Интегрируемые иерархии нелинейных дифференциальных уравнений и многочастичные системы тема диссертации и автореферата по ВАК РФ 01.01.03, кандидат наук Прокофьев Вадим Вячеславович

• Прокофьев Вадим Вячеславович
• кандидат наук
• 2022

Оглавление диссертации кандидат наук Прокофьев Вадим Вячеславович

Contents

Introduction

1 Historical remarks

1.1 Nonlinear differential hierarchies

1.2 Many body systems

2 Tau function and bilinear equation

2.1 Baker-Akhiezer function

2.2 Tau function

3 Further Generalizations

3.1 Modified KP

3.2 0Î((œ)) algebra and 2d Toda hierarchy

3.3 Multi-component KP hierarchy

4 Main Results

4.1 KP hierarchy

4.2 2d Toda hierarchy

4.3 Matrix KP

Bibliography

Appendix A. Paper 1. Toda lattice hierarchy and trigonometric

Ruijsenaars—Schneider hierarchy

Appendix B. Paper 2. Matrix Kadomtsev-Petviashvili Hierarchy and Spin Generalization of Trigonometric Calogero Moser Hierarchy

Appendix C. Paper 3. Elliptic solutions to the KP hierarchy and

elliptic Calogero—Moser model

Appendix D. Paper 4. Elliptic solutions to Toda lattice hierarchy

and elliptic Ruijsenaars-Schneider model

Appendix E. Paper 5. Elliptic solutions to matrix KP hierarchy and

spin generalization of elliptic Calogero-Moser model

Введение диссертации (часть автореферата) на тему «Интегрируемые иерархии нелинейных дифференциальных уравнений и многочастичные системы»

Introduction

One of the most astonishing quality characterising integrable systems is their nontrivial interconnections with each other. In particular, there is a connection between integrable spin chains, integrable hierarchies of nonlinear partial equations and classical many-bodies models.

In this thesis we study poles dynamics of singular solutions of integrable hierarchies of KP type and show that it is isomorphic to dynamics of particles in many-body integrable systems on the level of hierarchies. Such connection between two different types of integrable systems has been a long known conjecture. The connection between nonlinear integrable equations and many-body systems was first study in seminal paper (Airault et al. [1977]). After that in the works such as (Krichever [1978],Krichever [1980], Krichever and Zabrodin [1995]) it was established that for the first nontrivial times dynamics of poles correspond to the motion of particles in systems of Calogero-Moser type with standard Hamiltonians. After that in papers (Shiota [1994], Haine [2007], Zabrodin [2020]) such connection was extended to the level of whole hierarchies, however it was done only for rational or trigonometric solutions which are just a limits of the most general elliptic solutions.

In a series of the articles presented in this thesis authors extend a connection between integrable hierarchies and many-body systems of Calogero type for three different hierarchies such as KP, 2D Toda lattice and matrix KP up to the most general elliptic solutions. The main results of these paper is that authors establish a connection between spectral curves of elliptic many-body systems and Hamiltonians responsible for dynamics of poles in higher times of corresponding hierarchy. Besides that methods developed in these articles could be used to discover poles dynamics for singular solutions of other hierarchies.

My thesis presents the results of five articles in which I am one of co-authors. In these articles a connection between integrable hierarchies of nonlinear differential equations and integrable many-body systems was studied. These works contain most general results for KP 2d-Toda and matrix KP hierarchies.

Список литературы диссертационного исследования кандидат наук Прокофьев Вадим Вячеславович, 2022 год

REFERENCES

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