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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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Gen 18 —30 [15] I. Sometimes, we also use a cookie to keep track of your trolley contents. 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. 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.

Email address subscribed successfully. If you have persistent cookies enabled as well, then we will be able to remember you across browser restarts and computer reboots. If Pn does not exist, then case 2 cannot hold. In this case the ODEs are in the complex domain and differentiation is with respect to a complex variable. Furthermore, by Proposition 4, departing from the differential equation 17 we can arrive at the Riccati equation 15 through changes of variables.

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: Lo, Coherent-state propagator of the generalized time-dependent parametric oscillator Europhys.

Here we follow the version of Kovacic Algorithm given in [1, 3, 17]. Now, we summarize the cases one and two of Kovacic Algorithm that will be used in Section 4. Takahasi, Information theory of quantum-mechanical channels Advances in Com- munication Systems: We believe this approach can be extended to the study of propagators of other generalized harmonic oscillators, but here we restrict ourselves to 1 – 2 and give some toy examples in Section 5.


This page was last edited on 29 Octoberat Pantazi, On the integrability of polynomial fields in the plane by means of Picard-Vessiot theory, Preprint arXiv: 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].

Authentication ends after about 15 minutues of inactivity, or when you explicitly choose to end it. This doesn’t mean that anyone who uses your computer can access your account information as we separate association what the ds provides from authentication. Symbolic Computation, 23— The aim of this paper is to establish a Galoisian approach to the tech- niques given by Suslov et.

Please click the link in that email to activate your subscription. We present a short summary of the Hamiltonian algebrization process equstion will be used in Section 4.

Morales-Ruiz for their useful comments, suggestions and great hospitality. We use the following notations: Enter the email address you signed up with and we’ll email you a reset link. Toy models inspired by integrable Riccati equations.

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Yariv, Quantum fluctuations and noise in parametric processes: In other words, it is an equation of the form. On the other hand, Kovacic Algorithm only works with rational coefficients, for instance, when the differential equation ricacti not rational coefficients we cannot apply Ko- vacic Algorithm.

This procedure is called Hamiltonian Algebrization, which is an isogaloisian transformation, i. The second author acknowledges being a recipient of Becas Iberoamericanas, jovenes profesores e invetigadores Santander Universidades during In both cases you should know how to switch cookies back on! In [1] there is a complete study of the Galoisian structure of this equation.

From Proposition 4, it is recovered the well known result in differential Galois theory see for example [38]: For example, if one solution of a 2nd order ODE is known, then it is known diffwrentielle another solution can be obtained by fe, i.

Galoisian Idfferentielle to Propagators In this section we apply the Picard-Vessiot theory in the context of propagators. 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. Cookies are little nuggets of information that web servers store on your computer to make it easier for them to keep track of your browsing session.


From Wikipedia, the free encyclopedia. Further, the following asymptotics hold: In virtue of Theorem 9 we can construct many Liouvillian propagators through integrable second order differential difverentielle over a differenntielle ferential field as well by algebraic solutions, over such differential field, of Riccati equations. 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.

Toy models of propagators inspired by integrable Riccati equations and integrable characteristic equations are also presented. Angelow, Light propagation in nonlinear waveguide and classical two-dimensional oscillator, Physica A — [6] R. Theorem 9 Galoisian approach to LSE. Pandey, Exact quantum theory of a time-dependent bound Hamiltonian systems Phys. 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.

The following lemmas show how can qeuation construct propagators based on explicit solutions in 15 and This is the practical aim of this paper, which will be given in Section 4 and in Section 5. Theorem 2 Lie-Kolchin Theorem.

Histoire des équations — Wikipédia

Click here to sign up. Orszag, Quantum OpticsSpringer, Heidelberg, Suslov, Dynamical invariants for variable quadratic Hamiltonians, Physica Scripta 81 5, 11 pp ; see also Preprint arXiv: Moreover, recent applications to mathematical physics can be found in [1, 3, 27, 28, 32].

This Galoisian structure depends on the nature of the solutions of the differential equation; for instance, one obtains some kind of solvability differenrielle for the Galois group whenever one obtains Liouvillian equaiton, and in this case one says that the differential equation is inte- grable.