THREE-WAVE INTERACTION EQUATIONS: CLASSICAL AND NONLOCAL

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 thirdorder linear scattering problem. The inverse scattering transform (IST) is carried out in detail forall 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<br />nonlocal three-wave reductions are asymptotic limits of physically significant nonlinear equations, including water/gravity waves with surface tension.