161 research outputs found
Nondegeneracy and Stability of Antiperiodic Bound States for Fractional Nonlinear Schr\"odinger Equations
We consider the existence and stability of real-valued, spatially
antiperiodic standing wave solutions to a family of nonlinear Schr\"odinger
equations with fractional dispersion and power-law nonlinearity. As a key
technical result, we demonstrate that the associated linearized operator is
nondegenerate when restricted to antiperiodic perturbations, i.e. that its
kernel is generated by the translational and gauge symmetries of the governing
evolution equation. In the process, we provide a characterization of the
antiperiodic ground state eigenfunctions for linear fractional Schr\"odinger
operators on with real-valued, periodic potentials as well as a
Sturm-Liouville type oscillation theory for the higher antiperiodic
eigenfunctions.Comment: 46 pages, 2 figure
Periodic solutions and torsional instability in a nonlinear nonlocal plate equation
A thin and narrow rectangular plate having the two short edges hinged and the
two long edges free is considered. A nonlinear nonlocal evolution equation
describing the deformation of the plate is introduced: well-posedness and
existence of periodic solutions are proved. The natural phase space is a
particular second order Sobolev space that can be orthogonally split into two
subspaces containing, respectively, the longitudinal and the torsional
movements of the plate. Sufficient conditions for the stability of periodic
solutions and of solutions having only a longitudinal component are given. A
stability analysis of the so-called prevailing mode is also performed. Some
numerical experiments show that instabilities may occur. This plate can be seen
as a simplified and qualitative model for the deck of a suspension bridge,
which does not take into account the complex interactions between all the
components of a real bridge.Comment: 34 pages, 4 figures. The result of Theorem 6 is correct, but the
proof was not correct. We slightly changed the proof in this updated versio
Antiperiodic Problems for Nonautonomous Parabolic Evolution Equations
This work focuses on the antiperiodic problem of nonautonomous semilinear parabolic evolution equation in the form u′(t)=A(t)u(t)+f(t,u(t)), t∈R, u(t+T)=-u(t), t∈R, where (At)t∈R (possibly unbounded), depending on time, is a family of closed and densely defined linear operators on a Banach space X. Upon making some suitable assumptions such as the Acquistapace and Terreni conditions and exponential dichotomy on (At)t∈R, we obtain the existence results of antiperiodic mild solutions to such problem. The antiperiodic problem of nonautonomous semilinear parabolic evolution equation of neutral type is also considered. As sample of application, these results are applied to, at the end of the paper, an antiperiodic problem for partial differential equation, whose operators in the linear part generate an evolution family of exponential stability
Dissipative dynamics at first-order quantum transitions
We investigate the effects of dissipation on the quantum dynamics of many-body systems at quantum transitions, especially considering those of the first order. This issue is studied within the paradigmatic one-dimensional quantum Ising model. We analyze the out-of-equilibrium dynamics arising from quenches of the Hamiltonian parameters and dissipative mechanisms modeled by a Lindblad master equation, with either local or global spin operators acting as dissipative operators. Analogously to what happens at continuous quantum transitions, we observe a regime where the system develops a nontrivial dynamic scaling behavior, which is realized when the dissipation parameter u (globally controlling the decay rate of the dissipation within the Lindblad framework) scales as the energy difference Δ of the lowest levels of the Hamiltonian, i.e., u∼Δ. However, unlike continuous quantum transitions where Δ is power-law suppressed, at first-order quantum transitions Δ is exponentially suppressed with increasing the system size (provided the boundary conditions do not favor any particular phase)
Instability of coherent states of a real scalar field
We investigate stability of both localized time-periodic coherent states
(pulsons) and uniformly distributed coherent states (oscillating condensate) of
a real scalar field satisfying the Klein-Gordon equation with a logarithmic
nonlinearity. The linear analysis of time-dependent parts of perturbations
leads to the Hill equation with a singular coefficient. To evaluate the
characteristic exponent we extend the Lindemann-Stieltjes method, usually
applied to the Mathieu and Lame equations, to the case that the periodic
coefficient in the general Hill equation is an unbounded function of time. As a
result, we derive the formula for the characteristic exponent and calculate the
stability-instability chart. Then we analyze the spatial structure of the
perturbations. Using these results we show that the pulsons of any amplitudes,
remaining well-localized objects, lose their coherence with time. This means
that, strictly speaking, all pulsons of the model considered are unstable.
Nevertheless, for the nodeless pulsons the rate of the coherence breaking in
narrow ranges of amplitudes is found to be very small, so that such pulsons can
be long-lived. Further, we use the obtaned stability-instability chart to
examine the Affleck-Dine type condensate. We conclude the oscillating
condensate can decay into an ensemble of the nodeless pulsons.Comment: 11 pages, 8 figures, submitted to Physical Review
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