A non-trivial conservation law with a trivial characteristic

TL;DR

The paper demonstrates that non-topological conservation laws can be non-trivial even when their characteristics vanish on the system, by leveraging Vinogradov's $\mathcal{C}$-spectral sequence. It constructs a concrete overdetermined system $u_t - 4u_x^3 - u_{xxx} = 0$, $u_y = 0$ with a conserved density $u_x^4$ that defines a non-trivial element of $E_2^{0,2}$, while one of its characteristics is $(u_{xy},0)$. The work further shows a presymplectic structure derived from a stationary action for the potential mKdV that is not $d_1$-exact, and uses homotopy and symmetry reduction to transfer these ideas to an extended system where the cosymmetry vanishes yet the conservation law remains non-trivial. These results illustrate a richer landscape for conservation laws beyond Noether-type correspondences and point to future directions involving gauge structures and hidden internal Lagrangians.

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.