Toward a General Solution of the Three-Wave Partial Differential Equations

The three-wave, resonant interaction equations appear in many physical applications. These partial differential equations (PDEs) are known to be completely integrable, and have been solved with initial data that decay rapidly in space, using inverse scattering theory. We present a new way to solve these equations, which makes no use of inverse scattering theory, and which can be used with a wide variety of boundary conditions. A “general solution” of these PDEs would involve six free, real-valued functions of space. At this time, our “nearly general solution” accepts five free, real-valued functions of space, and embeds them in convergent series in a deleted neighborhood of a pole.