Progresses on some open problems related to infinitely many symmetries

TL;DR

This paper investigates the nature of infinite symmetries in integrable systems, proposing that known symmetries are linear combinations of finite wave parameter translations and exploring their implications for unifying various integrable systems.

Contribution

It introduces a new analytical approach to understanding infinite symmetries, conjectures the incompleteness of known symmetries, and suggests a unifying framework for different classes of integrable systems.

Findings

Finite wave parameter translations form the basis of known symmetries.

The conjecture that all known symmetries are linear combinations of these translations.

Potential for a unified hierarchical framework encompassing classical, supersymmetric, and ren-symmetric integrable systems.

Abstract

The quest to reveal the physical essence of the infinitely many symmetries and conservation laws that are intrinsic to integrable systems has historically posed a significant challenge at the confluence of physics and mathematics. This scholarly investigation delves into five open problems related to these boundless symmetries within integrable systems by scrutinizing their multi-wave solutions, employing a fresh analytical methodology. For a specified integrable system, there exist various categories of $n$-wave solutions. Each sub-wave comprising the $n$-wave solution may possess free parameters, including center, width, and periodic parameters. It is evident that these solutions are translation invariant with respect to all these free parameters. We postulate that the entirety of the recognized infinite symmetries merely constitute linear combinations of these finite wave parameter translation symmetries. The conjecture intimates that the currently known infinitely many symmetries are not exhaustive, and an indeterminate number of symmetries remain to be discovered. This conjecture further indicates that by imposing an infinite array of symmetry constraints, it becomes feasible to derive exact multi-wave solutions. By considering the renowned KdV equation and the Burgers equation as simple examples, the conjecture is substantiated for the $n$-soliton solutions. It is unequivocal that any linear combination of the wave parameter translation symmetries retains its status as a symmetry associated with the particular solution. This observation suggests that by introducing a ren-variable and a ren-symmetric derivative which serve as generalizations of the Grassmann variable and the super derivative, it may be feasible to unify classical integrable systems, supersymmetric integrable systems, and ren-symmetric integrable systems within a cohesive hierarchical framework.