Figure \[fig:f0\].*[]{data-label="fig:f4"}](figs/gstates/60x120_0_18_select.eps){width="99.00000%"}

![*A bi-periodic wave pattern $B(x,y)$, $\Omega = [-\frac{55}{2}, \frac{55}{2}]\times[-\frac{89}{2}, \frac{89}{2}]$ and $\omega = 0.51$.*[]{data-label="fig:f5"}](figs/gstates/55x89_0_51.eps){width="99.00000%"}

![*(Right) a bi-periodic wave pattern $B(x,y)$, $\Omega = [-105, 105]^2$ and $\omega = 1.3295$; (Left) zoom on the sub-region $[-40, -10] \times [30, 80]$.*[]{data-label="fig:f6"}](figs/gstates/210x210_1_3295_select.eps){width="99.00000%"}

Symbolic coding {#sec:descr}
===============

The theory of bi-dimensional words and tilings (see [@Berthe2000; @DeBruijn1981]) can be exploited to describe standing wave patterns of the equation. The key tool in tiling theory is the substitution operation that consists in replacing letters by group of letters and constructing iteratively words on a finite alphabet. For example, the Fibonacci substitution $\sigma(a) = ab$ and $\sigma(b) = a$ constructs a fixed point (limit of infinite substitutions) with a non-periodic structure ([@Arnoux1991; @Lind1995; @Allouche2003]. This word constitutes a model for 1-D quasicrystals in physics (see [@Senechal1995]) and a coding of a discrete line with irrational slope in discrete geometry (see [@Vuillon2003]).

The method of substitutions empowers the construction of either non-periodic or periodic words $G$ as fixed points of the mapping $G=\sigma^n(G)$ as $n\to +\infty$ (see [@Lind1995]). On the one hand, non-periodic infinite words can be constructed iteratively by the use of morphism properties. In particular, a substitution $\sigma$ applied to a word $w=w_1w_2\cdots w_n$ 