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Zbl 0741.35080
Gonçalves Ribeiro, J.M.
Instability of symmetric stationary states for some nonlinear Schrödinger equations with an external magnetic field. (English)
[J] Ann. Inst. Henri Poincaré, Phys. Théor. 54, No.4, 403-433 (1991). ISSN 0246-0211

Summary: This work is concerned with instability properties of solutions $u(t,x)=e\sp{-i\omega}\Phi(x)$ of the equation $id\sb tu=-L\sb Au+\vert u\vert \sp{p-1}u$, where $iL\sb A$ is the Schrödinger operator in the presence of a uniform magnetic field, defined by the solenoidal vector potential $A: L\sb Au=-\Delta u-2iA.\nabla u+\vert A\vert \sp 2u$. $\Phi$ is a solution of the nonlinear elliptic equation $L\sb A\Phi+\omega\Phi- \vert \Phi\vert \sp{p-1}\Phi=0$, invariant by rotations around the $z$- axis, and solving a certain variational problem. Put $\Sigma=\{e\sp{- i\theta}\Phi(\hat x-\zeta e\sb z): \theta,\ \zeta\in \bbfR\}$. We prove that $\Sigma$ is unstable by the flow of the evolution equation, for some values of $\omega$, $p$. Moreover, the trajectories used to exhibit instability are global and uniformly bounded.

MSC 2000:: *35Q55 NLS-like (nonlinear Schroedinger) equations
35B35 Stability of solutions of PDE
37C75 Stability theory
35Q51 Solitons
35J10 Schroedinger operator

Keywords: instability properties of solutions; variational problem

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