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Abstract Nonlocal reverse space‐time Sine/Sinh‐Gordon type equations were recently introduced. They arise from a remarkably simple nonlocal reduction of the well‐known AKNS scattering problem, hence, they constitute an integrable evolution equations. Furthermore, the inverse scattering transform (IST) for rapidly decaying data was also constructed. In this paper, the IST for these novel nonlocal equations corresponding to nonzero boundary conditions (NZBCs) at infinity is presented. The NZBC problem is more complex due to the intricate branching structure of the associated linear eigenfunctions. Two cases are analyzed, which correspond to two different values of the phase at infinity. Special soliton solutions are discussed and explicit 1‐soliton and 2‐soliton solutions are found. Both spatially independent and spatially dependent boundary conditions are considered.
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Inverse scattering transform for continuous and discrete space‐time‐shifted integrable equations
Abstract Nonlocal integrable partial differential equations possessing a spatial or temporal reflection have constituted an active research area for the past decade. Recently, more general classes of these nonlocal equations have been proposed, wherein the nonlocality appears as a combination of a shift (by a real or a complex parameter) and a reflection. This new shifting parameter manifests itself in the inverse scattering transform (IST) as an additional phase factor in an analogous way to the classical Fourier transform. In this paper, the IST is analyzed in detail for several examples of such systems. Particularly, time, space, and space‐time‐shifted nonlinear Schrödinger (NLS) and space‐time‐shifted modified Korteweg‐de Vries equations are studied. Additionally, the semidiscrete IST is developed for the time, space, and space‐time‐shifted variants of the Ablowitz–Ladik integrable discretization of the NLS. One‐soliton solutions are constructed for all continuous and discrete cases.
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- Award ID(s):
- 2306290
- PAR ID:
- 10542451
- Publisher / Repository:
- Wiley-Blackwell
- Date Published:
- Journal Name:
- Studies in Applied Mathematics
- ISSN:
- 0022-2526
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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