Simplest potential conservation laws of linear evolution equations

TL;DR

The paper demonstrates that all simplest potential conservation laws of linear (1+1)-dimensional evolution equations are induced by local conservation laws, providing a criterion for identifying purely potential conservation laws in odd-order cases.

Contribution

It establishes a comprehensive link between potential and local conservation laws for linear evolution equations and introduces an effective criterion for quadratic conservation laws.

Findings

Potential conservation laws are induced by local laws for even and odd orders.

A criterion is provided to identify purely potential conservation laws.

The results unify the understanding of conservation laws in linear evolution equations.

Abstract

Every simplest potential conservation law of any (1+1)-dimensional linear evolution equation of even order proves induced by a local conservation law of the same equation. This claim is true also for linear simplest potential conservation laws of (1+1)-dimensional linear evolution equations of odd order, which are related to linear potential systems. We also derive an effective criterion for checking whether a quadratic conservation law of a simplest linear potential system is a purely potential conservation law of a (1+1)-dimensional linear evolution equation of odd order.