Non-homogeneous initial boundary value problems and small data scattering of the Hirota equations posed on the half line

Abstract This paper deals with the well-posedness problems and scattering property of the Hirota equations posed on the right half line. We first prove the global well-posedness of the Hirota equations in three different solution spaces, including L p - L q and Bourgain related spaces, for the initial data in H s (R +) with the index s being in [ 1 4, + ∞) ∖ Z + 1 2, [ 0, 1 4) and (− 1 4, 0), separately. The global solution is proved in different solution spaces with different techniques based on different ranges of s. Then we prove that the Hirota equation posed on the half line admits the scattering property in L 2 (R +) if the initial data exponentially decays at infinity. As far as we know, this is the first scattering result for dispersive type equations posed on the half-line although there are numerous and wonderful scattering results for dispersive equations posed on the whole domain. The key and new ingredient in this paper is to construct rotation-rescaling transformations on the Hirota equations which lead to complex-valued gKdV equations and KdV-Burgers equations. We apply the inhomogeneous and homogeneous boundary operator estimates as well as the regularity of the solution space for complex-valued gKdV equations and KdV-Burgers equations to obtain well-posedness and scattering property.