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

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