 11186 Riccardo Adami, Diego Noja
 Stability and symmetrybreaking bifurcation for the ground states of a NLS with a $\delta^\prime$ interaction
(237K, LaTeX 2e with 5 PS Figures)
Dec 6, 11

Abstract ,
Paper (src),
View paper
(auto. generated ps),
Index
of related papers

Abstract. We determine and study the ground states of a focusing
Schr\"odinger equation in dimension one with a power nonlinearity
$\psi^{2\mu} \psi$ and a strong inhomogeneity represented
by a singular point perturbation, the socalled (attractive) $\delta^\prime$ interaction, located at the origin.
The timedependent
problem turns out to be globally well posed in the subcritical regime,
and locally well posed in the supercritical and critical regime in the
appropriate energy space. The set of the (nonlinear) ground states
is completely determined.
For any value of the nonlinearity power,
it exhibits a symmetry breaking bifurcation structure as a function of
the frequency (i.e., the nonlinear eigenvalue) $\omega$. More precisely,
there exists a critical value $\om^*$ of the nonlinear eigenvalue $\om$, such that: if $\om_0 < \om < \om^*$, then there is a single
ground state and it is an odd function; if $\om > \om^*$ then
there exist two nonsymmetric ground states.
We prove that before bifurcation (i.e., for $\om < \om^*$) and for any
subcritical power, every ground state is orbitally stable. After bifurcation ($\om =\om^*+0$), ground states are stable if $\mu$ does not exceed a value $\mu^\star$ that lies between $2$ and $2.5$,
and become unstable for $\mu > \mu^*$. Finally, for $\mu > 2$ and $\om \gg \om^*$, all ground states are unstable. The branch of odd ground states for $\om < \om^*$ can be continued at any $\om > \om^*$, obtaining a family of orbitally unstable stationary states.
Existence of ground states is proved by variational techniques,
and the stability properties of stationary states are investigated by means of the GrillakisShatahStrauss framework, where some non standard
techniques have to be used to establish the needed properties
of linearization operators.
 Files:
11186.src(
11186.keywords ,
adaminojanlsbif.tex ,
bif2tris.eps ,
bifstabbis.eps ,
bifstabtris.eps ,
bifuncertainrightbis2.eps ,
bifuncertainrighttris2.eps )