Inverse scattering transform for the defocusing Ablowitz–Ladik system with arbitrarily large nonzero background. Issue 4 (19th September 2019)

• Record Type: Journal Article
• Authors: Ortiz, Alyssa K.; Prinari, Barbara
• Is Part Of: Studies in applied mathematics. Volume 143:Issue 4(2019)
• Page Start: 373
• Page End: 403
• Publication Date: 2019-09-19
• Subjects: Ablowitz–Ladik equations -- discrete rogue waves -- solitons and integrable systems; Mathematics -- Periodicals; Mathématiques -- Périodiques; Mathématique; Mathématique appliquée; Ressource Internet (Descripteur de forme); Périodique électronique (Descripteur de forme); 519
• DOI: 10.1111/sapm.12282
• Languages: English
• ISSNs: 0022-2526

Abstract

Abstract: In this work we develop the inverse scattering transform (IST) for the defocusing Ablowitz–Ladik (AL) equation with arbitrarily a large nonzero background at space infinity. The IST was developed in previous works under the assumption that the amplitude of the background Q o satisfies a "small norm" condition 0 < Q o < 1 . On the other hand, Ohta and Yang recently showed that the defocusing AL system, which is modulationally stable for 0 ⩽ Q o < 1, becomes unstable if Q o > 1, and exhibits discrete rogue wave solutions, some of which are regular for all times. Here, we construct the IST for the defocusing AL with Q o > 1, analyze the spectrum, and characterize the soliton and rational solutions from a spectral point of view. We formulate the direct and inverse problems by using a suitable uniformization variable, and pose the inverse problem as an RHP across a simple contour in the complex plane of the uniform variable. As a by‐product of the IST, we also obtain explicit soliton solutions, which are the discrete analog of the celebrated Kuznetsov–Ma, Akhmediev, Peregrine solutions, and which mimic the corresponding solutions for the focusing AL equation. Soliton solutions that are the analog of the dark soliton solutions of the defocusing AL equation in the case 0 < Q o < 1 are also presented.