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

Abstract

We are interested in the long time asymptotic behavoir of solutions to the scalar Zakharov system iut + ∆u = nu, ntt − ∆n = ∆|u| 2 and the Klein-Gordon Zakharov systemin 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 asuming 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 appropiate for the equations and follow the technics of [20, 23] and [32].

On long-time behavior of solutions of the Zakharov–Rubenchik/Benney–Roskes system

“…Remark 1.4. One important difference between the results above and those in [11,12] is that, in both of those works, the equations under study preserve the oddness of the initial data (for the Schrödinger component ψ), while system (1.1) does not. Hence, the analysis presented there assuming parity conditions on the initial data cannot be applied to system (1.1).…”

“…Note that none of the above theorems require any smallness assumption in terms of the initial data ψ 0 H 1 1. Moreover, they do not require any parity assumption either (as their counterparts founded in [11,12]), nor any extra decay hypotheses in terms of weighted Sobolev norms, such as xψ L 2 1 for example.…”

“…Finally, it is worth mentioning that the techniques involved in the proof of theorems 1.1 and 1.2 have already been used before in some other contexts. We refer to [11] for the use of some of these ideas in context of the one-dimensional Schrödinger equation, and to [12] for scalar Zakharov system (as well as the Klein-Gordon Zakharov system). On the other hand, for other type of systems that have served us for motivations we refer to [4,6,14].…”