On the Construction and Integration of a Hierarchy for the Periodic Toda Lattice with a Self-Consistent Source

In this paper, it is derived a rich hierarchy for the Toda lattice with a selfconsistent source in the class of periodic functions. We discuss the complete integrability of the constructed systems that is based on the transformation to the spectral data of an associated discrete Hill‘s equation with periodic coefficients. In particular, Dubrovintype equations are derived for the time-evolution of the spectral data corresponding to the solutions of any system in the hierarchy. At the end of the paper, we illustrate our theory on concrete example with analytical and numerical results.

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ON THE INTEGRATION OF A HIRERARCHY FOR THE PEREODIC TODA LATTICE WITH A FREE TERM

Central Asian Problems of Modern Science and Education ◽

10.51348/campse0024 ◽

2021 ◽

pp. 101-110

Keyword(s):

Spectral Problem ◽

Spectral Data ◽

Time Evolution ◽

Toda Lattice ◽

Inverse Spectral Problem ◽

Complete Integrability ◽

Free Term ◽

Periodic Functions ◽

Periodic Coefficients ◽

Toda Lattice Hierarchy

In this article, we have explored the Toda lattice hierarchy in the class of periodic functions with a free term. We have given an effective method of constructing of the periodic Toda lattice hierarchy with a free term. We have discussed the complete integrability of the constructed systems that is based on the inverse spectral problem of an associated discrete Hill`s equation with periodic coeﬃcients. In particular, Dubrovin-type equations are derived for the time-evolution of the spectral data corresponding to the solutions of any system in the hierarchy.

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Analogue of Inverse Scattering Theory for the Discrete Hill's Equation and Exact Solutions for the Periodic Toda Lattice

Progress of Theoretical Physics ◽

10.1143/ptp.55.457 ◽

1976 ◽

Vol 55 (2) ◽

pp. 457-465 ◽

Cited By ~ 71

Author(s):

E. Date ◽

S. Tanaka

Keyword(s):

Exact Solutions ◽

Inverse Scattering ◽

Scattering Theory ◽

Toda Lattice ◽

Hill’S Equation ◽

Hill's Equation ◽

Inverse Scattering Theory

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Periodic solutions of a linear differential equation of the second order with periodic coefficients

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100012056 ◽

1926 ◽

Vol 23 (1) ◽

pp. 44-46 ◽

Cited By ~ 9

Author(s):

E. L. Ince

Keyword(s):

Differential Equation ◽

Linear Differential Equation ◽

Periodic Solutions ◽

Second Order ◽

Hill’S Equation ◽

Periodic Coefficients ◽

Hill's Equation ◽

Linear Differential ◽

Image Position

The equation to be considered is of the typewhere p (x) is continuous for all real values of x, even, and periodic. It is no restriction to suppose that the period is π, and this assumption will be made, so that the equation is virtually Hill's equation.

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Time Evolution of the Spectral Data Associated with the Finite Complex Toda Lattice

Dynamical Systems and Methods ◽

10.1007/978-1-4614-0454-5_19 ◽

2011 ◽

pp. 323-334 ◽

Cited By ~ 1

Author(s):

Aydin Huseynov ◽

Gusein Sh. Guseinov

Keyword(s):

Spectral Data ◽

Time Evolution ◽

Toda Lattice ◽

Finite Complex

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Passive static stretching alters the characteristics of the force-velocity curvature differently for fast and slow muscle groups—A practical application of Hill's equation

Human Movement Science ◽

10.1016/j.humov.2021.102852 ◽

2021 ◽

Vol 79 ◽

pp. 102852

Author(s):

Junhai Xu ◽

Arnold G. Nelson ◽

Jan M. Hondzinski

Keyword(s):

Slow Muscle ◽

Static Stretching ◽

Practical Application ◽

Hill’S Equation ◽

Fast And Slow Muscle ◽

Hill's Equation ◽

Force Velocity ◽

Muscle Groups

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On the solution of Hill's equation using Milne's method

Journal of Physics A General Physics ◽

10.1088/0305-4470/24/9/017 ◽

1991 ◽

Vol 24 (9) ◽

pp. 2069-2081 ◽

Cited By ~ 13

Author(s):

J A Nunez ◽

F Bensch ◽

H J Korsch

Keyword(s):

Hill’S Equation ◽

Hill's Equation

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Stability Regions of Hill's Equation

IMA Journal of Applied Mathematics ◽

10.1093/imamat/19.3.253 ◽

1977 ◽

Vol 19 (3) ◽

pp. 253-259 ◽

Cited By ~ 3

Author(s):

N. MOSTAGHEL

Keyword(s):

Hill’S Equation ◽

Stability Regions ◽

Hill's Equation

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Vibration and Stability of an Elastic Beam Subjected to a Periodic Axial Load With Time-Dependent Displacement Excitation at Both Ends

13th Biennial Conference on Mechanical Vibration and Noise: Vibration Analysis — Analytical and Computational ◽

10.1115/detc1991-0334 ◽

1991 ◽

Author(s):

Xiao-Feng Wu ◽

Adnan Akay

Keyword(s):

Axial Load ◽

Perturbation Technique ◽

Harmonic Balance Method ◽

Elastic Beam ◽

Time Dependent ◽

Transverse Load ◽

Order Partial Differential Equation ◽

Weighted Residual Method ◽

Hill’S Equation ◽

Hill's Equation

Abstract This paper concerns the transverse vibrations and stabilities of an elastic beam simultaneously subjected to a periodic axial load, a distributed transverse load, and time-dependent displacement excitations at both ends. The equation of motion derived from Bernoulli-Euler beam theory is a fourth-order partial differential equation with periodic coefficients. To obtain approximate solutions, the method of assumed-modes is used. The unknown time-dependent function in the assumed-modes method is determined by a generalized inhomogeneous Hill’s equation. The instability regions possessed by this generalized Hill’s equation are obtained by both the perturbation technique up to the second order and the harmonic balance method. The dynamic response and the corresponding spectrum of the transversely oscillating elastic beam are calculated by the weighted-residual method.

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