FURTHER EXPLORATION ON THE EFFECT OFNEGATIVE-ENERGY MODES IN THENONLINEAR INSTABILITIES

Abstract

The nonlinear variations of the eigenvectors are discussed on the basis of Ref. [1, 2]. It is shown that relative to a steady wave the nonlinear eigen frequency allows for both forward and backward transports. Their eigenvectors deflect in the Fourier space as compared to the linear case. The positive (P-) and negative (N-) energy modes can be defined in the nonlinear case according to the relative strength of the forward and backward eigenvectors of the perturbations. Based on this definition the effect of N-mode on the nonlinear instabilities is further studied. The results are in consistency with the conclusion in Ref. [1,2]. The bista-bility of wavepacket is associated with the transition of a resonance N-mode to a p-mode, in particular its eigenvector is discontinuous in the parameter space; The Hop f bifurcation of a steady wavepacket is associated with the resonance of a N-mode to a p-mode. As a result re-connections occur between their eigenvectors of forward and backward perturbations respec-tively. With the increase of nonlinearity, the p-mode first and the N-mode following next change their energy types. Consequently there exists a parameter regime where both modes become N-type. This regime coincides with that of the Hopf bifurcation. This fact strongly supports the conjecture that the excitation of N-modes induces the nonlinear (Hopf) bifurcation.