Preprint A258/2003
Spatial Analyticity of Solutions of a Nonlocal Perturbation of the KdV Equation
Borys Alvarez Samaniego
Keywords: Spatial Analyticity | Hilbert transform | KdV equation.
Let $\H$ 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 decrease with time. We also study the case when the initial data of the first problem is complex-valued.