arXiv subjects

María E. Martínez

Publications and source records attributed to María E. Martínez.

3 recordsLinked to original sources

arXiv2021

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

We study decay properties for solutions to the initial value problem associated with the one-dimensional Zakharov-Rubenchik/Benney-Roskes system. We prove time-integrability in growing compact intervals of size $t^{r}$, $r<2/3$, centered on some characteristic curves coming from the underlying transport equations associated with the ZR/BR system. Additionally, we prove decay to zero of the local energy-norm in so-called far-field regions. Our results are independent of the size of the initial data and do not require any parity condition.

math.AP↗

arXiv2020

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

We are interested in the long time asymptotic behavoir of solutions to the scalar Zakharov system \[ i u_{t} + Δu = nu,\] \[n_{tt} - Δn= Δ|u|^2\] and the Klein-Gordon Zakharov system \[ u_{tt} - Δu + u = - nu,\] \[ n_{tt} - Δn= Δ|u|^2\] 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 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 Kowalczyk-Martel-Muñoz and Muñoz-Ponce-Saut.

math.AP↗

arXiv2019

Decay of small odd solutions for long range Schrödinger and Hartree equations in one dimension

We consider the long time asymptotics of (not necessarily small) odd solutions to the nonlinear Schrödinger equation with semi-linear and nonlocal Hartree nonlinearities, in one dimension of space. We assume data in the energy space $H^1(\mathbb{R})$ only, and we prove decay to zero in compact regions of space as time tends to infinity. We give three different results where decay holds: semilinear NLS, NLS with a suitable potential, and defocusing Hartree. The proof is based on the use of suitable virial identities, in the spirit of nonlinear Klein-Gordon models as in Kowalczyk-Martel-Muñoz, and covers scattering sub, critical and supercritical (long range) nonlinearities. No spectral assumptions on the NLS with potential are needed.

math.AP↗