Stability of some waves in the Boussinesq system (Q2464576)

The variant of the classical Boussinesq system with small parameter \(\varepsilon\) is considered (\(\alpha>0\) is a constant) \[ \partial_t u=-\partial_x v-\alpha \partial_{xxx}v-\varepsilon\partial_x (uv),\quad \partial_t v =-\partial_x u-\varepsilon v\partial_x v.\tag{1} \] It describes the two-dimensional surface waves propagation in a uniform horizontal channel of finite length, filled with an incompressible inviscid fluid, and some other physical systems. The initial problem for this system is ill-posed, however the author shows that for some values of \(\alpha\) it has solutions defined for large time-values which are close of order \(O(\varepsilon)\) to a linear torus for long times of order \(O(\varepsilon^{-1})\). In the previous author's article [\textit{C. Valls}, ``The Boussinesq system: dynamics on the center manifold'', Commun. Pure Appl. Anal., to appear] the existence of periodic and quasiperiodic solutions with two frequencies was proved. Here the stability of such solutions and the transfer of energy between the various modes is investigated at the usage of center manifold, normal forms techniques and perturbation theory for Hamiltonian formulation of the problem. The existence of a finite-dimensional center manifold is proved and the dynamics of the reduction of (1) to it is studied, overcoming technical difficulties connected with the presence of small divisors at each step of the normal form procedure.