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.
The inverse scattering transform for the focusing nonlinear Schrödinger equation is presented for a general class of initial conditions whose asymptotic behavior at infinity consists of counterpropagating waves. The formulation takes into account the branched nature of the two asymptotic eigenvalues of the associated scattering problem. The Jost eigenfunctions and scattering coefficients are defined explicitly as single‐valued functions on the complex plane with jump discontinuities along certain branch cuts. The analyticity properties, symmetries, discrete spectrum, asymptotics, and behavior at the branch points are discussed explicitly. The inverse problem is formulated as a matrix Riemann‐Hilbert problem with poles. Reductions to all cases previously discussed in the literature are explicitly discussed. The scattering data associated to a few special cases consisting of physically relevant Riemann problems are explicitly computed.
more » « less- Award ID(s):
- 2009487
- NSF-PAR ID:
- 10454248
- Publisher / Repository:
- Wiley-Blackwell
- Date Published:
- Journal Name:
- Studies in Applied Mathematics
- Volume:
- 146
- Issue:
- 2
- ISSN:
- 0022-2526
- Page Range / eLocation ID:
- p. 371-439
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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