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A discussion of three-wave interaction systems with rapidly decaying data is provided. Included are the classical and two nonlocal three-wave interaction systems. These three-wave equations are formulated from underlying compatible linear systems and are connected to a third order linear scattering problem. The inverse scattering transform (IST) is carried out in detail for all these three-wave interaction equations. This entails obtaining and analyzing the direct scattering problem, discrete eigenvalues, symmetries, the inverse scattering problem via Riemann--Hilbert methods, minimal scattering data, and time dependence. In addition, soliton solutions illustrating energy sharing mechanisms are also discussed. A crucial step in the analysis is the use of adjoint eigenfunctions which connects the third order scattering problem to key eigenfunctions that are analytic in the upper/lower half planes. The general compatible nonlinear wave system and its classical and nonlocal three-wave reductions are asymptotic limits of physically significant nonlinear equations, including water/gravity waves with surface tension.
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Inverse scattering transform for the focusing nonlinear Schrödinger equation with counterpropagating flows
Abstract 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.
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- Award ID(s):
- 2009487
- 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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