Dynamics of 2D fluid in bounded domain via conformal variables

In the present work, we compute numerical solutions of an integro-differential equation for traveling waves on the boundary of a 2D blob of an ideal fluid in the presence of surface tension. We find that solutions with multiple lobes tend to approach Crapper capillary waves in the limit of many lobes. Solutions with a few lobes become elongated as they become more nonlinear. It is unclear whether there is a limiting solution for small number of lobes, and what are its properties. Solutions are found from solving a nonlinear pseudodifferential equation by means of the Newton conjugate-residual method. We use Fourier basis to approximate the solution with the number of Fourier modes up to $N=65536$.