Improved effective linearization of nonlinear Schrödinger waves by increasing nonlinearity

From among the waves whose dynamics are governed by the nonlinear Schrödinger equation, we find a robust, spatiotemporally disordered family, in which waves initialized with increasing amplitudes, on average, over long timescales, effectively evolve as ever more weakly coupled collections of plane waves. In particular, the relative amount of energy contained in their coupling decays to zero with increasing wave amplitude.

Figure 1

Scatter plot of the frequency shift $\mathrm{\Delta }{\omega }_{\psi }$, predicted by the EDR in Eq. (3), vs the corresponding frequency shift ${\widetilde{\mathrm{\Delta }\omega }}_{\psi }$, obtained using the WFS method, for increasing values of the wave norm.

Figure 2

The values of the renormalization parameter $\beta$ that minimize the time-averaged norm of the trial renormalized nonlinear term ${\mathcal{N}}_{\beta }$ in the NLS for increasing norms of the NLS waves.

Figure 3

Ratio between the time-averaged norms of the effective nonlinear and linear NLS terms for waves with increasing norms. The values of the wave norms are listed in Table 1.

Figure 4

The values of the renormalization parameter $\beta$ that minimize the time-averaged size of the trial renormalized superquadratic part of the NLS Hamiltonian $H_{\beta }^s$ for increasing norms of the NLS waves.

Figure 5

Ratio between the time-averaged sizes of the effective superquadratic and quadratic parts of the NLS Hamiltonian for waves with increasing norms. The values of the wave norms are listed in Table 1.