Recurrence in the high-order nonlinear Schrödinger equation: A low-dimensional analysis

We study a three-wave truncation of the high-order nonlinear Schrödinger equation for deep-water waves (also named Dysthe equation). We validate the model by comparing it to numerical simulation; we distinguish the impact of the different fourth-order terms and classify the solutions according to their topology. This allows us to properly define the temporary spectral upshift occurring in the nonlinear stage of Benjamin-Feir instability and provides a tool for studying further generalizations of this model.

Figure 1

(a) Numerical results of HONLSe simulations. Evolution of the squared amplitude of the Stokes wave (blue solid line), its unstable sideband pair ($\pm \mathrm{\Omega }$, dash-dotted lines) and second-order sidebands ($\pm 2\mathrm{\Omega }$, dashed lines). Red (green with asterisks) lines represent the low-frequency (high-frequency) sideband. The black dotted line represent the sum of square amplitudes of Stokes wave and its unstable sidebands. (b) The corresponding relative unbalance of sideband amplitudes $\mathrm{\Pi }$ (solid blue for HONLSe, dash-dotted red in the HONLSp) and spectral shift $P/N$ (dashed black, full HONLSe; not reported for HONLSp). The labels (i–iii) correspond to three stages of recurrence in HONLSe mentioned in the main text.

Figure 2

Same as Fig. 1, but with $\mathrm{\Omega }=1.6$. Notice that the growth of unstable modes is much smaller and the kinematic peak downshift (still noticeable in HONLSp) cannot compensate the mean upshift in HONLSe.

Figure 3

Representation of the curves of constant $H$ [level sets, Eq. (6)] in phase plane $(\psi ,\eta )$. The dotted blue line represent the separatrix ($H=0$), the blue cross identifies the center fixed point; the level set corresponding to $\eta \left(0\right)=0.01, \psi \left(0\right)=0, \epsilon =\sqrt{2}/20$ is shown in green, while the dashed black line represents the actual solution of Eq. (7). Notice that the superposition of the curves is almost perfect everywhere.

Figure 4

Representation of solutions of HONLSe on the phase plane of the one degree-of-freedom model of Fig. 3. The red solid line correspond to the inside of the separatrix ($\psi \left(0\right)=0$, period-one), the blue dashed line to the outside ($\psi \left(0\right)=\pi /2$, period-two). In spite of a deviation from three-wave truncation, the topology is conserved. The two insets represent ${\left|a\right|}^2$ in the $(\tau ,\xi )$ plane. As indicated by arrows, the “phase-shifted” solution (top-left) corresponds to the blue dashed curve on the phase space; the top-right to the red solid curve, i.e., nonphase shifted solution.

Figure 5

Comparison of the peak spectral shifts $\mathrm{\Pi }$ and ${\mathrm{\Pi }}_3$ obtained, respectively, from the full HONLS simulation (blue solid line) and the truncated three-wave model (red dash-dotted line). The spectral mean $P/N$ is also shown (black dashed line). $\psi \left(0\right)=0$ while all other parameters are as above.