Mechanism of Kuznetsov-Ma breathers

We discuss how to understand the dynamical process of Kuznetsov-Ma breather, based on some basic physical mechanisms. It is shown that the dynamical process of Kuznetsov-Ma breather involves at least two distinctive mechanisms: modulational instability and the interference effects between a bright soliton and a plane-wave background. Our analysis indicates that modulational instability plays dominant roles in the mechanism of Kuznetsov-Ma breather admitting weak perturbations, and the interference effect plays a dominant role for the Kuznetsov-Ma breather admitting strong perturbations. For intermediate cases, the two mechanisms are both greatly involved. These characters provide a possible way to understand the evolution of strong perturbations on a plane-wave background.

Figure 1

A qualitative description on the mechanisms of KMB. MI plays a dominant role for KMB admitting weak perturbations, and the interference effects (IE) play a dominant role for KMB admitting strong perturbations. The weak and strong perturbations are clarified by the ratio $p/s\ll 1$ and $p/s\gg 1$, respectively ($p$ and $s$ denote perturbation amplitude and background amplitude, respectively). For intermediate cases ($p$ is comparable with $s$), it is expected that the two mechanisms are both greatly involved. For the weak perturbation case and intermediate cases, the breathing period can be explained well by nonlinear interference effects.

Figure 2

The oscillation amplitude characterization parameter $\eta$ vs $p/s$ (the ratio of perturbation soliton amplitude and background amplitude). The solid red line and blue dashed line correspond to the results of KMB and the linear interference case, respectively. It is shown that the amplitude oscillation behavior of KMB does not agree with linear interference effects for weak soliton perturbations and the intermediate cases, but the behavior agrees perfectly with the prediction of linear interference effects for strong soliton perturbations.

Figure 3

The oscillation period $T$ vs $p/s$ (the ratio of perturbation soliton amplitude and background amplitude). The solid red line and blue dashed line correspond to the results of KMB and the linear interference case, respectively. It is shown that the amplitude oscillation behavior of KMB does not agree with linear interference effects for weak soliton perturbations and the intermediate cases, but the behavior agrees perfectly with the prediction of linear interference effects for strong soliton perturbations.

Figure 4

The evolution of KMB (a) and the linear interference form (b) with an identical strong perturbation strength $p/s=4\sqrt{6}$. It is shown that their evolution processes agree well for large perturbation soliton amplitude, which indicates the linear interference effect can be used to explain the dynamical process of KMB with strong perturbation cases. The plane-wave background amplitude is $s=1$.