Asymptotic-numerical solvers for highly oscillatory ordinary differential equations and Hamiltonian systems

2021 ◽  
Vol 40 (8) ◽  
Author(s):  
Zhongli Liu ◽  
Xiaoxue Sa ◽  
Hongjiong Tian
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Hamiltonian Systems ◽  
Highly Oscillatory

The HELP inequality for lim-p Hamiltonian systems

2000 ◽  
Vol 130 (4) ◽  
pp. 699-720 ◽  
Author(s):  
B. M. Brown ◽  
M. Marletta
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Hamiltonian Systems ◽  
Wide Class ◽  
Limit Point ◽  
Type Singularity ◽  
Linear Hamiltonian Systems ◽  
Infinite Intervals ◽  
Point Type

In a recent paper, Brown, Evans and Marletta extended the HardyEverittLittlewoodPolya inequality from 2nth-order formally self-adjoint ordinary differential equations to a wide class of linear Hamiltonian systems in 2n variables. The paper considered only problems on semi-infinite intervals [a, ∞) with a limit-point type singularity at infinity. In this paper we extend the theory to cover all types of endpoint ( lim-p for n ≤ p ≤ 2n ).

scholarly journals New Neumann System Associated with a 3 × 3 Matrix Spectral Problem

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Fang Li ◽  
Liping Lu
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Spectral Problem ◽  
Hamiltonian Systems ◽  
Lax Pair ◽  
Completely Integrable ◽  
Neumann System ◽  
Matrix Spectral Problem

The nonlinearization approach of Lax pair is applied to the case of the Neumann constraint associated with a 3 × 3 matrix spectral problem, from which a new Neumann system is deduced and proved to be completely integrable in the Liouville sense. As an application, solutions of the first nontrivial equation related to the 3 × 3 matrix spectral problem are decomposed into solving two compatible Hamiltonian systems of ordinary differential equations.

Finite Genus Solutions to the Generalised Kaup-Newell Hierarchy and Two 2+1 Dimensional Modified Korteweg-de Vries Equations

2017 ◽  
Vol 72 (7) ◽  
pp. 589-594
Author(s):  
Xiao Yang ◽  
Jiayan Han
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Hamiltonian Systems ◽  
Algebraic Curve ◽  
Inversion Theorem ◽  
First Order ◽  
Hamiltonian Flows ◽  
De Vries ◽  
Jacobi Inversion ◽  
Finite Genus

AbstractA generalised Kaup-Newell (gKN) hierarchy is introduced, which starts with a system of first-order ordinary differential equations and includes the Gerdjikov-Ivanov equation. By introducing an appropriate generating function, its related Hamiltonian systems and algebraic curve are given. The Hamiltonian systems are proved to be integrable, then the gKN hierarchy is solved by Hamiltonian flows. The algebraic curve is provided with suitable genus, then based on the trace formula and Riemann-Jacobi inversion theorem, finite genus solutions of the gKN hierarchy are obtained. Besides, two 2+1 dimensional modified Korteweg-de Vries (mKdV) equations are also solved.

A generalization of the Liouville–Arnol'd theorem

1995 ◽  
Vol 117 (2) ◽  
pp. 353-370 ◽  
Author(s):  
G. E. Prince ◽  
G. B. Byrnes ◽  
J. Sherring ◽  
S. E. Godfrey
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Hamiltonian Systems ◽  
First Integrals ◽  
Second Order ◽  
Lagrangian Mechanics ◽  
Original Theorem

AbstractWe show that the Liouville-Arnol'd theorem concerning knowledge of involutory first integrals for Hamiltonian systems is available for any system of second order ordinary differential equations. In establishing this result we also provide a new proof of the standard theorem in the setting of non-autonomous, regular Lagrangian mechanics on the evolution space ℝ × TM of a manifold M. Both the original theorem and its generalization rely on a certain bijection between symmetries of the system and its first integrals. We give two examples of the use of the theorem for systems on ℝ2 which are not Euler-Lagrange.

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