Isoperiodic deformations of Toda curves and chains, the difference Korteweg - de Vries equation, and $SU(N)$ Seiberg-Witten theories

TL;DR

This paper explores the dynamics and deformations of Toda curves, chains, and related spectral problems, linking integrable systems with $SU(N)$ Seiberg-Witten theories and providing explicit solutions to complex differential systems.

Contribution

It introduces new differential equations for Toda curves, describes isoperiodic deformations of Toda chains and KdV equations, and analyzes spectral deformations in $SU(N)$ Seiberg-Witten theories.

Findings

Derived differential equations for Toda curve dynamics

Described isoperiodic deformations of Toda chains and KdV equations

Provided explicit solutions to constrained Schlesinger systems

Abstract

We introduce the dynamics of Toda curves of order $N$ and derive differential equations governing this dynamics. We apply the obtained results to describe isoperiodic deformations of $N$-periodic Toda chains and periodic difference Korteweg-de Vries equation. We describe deformations of the essential spectra of $N$-periodic two-sided Jacobi matrices. We also study singular regimes of $SU(N)$ Seiberg-Witten theory and describe their deformations preserving the number of singularities where new massless particles may occur. We introduce and describe isoequilibrium deformations of arbitrary collections of $d$ real disjoint closed intervals. We conclude by providing explicit triangular solutions to constrained Schlesinger systems.