Searcharxiv⌕ Search

arXiv subjects

Eddye Bustamante

Publications and source records attributed to Eddye Bustamante.

11 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↗