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Análise / Equações Diferenciais Parciais

Adiabatic limit of Benney type systems
Juan C. Cordero C.

Universidad Nacional de Colombia - Sede Manizales
Quinta-feira, 13 de fevereiro de 2020, 15:30
Sala 232

We consider the nonlinear system: 
i\partial_tu+ \partial_{xx}u= (\tau \vert u\vert^2+\alpha v+  \nu w)u, & (x,t)\in \mathbb{R}\times \mathbb{R}^+,\\
\varepsilon \partial_tv + \lambda \partial_xv= \beta\partial_x|u|^2,&\\
\varepsilon \partial_tw + \gamma\partial_xw= \eta \partial_x|u|^2,
where $u$ is a complex-valued function, $v$ and $w$ are real-valued functions, all parameters are real, and $0 < \varepsilon \ll 1$.

This model contains the 1d-Zakharov-Rubenchik type system which describes the dynamics (non resonant) of small amplitude Alfven waves propagating in a plasma,  the 1d-Zakharov system which models Langmuir turbulence, and  the Benney system used to study general theory of water wave interactions in a nonlinear medium.

Our main purpose is to show that solutions of the cubic nonlinear Schrödinger equation are asymptotic limit of  solutions to the Benney system, in the natural topology given by the energy space $H^1\times L^2$. Due to the special feature of the transport equation same result is obtain for solutions  to the 1d-Zakharov system and 1d-Zakharov-Rubenchik system. 

Joint work with Adan Corcho (UFRJ)