Nonlinear Evolution Equations and Dynamical Systems

Three–wave nonlinear resonant interaction (TWRI) is typically encountered in the description of any conservative nonlinear medium where the nonlinear dy-namics can be considered as a perturbation of the linear wave solution, the lowest–order nonlinearity is quadratic in the field amplitudes, the three–wave resonance can be satisfied [1]. Indeed TWRI is the lowest–order nonlinear effect for a system approximately described by a linear superposition of discrete waves. TWRI has been extensively studied alongside with the development of plasma physics, since it applies to the saturation of parametric decay instabilities, non-linear collisions of large–amplitude wave packets, radio frequency heating, and laser–plasma interactions [2, 3, 4]. In the domain of nonlinear optics, TWRI de-scribes parametric amplification, frequency conversion, stimulated Raman and Brillouin scattering [5, 6, 7]. TWRIs have also been studied in the context of interactions of water waves [8, 9], interactions of bulk acoustic waves, surface acoustic waves [10] and wave–wave scattering in solid state physics. The physical model describing TWRI is completely integrable [11]. Integra