Searcharxiv⌕ Search

arXiv subjects

Jorge Mejía

Publications and source records attributed to Jorge Mejía.

12 recordsLinked to original sources
arXiv2025

A note on the uniqueness properties of solutions for the Schrödinger-Korteweg de Vries system

In this work we prove that if $(u_i,v_i)$, $i=1,2$, are smooth enough solutions of the coupled Schrödinger-Korteweg-de Vries system \begin{align*} \left. \begin{array}{rl} i u_t+\partial_x^2 u &\hspace{-2mm}=βuv - |u|^2 u,\\ \partial_t v + \partial_x^3 v &\hspace{-2mm}=γ\partial_x |u|^2-\frac12\partial_x (v^2) \end{array} \right\} \end{align*} with appropriate decay at infinity such that at two different times $t_0=0$ and $t_1=1$ satisfy that $$u_1(0)-u_2(0),u_1(1)-u_2(1),v_1(0)-v_2(0),v_1(1)-v_2(1)\in H^1(e^{ax^{2}}dx),$$ for $a>0$ big enough, then $u_1=u_2$ and $v_1=v_2$. (Let us recall that $f\in H^1(e^{ax^{2}} dx)$ iff $f\in L^2(e^{ax^{2}}dx)$ and $\partial_x f\in L^2(e^{ax^{2}}dx)$).

math.AP↗
arXiv2024

Dispersive blow-up for a coupled Schrödinger-fifth order KdV system

In this work we establish a dispersive blow-up result for the initial value problem (IVP) for the coupled Schrödinger-fifth order Korteweg-de Vries system \begin{align*} \left. \begin{array}{rl} i u_t+\partial_x^2 u &\hspace{-2mm}=αuv + γ|u|^2 u, \quad x\in\mathbb R,\quad t\in\mathbb R,\\ \partial_t v + \partial_x^5 v + \partial_x v^2&\hspace{-2mm}=ε\partial_x |u|^2, \quad x\in\mathbb R,\quad t\in\mathbb R,\\ u(x,0)&\hspace{-2mm}= u_0(x), \quad v(x,0)=v_0(x). \end{array} \right\} \end{align*} To achieve this, we prove a local well-posedness result in Bourgain spaces of the type $X^{s+β,b}\times Y^{s,b}$, along with a regularity property for the nonlinear part of the IVP solutions. This property enables the construction of initial data that leads to the dispersive blow-up phenomenon.

math.AP↗
arXiv2024

Dispersive blow-up for the fifth order Korteweg-de Vries equation on the line

In this work we establish a dispersive blow-up result for the initial value problem (IVP) for the fifth order Korteweg-de Vries equation \begin{align*} \left. \begin{array}{rlr} u_t+\partial_x^5 u+u\partial_x u&\hspace{-2mm}=0,&\quad x\in\mathbb R,\; t>0,\\ u(x,0)&\hspace{-2mm}=u_0(x),& \end{array} \right\} \end{align*} To achieve this, we prove a local well-posedness result in Bourgain spaces of the type $X^{s,b}$ for appropriate values of $s$ and $b$, along with a regularity property for the nonlinear part of that solution. This property enables the construction of initial data that leads to the dispersive blow-up phenomenon.

math.AP↗
arXiv2024

Local well-posedness and regularity properties for an initial-boundary value problem associated to the fifth order Korteweg-de Vries equation

In this work we prove that the initial-boundary value problem (IBVP) for the fifth order Korteweg-de Vries equation \begin{align*} \left. \begin{array}{rlr} u_t+\partial_x^5 u+u\partial_x u&\hspace{-2mm}=0,&\quad x\in\mathbb R^+,\; t\in\mathbb R^+,\\ u(x,0)&\hspace{-2mm}=g(x),&\\ u(0,t)=h_1(t),\, \partial_x u(0,t)&\hspace{-2mm}=h_2(t),\,\partial_x^2 u(0,t)=h_3(t), \end{array} \right\} \end{align*} is locally well posed, when the data $g$, $h_1$, $h_2$, $h_3$ are taken in such a way that $g\in H^s(\mathbb R_x^+)$, and $h_{j+1}\in H^{\frac{s+2-j}5}(\mathbb R_t^+)$, $j=0,1,2$, $s\in [0,\frac{11}4)\setminus \{\frac12,\frac32,\frac52\}$, and satisfy the following compatibility conditions: \begin{align*} g(0)=h_1(0) \text{ if } \frac12<s<\frac32;\\ g(0)=h_1(0),\; g'(0)=h_2(0) \text{ if } \frac32<s<\frac52;\\ g(0)=h_1(0), \; g'(0)=h_2(0),\; g''(0)=h_3(0) \text{ if } \frac52<s<\frac{11}4. \end{align*} Besides, we prove that the nonlinear part of the solution is smoother than the initial datum $g$.

math.AP↗
arXiv2019

Periodic Cauchy Problem for one Two-dimensional Generalization of the Benjamin-Ono Equation in Sobolev Spaces of Low Regularity

In this work we prove that the initial value problem (IVP) associated to the two-dimensional Benjamin-Ono equation $$\left. \begin{array}{rl} u_t+\mathcal H Δu +uu_x &\hspace{-2mm}=0,\qquad\qquad (x,y)\in\mathbb T^2,\; t\in\mathbb R,\\ u(x,y,0)&\hspace{-2mm}=u_0(x,y), \end{array} \right\}\,,$$ where $\mathcal H$ denotes the Hilbert transform with respect to the variable $x$ and $Δ$ is the Laplacian with respect to the spatial variables $x$ and $y$, is locally well-posed in the periodic Sobolev space $H^s(\mathbb T^2)$, with $s>7/4$.

math.AP↗
arXiv2017

The Cauchy problem for a family of two-dimensional fractional Benjamin-Ono equations

In this work we prove that the initial value problem (IVP) associated to the fractional two-dimensional Benjamin-Ono equation $$\left. \begin{array}{rl} u_t+D_x^α u_x +\mathcal Hu_{yy} +uu_x &=0,\qquad\qquad (x,y)\in\mathbb R^2,\; t\in\mathbb R, u(x,y,0)&=u_0(x,y), \end{array} \right\}\,,$$ where $0<α\leq1$, $D_x^α$ denotes the operator defined through the Fourier transform by \begin{align} (D_x^αf)\widehat{\;}(ξ,η):=|ξ|^α\widehat{f}(ξ,η)\,, \end{align} and $\mathcal H$ denotes the Hilbert transform with respect to the variable $x$, is locally well posed in the Sobolev space $H^s(\mathbb R^2)$ with $s>\dfrac32+\dfrac14(1-α)$.

math.AP↗
arXiv2016

A note on the Ostrovsky equation in weighted Sobolev spaces

In this work we consider the initial value problem (IVP) associated to the Ostrovsky equations $$\left. \begin{array}{rl} u_t+\partial_x^3 u\pm \partial_x^{-1}u +u \partial_x u &\hspace{-2mm}=0,\qquad\qquad x\in\mathbb R,\; t\in\mathbb R,\\ u(x,0)&\hspace{-2mm}=u_0(x). \end{array} \right\}$$ We study the well-posedness of the IVP in the weighted Sobolev spaces $$Z_{s,\frac{s}2}:=\{u\in H^s(\mathbb R):D_x^{-s} u\in L^2(\mathbb R)\}\cap L^2(|x|^s dx ),$$ with $\frac34<s\leq 1$.

math.AP↗
arXiv2014

The Zakharov-Kuznetsov equation in weighted Sobolev spaces

In this work we consider the initial value problem (IVP) associated to the two dimensional Zakharov-Kuznetsov equation $$\left. \begin{array}{rl} u_t+\partial_x^3 u+\partial_x \partial_y^2 u +u \partial_x u &\hspace{-2mm}=0,\qquad\qquad (x,y)\in\mathbb R^2,\; t\in\mathbb R,\\ u(x,y,0)&\hspace{-2mm}=u_0(x,y). \end{array} \right\}$$ We study the well-posedness of the IVP in the weighted Sobolev spaces $$H^s(\mathbb R^2) \cap L^2((1+x^2+y^2)^{r} dx dy),$$ with $s,r\in\mathbb R$.

math.AP↗
arXiv2013

The Cauchy problem for a fifth order KdV equation in weighted Sobolev spaces

In this work we study the initial value problem (IVP) for the fifth order KdV equations, \begin{align*} \partial_{t}u+\partial_{x}^{5}u+u^k\partial_{x}u=0,\text{} & \quad x,t\in \mathbb R, \quad k=1,2, \end{align*} in weighted Sobolev spaces $H^s(\mathbb R)\cap L^2(\langle x \rangle^{2r}dx)$. We prove local and global results. In the case $k=2$ we point out the relation between decay and regularity of the solution of the IVP.

math.AP↗
arXiv2004

Galactic foreground contribution to the BEAST CMB Anisotropy Maps

We report limits on the Galactic foreground emission contribution to the Background Emission Anisotropy Scanning Telescope (BEAST) Ka- and Q-band CMB anisotropy maps. We estimate the contribution from the cross-correlations between these maps and the foreground emission templates of an H$α$ map, a de-striped version of the Haslam et al. 408 MHz map, and a combined 100 $μ$m IRAS/DIRBE map. Our analysis samples the BEAST $\sim10^\circ$ declination band into 24 one-hour (RA) wide sectors with $\sim7900$ pixels each, where we calculate: (a) the linear correlation coefficient between the anisotropy maps and the templates; (b) the coupling constants between the specific intensity units of the templates and the antenna temperature at the BEAST frequencies and (c) the individual foreground contributions to the BEAST anisotropy maps. The peak sector contributions of the contaminants in the Ka-band are of 56.5% free-free with a coupling constant of $8.3\pm0.4$ $μ$K/R, and 67.4% dust with $45.0\pm2.0$ $μ$K/(MJy/sr). In the Q-band the corresponding values are of 64.4% free-free with $4.1\pm0.2$ $μ$K/R and 67.5% dust with $24.0\pm1.0$ $μ$K/(MJy/sr). Using a lower limit of 10% in the relative uncertainty of the coupling constants, we can constrain the sector contributions of each contaminant in both maps to $< 20$% in 21 (free-free), 19 (dust) and 22 (synchrotron) sectors. At this level, all these sectors are found outside of the $\mid$b$\mid = 14.6^\circ$ region. By performing the same correlation analysis as a function of Galactic scale height, we conclude that the region within $b=\pm17.5^{\circ}$ should be removed from the BEAST maps for CMB studies in order to keep individual Galactic contributions below $\sim 1$% of the map's rms.

astro-ph↗