periodic and non-periodic infinite patterns using 2-D words and tilings was done in [@Berthe2000; @DeBruijn1981]. The numerical standing waves investigated in this work are periodic in space, thus the associated patterns are described up to toric constraints. This is the first step for developing a theory for the description of periodic or quasi-periodic patterns associated to trajectories $\{a_n(t)\}$ of the dynamics in the infinite phase space spanned by, for example, generalized Fourier basis $\phi_n(x,y)$ associated to the formal series for $A(x,y,t) = \sum_n a_n(t)\phi_n(x,y)$ on a periodic domain.

The present study is organized as follows. First, the equation is introduced in the context of deep water waves. Then, the Petviashvili method [@Petviashvili1976] used to compute a class of special standing wave solutions is presented. Symbolic dynamics is then applied to describe the associated spatial patterns using words and substitutions. Finally, a highly accurate Fourier-based solver is exploited to investigate the dynamics of a perturbed standing wave.

The Mathematical Model {#sec:math}
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Consider a three-dimensional fluid domain with a free surface. The water is assumed to be infinitely deep. The Cartesian coordinate system $Oxyz$ is chosen such that the undisturbed water level corresponds to $z = 0$, and the free surface 