Integration Of Linear Differential Equations Research Articles

Symmetric-conjugate splitting methods for linear unitary problems

We analyze the preservation properties of a family of reversible splitting methods when they are applied to the numerical time integration of linear differential equations defined in the unitary group. The schemes involve complex coefficients and are conjugated to unitary transformations for sufficiently small values of the time step-size. New and efficient methods up to order six are constructed and tested on the linear Schrödinger equation.

ON THE CONSTRUCTION OF SOLUTIONS OF DIFFERENTIAL EQUATIONS ACCORDING TO GIVEN SEQUENCES OF ZEROS AND CRITICAL POINTS

A part of the theory of differential equations in the complex plane $\mathbb C$ is the study of their solutions. To obtain them sometimes researchers can use local expand of solution in the integer degrees of an independent variable. In more difficult cases received local expand in fractional degrees of an independent variable, on so-called Newton - Poiseux series. A row of mathematicians for integration of linear differential equations applied a method of so-called generalized degree series, where meets irrational, in general real degree of an independent variable. One of the directions of the theory of differential equations in the complex plane $\mathbb C$ is the construction a function $f$ according given sequence of zeros or poles, zeros of the derivative $f'$ and then find a differential equation for which this function be solution. Some authors studied sequences of zeros of solutions of the linear differential equation \begin{equation*} f''+Af=0, \end{equation*} where $A$ is entire or analytic function in a disk ${\rm \{ z:|z| < 1\} }$. In addition to the case when the above-mentioned differential equation has the non-trivial solution with given zero-sequences it is possible for consideration the case, when this equation has a solution with a given sequence of zeros (poles) and critical points. In this article we consider the question when the above-mentioned differential equation has the non-trivial solution $f$ such that $f^{1/\alpha}$, $\alpha \in {\mathbb R}\backslash \{ 0;-1\} $ is meromorphic function without zeros with poles in given sequence and the derivative of solution $f'$ has zeros in other given sequence, where $A$ is meromorphic function. Let's note, that representation of function by Weierstrass canonical product is the basic element for researches in the theory of the entire functions. Further we consider the question about construction of entire solution $f$ of the differential equation \begin{equation*} f^{(n)} +Af^{m} =0, \quad n,m\in {\mathbb N}, \end{equation*} where $A$ is meromorphic function such that $f$ has zeros in given sequence and the derivative of solution $f'$ has zeros in other given sequence.

Symmetry algebra of the time-dependent Schrödinger equation in constant and uniform fields

The problems of constructing the symmetry operators of the time-dependent Schrödinger equation in the constant and uniform electric and magnetic fields with the use of the obtained operators in the method of non-commutative integration of linear differential equations are considered. A system of defining equations for the first-order symmetry operators is written. The paper investigates the structure of the symmetry algebras. It is shown that in the case of electric field the resulting algebra is the deformation of the Schrödinger algebra sch(1, 3).

Some tastings in Morales-Ramis theory

In this paper we present a short material concerning to some results in Morales-Ramis theory, which relates two different notions of integrability: Integrability of Hamiltonian systems through Liouville Arnold theorem and integrability of linear differential equations through differential Galois theory. As contribution, we obtain the abelian differential Galois group of the variational equation related to a bi-parametric Hamiltonian system.

Асимптотическое интегрирование линейных дифференциальных уравнений второго порядка

Асимптотическое интегрирование линейных дифференциальных уравнений второго порядка

DIFFERENCE ALGEBRAIC RELATIONS AMONG SOLUTIONS OF LINEAR DIFFERENTIAL EQUATIONS

We extend and apply the Galois theory of linear differential equations equipped with the action of an endomorphism. The Galois groups in this Galois theory are difference algebraic groups, and we use structure theorems for these groups to characterize the possible difference algebraic relations among solutions of linear differential equations. This yields tools to show that certain special functions are difference transcendent. One of our main results is a characterization of discrete integrability of linear differential equations with almost simple usual Galois group, based on a structure theorem for the Zariski dense difference algebraic subgroups of almost simple algebraic groups, which is a schematic version, in characteristic zero, of a result due to Z. Chatzidakis, E. Hrushovski, and Y. Peterzil.

On the algebraic properties of integrals of linear differential equations with constant coefficients

On the algebraic properties of integrals of linear differential equations with constant coefficients

What does integrability of finite-gap or soliton potentials mean?

In the example of the Schrödinger/KdV equation, we treat the theory as equivalence of two concepts of Liouvillian integrability: quadrature integrability of linear differential equations with a parameter (spectral problem) and Liouville's integrability of finite-dimensional Hamiltonian systems (stationary KdV equations). Three key objects in this field-new explicit Psi-function, trace formula and the Jacobi problem-provide a complete solution. The Theta-function language is derivable from these objects and used for ultimate representation of a solution to the inversion problem. Relations with non-integrable equations are also discussed.

IntegrableN-dimensional systems on the Hopf algebra andq-deformations

We construct the class of integrable classical and quantum systems on the Hopf algebras describing n interacting particles. We obtain the general structure of an integrable Hamiltonian system for the Hopf algebra A(g) of a simple Lie algebra g and prove that the integrals of motion depend only on linear combinations of k coordinates of the phase space, 2·ind g≤k≤g·ind g, whereind g andg are the respective index and Coxeter number of the Lie algebra g. The standard procedure of q-deformation results in the quantum integrable system. We apply this general scheme to the algebras sl(2), sl(3), and o(3, 1). An exact solution for the quantum analogue of the N-dimensional Hamiltonian system on the Hopf algebra A(sl(2)) is constructed using the method of noncommutative integration of linear differential equations.

Noncommutative solutions of the d'Alembert equation

All the subalgebras of first-order symmetry operators for the d'Alembert equation, generating the bases of solutions in the method of noncommutative integration of linear differential equations, which cannot be constructed in the method of separation of variables, are found. These bases themselves are then given in explicit form. The complete systems of solutions of the d'Alembert equation, determined by noncommutative sets of first-order symmetry operators, are thereby classified.

Quantum projection. Integration of the open Tod quantum chain

A quantum projection method is developed on the basis of noncommutative integration of linear differential equations and the results of M. A. Ol’shanetskii and A. M. Perelomov on the integration of classical Hamiltonian systems (projection method). The method proposed makes it possible to obtain in explicit form solutions of the quantum equations whose classical analogs can be integrated by projection. Then the semisimplicity property of the symmetry algebra of the original equation is no longer a factor. The solution basis of a Schrodinger equation with the potential of an open three-particle Tod chain is constructed as a nontrivial example.

Reduction of quantum analogs of Hamiltonian systems described by Lie algebras to orbits in a coadjoint representation

A study is made of the possibility of reducing quantum analogs of Hamiltonian systems to Lie algebras. The procedure of reducing classical systems to orbits in a coadjoint representation based on Lie algebra is well-known. An analog of this procedure for quantum systems described by linear differential equations (LDEs) in partial derivatives is proposed here on the basis of the method of noncommutative integration of LDEs. As an example illustrating the procedure, an examination is made of nontrivial systems that cannot be integrated by separation of variables: the Gryachev-Chaplygin hydrostat and the Kovalevskii gyroscope. In both cases, the problem is reduced to a system with a smaller number of variables.

Dichotomies and asymptotic integration of linear differential equations of order n

Dichotomies and asymptotic integration of linear differential equations of order n

Asymptotic Integration of Linear Differential Equations

We establish a new result on the asymptotic integration of linear differential systems of the formy′=(Λ+R)y, where Λ is diagonal andRis small. More specifically, we allow for different scales of asymptotic behaviour of the individual matrix elements of Λ andR. We also discuss an application to a self-adjoint fourth order equation.

Integration of the dirac equation, which does not presume complete separation of variables, in Stäckel spaces

The method of noncommutative integration of linear differential equations is used to construct an exact solution of the Dirac equation, which does not presume complete separation of variables, in Stackel spaces. The Dirac equation in an external electromagnetic field is integrated by this method, using one example. The Stackel space under consideration does not enable one to solve this equation exactly within the framework of the theory of separation of variables.

Noncommutative integration of linear differential equations

A method of noncommutative integration of linear partial differential equations that is analogous to noncommutative integration of finite-dimensional Hamiltonian systems is proposed. The method is based on the concept, introduced in the paper, of a λ representation of Lie algebras. The method can be applied to the integration of the Klein-Gordon equation in Riemannian spaces of non-Stackel type (i.e., in spaces that do not admit complete separation of the variables).

Quadratic algebras in the method of noncommutative integration of the wave equation

In this paper, we continue an investigation of applications of the method of noncommutative integration of linear differential equations in partial derivatives (A. V. Shapovalov and I. V. Shirokov, Izv. Vyssh. Uchebn. Zaved., Fizika., No. 4, 116 (1991); ibid., No. 5, 100 (1991)). We demonstrate the application of quadratic algebras (allowing for second-order operators) to the problem of constructing an exact basis for solutions of the wave equation in unseparated variables. For a nontrivial example, we have integrated the three-dimensional wave equation using a nonabelian quadratic algebra.

Reduction and noncommutative integration of linear differential equations

A new method is proposed for derivation of exactly integrable linear differential equations based on the theory of noncommutative integration. The equations are obtained by reduction from original equations which are integrable in the noncommutative sense, with a large number of independent variables. It is shown that the reduced equations cannot be solved by traditional methods, since they do not possess the required algebraic symmetry.

Classification of F algebras and noncommutative integration of the Klein-Gordon equation in Riemannian spaces

The method of noncommutative integration of linear differential equations [A. V. Shapovalov and I. V. Shirokov, Izv. Vyssh. Uchebn. Zaved. Fiz., No. 4, 116; No. 5, 100 (1991)] is used to integrate the Klein-Gordon equation in Riemannian spaces. The situation is investigated where the set of noncommuting symmetry operators of the Klein-Gordon equation consists of first-order operators and one second-order operator and forms a so-called F algebra, which generalizes the concept of a Lie algebra. The F algebra is a quadratic algebra in the given situation. A classification of four- and five-dimensional F algebras is given. The integration of the Klein-Gordon equation in a Riemannian space, which does not admit separation of variables, is demonstrated in a nontrivial example.

Nonlinear Poisson bracket, F-algebras, and noncommutative integration of linear differential equations

We study some properties of the nonlinear Poisson bracket and its analog for linear differential equations in partial derivatives (so-called F-algebras). We propose a method for noncommutative integration of linear differential equations for the case when the equation operator is embedded in an F-algebra. The method is based on the exact infinite irreducible representation of an F-algebra (λ-representation), which is introduced in the present paper.