Some regularity properties of scattering data for the derivative nonlinear Schrödinger equation

Abstract In this paper, we present some properties of scattering data for the derivative nonlinear Schrödinger equation in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msup> <mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>s</mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="double-struck">R</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>≥</mml:mo> <mml:mstyle displaystyle="false"> <mml:mfrac> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:mfrac> </mml:mstyle> <mml:mo stretchy="false">)</mml:mo> </mml:math> starting from the Lax pair. We show that the reciprocal of the transmission coefficient can be expressed as the sum of some iterative integrals, and its logarithm can be written as the sum of some connected iterative integrals. We provide the asymptotic properties of the first few iterative integrals of the reciprocal of the transmission coefficient. Moreover, we provide some regularity properties of the reciprocal of the transmission coefficient related to scattering data in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msup> <mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>s</mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="double-struck">R</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> .