| Summary: | Let $\mathcal{H}$ denote the Hilbert transform and $\eta \ge 0$. We show that if the initial data of the following problems
$ u_t + u u_x + u_{xxx} + \eta(\mathcal{H} u_x + \mathcal{H} u_{xxx}) = 0, \,
u(\cdot , 0) = \phi (\cdot)$ and
$ v_t + \frac{1}{2} (v_x)^2 + v_{xxx} + \eta(\mathcal{H} v_x + \mathcal{H} v_{xxx}) = 0, \,
v(\cdot , 0) = \psi (\cdot)$
has an analytic continuation to a strip containing the real axis, then the solution has the same property, although the width of the strip might diminish with time. When $\eta>0$ and the initial data is complex-valued we prove local well-posedness of the two problems above in spaces of analytic functions, which implies the constancy over time of the radius of the strip of analyticity in the complex plane around the real axis.
|
|---|