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Stability and Symmetry-Breaking Bifurcation for the Ground States of a NLS with a δ′ Interaction
No full textWe determine and study the ground states of a focusing Schrödinger equation in dimension one with a power nonlinearity |ψ|2μψ and a strong inhomogeneity represented by a singular point perturbation, the so-called (attractive) δ interaction, located at the origin. The time-dependent 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) ω. More precisely, there exists a critical value ω ∗ of the nonlinear eigenvalue ω, such that: if ω0 < ω < ω ∗, then there is a single ground state and it is an odd function; ifω > ω ∗ then there exist two non-symmetric ground states.We prove that before bifurcation (i.e., for ω < ω ∗) and for any subcritical power, every ground state is orbitally stable. After bifurcation (ω = ω ∗ + 0), ground states are stable if μ does not exceed a value μ that lies between 2 and 2.5, and become unstable for μ > μ ∗. Finally, for μ > 2 and ω ω ∗, all ground states are unstable. The branch of odd ground states for ω < ω ∗ can be continued at any ω > ω ∗ , 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 Grillakis-Shatah-Strauss framework, where some non-standard techniques have to be used to establish the needed properties of linearization operator
Nonlinearity-defect interaction: symmetry breaking bifurcation in a NLS with a delta' impurity.
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The wave equation with one point interaction and the (linearized) classical electrodynamics of a point particle
Full text linkWell posedness of the nonlinear Schrödinger equation with isolated singularities
No full textWe study the well posedness of the nonlinear Schrödinger (NLS) equation with a point interaction and power nonlinearity in dimension two and three. Behind the autonomous interest of the problem, this is a model of the evolution of so called singular solutions that are well known in the analysis of semilinear elliptic equations. We show that the Cauchy problem for the NLS considered enjoys local existence and uniqueness of strong (operator domain) solutions, and that the solutions depend continuously from initial data. In dimension two well posedness holds for any power nonlinearity and global existence is proved for powers below the cubic. In dimension three local and global well posedness are restricted to low powers
Orbital and asymptotic stability for standing waves of a NLS equation with concentrated nonlinearity in dimension three
No full textWe begin to study in this paper orbital and asymptotic stability of standing waves for a model of Schr\"odinger equation with concentrated nonlinearity in dimension three. The nonlinearity is obtained considering a {point} (or contact) interaction with strength , which consists of a singular perturbation of the laplacian described by a selfadjoint operator , where the strength depends on the wavefunction: , . If is the so-called charge of the domain element , i.e. the coefficient of its singular part, we let the strength depend on according to the law , with . This characterizes the model as a focusing NLS with concentrated nonlinearity of power type. For such a model we prove the existence of standing waves of the form , which are orbitally stable in the range , and orbitally unstable for Moreover, we show that for every standing wave is asymptotically stable in the following sense. Choosing initial data close to the stationary state in the energy norm, and belonging to a natural weighted space which allows dispersive estimates, the following resolution holds: , where is the free Schr\"odinger propagator, and , with . Notice that in the present model the admitted nonlinearity for which asymptotic stability of solitons is proved is subcritical
On the mathematical description of the effective behaviour of one-dimensional Bose-Einstein condensates with defect.
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