On the optimal conditions for the focusing of giant sea waves

The dynamics of a wave packet on a two-dimensional sea surface, which is described by the nonlinear Schrödinger equation $2i\psi_t+\psi_{xx}-\psi_{yy}+|\psi|^2\psi=0$, has been analyzed within the Gaussian variational ansatz in application to the problem of the formation of rogue waves. The longitudinal $(X(t))$ and transverse $(Y(t))$ sizes of the packet are described by a system of differential equations: $\ddot X=1/X^3-N/(X^2Y)$ and $Y=1/Y^3+N/(Y^2X)$, where the parameter $N$ is proportional to the integral of motion $\int|\psi|^2dx dy$. This system is interated in quadratures at an arbitrary $N$ value, which makes it possible to understand the linear and nonlinear regimes of the focusing of a wavepacket and to formulate the optimal initial conditions under which the amplitude of the wave at the maximum is much larger than that in the linear case.