Inverse nonstationary scattering for the linear system of the 3-wave interaction problem in the case of two incident waves with the same velocity

In this paper the nonstationary scattering problems for the hyperbolic system on the whole plane and on the half-plane in the case of two incident waves with the same velocity are considered. The scattering operators for the hyperbolic system on the whole plane and on the half-plane are defined and by using the Gelfand-Levitan-Marchenko (GLM) method the uniqueness of the recovery of the potential from the scattering operators is proved. Moreover, a nonlinear evolution equation in 2 + 1 dimensions which can be integrated by the inverse scattering transform method is presented in association with the considered scattering problems.