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

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.

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

2021 ◽  
Vol 38 ◽  
pp. 3-18
Author(s):  
B. A. Babajanov ◽  
◽  
M. M. Ruzmetov ◽  
Keyword(s):  
Spectral Data ◽  
Time Evolution ◽  
Toda Lattice ◽  
Complete Integrability ◽  
Numerical Results ◽  
Periodic Functions ◽  
Hill’S Equation ◽  
Periodic Coefficients ◽  
Hill's Equation ◽  
Self Consistent

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.

scholarly journals An inverse spectral problem for a star graph of Krein strings

2016 ◽  
Vol 2016 (715) ◽  
Author(s):  
Jonathan Eckhardt
Keyword(s):  
Spectral Problem ◽  
Spectral Data ◽  
Inverse Spectral Problem ◽  
Trace Class ◽  
Star Graph ◽  
Borel Measures ◽  
Inverse Spectral ◽  
The Individual

AbstractWe solve an inverse spectral problem for a star graph of Krein strings, where the known spectral data comprises the spectrum associated with the whole graph, the spectra associated with the individual edges as well as so-called coupling matrices. In particular, we show that these spectral quantities uniquely determine the weight within the class of Borel measures on the graph, which give rise to trace class resolvents. Furthermore, we obtain a concise characterization of all possible spectral data for this class of weights.

On a Toda lattice hierarchy: Lax pair, integrable symplectic map and algebraic–geometric solution

2018 ◽  
Vol 32 (28) ◽  
pp. 1850344
Author(s):  
Xiao Yang ◽  
Dianlou Du
Keyword(s):  
Spectral Problem ◽  
Theta Function ◽  
Toda Lattice ◽  
Burgers Equation ◽  
Lax Pair ◽  
Discrete Variable ◽  
Riemann Theta Function ◽  
Symplectic Map ◽  
Discrete Counterpart ◽  
Toda Lattice Hierarchy

A Toda lattice hierarchy is studied by introducing a new spectral problem which is a discrete counterpart of the generalized Kaup–Newell spectral problem. Based on the Lenard recursion equation, Lax pair of the hierarchy is given. Further, the discrete spectral problem is nonlinearized into an integrable symplectic map. As a result, an algebraic–geometric solution in Riemann theta function of the hierarchy is obtained. Besides, two equations, the Volterra lattice and a (2[Formula: see text]+[Formula: see text]1)-dimensional Burgers equation with a discrete variable, yielded from the hierarchy are also solved.

scholarly journals A Toda lattice hierarchy with variable coefficients and its multi-wave solutions

2014 ◽  
Vol 18 (5) ◽  
pp. 1563-1566 ◽  
Author(s):  
Sheng Zhang ◽  
Wang Di
Keyword(s):  
Spectral Problem ◽  
Toda Lattice ◽  
Variable Coefficients ◽  
Curvature Equation ◽  
Wave Solutions ◽  
Zero Curvature ◽  
N Wave ◽  
Special Case ◽  
Toda Lattice Hierarchy

Starting from the Toda spectral problem, a new Toda lattice hierarchy of isospectral equations with variable coefficients is constructed through the discrete zero curvature equation. In order to solve one special case of the derived Toda lattice hierarchy, a series of appropriate transformations are utilized. As a result, a new uniform formula of N-wave solutions is obtained.

scholarly journals Inverse spectral problem for Dirac operators by spectral data

Filomat ◽  
2017 ◽  
Vol 31 (4) ◽  
pp. 1065-1077
Author(s):  
Ozge Akcay ◽  
Khanlar Mamedov
Keyword(s):  
Boundary Condition ◽  
Inverse Problem ◽  
Spectral Problem ◽  
Spectral Data ◽  
Spectral Parameter ◽  
Inverse Spectral Problem ◽  
Sufficient Conditions ◽  
Dirac Operators ◽  
Piecewise Continuous ◽  
Necessary And Sufficient

This work deals with the solution of the inverse problem by spectral data for Dirac operators with piecewise continuous coefficient and spectral parameter contained in boundary condition. The main theorem on necessary and sufficient conditions for the solvability of inverse problem is proved. The algorithm of the reconstruction of potential according to spectral data is given.

Sign in / Sign up

Export Citation Format

Share Document