On the decay problem for the Zakharov and Klein–Gordon–Zakharov systems in one dimension

Abstract

We are interested in the long time asymptotic behaviour of solutions to the scalar Zakharov system $$\begin{aligned} \begin{array}{ll} i u_{t} + \Delta u = nu,\\ n_{tt} - \Delta n= \Delta |u|^2 \end{array} \end{aligned}$$and the Klein–Gordon–Zakharov system $$\begin{aligned} \begin{array}{ll} u_{tt} - \Delta u + u = - nu,\\ n_{tt} - \Delta n= \Delta |u|^2 \end{array} \end{aligned}$$in one dimension of space. For these two systems, we give two results proving decay of solutions for initial data in the energy space. The first result deals with decay over compact intervals assuming smallness and parity conditions (u odd). The second result proves decay in far field regions along curves for solutions whose growth can be dominated by an increasing \(C^1\) function. No smallness condition is needed to prove this last result for the Zakharov system. We argue relying on the use of suitable virial identities appropriate for the equations and follow the technics of [22, 24] and [33].