Session S18 - Recent progress in non-linear PDEs and their applications

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

Isnaldo Isaac Barbosa

UFAL, Brazil   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

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 [1], [2], [3] and [4].

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

[1] {\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}

[2] {\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

[3] {\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.

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

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