Three-Wave Interactions and Spatiotemporal Chaos

Abstract

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 spatio-temporal chaos. Two length scales arise naturally in the Faraday wave experiment with multi-frequency forcing, and our results enable some previously unexplained experimental observations of spatio-temporal chaos to be interpreted in a new light. Our predictions are illustrated with numerical simulations of a model partial differential equation. Patterns arise in many non-equilibrium physical, chemical and biological systems, often when a uniform state is subjected to external driving and becomes unstable to modes with a finite wavelength. In some systems, modes with a second wavelength can play an important role in pattern formation if these modes are either unstable or only weakly damped. The interaction between two waves of one wavelength with a third wave of the other wavelength is known both experimentally and theoretically to play a key role in producing a rich variety of interesting phenomena such as quasipatterns, superlattice patterns and spatio-temporal chaos (STC) [1][2][3][4][5][6][7][8].In this paper, we focus on three-wave interactions (3WIs) involving two comparable wavelengths and develop a criterion for when such interactions are likely to lead to STC, as opposed to steady patterns and quasipatterns. The mechanism we describe is generic, and will apply to any system in which such 3WIs can occur, such as the Faraday wave experiment [1-3], coupled Turing systems [4] and some optical systems [5]. In order to illustrate our criterion we focus on two examples. The first is the Faraday experiment, in which patterns of standing waves are excited on the surface of a fluid by periodically forcing the fluids' container up and down. Using a multi-component forcing f (t) enables the excitation of waves with comparable wavelengths. Experimentally, the phases and amplitudes of the different components of f (t) have been shown to determine whether simple pattterns, superlattice patterns, quasipatterns or STC are seen [1][2][3]. A theoretical understanding of the stabilization of some superlattice patterns has been developed using a single 3WI [2,6,7], but up until this point there has been no explanation for the presence of STC. We show how our extension of the notion of 3WIs can explain the (a) A vector q1 on the inner circle can be written as the sum of two vectors k1 and k2 on the outer circle; this defines the angle θz = 2 arccos(q/2). (b) The vector k1 is the sum of two inner vectors q2 and q3; this defines the angle θw = 2 arccos(1/2q). (c) Similarly, k2 is the sum of two inner vectors, and each of t...

Three-Dimensional Icosahedral Phase Field Quasicrystal

“…The ratio k 2 =k 1 must be close to certain special values; e.g., for dodecagonal QCs the value is 2 cosðπ=12Þ. Second, strong reinforcing (i.e., resonant) nonlinear interactions between these two characteristic density waves are required [17,19,20]. Earlier work on quasipatterns observed in Faraday wave experiments reveals similar requirements [19,[21][22][23].…”

“…However, the growth rates of the two length scales in their models were constrained to be in a fixed ratio. In our model, we choose the linear operator L (based on the one introduced by Rucklidge et al [20]) to allow marginal instability at two wave numbers k ¼ 1 and k ¼ q, with the growth rates of the two length scales determined by two independent parameters μ and ν, respectively. The resulting growth rate σðkÞ of a mode with wave number k is given by a tenth-order polynomial:…”

Soft-core particles freezing to form a quasicrystal and a crystal-liquid phase

“…This configuration leads to dodecagonal quasicrystals; with other radius ratios, the situation can be yet more complicated [25]. In fact, arguments based on the contribution to the free energy from resonant triads only, important though these are, overlook the potential importance of higher-order harmonics, whose coefficients may become arbitrarily large owing to the problem of small divisors that inevitably appears whenever quasiperiodicity and nonlinearity occur together [62].…”

“…One mechanism relies on strong self-coupling to downplay the effect of waves with different orientations [18][19][20], so permitting 8-, 10-, 12-, 14-, 16-, 18-, 20-fold, or higher quasipatterns [17]. The second mechanism invokes nonlinear coupling between the primary waves with secondary weakly damped (or weakly excited) waves such that primary waves with wave vectors separated by a certain angle determined by the ratio of the primary to secondary wave number are favored [16,[21][22][23][24][25][26][27]. We invoke here this second mechanism, as done in [28,29], and select the length scale ratio for our investigation to be R s /R = 1.855 (Sec.…”

Instabilities and patterns in coupled reaction-diffusion layers

“…[27][28][29][30], which study resonant triads in Faraday waves. Recent work connecting the understanding of resonant triads to complex nonlinear patterns and to spatiotemporal chaos appears in [31]. Resonant triad interactions, the lowest order nonlinear interactions, involve three modes with wave vectors Q 1 , Q 2 , and Q 3 satisfying the condition…”