On the support of solutions to the NLS-KdV system. (English) Zbl 1265.35305

The paper investigates support properties of smooth solutions of a coupled NLS-KdV system in one spatial dimension which describes the interaction of long and short waves. The theorem proved is a type of a unique continuation result. In detail, “if for a sufficiently smooth solution \((u,v)\) there exist \(a,b\in \mathbb {R}\) with \(\text{ supp }u(t_j)\subseteq (a,\infty )\) (or \((-\infty ,a)\)) and \(\text{ supp }v(t_j)\subseteq (b,\infty )\) (or \((-\infty ,b)\)) for \(j=1,2\) (\(t_1\neq t_2\)), then \(u\equiv v\equiv 0\)”. The smoothness condition is \(C([0,a]\: H^3(\mathbb {R})\times H^4(\mathbb {R}))\cap C^1([0,a]\: L^2(\mathbb {R})\times H^1(\mathbb {R}))\). The proof uses analogous results for a scalar generalized KdV due to C. E. Kenig, G. Ponce and L. Vega [Ann. I. H. Poincare Anal. Non Lineaire 19, 191–208 (2002; Zbl 1001.35106)] and for the scalar linear Schrödinger equation with a potential satisfying a decay property also due to C. E. Kenig, G. Ponce and L. Vega [Comm. Pure Appl. Math. 56, 1247–1262 (2002; Zbl 1041.35072)].


35Q53 KdV equations (Korteweg-de Vries equations)
35Q55 NLS equations (nonlinear Schrödinger equations)
35B60 Continuation and prolongation of solutions to PDEs
35G25 Initial value problems for nonlinear higher-order PDEs