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. 

Publication: 

SIAM Journal on Mathematical Analysis, Vol. 55, Iss. 4 (2023) 

http://dx.doi.org/10.1137/22M1488880 

Author: 

Mark J. Ablowitz 

Department of Applied Mathematics, University of Colorado at Boulder, Boulder, CO 80309-0526 USA. 

Xu-Dan Luo 

Corresponding author. Key Laboratory of Mathematics Mechanization, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China. 

Email: lxd@amss.ac.cn 

Ziad H. Musslimani 

Department of Mathematics, Florida State University, Tallahassee, FL 32306-4510 USA.