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

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.