共查询到20条相似文献,搜索用时 15 毫秒
1.
Jaume Giné 《Applied Mathematics Letters》2012,25(11):1862-1865
The integrability problem consists in finding the class of functions, a first integral of a given planar polynomial differential system must belong to. We recall the characterization of systems which admit a Darboux, elementary, Liouvillian or Weierstrass first integral. The reduction problem of an integrable planar system consists in finding the class of functions, a map that reduces the original system (transforms into a simple system or equation) must belong to. We identify the class of functions of this map for polynomial, rational, Darboux, elementary, Liouvillian and Weierstrass integrable systems. 相似文献
2.
H. Giacomini 《Journal of Differential Equations》2005,213(2):368-388
We consider a planar differential system , , where P and Q are C1 functions in some open set U⊆R2, and . Let γ be a periodic orbit of the system in U. Let f(x,y):U⊆R2→R be a C1 function such that
3.
In Cheb-Terrab and Roche (Comput Phys Commun 130(1–2):204–231, 2000) a classification of the Abel equations known as solvable
in the literature was presented. In this paper, we show that all the integrable rational Abel differential equations that
appear in Cheb-Terrab and Roche (Comput Phys Commun 130(1–2):204–231, 2000) and consequently in Cheb-Terrab and Roche (Eur
J Appl Math 14(2):217–229, 2003) can be reduced to a Riccati differential equation or to a first-order linear differential
equation through a change with a rational map. The change is given explicitly for each class. Moreover, we have found a unified
way to find the rational map from the knowledge of the explicitly first integral. 相似文献
4.
This paper is devoted to a systematic study of quantum completely integrable systems (i.e., complete systems of commuting differential operators) from the point of view of algebraic geometry. We investigate the eigenvalue problem for such systems and the correspondingD-module when the eigenvalues are in generic position. In particular, we show that the differential Galois group of this eigenvalue problem is reductive at generic eigenvalues. This implies that a system is algebraically integrable (i.e., its eigenvalue problem is explicitly solvable in quadratures) if and only if the differential Galois group is commutative for generic eigenvalues. We apply this criterion of algebraic integrability to two examples: finite-zone potentials and the elliptic Calogero-Moser system. In the second example, we obtain a proof of the Chalyh-Veselov conjecture that the Calogero-Moser system with integer parameter is algebraically integrable, using the results of Felder and Varchenko. 相似文献
5.
Maoan Han 《中国科学A辑(英文版)》1997,40(12):1247-1258
A general method for a homoclinic loop of planar Hamiltonian systems to bifurcate two or three limit cycles under perturbations
is established. Certain conditions are given under which the cyclicity of a homoclinic loop equals 1 or 2. As an application
to quadratic systems, it is proved that the cyclicity of homoclinic loops of quadratic integrable and non-Hamiltonian systems
equals 2 except for one case.
Project supported by the National Natural Science Foundation of China. 相似文献
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When both Hamiltonian operators of a bi-Hamiltonian system are pure differential operators, we show that the generalized Kupershmidt deformation (GKD) developed from the Kupershmidt deformation in [10] offers an useful way to construct new integrable system starting from the bi-Hamiltonian system. We construct some new integrable systems by means of the generalized Kupershmidt deformation in the cases of Harry Dym hierarchy, classical Boussinesq hierarchy and coupled KdV hierarchy. We show that the GKD of Harry Dym equation, GKD of classical Boussinesq equation and GKD of coupled KdV equation are equivalent to the new integrable Rosochatius deformations of these soliton equations with self-consistent sources. We present the Lax pair for these new systems. Therefore the generalized Kupershmidt deformation provides a new way to construct new integrable systems from bi-Hamiltonian systems and also offers a new approach to obtain the Rosochatius deformation of soliton equation with self-consistent sources. 相似文献
9.
Taking linear hyperbolic partial differential equations as an illustration, we attempt to construct weak solutions with higher
integrable gradients, in the sense of Gehring, to hyperbolic diffeential equations with initial and boundary conditions. We
adopt Rothe's method and follow the calculation which has been expanded by Giaquinta and Struwe in dealing with parabolic
equations. To establish the scheme, we evaluate some local estimates for solutions to Rothe's approximations to hyperbolic
differential equations. Bibliography: 6 titles.
Published inZapiski Nauchnykh Seminarov POMI, Vol. 233, 1996, pp. 30–52. 相似文献
10.
A. V. Marshakov 《Theoretical and Mathematical Physics》1997,112(1):791-826
We discuss the properties ofN=2 supersymmetric gauge theories underlying the Seiberg-Witten hypothesis. We consider the main points of the theory that
describes the finite-gap solutions to integrable equations in terms of complex curves and generating differentials. We clarify
the invariant meaning of these definitions. This formalism is applied to the exact nonperturbative solutions found recently
in theN=2 supersymmetric non-Abelian gauge theories. In the known cases, we compare this formalism with the results that can be obtained
using standard quantum field-theory methods.
The paper was written at the request of the Editorial Board.
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 112, No. 1, pp. 3–46, July, 1997. 相似文献
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Franz Spirig 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》1988,39(4):504-517
An oscillator with a small linear damping forced by a non-linear autonomous perturbation is considered. Conditions for the existence of small periodic solutions are derived. This was already done by Flockerzi [2], but the method of averaging used thereby is rather demanding. Here instead, a more elementary and more tractable approach is presented. In particular an algorithm is given by which all small oscillations as well as their stability are established.
Zusammenfassung Es wird ein Oszillator mit einer kleinen linearen Dämpfung und einer nicht-linearen zeitunabhängigen Erregung betrachtet. Die Bedingungen für die Existenz kleiner periodischer Schwingungen werden angegeben. Eine Lösung dieses Problems wurde schon von Flockerzi [2] vorgelegt. Die dabei angewandte Mittelungsmethode ist aber ziemlich aufwendig. In dieser Arbeit wird ein elementarerer und handlicherer Zugang aufgezeigt. Insbesondere wird ein Algorithmus angegeben, welcher es erlaubt, alle kleinen periodischen Schwingungen sowie deren Stabilitätsverhalten zu bestimmen.相似文献
15.
S. I. Mitrokhin 《Differential Equations》2011,47(12):1833-1836
We consider a boundary value problem with irregular boundary conditions for a differential operator of arbitrary odd order.
The potential in this operator is assumed to be an integrable function. We suggest a method for studying the spectral properties
of differential operators with integrable coefficients. We analyze the asymptotic behavior of solutions of the differential
equation in question for large values of the spectral parameter. The eigenvalue asymptotics for the considered differential
operator is obtained. 相似文献
16.
For an analytic differential system in Rn with a periodic orbit, we will prove that if the system is analytically integrable around the periodic orbit, i.e. it has n−1 functionally independent analytic first integrals defined in a neighborhood of the periodic orbit, then the system is analytically equivalent to its Poincaré–Dulac type normal form. This result is an extension of analytically integrable differential systems around a singularity to the ones around a periodic orbit. 相似文献
17.
Yu Qiu 《Journal of Differential Equations》2009,246(8):3361-3379
This paper is devoted to finding the highest possible focus order of planar polynomial differential equations. The results consist of two parts: (i) we explicitly construct a class of concrete systems of degree n, where n+1 is a prime p or a power of a prime pk, and show that these systems can have a focus order n2−n; (ii) we theoretically prove the existence of polynomial systems of degree n having a focus order n2−1 for any even number n. Corresponding results for odd n and more concrete examples having higher focus orders are given too. 相似文献
18.
In this paper we establish bifurcation theory of limit cycles for planar smooth autonomous differential systems, with . The key point is to study the smoothness of bifurcation functions which are basic and important tool on the study of Hopf bifurcation at a fine focus or a center, and of Poincaré bifurcation in a period annulus. We especially study the smoothness of the first order Melnikov function in degenerate Hopf bifurcation at an elementary center. As we know, the smoothness problem was solved for analytic and differential systems, but it was not tackled for finitely smooth differential systems. Here, we present their optimal regularity of these bifurcation functions and their asymptotic expressions in the finite smooth case. 相似文献
19.
Zhengxin Zhou 《Journal of Applied Analysis & Computation》2012,2(2):241-249
In this article. First, we construct some nonlinear differential systems which are equivalent to some known systems. Second, we discuss, in a different method, the equivalence between some linear differential systems. And then we apply the obtained results to the study of the qualitative properties of these systems simultaneously. 相似文献