Higher-order conservation laws for the non-linear Poisson equation via the characteristic cohomology

TL;DR

This paper investigates higher-order conservation laws for a nonlinear elliptic Poisson equation, establishing conditions for their existence, normal forms, and explicit forms in special cases, linking them to symmetries and Jacobi fields.

Contribution

It introduces a normal form for conservation laws via characteristic cohomology and characterizes when higher-order laws exist, relating them to symmetries and explicit solutions.

Findings

Higher-order conservation laws exist only if f satisfies a specific second order ODE.

An at most two-dimensional space of conservation laws appears at each even prolongation.

Explicit forms of conservation laws are obtained when f_{uu} = βf, using Pinkall and Sterling's work.

Abstract

We study higher-order conservation laws of the non-linearizable elliptic Poisson equation $ \frac{{\partial}^2 u}{\partial z \partial \bar{z}} = -f(u) $ as elements of the characteristic cohomology of the associated exterior differential system. The theory of characteristic cohomology determines a normal form for differentiated conservation laws by realizing them as elements of the kernel of a linear differential operator. The S^1--symmetry of the PDE leads to a normal form for the undifferentiated conservation law as well. We show that for higher-order conservation laws to exist, it is necessary that $f$ satisfies a linear second order ODE. In this case, an at most real two--dimensional space of new conservation laws in normal form appears at each even prolongation. When $f_{uu}=\beta f$ this upper bound is attained and the work of Pinkall and Sterling allows them to be written explicitly. We relate higher-order conservation laws to generalized symmetries of the exterior differential system by identifying their generating functions. This Noether correspondence provides the connection between conservation laws and the canonical Jacobi fields of Pinkall and Sterling.