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On multicomponent MKdV equations on symmetric spaces of DIII-type and their reductions. (English) Zbl 1193.35184

Mladenov, Ivaïlo M. (ed.), Proceedings of the 9th international conference on geometry, integrability and quantization, Sts. Constantine and Elena, Bulgaria, June 8–13, 2007. Sofia: Bulgarian Academy of Sciences (ISBN 978-954-8495-42-4/pbk). 36-65 (2008).
Summary: New reductions for the multicomponent modified Korteweg-de Vries (MMKdV) equations on the symmetric spaces of DIII-type are derived using the approach based on the reduction group introduced by A. Mikhailov. The relevant inverse scattering problem is studied and reduced to a Riemann-Hilbert problem. The minimal sets of scattering data \({\mathcal T}_i\), \(i=1,2\), which allow one to reconstruct uniquely both the scattering matrix and the potential of the Lax operator are defined. The effect of the new reductions on the hierarchy of Hamiltonian structures of MMKdV and on \({\mathcal T}_i\) are studied. We illustrate our results by the MMKdV equations related to the algebra \({\mathfrak g}\simeq{\mathfrak{so}}(8)\) and derive several new MMKdV-type equations using group of reductions isomorphic to \(\mathbb Z_2,\mathbb Z_3,\mathbb Z_4\).
For the entire collection see [Zbl 1154.17001].

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35A08 Fundamental solutions to PDEs
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
30E25 Boundary value problems in the complex plane
22E70 Applications of Lie groups to the sciences; explicit representations
37K30 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures
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