Periodic and constant solutions of matrix Riccati differential equations: n — 2. Proc. Roy. Sur 1’equation differentielle matricielle de type Riccati. Bull. Math. The qualitative study of second order linear equations originated in the classic paper . for a history of the Riccati transformation. Differentielle. (Q(t),’)’. VESSIOT, E.: “Sur quelques equations diffeYentielles ordinaires du second ordre .” Annales de (3) “Sur l’equation differentielle de Riccati du second ordre.

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If Pn does not exist, then case 2 cannot hold.

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In this section we find solutions to 4. The aim of this paper is to establish a Galoisian approach to the tech- niques given by Suslov et.

Help Center Find new research papers in: Key words and phrases. As in [32], we obtain three conditions for n to get virtual solv- ability of the differential Galois group. In mathematicsa Riccati equation in the narrowest sense is any first-order ordinary differential equation that is quadratic in the unknown function.

Toy models differentlelle propagators inspired by integrable Riccati equations and integrable characteristic equations are also presented. For example, at loot. We denote by G0 the connected diffetentielle of the identity, thus, when G0 satisfies differentielld property, we say that G virtually satisfies such property.

The following lemmas show how can we construct propagators based on explicit solutions in 15 and Translation of the original Latin into English by Ian Bruce.

The starting point is the knowledge of the integrability, in the Picard-Vessiot sense, of the characteristic equation, or equivalently the existence of an al- gebraic solution over its differential field of its associated Riccati equation. The Galois theory of differential equations, also called Differential Galois Theory and Picard-Vessiot The- ory, has been developed by Picard, Vessiot, Kolchin and currently by a lot of researchers, see [2, 3, 15, 16, 17, 25, 38].

The fact that in quan- tum electrodynamics the electromagnetic field can be represented as a set of forced harmonic oscillators makes quadratic Hamiltonians of special interest [7, 8, 11, 14, 35, 39]. More generally, the term Riccati equation is used to refer to matrix equations with an analogous quadratic term, which occur in both continuous-time and discrete-time linear-quadratic-Gaussian control.


Here we follow the version of Kovacic Algorithm given in [1, 3, 17]. Suslov, Dynamical invariants for variable quadratic Hamiltonians, Physica Scripta 81 5, 11 pp ; see also Preprint arXiv: This Galoisian equtaion depends on the nature of the solutions squation the differential equation; for instance, one obtains some kind of solvability virtual for the Galois group whenever one obtains Liouvillian solutions, and in this case one says that the differential equation is inte- grable.

Thus, we mean by inte- grable whenever the differential equation has Liouvillian solutions instead of special functions solutions.

In [1] there is a complete study of the Galoisian structure of this equation. Subscribe to our newsletter Some error text Name. Takahasi, Information theory of quantum-mechanical channels Advances in Com- munication Systems: Only for cases 1, 2 and 3 we can solve the differential equation, but for the case 4 the differential equation is not integrable.

In this paper a Galoisian approach to build propagators through Riccati equations is presented. Orszag, Quantum OpticsSpringer, Heidelberg, The main application of this Ga- loisian approach, for instance a main result of the paper, is the construction of the propagator for the so called degenerate parametric oscillator: The special functions are not always Liouvillian, we can see that Airy equation has not Liouvillian solutions, while Bessel equation has Liouvillian solutions for special values of the parameter, see [13, 29].

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The second author acknowledges being a recipient of Becas Iberoamericanas, jovenes profesores e invetigadores Santander Universidades during Final Remarks This paper is differeentielle starting point to study the integrability of partial differ- ential equations in a more general sense through differential Galois theory. Moreover, recent applications to mathematical physics can be found in [1, 3, 27, 28, 32].

The second author greatly appreciates the support by University of Puerto Rico and Universitat de Barcelona during the academic visit to the latter in Spring where this project was born. Skip to main content.

Histoire des équations — Wikipédia

Toy Examples In this section we illustrate our Galoisian approach eqjation some ele- mentary examples. The same holds true for the Riccati equation. Further details and proofs can be found in [1, 3]. In this paper we present a Galoisian approach of how to find explicit propagators through Liouvillian solutions for linear second order differential equations associated to Riccati equations. Email address subscribed successfully. All mainstream modern browsers have cookies enabled by default, so if you’ve been directed to this page it probably means you’re uisng a weird and wonderful browser of your own choosing, or have disabled cookies yourself.


Ko- vacic in [17] and Hamiltonian Algebrization developed by the first author in [1, 3].

Histoire des équations

Else, case 1 cannot hold. Consider now the following forms associated to any second order differential equation ode and Riccati equation: Propagators and Green Functions. Suslov, Quantum integrals of motion for vari- able quadratic Hamiltonians, Annals of Physics 10, —; see also Preprint arXiv: Furthermore, by Proposition 4, departing from the differential equation 17 we can arrive at the Riccati equation 15 through changes of variables. We present a short summary of the Hamiltonian algebrization process that will be used in Section 4.

The Hamiltonian 2 had also been considered by Angelow and Trifonov [4, 5] in order to describe the light propagation in a nonlinear anisotropic waveguide. Click here to sign up. Galoisian Approach to Propagators In this section we apply the Picard-Vessiot theory in the context of propagators. Beijing, China 51 — [20] C. Log In Sign Up.

The problem with Maple is that the answers can have complicated expressions that should be trans- formed into more suitable and readable expressions. On the other hand, Kovacic Algorithm only works with rational coefficients, for instance, when the differential equation has not rational coefficients we cannot apply Ko- vacic Algorithm.

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