Nonlinear spatial evolution of degenerate quartets of water waves

Abstract

In this manuscript we investigate the Benjamin-Feir (or modulation) instability for the spatial evolution of water waves from the perspective of the discrete, spatial Zakharov equation, which captures cubically nonlinear and resonant wave interactions in deep water without restrictions on spectral bandwidth. Spatial evolution, with measurements at discrete locations, is pertinent for laboratory hydrodynamic experiments, such as in wave flumes, which rely on time-series measurements at a series of fixed gauges installed along the facility. This setting is likewise appropriate for experiments in electromagnetic and plasma waves. Through a reformulation of the problem for a degenerate quartet, we bring to bear techniques of phase-plane analysis which elucidate the full dynamics without recourse to linear stability analysis. In particular we find hitherto unexplored breather solutions and discuss the optimal transfer of energy from carrier to sidebands. Finally, we discuss the observability of such discrete solutions in light of numerical simulations.

Figures (9)

• Figure 1: Phase portraits for $(\Theta,\eta)$ for various values of the mode separation parameter $p$ with $f=1$ and $\epsilon=0.2$. Panels (a)--(f) show $p=0, \, 0.6p_c, \, p_c, \, 1.1p_c, \, 2p_c$ and $3p_c$ respectively for $p_c \approx 0.5791.$ Fixed points of the dynamical system are denoted by black circles, separatrices are denoted by dashed curves connecting pairs of fixed points.
• Figure 2: Plot in $\epsilon, \, p$ parameter space of the existence of fixed points (i.e. instability) at $\eta=1$ (monochromatic waves, left panel) and $\eta=0$ (bichromatic waves, right panel). Colours denote the growth rate, calculated from the eigenvalues of the Jacobi matrix. The red curve shows the location in $\epsilon, \,p$ space of the maximum depletion of the carrier, associated with the vertical separatrix shown in panel (c) of Figure \ref{['fig:phase portraits']}.
• Figure 3: Three views of a period solution: (top panel) free-surface envelope in space and time. (Bottom left) solution depicted in phase space (red curve). (Bottom right) Fourier amplitudes $|B_i(x)|$ and dynamic phase $\Theta(x)$ plotted with distance $x$.
• Figure 4: The free surface envelope of a breather with monochromatic background, corresponding to $f=1$ Hz, $\epsilon = 0.2,$ and $p=1.4.$ The focusing occurs at $x=0,$ which is the minimum of the energy scale parameter $\eta$ along the separatrix. As $x$ tends to $\pm \infty,$$\eta$ tends toward 1 and the wave field asymptotically becomes monochromatic.
• Figure 5: The free surface envelope of a breather with bichromatic background, corresponding to $f=1$ Hz, $\epsilon = 0.2,$ and $p=0.4.$ The focusing (in the form of a "demodulation") occurs at $x=0,$ which is the maximum of the energy scale parameter $\eta$ along the separatrix. At this location $x=0$ the wave field is close to monochromatic, as components $b$ and $c$ are very small. As $x$ tends to $\pm \infty$, $\eta$ tends toward 0 and the wave field asymptotically becomes bichromatic.