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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mi>H</mml:mi> <mml:mi>s</mml:mi> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>+</mml:mo> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> with the index s being in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>4</mml:mn> </mml:mfrac> <mml:mo>,</mml:mo> <mml:mo>+</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>∖</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mrow> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mo>+</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> </mml:math> , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>4</mml:mn> </mml:mfrac> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mfrac> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mn>4</mml:mn> </mml:mfrac> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> , 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>+</mml:mo> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> 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.