On the Cauchy problem for the Zakharov-Schulman systems
Filipe Oliveira
Departamento de Matemática
Universidade Nova de Lisboa

Abstract

In this communication we will consider the Zakharov-Schulman (Z-S) system in dimension n≤3

iu_t+L₁u=|u|²u+uv
L₂v=L₃(|u|²),

where L_j denote spatial differential operators of order 2, with L₁ non-degenerated and L₂ elliptic.

This system is a model for the interaction of small-amplitude high-frequency waves with low-frequency acoustic-type waves.

In dimension n=2, by choosing L₃=∂_xx, (Z-S) reduces to the well-known Davey-Stewartson system, which describes the propagation of water waves in one main direction with slow transverse modulation.

We will derive the local well-posedness of (Z-S) in Sobolev spaces H^s(Rⁿ) below the energy space. The main tools are Strichartz-type estimates for the group {e^{it L₁}}_t∈R and L^p commutator estimates for fractional derivatives D^s, 0‹s‹1.

(Joint work with Jorge D. Silva and Mahendra Panthee.)

Jornada SPM/CIM/CMAT - "Dia das Equações"