2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/70675We present the necessary and sufficient condition for the $L^2$-well-posedness of the initial problem for a third order linear dispersive equation on the two dimensional torus. Birkhoff's method of asymptotic solutions is used to prove the necessity. Some properties of a system for quadratic algebraic equations associated to the principal symbol play crucial role in proving the sufficiency.7 pages, no figureAnalysis of PDEsClassical Analysis and ODEs35G10text