Integrable spinor models in two dimensional space-time

by Владимир Герджиков (Institute for Advanced Physical Studies)

Лабораторен комплекс (София тех парк)

Лабораторен комплекс

София тех парк

astrophotography by Nikola AntonovSeveral spinor models were proposed several decades ago in the efforts to explain the nature of strong interactions, see [1]. These are
Nambu--Jona-Lasinio--Vaks--Larkin models (1961), related to SU(n)
Gross–Neveu models (1974), related to SP(2n).
Zakharov– Mikhailov spinor models (1980) related to SO(n).

Following [1] we will analyze their reductions to two-dimensional space-time and will explain how they could be integrated explicitly. We will start by constructing their Lax representations. Then the spectral properties of their Lax operators will be outlined. Using the dressing Zakharov-Shabat method we will construct their soliton solutions. Other integrable deformations of these models will be formulated following [2].

The lectures will take place each Tuesday, 18:30 EET. For participants residing in Sofia it is good to attend the lectures in person at Sofia Tech Park, Laboratory complex. The online participation is at

[1] V. E. Zakharov and A. V. Mikhailov. On the Integrability of Classical Spinor Models in Two- Dimensional Space-Time. Commun. Math. Phys. 74, 2140 (1980).
[2] V. S. Gerdjikov. Z2-reductions of spinor models in two dimensions. arXiv:1210.3722v1 [nlin.SI]

[3]  В. Е. Захаров, А. Б. Шабат, “Интегрирование нелинейных уравнений математической физики методом обратной задачи рассеяния. II”, Функц. анализ и его прил., 13:3 (1979), 13–22; Funct. Anal. Appl., 13:3 (1979), 166–174




Lecture 1. July 26th, 2022. Lax representation and Nambu-Jona-Lasinio-Vaks-Larkin, Gross-Neveu and Zakharov-Mikhailov models, slides, video (damaged recording; change the video player if interruptions occur) 

Lecture 2. August 2nd, 2022, The inverse scattering method, slides, video

Lecture 3. August 9th, 2022, Zakharov -- Shabat dressing method and soliton solutions. Soliton solutions of the Nambu--Jona-Lasinio--Vaks--Larkin model, slides, video

Lecture 4. August 16th, 2022, Lax representations and simple Lie algebras, slides, video

Lecture 5. August 23rd, 2022, Lax representations, simple Lie algebras and spinor models, slides, video (damaged recording; change the video player if interruptions occur)

Lecture 6. August 30th, 2022, Riemann-Hilbert problems and soliton solutions of Zakharov--Mikhailov models, slides, video.

Organized by

Институт за съвременни физически изследвания и Квантерал

Антон Андонов, Диана Тодорова, Стоян Мишев