Four-Wave Semidiscrete Nonlinear Integrable System with PT-Symmetry

The new type of third-order spectral operator suitable to generate new multifield semidiscrete nonlinear systems with two coupling parameters in the framework of zero-curvature equation is proposed. The evolution operator corresponding to the first integrable system in an infinite hierarchy is explicitly recovered and the general form of first integrable system is isolated. The generalized procedure for the direct recursive development of infinite hierarchy of local conservation laws is presented and several lowest local conservation laws and local conserved densities are found. The reduction to the real field amplitudes in general system with unfixed sampling functions is made and the symmetric parametrization of field amplitudes allowing to exclude the redundant field function and to resolve the problem of sampling fixation for the particular realization of reduced integrable system is considered. This parametrization gives rise to the four field nonlinear model which in certain intervals of adjustable coupling parameter could serve as a semidiscrete analogy to the beam-plasma interaction system.