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In this paper we study the generalized Korteweg de Vries (KdV) equation withthe nonlinear term of order three: $(u^{3+1})_x$. We prove sharp localwell--posedness for the initial and boundary value problem posed on the righthalf line. We thus close the gap in the well--posedness theory of thegeneralized KdV which remained open after the seminal work of Colliander andKenig in \cite{CK}.

Sharp well--posedness for the generalized KdV of order three on the half line