| We prove short-time existence of smooth solutions of the asymptotic equation derived for the Burgers-Hilbert equation. This is an cubically nonlinear evolution equation that describes weakly nonlinear waves which are called constant-frequency waves. Following the work done by Bona, J., we prove the continuous dependence on the initial data, concluding the local well-posedness for the same wave equation.;We consider an initial value problem for a quadratically nonlinear inviscid Burgers-Hilbert equation that models the motion of vorticity discontinuities. We use a normal form transformation, which is implemented by means of a near-identity coordinate change of the independent spatial variable, to prove the existence of small, smooth solutions over cubically nonlinear time-scales. For vorticity discontinuities, this result means that there is a cubically nonlinear time-scale before the onset of filamentation.;Lastly, using the method of multiple scale, we derive an asymptotic equation for the motion of the boundary of a circular vortex patch, which generalizes the planar problem. |
