Infinitely many local higher symmetries without recursion operator or master symmetry: integrability of the Foursov--Burgers system revisited

TL;DR

This paper demonstrates that a specific Burgers-type system possesses infinitely many local symmetries generated through a novel two-term recursion relation, challenging previous assumptions about the necessity of recursion operators or master symmetries for integrability.

Contribution

The authors establish the integrability of the Foursov--Burgers system by constructing infinitely many local symmetries without relying on recursion operators or master symmetries.

Findings

Existence of infinitely many local symmetries for the system

Construction of symmetries via a two-term recursion relation

Revisiting integrability criteria for Burgers-type systems

Abstract

We consider the Burgers-type system studied by Foursov, w_t &=& w_{xx} + 8 w w_x + (2-4\alpha)z z_x, z_t &=& (1-2\alpha)z_{xx} - 4\alpha z w_x + (4-8\alpha)w z_x - (4+8\alpha)w^2 z + (-2+4\alpha)z^3, (*) for which no recursion operator or master symmetry was known so far, and prove that the system (*) admits infinitely many local generalized symmetries that are constructed using a nonlocal {\em two-term} recursion relation rather than from a recursion operator.