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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. 

Three wave resonant triad interactions in two space and one time dimensions form a well-known system of first-order quadratically nonlinear evolution equations that arise in many areas of physics. In deep water waves, they were first derived by Simmons in 1969 and later shown to be exactly solvable by Ablowitz & Haberman in 1975. Specifically, integrability was established by introducing a system of six wave interactions whose symmetry reduction leads to the well-known three wave equations. Here, it is shown that the six wave interaction and classical three wave equations satisfying triad resonance conditions in finite-depth gravity waves can be derived from the non-local integro-differential formulation of the free surface gravity wave equation with surface tension. These quadratically nonlinear six wave interaction equations and their reductions to the classical and non-local complex as well as real reverse space–time three wave interaction equations are integrable. Limits to infinite and shallow water depth are also discussed. 

Abstract Integrable standard and nonlocal derivative nonlinear Schrödinger equations are investigated. The direct and inverse scattering are constructed for these equations; included are both the Riemann–Hilbert and Gel’fand–Levitan–Marchenko approaches and soliton solutions. As a typical application, it is shown how these derivative NLS equations can be obtained as asymptotic limits from a nonlinear Klein–Gordon equation. 

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.