and scalar oscillation fields in a general physically anisotropic case are studied by the boundary integral equation methods. Uniqueness and existence theorems are proved by the reduction of the original interface problems to equivalent systems of boundary pseudodifferential equations. In the non-resonance case the invertibility of the corresponding matrix pseudodifferential operators in appropriate functional spaces is shown on the basis of the generalized Sommerfeld-Kupradze type thermoradiation conditions for anisotropic bodies. In the resonance case the co-kernels of the pseudodifferential operators are analysed and the efficient conditions of solvability of the original interface problems are established.

The generalized KP hierarchy, Hamiltonian structure and non-abelian two-boson realization

A matrix generalization of the KP hierarchy is formulated with the help of matrix pseudodifferential operator. While the first Gelfand-Dickey bracket is found to be a non-abelian generalization of the W1 + ∞ algebra, the corresponding second one is found to be nonlinear and nonlocal. Finally, the constrained system corresponding to this hierarchy is considered. In a particular case the resulting equations turn out to be a multicomponent generalization of the Nonlinear Schrödinger equation. It is also demonstrated that a matrix generalization of the usual two-boson realization exists in the present case.

Extensions of the matrix Gelfand-Dickey hierarchy from generalized Drinfeld-Sokolov reduction

The $p\times p$ matrix version of the $r$-KdV hierarchy has been recently treated as the reduced system arising in a Drinfeld-Sokolov type Hamiltonian symmetry reduction applied to a Poisson submanifold in the dual of the Lie algebra $\widehat{gl}_{pr}\otimes {\Complex}[\lambda, \lambda^{-1}]$. Here a series of extensions of this matrix Gelfand-Dickey system is derived by means of a generalized Drinfeld-Sokolov reduction defined for the Lie algebra $\widehat{gl}_{pr+s}\otimes {\Complex}[\lambda,\lambda^{-1}]$ using the natural embedding $gl_{pr}\subset gl_{pr+s}$ for $s$ any positive integer. The hierarchies obtained admit a description in terms of a $p\times p$ matrix pseudo-differential operator comprising an $r$-KdV type positive part and a non-trivial negative part. This system has been investigated previously in the $p=1$ case as a constrained KP system. In this paper the previous results are considerably extended and a systematic study is presented on the basis of the Drinfeld-Sokolov approach that has the advantage that it leads to local Poisson brackets and makes clear the conformal ($\cal W$-algebra) structures related to the KdV type hierarchies. Discrete reductions and modified versions of the extended $r$-KdV hierarchies are also discussed.

The dressing techniques for intermediate hierarchies

A generalized AKNS systems introduced and discussed recently in \cite{dGHM} are considered. It was shown that the dressing technique both in matrix pseudo-differential operators and formal series with respect to the spectral parameter can be developed for these hierarchies.

An algebraic model of graded calculus of variations

The modern theory of integrable systems rests on two fundamental pillars: the classification of Lax [13] and zero-curvature equations [14, 1, 2]; and algebraic models of the classical calculus of variations [9,5] specialized to the residue calculus in modules of differential forms over rings of matrix pseudo-differential operators [9, 6]. Both these aspects of the theory are by now very well understood for integrable systems in one space dimension.