Multi-Dimensional Conservation Laws and Integrable Systems

We introduce and investigate a new phenomenon in the Theory of Integrable Systems – the concept of multi-dimensional conservation laws for two- and three-dimensional integrable systems.
Existence of infinitely many local two-dimensional conservation laws is a well-known property of two-dimensional integrable systems.
We show that pairs of commuting two-dimensional integrable systems possess infinitely many three-dimensional conservation laws.
Examples: the Benney hydrodynamic chain, the Korteweg de Vries equation.
Simultaneously three-dimensional integrable systems (like the Kadomtsev-Petviashvili equation) have infinitely many three-dimensional quasi-local conservation laws.
We illustrate our approach considering the dispersionless limit of the Kadomtsev-Petviashvili equation and the Mikhalev equation.
Applications in three-dimensional case: the theory of shock waves, the Whitham averaging approach.