Three-Wave Interactions and Spatiotemporal Chaos

Three-wave interactions form the basis of our understanding of many pattern-forming systems because they encapsulate the most basic nonlinear interactions. In problems with two comparable length scales, it is possible for two waves of the shorter wavelength to interact with one wave of the longer, as well as for two waves of the longer wavelength to interact with one wave of the shorter. Consideration of both types of three-wave interactions can generically explain the presence of complex patterns and spatiotemporal chaos. Two length scales arise naturally in the Faraday wave experiment, and our results enable some previously unexplained experimental observations of spatiotemporal chaos to be interpreted in a new light. Our predictions are illustrated with numerical simulations of a model partial differential equation.

Figure 1

Three-wave interactions between waves on two critical circles, with outer radius 1 and inner radius $q>\frac{1}{2}$. (a) A vector ${\mathit{q}}_1$ on the inner circle can be written as the sum of two vectors ${\mathit{k}}_1$ and ${\mathit{k}}_2$ on the outer circle; this defines the angle ${\theta }_z=2\arccos (q/2)$. (b) The vector ${\mathit{k}}_1$ is the sum of two inner vectors ${\mathit{q}}_2$ and ${\mathit{q}}_3$; this defines the angle ${\theta }_w=2\arccos (1/2q)$. (c) Similarly, ${\mathit{k}}_2$ is the sum of two inner vectors, and each of these is in turn the sum of two outer vectors, and so on.

Figure 2

(a),(b) Experimental results of Ding and Umbanhowar [3], reproduced with permission. (a) The 30° angular autocorrelation function with (4, 5, 2) forcing: white indicates twelvefold quasipatterns, as a function of (${\varphi }_5$, ${\varphi }_2$). Black indicates disordered patterns. (b) The 22° angular autocorrelation function with (6, 7, 2) forcing: white indicates 22° superlattice patterns, as a function of (${\varphi }_7$, ${\varphi }_2$). (c),(d),(e) Products of quadratic coefficients. (c),(d) Parameter values as in the experiment in (a), with (c) $Q_{zz}{Q}_{zw}$ (contours in steps of 0.2) and (d) $Q_{ww}{Q}_{wz}$ (contours in steps of 0.05). (e) Parameter values as in (b), showing $Q_{zz}{Q}_{zw}$ (contours in steps of 0.1). In all cases, the thick line is the zero contour, solid (dashed) lines are positive (negative) contours. In (c),(e), $Q_{zz}{Q}_{zw}$ is negative only in very narrow diagonal bands; in (d), $Q_{ww}{Q}_{wz}$ is positive in two horizontal bands.

Figure 3

(a) Bifurcation set showing the patterns seen for $q=0.66$, ${\sigma }_0=-2$, $Q_1=0.3$, $Q_2=1.3$, $Q_3=1.7$ (so $Q_{zz}{Q}_{zw}<0$ and $Q_{ww}{Q}_{wz}<0$). We find $z$ hexagons ($k=1$), $w$ hexagons ($k=q$), spatiotemporal chaos (STC), mixed patterns ($w$ stripes with patches of $z$ rectangles) and two superlattice patterns: one with 6 modes on the outer circle and 12 on the inner, separated by ${\theta }_w$, the other with 6 modes on the inner circle and 12 on the outer, separated by ${\theta }_z$. Steady (time-dependent) patterns are indicated with + (◇). The ($\mu$, $\nu$) scale is not uniform. (b) Pattern in the STC region, with $\mu =\nu =0.00707$. The correlation length is about 1–2 wavelengths. (c) The two critical circles are clearly seen in its power spectrum.