in nonlinear science [@Sulem1999]. For example, it arises in plasma physics [@Sen1978] and in normally dispersive optical waveguide arrays modeling [@Conti2003; @Lahini2007]. In the context of water waves the equation is the leading order model of the wave envelope evolution. It was derived for the first time by V. <span style="font-variant:small-caps;">Zakharov</span> (1968) [@Zakharov1968] and rediscovered later by several other authors [@Chu1970; @Hasimoto1972]. In one dimension (1-D), the NLS equation for unidirectional water waves is integrable [@Zakharov1972] and of focusing type. As a result, a steady or periodic balance of the cubic nonlinearities and wave dispersion can be attained, and this yields the formation of localized traveling waves (solitons) or homoclinic orbits to a plane wave (breathers). Analytical solutions for solitons follow via the inverse scattering transform [@Zakharov1972; @John; @Ablowitz1979; @Pelinovsky2001a] and breathers can be easily obtained via the Darboux transformation [@Osborne2010]. The two-dimensional (2-D) propagation of deep water narrowband waves is instead governed by the 2-D Hyperbolic NLS equation [@Ablowitz1979; @Yuen1982; @Sulem1999; @Pelinovsky2003]. In this case, energy can spread along the transversal direction to the main propagation. A consequence of this defocusing is that localized traveling waves cannot occur. Indeed, their non-existence was proved in ([@Ghidaglia1996a], see also [@Ghidaglia1993]). However, 