Darboux integrability of determinant and equations for principal minors

TL;DR

This paper establishes Darboux integrability for certain determinantal equations related to 2D Wronskians and Casoratians, linking them to Toda-type equations and providing explicit solutions for cases of vanishing determinants.

Contribution

It demonstrates Darboux integrability of determinantal equations and connects them to Toda equations, offering explicit solutions and recurrent formulas for integrals.

Findings

Determinantal equations have as many independent integrals as their order.

Recurrent formulas for principal minors relate to 2D Toda equations.

Explicit solutions are provided for cases with vanishing determinants.

Abstract

We consider equations that represent a constancy condition for a 2D Wronskian, mixed Wronskian-Casoratian and 2D Casoratian. These determinantal equations are shown to have the number of independent integrals equal to their order - this implies Darboux integrability. On the other hand, the recurrent formulas for the leading principal minors are equivalent to the 2D Toda equation and its semi-discrete and lattice analogues with particular boundary conditions (cut-off constraints). This connection is used to obtain recurrent formulas and closed-form expressions for integrals of the Toda-type equations from the integrals of the determinantal equations. General solutions of the equations corresponding to vanishing determinants are given explicitly while in the non-vanishing case they are given in terms of solutions of ordinary linear equations.