Formula omitted]-Sobolev space bijectivity of the scattering-inverse scattering transforms related to defocusing Ablowitz–Ladik systems

Abstract

In this paper, we establish the l2-Sobolev space bijectivity of the inverse scattering transform related to the defocusing Ablowitz–Ladik system. On the one hand, based on the spectral problem, we establish the reflection coefficient and the corresponding Riemann–Hilbert problem for the direct scattering problem. And we prove that if the potential belongs to l2,k space, then the reflection coefficient belongs to Hθk(Σ). On the other hand, based on the Riemann–Hilbert problem, we obtain the corresponding reconstruction formula and recover the potential from the reflection coefficient. Furthermore, we confirm that if the reflection coefficient is in Hθk(Σ), then we show that the corresponding potential also belongs to l2,k for the inverse scattering problem. This study also confirms that for the initial-value problem of the defocusing Ablowitz–Ladik system, if the initial potential qn(0) belongs to l2,k and satisfies ∥qn∥∞<1, then the solution qn(t) for t≠0 also belongs to l2,k.