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Young Women in Harmonic Analysis and PDE

December 2-4, 2016

Janina Gärtner (Karlsruhe Institute of Technology)

Existence of solutions of the Lugiato-Lefever equation on $\mathbb{R}$

The stationary Lugiato-Lefever equation is given by \begin{align*} -du''+(\zeta-\mathrm i)u-|u|^2u+\mathrm i f\,=\,0 \end{align*} for $d\in\mathbb{R}$, $\zeta,f>0$. Here, we are interested in solutions in $\{u=u^\star+\widetilde{u}: u^\star=const., \widetilde{u}\in H^2(\mathbb{R}), u'(0)=0\}$. It is well known that for $d>0$, $\zeta>0$ the solutions of the nonlinear Schr\"odinger equation \begin{align*} \left\{ \begin{array}{l l} u\in H^1(\mathbb{R}), u\neq0, \\ -du''+\zeta u-|u|^2u\,=\,0 \end{array} \right. \end{align*} are given by $u(x)=\mathrm e^{\mathrm i \alpha}\varphi(x)$, $\alpha\in\mathbb{R}$, where $\varphi(x)=\sqrt{2\zeta}\frac{1}{\cosh\sqrt{\frac{\zeta}{d}}x}$. Using a theorem of Crandall-Rabinowitz, we can show that bifurcation with respect to the parameter $f$ only arises for $\alpha\in\left\{\frac{\pi}{2},\frac{3\pi}{2}\right\}$. \noindent Afterwards, we are interested in solutions $u_\varepsilon$ of \begin{align*} -du''+(\widetilde{\zeta}-\varepsilon \mathrm i)u-|u|^2 u+\mathrm i \widetilde{f}\,=\,0 \end{align*} for small $\varepsilon>0$. Then $a(x):=\frac{1}{\sqrt{\varepsilon}}\cdot u_{\varepsilon}\left(\frac{1}{\sqrt{\varepsilon}}x\right)$ solves the stationary Lugiato-Lefever equation for $\zeta=\frac{\widetilde{\zeta}}{\varepsilon}$ and $f=\frac{\widetilde{f}}{\varepsilon^{\frac{3}{2}}}$. Using a reformulation of this equation, Sturm's oscillation and comparison theorem, Agmon's principle and a suitable version of the implicit function theorem, we can find a quantitative neighborhood where the reformulated equation is uniquely solvable.