Seminar talk, 22 March 2017

From Geometry of Differential Equations
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Speaker: Maxim Pavlov

Title: 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.