Unique Continuation for the Korteweg–de Vries Equation

Abstract

Unique continuation problems are considered for the Korteweg–de Vries (KdV) equation \[ u_t + uu_x + u_{xxx} = 0,\quad - \infty < x,\quad t < + \infty .\] By using the inverse scattering transform and some results from the Hardy function theory, it is proven that if $u \in L_{{\text{loc}}}^\infty (R,H^s (R))(s > \frac{3}{2})$ is a solution of the KdV equation, then it cannot have compact support at two different moments unless it vanishes identically. In addition, it is shown under certain conditions that if u is a solution of the KdV equation, then u must vanish everywhere if it vanishes on two horizontal half lines in the x-t space. This implies that the solution u must vanish everywhere if it vanishes on an open subset in the x-t space. As a consequence of the Miura transformation, the above results for the KdV equation are also true for the modified Korteweg–de Vries equation \[ v_t - 6v^2 v_x + v_{xxx} = 0,\quad - \infty < x,\quad t < + \infty .\]