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## Uniquennes continuation for the Cauchy problem of nonlinear interactions of Schrondinger type

### Isnaldo Isaac Barbosa

This work concerns with the study of the Uniqueness Continuation problem of Cauchy's smooth solutions of a system of equations which arises in the context of nonlinear optics problems. The main results are prove that if $(u, v)$ are sufficiently smooth solutions to the Cauchy problem associated with a system of Schrodinger equations with quadratic nonlinear interactions such that $a,\ b\in \mathbb{R}$ with supp $u(t_j) \subset (a,\ \infty)$ e supp $v(t_j) \subset (b,\ \infty)$ for $j=1$ or $2$ ($t_1\neq t_2$) then $u\equiv v\equiv 0$.}

Basically, we will study the following mathematical model:

$$\begin{cases} i\partial_{t} u(x,t)+p\partial^2_{x} u(x,t) -\theta u(x,t)+ \bar{u}(x,t)v(x,t)=0, \ \ x\in \mathbb{R},\; t\ge 0,\\ i\sigma\partial_{t} v(x,t)+q\partial^2_{x} v(x,t) -\alpha v(x,t)+\tfrac{a}{2}u^2(x,t)=0,\ \ \\ u(x,0)=u_{0}(x),\quad v(x,0)=v_{0}(x), \end{cases}$$ where $u$ and $v$ are functions that assume complex values, $\alpha$, $\theta$ and $a: =1/\sigma$ are real numbers that play a physical parametrical role of the system, for which $\sigma >0$ e $p, \ q\ =\pm1$.

The main result of this work is the following:

\textbf{Theorem}

Let $(u,v)\in C([0,T]: H^3\times H^3)\cap C^1([0,T]:L^2(\mathbb{R})\times L^2(\mathbb{R}))$ be a strong solution of above system. Suppose that there exist $a,b \in \mathbb{R}$ with supp $u(0)$, supp $u(T)\subset (a,\infty)$ and supp $v(0)$, supp $v(T)\subset (b,\infty)$, then $u\equiv v\equiv 0.$ in $[0,T]\times \mathbb{R}$.

\tectbf{Sketch of the proof:}

The proof follows the ideas of the works , ,  and .

This result is part of a more complete work aimed at establishing uniqueness continuation results for systems of dispersive equations whose Schrodinger equation is one of the coupled equations.

For instance: Benney System

$$\left\{\begin{array}{ll}{i \partial_t u+\partial_x^2 u=u v+\beta|u|^{2} u,} & \quad t \in \mathbb{R}, \quad x \in \mathbb{R} \\ {v_{t}=\partial_x\left(|u|^{2}\right)} & {} \\ {u(x, 0)=u_{0}(x),} & {v(x, 0)=v_{0}(x)}\end{array}\right. ,$$

and Schrodinger-Debye System $$\left\{\begin{array}{l}{i \partial_t u+\frac{1}{2} \partial_{x}^{2} u=u v, \quad t \in \mathbb{R}, \quad x \in \mathbb{R}} \\ {\sigma \partial_{t} v+v=\epsilon|u|^{2}} \\ {u(x, 0)=u_{0}(x), \quad v(x, 0)=v_0(x)}\end{array}\right. .$$

 {\sc Barbosa, I.I.} { The Cauchy problem for nonlinear quadratic interactions of the Schr{\"o}dinger type in one dimensional space}, {\it Journal of Mathematical Physics}, {\bf 59}, {7},{2018}

 {\sc Angulo, J. and Linares, F.} {Periodic pulses of coupled nonlinear Schr\"odinger equations in optics.}, {\it Indiana University Mathematics Journal}, 56(2):847?878. 2007

 {\sc Kenig, C.E., Ponce, G. and Vega, L.} - On unique continuation for nonlinear Schrodinger equations. {\it Comm. Pure Appl. Math.,}, {\bf 56}, 1247-1262, 2002.

 {\sc Urrea, J. J.} - On the support of solutions to the NLS-KDV system. {\it Differential and Integral Equations}, {\bf 25}, 611--618, 2012.