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On the integrable variant of the Boussinesq system: Painlevé property, rational solutions, a related many-body system, and equivalence with the AKNS hierarchy. (English) Zbl 0694.35207

Summary: An infinite family of rational solutions of the integrable version of the Boussinesq system

$\left(*\right)\phantom{\rule{1.em}{0ex}}{u}_{t}+{w}_{x}+u{u}_{x}=0,\phantom{\rule{1.em}{0ex}}{w}_{t}+{u}_{xxx}+\left(uw\right)x=0$

and the associated higher-order flow is constructed. The differential equations governing the motion of the poles are derived and their complete integrability demonstrated. This system embeds into two disjoint Calogero-Moser systems coupled only through a constraint. A novel treatment of this constraint is given. Using the equivalence of (*) to the real version of the nonlinear Schrödinger system, it is seen that rational solutions of the AKNS hierarchy have been constructed as well. Finally, it is shown that all solutions of these hierarchies, including the rational ones, also solve the first- modified KP equation.

MSC:
 35Q99 PDE of mathematical physics and other areas 37J35 Completely integrable systems, topological structure of phase space, integration methods 37K10 Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies