Search

In this paper, we present the inverse scattering transformation (IST) for discrete sine‐Gordon equation via the Riemann–Hilbert approach. In the direct scattering part, we establish analyticity, symmetries, and asymptotic properties of Jost solutions and reflection coefficients. In the inverse part, we solve the discrete sine‐Gordon equation by establishing a Riemann–Hilbert problem with simple and double poles, respectively. N‐soliton solutions are obtained via the reconstruction formula correspondent to the Riemann–Hilbert problem. As examples of N‐solitons, we present some explicit solutions such as breathers, degenerate solitons for discrete sine‐Gordon equation, and the dynamical properties of these solutions are further analyzed. [ABSTRACT FROM AUTHOR]

Recently, we have established the l 2 bijectivity for the defocusing Ablowitz-Ladik system in the discrete weighted space l 2 , k with k ∈ N + by the inverse spectral method. Based on these results, our study is to investigate the long-time asymptotics for the initial-value problem of the defocusing Ablowitz-Ladik system with initial potential in lower regularity without rapid decay. In this case, since the reflection coefficient is not smooth enough, it is impossible to directly apply the nonlinear steepest descent method. Our main idea is to transform the corresponding Riemann-Hilbert problem on the unit circle as the jump contour into a ∂ ¯ -Riemann-Hilbert problem to be handled. As a result, we show that the solution admits the Zakharov-Manakov type formula in the region | n 2 t | ≤ V 0 < 1 , while the solution decays fast to zero in the region | n 2 t | ≥ V 0 > 1. [ABSTRACT FROM AUTHOR]

In this paper, we establish the l 2 -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 l 2 , k space, then the reflection coefficient belongs to H θ k (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 H θ k (Σ) , then we show that the corresponding potential also belongs to l 2 , 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 q n (0) belongs to l 2 , k and satisfies ∥ q n ∥ ∞ < 1 , then the solution q n (t) for t ≠ 0 also belongs to l 2 , k. • Inverse scattering maps a weighted l 2 space to a Sobolev space. • Defocusing Ablowitz–Ladik system solvable in the weighted space l 2 , k. • Inverse scattering bijective from the potential to the reflection coefficient. [ABSTRACT FROM AUTHOR]