The Davey-Stewartson solution
\begin{cases}
\mathrm{i}\psi_{0,\tau} - 3\eta\psi_{0, \xi\xi}+ \frac{\lambda}{\eta}\psi_{0, yy} - \left( \frac{1}{6\eta}|\psi_0|^2 + \eta\phi \right)\psi_0 = 0,\\~\\
3\eta^2 \varphi_{\xi\xi} + \lambda\phi_{yy} + |\psi_0|^2_{\xi\xi} = 0.
\end{cases}describes amplitude modulation of weakly two dimensional waves, for instance waves in shallow water (see e.g. ph/0601025 and references therein).
Semiclassical defocusing DS
The video above shows a solution for the defocusing DS II equation for the initial data
u(x,y,0)=exp(-x^2-y^2)
for 𝜀 = 0.1.
Semiclassical focusing DS
The video above shows a solution for the focusing DS II equation for the initial data
u(x,y,0)=\exp(-x^2-0.1y^2)
for 𝜀 = 0.1.
DS Solution (multiscales expansion)
For small initial data of order 𝜀 with rapid oscillations in x-direction of order 1/𝜀, a multi-scales expansion of the KP equation shows that the amplitude of the solution is asymptotically described by a solution of the Davey-Stewartson equation. The video shows the absolute value of the DS solution corresponding to the situation in Small dispersion KP.
See ph/0601025 for references and further details. For Semiclassical focusing DS, see also 1401.4745.