wave solutions to .

Standing wave patterns {#sec:gs}
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Consider the ansatz for standing wave solutions that oscillate harmonically in time $$A(x,y,t) = {\mathrm{e}}^{-{\mathrm{i}}\omega t}B(x,y),$$ where the real function $B$ describes the spatial pattern of a standing wave. According to the unscaled equation , $B$ satisfies the following real nonlinear hyperbolic PDE $$\label{eq:Beq}
  \omega B + B_{xx} - B_{yy} = 2B^3,$$ which can be solved numerically using the classical Petviashvili method [@Petviashvili1976; @Pelinovsky2004; @Lakoba2007]. To do so, is rewritten in the operator form $${\mathcal{L}}\cdot B = {\mathcal{N}}(B), \qquad {\mathcal{L}}:= \omega + \partial_{xx} - \partial_{yy}, \qquad {\mathcal{N}}(B) := 2B^3.$$ and the iteration scheme is given by $$B_{n+1} = {\ensuremath{\mathcal{S}}}^\gamma {\mathcal{L}}^{-1}\cdot {\mathcal{N}}(B_n), \quad
  {\ensuremath{\mathcal{S}}}= \frac{\langle B_n, {\mathcal{L}}\cdot B_n\rangle}{\langle B_n, {\mathcal{N}}(B_n)\rangle},$$ where ${\ensuremath{\mathcal{S}}}$ is the so-called stabilyzing factor and the exponent $\gamma$ is usually defined as a function of the degree of nonlinearity $p$ ($p = 3$ for the equation). The rule of thumb prescribes the following formula $\gamma = \frac{p}{p-1}$. The scalar product is defined in the $L_2$ space. The inverse operator ${\mathcal{L}}^{-1}$ can be efficiently computed in Fourier space using the Fast Fourier Transform (FFT) (see, for example, [@Frigo2005]), and spatial periodicity across the 2D computational box boundaries is implicitly imposed. 