elevation is $z = \eta(x,y,t)$.

The Euler equations that describe the irrotational flow of an ideal incompressible fluid of infinite depth with a free surface are of fundamental relevance in fluid mechanics, ocean sciences and both pure and applied mathematics (see for example, [@Stoker1957; @Zakharov1968; @Johnson1997]). The structure of the Euler equations is given in terms of the free-surface elevation $\eta(x, y, t)$ and the velocity potential $\varphi(x, y, t) = \phi(x, y, z = \eta(x,y,t), t)$ evaluated at the free surface of the fluid. In late 70s Dysthe used the method of multiple scales to derive from the Euler equations a modified Nonlinear Schrödinger (NLS) equation [@Dysthe1979] for the time evolution of the unidirectional narrowband envelope $A$ of the velocity potential $\varphi$ with carrier wave ${\mathrm{e}}^{{\mathrm{i}}(k_0 x - \omega_0 t)}$, $k_0$ and $\omega_0$ being, respectively, the wavenumber and frequency of the carrier wave. The equation for $A$ can be formulated in a frame moving with the group velocity $c_g = \omega_0/(2k_0)$ as follows. Define ${\varepsilon}$ as a small parameter, $a_0$ as a characteristic wave amplitude and rescale space, time and the envelope as $x \to k_0 x-c_g t$, $y \to k_0 y$, $t \to \omega_0 t$ and $A \to 