Considerations on the equations of the three-wave interactions

As a typical case of higher-order matrix in two dimensions, we discuss the third-order eigenvalue problem developed for the inverse spectral transform solving the equations of three-wave interactions. The third-order matrix equation is reformulated by the new mathematical concept, ''Riemann-Hilbert problem'', by which we give a simpler way to construct the soliton solution. Defining the Green function for the associated eigenvalue problem, we obtain the completeness relation of jost functions. Through variational computations, squared jost vectors are naturally introduced and the corresponding completeness is also obtained. According to the classical Green formula, we define another type of Green function for the linearized partial differential equation of the three-wave interactions. By this type of Green function we solve the non-homogeneous linear differential equation, which appears in perturbations of the three-wave interactions. (author)