arXiv subjects

Renata O. Figueira

Publications and source records attributed to Renata O. Figueira.

4 recordsLinked to original sources

arXiv2026

Evolution of the radius of analyticity for mKdV-type equations

In this paper, we obtain new lower bounds for the evolution of the radius of analyticity of solutions to two initial value problems (IVPs) with initial data belonging to the class of analytic functions $H^{σ,s}(\mathbb{R})$ defined via a hyperbolic cosine weight. First, we consider the IVP for the modified Korteweg-de Vries (mKdV) equation. For this problem, we prove that the evolution of the radius of analyticity $σ(T)$ of the solution admits an algebraic lower bound $cT^{-\frac 12}$ for some $c>0$ and given arbitrarily large $T>0$. Next, we analyze the IVP for the mKdV equation with generalized dispersion (mKdVm) and a damping term. For this problem, we guarantee the local well-posedness in $H^{σ,s}(\mathbb{R})$ and demonstrate that the local solution can be extended globally in time and admits constant lower bounds for the radius of analyticity $σ(t)$ as time goes to infinity. The outcome of this paper concerning the mKdV equation represents an improvement on that achieved by the authors' previous work in [R. O. Figueira and M. Panthee, New lower bounds for the radius of analyticity for the mKdV equation and a system of mKdV-type equations, J. Evol. Equ. 24 No. 42 (2024)]. As far as we know, the results for the mKdVm with damping are new.

math.AP↗

arXiv2024

Improved lower bound for the radius of analyticity for the modified KdV equation

We investigate the initial value problem (IVP) associated to the modified Korteweg-de Vries equation (mKdV) in the defocusing scenario: \begin{equation*} \left\{\begin{array}{l} \partial_t u+ \partial_x^3u-u^2\partial_x(u) = 0, \quad x,t\in\mathbb{R}, \\ u(x,0) = u_0(x), \end{array}\right. \end{equation*} where $u$ is a real valued function and the initial data $u_0$ is analytic on $\mathbb{R}$ and has uniform radius of analyticity $σ_0$ in the spatial variable. It is well-known that the solution $u$ preserves its analyticity with the same radius $σ_0$ for at least some time span $0<T_0\le 1$. This local result was obtained in [Nonlinear Differ. Equ. Appl. (2024), 31--68] by proving a trilinear estimate in the Gevrey spaces $G^{σ, s}$, $s\geq \frac14$. Global in time behaviour of the solution and algebraic lower bound of the evolution of the radius of analyticity was also studied in authors' earlier works in [Nonlinear Differ. Equ. Appl. (2024), 31--68] and [J. Evol. Equ. 24 No. 42 (2024)] by constructing almost conserved quantities in the classical Gevrey space with $H^1$ and $H^2$ levels of Sobolev regularities. The present study aims to construct a new almost conservation law in the Gevrey space defined with a weight function $\cosh(σ|ξ|)$ and use it demonstrate that the local solution $u$ extends globally in time, and the radius of spatial analyticity is bounded from below by $c T^{-\frac{1}{2}}$, for any time $T\geq T_0$. The outcome of this paper represents an improvement on the one achieved by the authors' previous work in [J. Evol. Equ. 24 No. 42 (2024)].

math.AP↗

arXiv2024

Decay of the radius of spatial analyticity for the modified KdV equation and the nonlinear Schrödinger equation with third order dispersion

We consider the initial value problems (IVPs) for the modified Korteweg-de Vries (mKdV) equation \begin{equation*} \label{mKdV} \left\{\begin{array}{l} \partial_t u+ \partial_x^3u+μu^2\partial_xu =0, \quad x\in\mathbb{R},\; t\in \mathbb{R} , \\ u(x,0) = u_0(x), \end{array}\right. \end{equation*} where $u$ is a real valued function and $μ=\pm 1$, and the cubic nonlinear Schrödinger equation with third order dispersion (tNLS equation in short) \begin{equation*} \label{t-NLS} \left\{\begin{array}{l} \partial_t v+iα\partial_x^2v+β\partial_x^3v+iγ|v|^2v = 0, \quad x\in\mathbb{R},\; t\in\mathbb{R} , \\ v(x,0) = v_0(x), \end{array}\right. \end{equation*} where $α, β$ and $γ$ are real constants and $v$ is a complex valued function. In both problems, the initial data $u_0$ and $v_0$ are analytic on $\mathbb{R}$ and have uniform radius of analyticity $σ_0$ in the space variable. We prove that the both IVPs are locally well-posed for such data by establishing an analytic version of the trilinear estimates, and showed that the radius of spatial analyticity of the solution remains the same $σ_0$ till some lifespan $0 0$, for the tNLS equation. The result for the mKdV equation improves the one obtained in [ J. L. Bona, Z. Grujić and H. Kalisch, Algebraic lower bounds for the uniform radius of spatial analyticity for the generalized KdV equation, Ann Inst. H. Poincaré 22 (2005) 783--797] and, as far as we know, the result for the tNLS equation is the new one.

math.AP↗

arXiv2022

Evolution of the radius of analyticity for the generalized Benjamin equation

In this work we consider the initial value problem for the generalized Benjamin equation \begin{equation}\label{Benj-IVP} \begin{cases} \partial_t u-l\mathcal{H} \partial_x^2u-\partial_x^3u+u^p\partial_xu = 0, \quad x,\; t\in \mathbb{R};\;\;,\; p\geq 1, \\ u(x,0) = u_0(x), \end{cases} \end{equation} where $u=u(x,t)$ is a real valued function, $0<l<1$ and $\mathcal{H}$ is the Hilbert transform. This model was introduced by T. B. Benjamin (J. Fluid Mech. 245 (1992) 401--411) and describes unidirectional propagation of long waves in a two-fluid system where the lower fluid with greater density is infinitely deep and the interface is subject to capillarity. We prove that the local solution to the IVP associated with the generalized Benjamin equation for given data in the spaces of functions analytic on a strip around the real axis continue to be analytic without shrinking the width of the strip in time. We also study the evolution in time of the radius of spatial analyticity and show that it can decrease as the time advances. Finally, we present an algebraic lower bound on the possible rate of decrease in time of the uniform radius of spatial analyticity.

math.AP↗