Infinitely many radial solutions of a variational problem related to dispersion-managed optical fibers

Kunze, Markus

doi:10.1090/S0002-9939-02-06780-1

Infinitely many radial solutions of a variational problem related to dispersion-managed optical fibers
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Proc. Amer. Math. Soc. 131 (2003), 2181-2188 Request permission

Abstract:

We consider a non-local variational problem whose critical points are related to bound states in certain optical fibers. The functional is given by $\varphi (u)=\frac {1}{2}|u|^2_{H^1}-\int _0^1\int _{\mathbb {R}^2}|e^{it\Delta }u|^4 dxdt$, and relying on the regularizing properties of the solution $e^{it\Delta }$ to the free Schrödinger equation, it will be shown that $\varphi$ has infinitely many critical points.

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