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Filipe Oliveira

Publications and source records attributed to Filipe Oliveira.

18 recordsLinked to original sources
arXiv2023

Sharp local well-posedness and nonlinear smoothing for dispersive equations through frequency-restricted estimates

We consider the problem of establishing nonlinear smoothing as a general feature of nonlinear dispersive equations, i.e. the improved regularity of the integral term in Duhamel's formula, with respect to the initial data and the corresponding regularity of the linear evolution, and how this property relates to local well-posedness. In a first step, we show how the problem generally reduces to the derivation of specific frequency-restricted estimates, which are multiplier estimates in the spatial frequency alone. Then, using a precise methodology, we prove these estimates for the specific cases of the modified Zakharov-Kuznetsov equation, the cubic and quintic nonlinear Schrödinger equation and the quartic Korteweg-de Vries equation.

math.AP↗
arXiv2022

Quadratic Symmetric Polynomials and an analogue of the Davenport Constant

In this paper, we define the constant $D(φ, p)$, an analogue for the Davenport constant, for sequences on the finite field $\mathbb{F}_p$, defined via quadratic symmetric polynomials. Next, we state a series of results presenting either the exact value of $D(φ, p)$, or lower and upper bounds for this constant.

math.NT↗
arXiv2020

On the nonlinear Schrödinger equation in spaces of infinite mass and low regularity

We study the nonlinear Schrödinger equation with initial data in $\mathcal{Z}^s_p(\mathbb{R}^d)=\dot{H}^s(\mathbb{R}^d)\cap L^p(\mathbb{R}^d)$, where $0<s<\min\{d/2,1\}$ and $2<p<2d/(d-2s)$. After showing that the linear Schrödinger group is well-defined in this space, we prove local well-posedness in the whole range of parameters $s$ and $p$. The precise properties of the solution depend on the relation between the power of the nonlinearity and the integrability $p$. Finally, we present a global existence result for the defocusing cubic equation in dimension three for initial data with infinite mass and energy, using a variant of the Fourier truncation method.

math.AP↗
arXiv2019

Mass-transfer instability of ground-states for Hamiltonian Schrödinger systems

We study generic semilinear Schrödinger systems which may be written in Hamiltonian form. In the presence of a single gauge invariance, the components of a solution may exchange mass between them while preserving the total mass. We exploit this feature to unravel new orbital instability results for ground-states. More precisely, we first derive a general instability criterion and then apply it to some well-known models arising in several physical contexts. In particular, this mass-transfer instability allows us to exhibit $L^2$-subcritical unstable ground-states.

math.AP↗
arXiv2018

Weighted EGZ Constant for p-groups of rank 2

Let $G$ be a finite abelian group of exponent $n$, written additively, and let $A$ be a subset of $\mathbb{Z}$. The constant $s_A(G)$ is defined as the smallest integer $\ell$ such that any sequence over $G$ of length at least $\ell$ has an $A$-weighted zero-sum of length $n$ and $η_A(G)$ defined as the smallest integer $\ell$ such that any sequence over $G$ of length at least $\ell$ has an $A$-weighted zero-sum of length at most $n$. Here we prove that, for $α\geq β$, and $A=\left\{x\in\mathbb{N}\; : \; 1 \le a \le p^α \; \mbox{ and }\; \gcd(a, p) = 1\right \}$, we have $s_{A}(\mathbb{Z}_{p^α}\oplus \mathbb{Z}_{p^β}) = η_A(\mathbb{Z}_{p^α}\oplus \mathbb{Z}_{p^β}) + p^α-1 = p^α + α+β$ and classify all the extremal $A$-weighted zero-sum free sequences.

math.NT↗
arXiv2018

On a Schrödinger system arizing in nonlinear optics

We study the nonlinear Schrödinger system \[ \begin{cases} \displaystyle iu_t+Δu-u+(\frac{1}{9}|u|^2+2|w|^2)u+\frac{1}{3}\overline{u}^2w=0,\\ i\displaystyle σw_t+Δw-μw+(9|w|^2+2|u|^2)w+\frac{1}{9}u^3=0, \end{cases} \] for $(x,t)\in \mathbb{R}^n\times\mathbb{R}$, $1\leq n\leq 3$ and $σ,μ>0$. This system models the interaction between an optical beam and its third harmonic in a material with Kerr-type nonlinear response. We prove the existence of ground state solutions, analyse its stability, and establish local and global well-posedness results as well as several criteria for blow-up.

math.AP↗
arXiv2018

Typed Linear Algebra for Efficient Analytical Querying

This paper uses typed linear algebra (LA) to represent data and perform analytical querying in a single, unified framework. The typed approach offers strong type checking (as in modern programming languages) and a diagrammatic way of expressing queries (paths in LA diagrams). A kernel of LA operators has been implemented so that paths extracted from LA diagrams can be executed. The approach is validated and evaluated taking TPC-H benchmark queries as reference. The performance of the LA-based approach is compared with popular database competitors (PostgreSQL and MySQL).

cs.DB↗
arXiv2018

On a coupled system of a Ginzburg-Landau equation with a quasilinear conservation law

We study the Cauchy problem for a coupled system of a complex Ginzburg-Landau equation with a quasilinear conservation law $$ \left\{\begin{array}{rlll} e^{-iθ}u_t&=&u_{xx}-|u|^2u-αg(v)u& v_t+(f(v))_x&=&α(g'(v)|u|^2)_x& \end{array}\right. \qquad x\in\mathbb{R},\, t \geq 0, $$ which can describe the interaction between a laser beam and a fluid flow (see [Aranson, Kramer, Rev. Med. Phys. 74 (2002)]). We prove the existence of a local in time strong solution for the associated Cauchy problem and, for a certain class of flux functions, the existence of global weak solutions. Furthermore we prove the existence of standing waves of the form $(u(t,x),v(t,x))=(U(x),V(x))$ in several cases.

math.AP↗
arXiv2017

Scattering theory for the Schrödinger-Debye System

We study the Schrödinger-Debye system over $\mathbb{R}^d$ iu_t+\frac 12Δu=uv,\quad μv_t+v=λ|u|^2 and establish the global existence and scattering of small solutions for initial data in several function spaces in dimensions $d=2,3,4$. Moreover, in dimension $d=1$, we prove a Hayashi-Naumkin modified scattering result.

math.AP↗
arXiv2017

On a nonlinear Schrödinger system arising in quadratic media

We consider the quadratic Schrödinger system $$iu_t+Δ_{γ_1}u+\overline{u}v=0$$ $$2iv_t+Δ_{γ_2}v-βv+\frac 12 u^2=0,$$ where $t\in\mathbf{R},\,x\in \mathbf{R}^d\times \mathbf{R}$, in dimensions $1\leq d\leq 4$ and for $γ_1,γ_2>0$, the so-called elliptic-elliptic case. We show the formation of singularities and blow-up in the $L^2$-(super)critical case. Furthermore, we derive several stability results concerning the ground state solutions of this system.

math.AP↗
arXiv2015

On a quasilinear non-local Benney System

We study the quasilinear non-local Benney System $$\left\{\begin{array}{llll} iu_t+u_{xx}=|u|^2u+buv\\ v_t+a(\int_{\mathbf{R}^+}v^2dx)v_x=-b(|u|^2)_x,\quad (x,t)\in\mathbf{R}^+\times [0,T],\, T>0. \end{array}\right.$$ We establish the existence and uniqueness of strong local solutions to the corresponding Cauchy problem and show, under certain conditions, the blow-up of such solutions in finite time. Furthermore, we prove the existence of global weak solutions and exhibit bound-state solutions to this system.

math.AP↗
arXiv2015

Semitrivial vs. fully nontrivial ground states in cooperative cubic Schrödinger systems with $d\ge3$ equations

In this work we consider the weakly coupled Schrödinger cubic system \[ \begin{cases} \displaystyle -Δu_i+λ_i u_i= μ_i u_i^{3}+ u_i\sum_{j\neq i}b_{ij} u_j^2 \\ u_i\in H^1(\mathbb{R}^N;\mathbb{R}), \quad i=1,\ldots, d, \end{cases} \] where $1\leq N\leq 3$, $λ_i,μ_i >0$ and $b_{ij}=b_{ji}>0$ for $i\neq j$. This system admits semitrivial solutions, that is solutions $\mathbf{u}=(u_1,\ldots, u_d)$ with null components. We provide optimal qualitative conditions on the parameters $λ_i,μ_i$ and $b_{ij}$ under which the ground state solutions have all components nontrivial, or, conversely, are semitrivial. This question had been clarified only in the $d=2$ equations case. For $d\geq 3$ equations, prior to the present paper, only very restrictive results were known, namely when the above system was a small perturbation of the super-symmetrical case $λ_i\equiv λ$ and $b_{ij}\equiv b$. We treat the general case, uncovering in particular a much more complex and richer structure with respect to the $d=2$ case.

math.AP↗
arXiv2015

Ground States for a nonlinear Schrödinger system with sublinear coupling terms

We study the existence of ground states for the coupled Schrödinger system \begin{equation} \left\{\begin{array}{lll} \displaystyle -Δu_i+λ_i u_i= μ_i |u_i|^{2q-2}u_i+\sum_{j\neq i}b_{ij} |u_j|^q|u_i|^{q-2}u_i \\ u_i\in H^1(\mathbb{R}^n), \quad i=1,\ldots, d, \end{array}\right. \end{equation} $n\geq 1$, for $λ_i,μ_i >0$, $b_{ij}=b_{ji}>0$ (the so-called "symmetric attractive case") and $1<q<n/(n-2)^+$. We prove the existence of a nonnegative ground state $(u_1^*,\ldots,u_d^*)$ with $u_i^*$ radially decreasing. Moreover we show that, for $1<q<2$, such ground states are positive in all dimensions and for all values of the parameters.

math.AP↗
arXiv2015

Ground states for a coupled nonlinear Schrödinger system

We study the existence of ground states for the coupled Schrödinger system \begin{equation} \label{ellipticabstract} \left\{ \begin{array}{llll} -Δu+u&=&|u|^{2q-2}u+b|v|^q|u|^{q-2}u\\ -Δv+ω^2v&=&|v|^{2q-2}v+b|u|^q|v|^{q-2}v \end{array}\right. \end{equation} in $\mathbf{R}^n$, for $ω\geq 1$, $b>0$ (the so-called "attractive case") and $q>1$ ($q<\frac n{n-2}$ if $n\geq 3$). We improve for several ranges of $(q,n,ω)$ the known results concerning the existence of positive ground state solutions with non-trivial components. In particular, we prove that for $1 2$.

math.AP↗
arXiv2014

A note on the existence of traveling-wave solutions to a Boussinesq system

We obtain a one-parameter family $$(u_μ(x,t),η_μ(x,t))_{μ\geq μ_0}=(ϕ_μ(x-ω_μ t),ψ_μ(x-ω_μ t))_{μ\geq μ_0}$$ of traveling-wave solutions to the Boussinesq system $$u_t+η_x+uu_x+cη_{xxx}=0,η_t+u_x+(ηu)_x+au_{xxx}=0$$ in the case $a,c<0$, with non-null speeds $ω_μ$ arbitrarily close to $0$ ($ω_μ\xrightarrow[μ\to+\infty]{} 0$). We show that the $L^2$-size of such traveling-waves satisfies the uniform (in $μ$) estimate $\|ϕ_μ\|_2^2+\|ψ_μ\|_2^2\leq C\sqrt{|a|+|c|},$ where $C$ is a positive constant. Furthermore, $ϕ_μ$ and $-ψ_μ$ are smooth, non-negative, radially decreasing functions which decay exponentially at infinity.

math.AP↗
arXiv2014

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

We prove the existence of solitary wave solutions to the quasilinear Benney system $$iu_{t}+u_{xx}=a|u|^pu+uv,\quad v_t+f(v)_x=(|u|^2)_x$$ where $f(v)=-γv^3$, $-1 0$. We establish, in particular, the existence of travelling waves with speed arbitrary large if $p<0$ and arbitrary close to $0$ if $p>\frac 23$. We also show the existence of standing waves in the case $-1<p\leq \frac 23$, with compact support if $-1<p<0$.\\ Finally, we obtain, under certain conditions, the linearized stability of such solutions.

math.AP↗
arXiv2012

Local and Global Well-Posedness for the Critical Schrodinger-Debye System

We establish local well-posedness results for the Initial Value Problem associated to the Schrödinger-Debye system in dimensions $N=2, 3$ for data in $H^s\times H^{\ell}$, with $s$ and $\ell$ satisfying $\max \{0, s-1\} \le \ell \le \min\{2s, s+1\}$. In particular, these include the energy space $H^1\times L^2$. Our results improve the previous ones obtained in \cite{Bidegaray1}, \cite{Bidegaray2} and \cite{Corcho-Linares}. Moreover, in the critical case (N=2) and for initial data in $H^1\times L^2$, we prove that solutions exist for all times, thus providing a negative answer to the open problem mentioned in \cite{Fibich-Papanicolau} concerning the formation of singularities for these solutions.

math.AP↗