# Davey-Stewartson solution

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.