arXiv subjects

Mário Figueira

Publications and source records attributed to Mário Figueira.

12 recordsLinked to original sources

arXiv2023

A note on bifurcations from eigenvalues of the Dirichlet-Laplacian with arbitrary multiplicity

In this short note, we consider the elliptic problem $$ λϕ+ Δϕ= η|ϕ|^σϕ,\quad ϕ\big|_{\partial Ω}=0,\quad λ, η\in \mathbb{C}, $$ on a smooth domain $Ω\subset \mathbb{R}^N$, $N\ge 1$. The presence of complex coefficients, motivated by the study of complex Ginzburg-Landau equations, breaks down the variational structure of the equation. We study the existence of nontrivial solutions as bifurcations from the trivial solution. More precisely, we characterize the bifurcation branches starting from eigenvalues of the Dirichlet-Laplacian of arbitrary multiplicity. This allows us to discuss the nature of such bifurcations in some specific cases. We conclude with the stability analysis of these branches under the complex Ginzburg-Landau flow.

math.AP↗

arXiv2020

A generalized Complex Ginzburg-Landau Equation: global existence and stability results

We consider the complex Ginzburg-Landau equation with two pure-power nonlinearities and a damping term. After proving a general global existence result, we focus on the existence and stability of several periodic orbits, namely the trivial equilibrium, bound-states and solutions independent of the spatial variable. In particular, we construct bound-states either explicitly in the real line or through a bifurcation argument for a double eigenvalue of the Dirichlet-Laplace operator on bounded domains.

math.AP↗

arXiv2018

Some stability results for the complex Ginzburg-Landau equation

Using some classical methods of dynamical systems, stability results and asymptotic decay of strong solutions for the complex Ginzburg-Landau equation (CGL), $$ \partial_t u = (a + iα) Δu - (b + i β) |u|^σu + k u, \,\, t > 0,\,\, x\in Ω, $$ with $a>0, α, b, β, k \in \mathbb{R}$, are obtained. Moreover, we show the existence of bound-states under certain conditions on the parameters and on the domain. We conclude with the proof of asymptotic stability of these bound-states when $Ω=\mathbb{R}$ and $-k$ large enough.

math.AP↗

arXiv2016

Some $L^\infty$ solutions of the hyperbolic nonlinear Schrödinger equation and their stability

Consider the hyperbolic nonlinear Schrödinger equation (HNLS) over $\mathbb{R}^d$ $$ iu_t + u_{xx} - Δ_{\textbf{y}} u + λ|u|^σu=0. $$ We deduce the conservation laws associated with (HNLS) and observe the lack of information given by the conserved quantities. We build several classes of particular solutions, including \textit{spatial plane waves} and \textit{spatial standing waves}, which never lie in $H^1$. Motivated by this, we build suitable functional spaces that include both $H^1$ solutions and these particular classes, and prove local well-posedness on these spaces. Moreover, we prove a stability result for both spatial plane waves and spatial standing waves with respect to small $H^1$ perturbations.

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arXiv2016

Existence and stability of spatial plane waves for the incompressible Navier-Stokes in $\mathbb{R}^3$

We consider the three-dimensional incompressible Navier-Stokes equation on the whole space. We observe that this system admits a $L^\infty$ family of global spatial plane wave solutions, which are connected with the two-dimensional equation. We then proceed to prove local well-posedness over a space which includes $L^3(\mathbb{R}^3)$ and these solutions. Finally, we prove $L^3$-stability of spatial plane waves, with no condition on their size.

math.AP↗

arXiv2016

Spatial plane waves for the nonlinear Schrödinger equation: local existence and stability results

We consider the Cauchy problem for the nonlinear Schrödinger equation on $\mathbb{R}^2$, $iu_t + u_{xx} + u_{yy} + λ|u|^σu =0$, $λ\in \mathbb{R}$, $σ>0$. We introduce new functional spaces over which the initial value problem is well-posed. Their construction is based on \textit{spatial plane waves} (cf. arXiv:1510.08745). These spaces contain $H^1(\mathbb{R}^2)$ and do not lie within $L^2(\mathbb{R}^2)$. We prove several global well-posedness and stability results over these new spaces, including a new global well-posedness result of $H^1$ solutions with indefinitely large $H^1$ and $L^2$ norms. Some of these results are proved using a new functional transform, the \textit{plane wave transform}. We develop a suitable theory for this transform, prove several properties and solve classical linear PDE's with it, highlighting its wide range of application.

math.AP↗

arXiv2014

Supercritical blowup in coupled parity-time-symmetric nonlinear Schrödinger equations

We prove finite time supercritical blowup in a parity-time-symmetric system of the two coupled nonlinear Schrödinger (NLS) equations. One of the equations contains gain and the other one contains dissipation such that strengths of the gain and dissipation are equal. We address two cases: in the first model all nonlinear coefficients (i.e. the ones describing self-action and non-linear coupling) correspond to attractive (focusing) nonlinearities, and in the second case the NLS equation with gain has attractive nonlinearity while the NLS equation with dissipation has repulsive (defocusing) nonlinearity and the nonlinear coupling is repulsive, as well. The proofs are based on the virial technique arguments. Several particular cases are also illustrated numerically.

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.

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arXiv2013

Finite-time blowup for a complex Ginzburg-Landau equation with linear driving

In this paper, we consider the complex Ginzburg--Landau equation $u_t = e^{iθ} [Δu + |u|^αu] + γu$ on ${\mathbb R}^N $, where $α>0$, $γ\in \R$ and $-π/2<θ<π/2$. By convexity arguments we prove that, under certain conditions on $α,θ,γ$, a class of solutions with negative initial energy blows up in finite time.

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arXiv2012

Convergence of a finite difference method for the KdV and modified KdV equations with $L^2$ data

We prove strong convergence of a semi-discrete finite difference method for the KdV and modified KdV equations. We extend existing results to non-smooth data (namely, in $L^2$), without size restrictions. Our approach uses a fourth order (in space) stabilization term and a special conservative discretization of the nonlinear term. Convergence follows from a smoothing effect and energy estimates. We illustrate our results with numerical experiments, including a numerical investigation of an open problem related to uniqueness posed by Y. Tsutsumi.

math.NA↗

arXiv2012

Convergence of a numerical scheme for a coupled Schrödinger--KdV system

We prove the convergence in a strong norm of a finite difference semi-discrete scheme approximating a coupled Schrödinger--KdV system on a bounded domain. This system models the interaction of short and long waves. Since the energy estimates available in the continuous case do not carry over to the discrete setting, we rely on a suitably truncated problem which we prove reduces to the original one. We present some numerical examples to illustrate our convergence result.

math.NA↗

arXiv2010

Convergence of numerical schemes for short wave long wave interaction equations

We consider the numerical approximation of a system of partial differential equations involving a nonlinear Schrödinger equation coupled with a hyperbolic conservation law. This system arises in models for the interaction of short and long waves. Using the compensated compactness method, we prove convergence of approximate solutions generated by semi-discrete finite volume type methods towards the unique entropy solution of the Cauchy problem. Some numerical examples are presented.

math.NA↗