Filipe Oliveira

Stability of the solitons for the one-dimensional Zakharov–Rubenchik equation

Abstract We prove the global well-posedness of the one-dimensional Zakharov–Rubenchik equation i B t +ωB xx −k(u− 1 2 vρ+q|B| 2 )B=0, θρ t +(u−vρ) x =−k|B| 2 x , θu t +(βρ−vu) x = 1 2 kv|B| 2 x in the space H 2 ( R )×H 1 ( R )×H 1 ( R ) . We also prove the existence and the orbital stability of solitary wave solutions to the above equations.

Adiabatic limit of the zakharov-rubenchik equation

The Zakharov-Rubenchik system { i ∂ T B + ω ∂ XX B − k ( u − υ 2 ρ + q | B | 2 ) B = 0 , ∈ ∂ T ρ + ∂ X ( u − υ ρ ) = − k ∂ X | B | 2 , ∈ ∂ T u + ∂ X ( β ρ − υ u ) = k 2 υ ∂ X | B | 2 appears in the context of Alfven waves propagating in a magnetized plasma. This system “contains” the well-known Zakharov equation and the Benney equation for the interaction of high and low frequency waves. We prove the pointwise convergence of the magnetic field B to a solution of the nonlinear Schrodinger equation, in the adiabatic limit ∈ → 0.

Ground States for a Nonlinear Schrödinger System with Sublinear Coupling Terms

Abstract We study the existence of ground states for the coupled Schrödinger system { - Δ ⁢ u i + λ i ⁢ u i = μ i ⁢ | u i | 2 ⁢ q - 2 ⁢ u i + ∑ j ≠ i b i ⁢ j ⁢ | u j | q ⁢ | u i | q - 2 ⁢ u i , u i ∈ H 1 ⁢ ( ℝ n ) , i = 1 , … , d ,

Existence and linearized stability of solitary waves for a quasilinear Benney system

\left\{\begin{aligned} &\displaystyle-\Delta u_{i}+\lambda_{i}u_{i}=\mu_{i}|u_% {i}|^{2q-2}u_{i}+\sum_{j\neq i}b_{ij}|u_{j}|^{q}|u_{i}|^{q-2}u_{i},\\ &\displaystyle u_{i}\in H^{1}(\mathbb{R}^{n}),\quad i=1,\ldots,d,\end{aligned}\right.

Scattering theory for the Schrödinger–Debye system

n ≥ 1

Structural and vibrational properties of SnxGe1-x: Modeling and experiments

{n\geq 1}

On a quasilinear nonlocal Benney system

, for λ i , μ i > 0

An H-Theorem for Chemically Reacting Gases

{\lambda_{i},\mu_{i}>0}

A class on non-local linear operators for vorticity waves

, b i ⁢ j = b j ⁢ i > 0

Existence of local strong solutions for a quasilinear Benney system

{b_{ij}=b_{ji}>0}