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Hamiltonian Mechanics and Hamilton-Jacobi Theory
Published in M.D.S. Aliyu, Nonlinear H∞-Control, Hamiltonian Systems and Hamilton-Jacobi Equations, 2017
The Toda lattice as a Hamiltonian system describes the motion of n particles moving in a straight line with “exponential interaction” between them. Mathematically, it is equivalent to a problem in which a single particle moves in ℜn. Let the positions of the particles at time t (in ℜ) be q1(t),…, qn(t), respectively. We assume that each particle has mass 1. The momentum of the i-th particle at time t is therefore pi=q˙i. The Hamiltonian function for the finite (or non-periodic) lattice is defined to be () H(q,p)=12∑j=1npj2+∑j=1n−1e2(qj−qj+1).
Introduction
Published in David S. Ricketts, Donhee Ham, Electrical Solitons, 2011
1967 (Anharmonic lattice) M. Toda introduced a spring-mass lattice similar to Fermi’s, except that the force between masses follows an exponential relationship [26,27]. This exponential relationship is key, as Toda shows that an exact soliton system is created. The Toda lattice, as it became known, supports soliton propagation and exhibits all of the dynamics of soliton systems. In the small amplitude and long-wavelength limit, the Toda lattice reduces to the KdV equation. The electrical soliton we will use in this book is an approximation between Toda lattice solitons and KdV solitons.
Nonlocal, Gradient and Local Models of Elastic Media: 1D Case
Published in Igor V. Andrianov, Vladyslav Danishevskyy, Jan Awrejcewicz, Linear and Nonlinear Waves in Microstructured Solids, 2021
Igor V. Andrianov, Vladyslav Danishevskyy, Jan Awrejcewicz
They construct soliton solutions in the following way: rapidly changing part of soliton is constructed using Toda lattice with constant coefficients, and for slowly part of solution continuous approximation is used. Then these solutions are matched.
The solitary waves, quasi-periodic waves and integrability of a generalized fifth-order Korteweg-de Vries equation
Published in Waves in Random and Complex Media, 2019
Pan-Li Ma, Shou-Fu Tian, Li Zou, Tian-Tian Zhang
In 1980s, Nakamura proposed a convenient way to construct a kind of quasi-periodic solutions of nonlinear equations in his paper [18]. Recently, Fan and Hon [19–21] extend this method to investigate the discrete Toda lattice, (2+1)-dimensional Bogoyavlenskii’s breaking soliton equation and the asymmetrical Nizhnik–Novikov–Veselov equation, etc. Ma [22–24] constructs one-periodic and two-periodic wave solutions to a class of (2+1)-dimensional Hirota bilinear equations. Chen et al. [25,26] investigate the bilinear forms, bilinear Bäcklund transformations, Lax pairs and conservation laws of some KdV-type equations. In Refs. [27–31], one of our authors Tian and Zhang investigate the integrability of Caudrey–Dodd–Gibbon–Sawada–Kotera equation, (2+1)-dimensional breaking soliton equation, Ito equation, generalized variable-coefficient Kadomtsev–Petviashvili and forced Korteweg-de Vries equations, including bilinear forms, bilinear Bäcklund transformations, Lax pairs, conservation laws, and periodic wave solutions, etc.