The Morales-Ramis theory provides an effective and powerful non-integrability criterion for complex analytic Hamiltonian systems via the differential Galoisian obstruction.In this paper,we give a new Morales-Ramis typ...The Morales-Ramis theory provides an effective and powerful non-integrability criterion for complex analytic Hamiltonian systems via the differential Galoisian obstruction.In this paper,we give a new Morales-Ramis type theorem on the meromorphic Jacobi non-integrability of general analytic dynamical systems.The key point is to show that the existence of Jacobian multipliers of a nonlinear system implies the existence of common Jacobian multipliers of Lie algebra associated with the identity component.In addition,we apply our results to the polynomial integrability of Karabut systems for stationary gravity waves in finite depth.展开更多
The main purpose of this paper is to investigate the connection between the Painlev′e property and the integrability of polynomial dynamical systems. We show that if a polynomial dynamical system has Painlev′e prope...The main purpose of this paper is to investigate the connection between the Painlev′e property and the integrability of polynomial dynamical systems. We show that if a polynomial dynamical system has Painlev′e property, then it admits certain class of first integrals. We also present some relationships between the Painlev′e property and the structure of the differential Galois group of the corresponding variational equations along some complex integral curve.展开更多
Let X denote a complex analytic manifold, and let Aut(X) denote the space of invertible maps of a germ (X, a) to a germ (X, b); this space is obviously a groupoid; roughly speaking, a 'Lie groupoid' is a subgr...Let X denote a complex analytic manifold, and let Aut(X) denote the space of invertible maps of a germ (X, a) to a germ (X, b); this space is obviously a groupoid; roughly speaking, a 'Lie groupoid' is a subgroupoid of Aut(X) defined by a system of partial differential equations.To a foliation with singularities on X one attaches such a groupoid, e.g. the smallest one whose Lie algebra contains the vector fields tangent to the foliation. It is called 'the Galois groupoid of the foliation'. Some examples are considered, for instance foliations of codimension one, and foliations defined by linear differential equations; in this last case one recuperates the usual differential Galois group.展开更多
基金supported by National Natural Science Foundation of China(Grant No.11771177),National Natural Science Foundation of China(Grant Nos.12001386 and 12090013)Science and Technology Development Project of Jilin Province(Grant No.YDZJ202101ZYTS141)+1 种基金supported by Sichuan University Postdoctoral Interdisciplinary Innovation Fund(Grant No.0020104153010)the Fundamental Research Funds for the Central Universities(Grant No.20826041E4168)。
文摘The Morales-Ramis theory provides an effective and powerful non-integrability criterion for complex analytic Hamiltonian systems via the differential Galoisian obstruction.In this paper,we give a new Morales-Ramis type theorem on the meromorphic Jacobi non-integrability of general analytic dynamical systems.The key point is to show that the existence of Jacobian multipliers of a nonlinear system implies the existence of common Jacobian multipliers of Lie algebra associated with the identity component.In addition,we apply our results to the polynomial integrability of Karabut systems for stationary gravity waves in finite depth.
基金The NSF (11371166,11301210)of ChinaNational 973 Project (2012CB821200) of ChinaJilin Province Youth Science Foundation
文摘The main purpose of this paper is to investigate the connection between the Painlev′e property and the integrability of polynomial dynamical systems. We show that if a polynomial dynamical system has Painlev′e property, then it admits certain class of first integrals. We also present some relationships between the Painlev′e property and the structure of the differential Galois group of the corresponding variational equations along some complex integral curve.
文摘Let X denote a complex analytic manifold, and let Aut(X) denote the space of invertible maps of a germ (X, a) to a germ (X, b); this space is obviously a groupoid; roughly speaking, a 'Lie groupoid' is a subgroupoid of Aut(X) defined by a system of partial differential equations.To a foliation with singularities on X one attaches such a groupoid, e.g. the smallest one whose Lie algebra contains the vector fields tangent to the foliation. It is called 'the Galois groupoid of the foliation'. Some examples are considered, for instance foliations of codimension one, and foliations defined by linear differential equations; in this last case one recuperates the usual differential Galois group.
