On time evolutions associated with the nonstationary Schr\"{o}dinger equation

TL;DR

This paper explores how the symmetries of the nonstationary Schrödinger equation can be decomposed to generate hierarchies of nonlinear integrable equations, linking linear symmetries to complex nonlinear evolutions.

Contribution

It demonstrates that the integrable symmetries of the nonstationary Schrödinger equation naturally decompose into commuting subsets, leading to hierarchies of nonlinear equations of KP type.

Findings

Symmetries form mutually commuting subsets.

Hierarchies lead to KP-type nonlinear equations.

Linear problems determine nonlinear integrable evolutions.

Abstract

The set of integrable symmetries of the nonstationary Schr\"{o}dinger equation is shown to admit a natural decomposition into subsets of mutually commuting symmetries. Hierarchies of time evolutions associated with each of these subsets ultimately lead to nonlinear (possibly, operator) equations of the Kadomtsev--Petviashvili I type or its higher analogues, thus demonstrating that the linear problem itself constructively determines the associated nonlinear integrable evolution equations and their hierarchies.