Instability of parametrically second- and third-subharmonic resonances governed by nonlinear Schrödinger equations with complex coefficients
A theoretical analysis of the parametric harmonic response of two resonant modes is made based on a cubic nonlinear system. Two types of modified nonlinear Schrödinger equations with complex coefficients are derived to govern the resonance wave. One of these equations contains the first derivatives in space for a complex-conjugate type as well as a linear complex-conjugate term that is valid in the second-harmonic resonance cases. The second parametric equation contains a complex-conjugate type which is valid at the third-subharmonic resonance case. The stability behavior at both parametric second- and third-harmonic resonance cases is discussed. Furthermore, the Benjamin-Feir instability is discussed for the sideband perturbation. The instability behavior at the sharp resonance is examined and the existence of the instability is proved.
modified nonlinear Schrödinger equations
scientific article; zbMATH DE number 1488262
scientific article