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Integrable dispersive chains and their multi-phase solutions. (English) Zbl 1420.37100

The authors describe the connection of global solutions of the Mikhalev system
\[ a_{1,t} = a_{2,x} ,\,\,\,a_1 a_{2,x} + a_{1,y} = a_2 a_{1,x} + a_{2,t} \]
and multi-phase solutions for integrable two-dimensional dispersive systems associated with the energy-dependent Schrödinger operator.
The presented approach is universal, i.e., applicable to any integrable system whose Lax pair contains coefficients which depend polynomially on the spectral parameter.
The authors’ approach allows reconsidering multi-phase solutions of multi-component dispersive reductions from a unified point of view.

MSC:

37K15 Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems
37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010)
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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References:

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