Toda lattice with a special self-consistent source

Abstract

We describe a method for integrating the Toda lattice with a self-consistent source using the inverse scattering method for a discrete Sturm-Liouville operator with moving eigenvalues.

On the Integration of the Higher Order Toda Lattice with a Self-Consistent Integral Type Source

“…The discrete soliton equations with self-consistent sources were first studied by Liu and Zeng [35], who investigated the Darboux transformation for formulating and calculating the Toda lattice with self-consistent sources. An inverse scattering method was also developed to find solutions for the Toda lattice with self-consistent sources [36][37][38]. Integrability of the periodic Toda lattice and its hierarchy with a source has been shown in previous works [39][40][41][42][43][44].…”

The Cauchy problem for a semi-infinite Volterra chain with an asymptotically periodic initial condition

“…(see [1][2][3][4][5][6][7][8][9][10][11][12][13]). As is well known, the method of the inverse scattering problem (MISP) allows us to study the Cauchy problem for the chains in the case of rapidly decreasing initial data [1-3, 5, 8, 14] and in the periodic case [6][7][8].…”

“…Note that for some chains connected with isospectral deformations of Jacobi matrices the question of global solvability was studied in [8,11] for a broader class of initial conditions. However, the method of this article can be used in the nonisospectral case too (see [12]). Note that the Cauchy problem for a Volterra chain infinite in both directions with an asymptotically periodic condition (of period 2) is stated in [9] and solved in [10].…”