b & a & b & a \cdots \\
  b & a & a & b & a & b \cdots \\
  a & b & b & a & b & a \cdots \\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots ~ \ddots
\end{pmatrix}$$

Hereafter, a discrete description of several spatial patterns $B(x,y)$ of the equation obtained via the Petviashvili method is presented. Each continuous pattern is coded on a finite alphabet using proper substitutions iterated on a discrete elementary pattern of letters.

Consider the bi-periodic pattern $B$ of the equation shown in Figure \[fig:f0\]. This can be easily described by repetition of the elementary 3x3 discrete pattern delimited by a box in the same Figure. To do so, coding with a three–letter alphabet is used since the pattern is characterized by a discrete structure with three different elementary spots. The letters $r$,$b$ and $w$ are used to code red spots (negative values far from zero), blue spots (positive values far from zero) and white spots (values around zero) respectively.

As clearly seen in Figure \[fig:f0\], the three spots are arranged on a rectangular sub-region of the ${\mathbb{Z}}^2$ grid. Thus, the continuous periodic pattern $B$ 