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Department of Mathematics

The KdV-hierarchy and integrability

Supervisor: Alexander Vasiliev, email: alexander.vasiliev math.uib.no

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Prerequisites: MAT213, MAT251, MAT236, together with MAT211, MAT234, MAT253, which can be taken in parallel with the project.

Abstract: The project is aimed at the complete integrability of the KdV equation.

Description: The most well-known “prototype” integrable system appeared solving the famous KdV equation in 1967 (Gardner, Greene, Kruskal, and Miura). Lax (1968) showed that the KdV equation is equivalent to the so-called “isospectral integrability condition” for pairs of linear operators, known as Lax pairs. It turned out that the KdV equation possesses a countable number of conserved quantities (first integrals) which are pairwise involutary. The KdV equation itself appeared as a third evolution equation in the hierarchy generated by these integrals treated as Hamiltonians (Zakharov, Faddeev, 1971). Students will learn definitions of classical Liouville integrability in connection with the conservative quantities of the KdV equation. The construction of the KdV hierarchy is a part of this project.

References:

[1] V.I. Arnold, Mathematical methods of classical mechanics, Springer-Verlag New-York, 1989. 
[2] V. Zakharov, L. Faddeev, The Korteweg-de Vries equation is a fully integrable Hamiltonian system, Funct. Anal. Appl. 5, 280-287 (1971) 
[3] P. Lax, Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math. 21 (1968), 467-490.