A non-trivial conservation law with a trivial characteristic

TL;DR

This paper demonstrates that a certain conservation law related to an overdetermined PDE system is non-trivial, even though its characteristic vanishes on the system, challenging typical assumptions about conservation laws.

Contribution

It reveals a non-trivial conservation law with a trivial characteristic for a specific overdetermined PDE system, providing new insights into conservation law structures.

Findings

Conservation law is non-trivial despite trivial characteristic.

The system involves a third-order PDE with a trivial characteristic.

Challenges assumptions about the relationship between characteristics and conservation laws.

Abstract

We show that the conservation law of the overdetermined system $u_t - 4u_x^3 - u_{xxx} = 0$, $u_y = 0$, associated with the characteristic $(u_{xy}, 0)$, is non-trivial despite the characteristic vanishing on the system.