Infinitely many nonlocal conservation laws for the $ABC$ equation with $A+B+C\neq 0$

TL;DR

This paper develops an infinite hierarchy of nonlocal conservation laws for the $ABC$ equation with specific conditions, using a nonisospectral Lax pair, and introduces new coverings that could apply to other integrable systems.

Contribution

It constructs an infinite hierarchy of nonlocal conservation laws for the $ABC$ equation with $A+B+C eq 0$ using a novel nonisospectral Lax pair approach.

Findings

Established an infinite hierarchy of conservation laws.

Presented new coverings for the $ABC$ equation.

Method applicable to other multidimensional integrable systems.

Abstract

We construct an infinite hierarchy of nonlocal conservation laws for the $ABC$ equation $A u_t\,u_{xy}+B u_x\,u_{ty}+C u_y\,u_{tx} = 0$, where $A,B,C$ are constants and $A+B+C\neq 0$, using a nonisospectral Lax pair. As a byproduct, we present new coverings for the ABC equation. The method of proof of nontriviality of the conservation laws under study is quite general and can be applied to many other integrable multidimensional systems.