We derive two-dimensional (2D) envelope equations for models based on linearly coupled Zakharov-Kuznetsov (ZK) and Kadomtsev-Petviashvili (KP) equations which describe the interaction between long nonlinear waves in fluid flows. The asymptotic equations coincide with those describing the second-harmonic generation (SHG) in a 2D optical waveguide, that take into regard both the spatial diffraction and temporal dispersion. The system derived from the ZK and KP equations turn out to be, respectively, fully elliptic and fully hyperbolic with respect to the spatial coordinates. The recently found "light-bullet" solutions to the elliptic SHG equations in nonlinear optics suggest the possible existence of fully localized 2D solitons in the corresponding ZK coupled system. Direct numerical simulations, for which the initial conditions are taken as suggested by an analytical variational approximation (VA), completely corroborate the existence and stability of the 2D solitons, with a shape fairly close to that predicted by VA. We also demonstrate that quasi-1D solitons are (numerically) stable against 2D perturbations in both the ZK and KP systems. The results suggest that 2D parametric spatio-temporal solitons, which are hard to generate experimentally in nonlinear optics, can be generated in certain fluid flows.
|Number of pages
|Physics Letters, Section A: General, Atomic and Solid State Physics
|Published - 9 Nov 1998