9,165 research outputs found
Blow up of Solutions to Semilinear Wave Equations with variable coefficients and boundary
This paper is devoted to studying the following two initial-boundary value
problems for semilinear wave equations with variable coefficients on exterior
domain with subcritical exponent in space dimensions:
u_{tt}-partial_{i}(a_{ij}(x)\partial_{j}u)=|u|^{p}, (x,t)\in
\Omega^{c}\times(0,+\infty), n\geq 3 and
u_{tt}-\partial_{i}(a_{ij}(x)\partial_{j}u)=|u_{t}|^{p}, (x,t)\in
\Omega^{c}\times (0,+\infty), n\geq 1, where p 1<p<p_{1}(n)p \leq p_{2}(n)p_{1}(n)$ is the larger root of the quadratic equation
(n-1)p^{2}-(n+1)p-2=0, and p_{2}(n)=\frac{2}{n-1}+1, respectively. It is
well-known that the numbers p_{1}(n) and p_{2}(n) are the critical exponents.
We will establish two blowup results for the above two initial-boundary value
problems, it is proved that there can be no global solutions no matter how
small the initial data are, and also we give the lifespan estimate of solutions
for above problems
Blow up for some semilinear wave equations in multi-space dimensions
In this paper, we discuss a new nonlinear phenomenon. We find that in space dimensions, there exists two indexes and such that the cauchy
problems for the nonlinear wave equations {equation} \label{0.1} \Box u(t,x) =
|u(t,x)|^{q}, \ \ x\in R^{n}, {equation} and {equation} \label{0.2} \Box u(t,x)
= |u_{t}(t,x)|^{p}, \ \ x\in R^{n} {equation} both have global existence for
small initial data, while for the combined nonlinearity, the solutions to the
Cauchy problem for the nonlinear wave equation {equation} \label{0.3} \Box
u(t,x) = | u_{t}(t,x)|^{p} + |u(t,x)|^{q}, \ \ x\in R^{n}, {equation} with
small initial data will blow up in finite time. In the two dimensional case, we
also find that if , the Cauchy problem for the equation \eqref{0.1} has
global existence, and the Cauchy problem for the equation {equation}
\label{0.4} \Box u(t,x) = u (t,x)u_{t}(t,x)^{2}, \ \ x\in R^{2} {equation} has
almost global existence, that is, the life span is at least for initial data of size . However, in the
combined nonlinearity case, the Cauchy problem for the equation {equation}
\label{0.5} \Box u(t,x) = u(t,x) u_{t}(t,x)^{2} + u(t,x)^{4}, \ \ x\in R^{2}
{equation} has a life span which is of the order of for
the initial data of size , this is considerably shorter in
magnitude than that of the first two equations. This solves an open optimality
problem for general theory of fully nonlinear wave equations (see
\cite{Katayama}).Comment: 13 page
Life-Span of Solutions to Critical Semilinear Wave Equations
The final open part of the famous Strauss conjecture on semilinear wave
equations of the form \Box u=|u|^{p}, i.e., blow-up theorem for the critical
case in high dimensions was solved by Yordanov and Zhang, or Zhou
independently. But the estimate for the lifespan, the maximal existence time,
of solutions was not clarified in both papers. Recently, Takamura and Wakasa
have obtained the sharp upper bound of the lifespan of the solution to the
critical semilinear wave equations, and their method is based on the method in
Yordanov and Zhang. In this paper, we give a much simple proof of the result of
Takamura and Wakasa by using the method in Y. Zhou for space dimensions n\geq
2. Simultaneously, this estimate of the life span also proves the last open
optimality problem of the general theory for fully nonlinear wave equations
with small initial data in the case n=4 and quadratic nonlinearity(One can see
Li and Chen for references on the whole history).Comment: 12 pages, no figure
Fermion superfluid with hybridized - and -wave pairings
Ever since the pioneering work of Bardeen, Cooper and Schrieffer in the
1950s, exploring novel pairing mechanisms for fermion superfluids has become
one of the central tasks in modern physics. Here, we investigate a new type of
fermion superfluid with hybridized - and -wave pairings in an ultracold
spin-1/2 Fermi gas. Its occurrence is facilitated by the co-existence of
comparable - and -wave interactions, which is realizable in a
two-component K Fermi gas with close-by - and -wave Feshbach
resonances. The hybridized superfluid state is stable over a considerable
parameter region on the phase diagram, and can lead to intriguing patterns of
spin densities and pairing fields in momentum space. In particular, it can
induce a phase-locked -wave pairing in the fermion species that has no
-wave interactions. The hybridized nature of this novel superfluid can also
be confirmed by measuring the -wave and -wave contacts, which can be
extracted from the high-momentum tail of the momentum distribution of each spin
component. These results enrich our knowledge of pairing superfluidity in Fermi
systems, and open the avenue for achieving novel fermion superfluids with
multiple partial-wave scatterings in cold atomic gases.Comment: 7 pages, 5 figures, with more discussions and references adde
Three-component Ultracold Fermi Gases with Spin-Orbit Coupling
We investigate the pairing physics in a three-component Fermi-Fermi mixture,
where a few impurities are immersed in a non-interacting spin-
Fermi gas with synthetic spin-orbit coupling (SOC), and interact attractively
with one spin species in the Fermi gas. Due to the interplay of SOC and
spin-selective interaction, the molecular state intrinsically acquires a
non-zero center-of-mass momentum, which results in a new type of Fulde-Ferrell
(FF) pairing in spin-orbit coupled Fermi systems. The existence of the Fermi
sea can also lead to the competition between FF-like molecular states with
different center-of-mass momenta, which corresponds to a first-order transition
between FF phases in the thermodynamic limit. As the interaction strength is
tuned, a polaron-molecule transition occurs in the highly imbalanced system,
where the boundary varies non-monotonically with SOC parameters and gives rise
to the reentrance of polaron states. The rich physics in this system can be
probed using existing experimental techniques.Comment: 5+1.5 pages, 5+1 figures; published versio
Global Instability of the Multi-dimensional Plane Shocks for the isothermal flow
In this paper, we are concerned with the long time behavior of the piecewise
smooth solutions to the generalized Riemann problem governed by the
compressible isothermal Euler equations in two and three dimensions.
Non-existence result is established for the fan-shaped wave structure solution,
including two shocks and one contact discontinuity and which is a perturbation
of plane waves. Therefore, unlike the one-dimensional case, the
multi-dimensional plane shocks are not stable globally. What is more, the sharp
lifespan estimate is established which is the same as the lifespan estimate for
the nonlinear wave equations in both two and three space dimensions.Comment: 26 page
Magnetic orders in a Fermi gas induced by cavity-field fluctuations
We study magnetic orders of fermions under cavity-assisted Raman couplings in
a one-dimensional lattice at half filling. The cavity-enhanced atom-photon
coupling introduces a dynamic long-range interaction between the fermions,
which competes with the short-range on-site interaction and leads to a variety
of magnetic orders. Adopting a numerical density-matrix-renormalization-group
method, we investigate the various magnetic orders and map out the steady-state
phase diagram. Interestingly, as all the phase transitions take place outside
the superradiant regime, the magnetic orders are associated with cavity-field
fluctuations with a vanishing number of photons on the mean-field level.Comment: 8 pages, 6 figure
Impurity-induced resonant states in topological nodal-line semimetals
Nodal-line semimetals are characterized by a kind of topologically nontrivial
bulk-band crossing, giving rise to almost flat surface states. Yet, a direct
evidence of the surface states is still lacking. Here we study theoretically
impurity effects in topological nodal-line semimetals based on the T-matrix
method. It is found that for a bulk impurity, some in-gap states may be induced
near the impurity site, while the visible resonant impurity state can only
exist for certain strength of the impurity potentials. For a surface impurity,
robust resonant impurity states exist in a wide range of impurity potentials.
Such robust resonant states stem from the topological protected weak dispersive
surface states, which can be probed by scanning tunneling microscopy, providing
a strong signature of the topological surface states in the nodal-line
semimetals.Comment: 7 pages, 5 figure
Shadow of noncommutative geometry inspired black hole
In this paper, the shadow casted by the rotating black hole inspired by
noncommutative geometry is investigated. In addition to the dimensionless spin
parameter with black hole mass and inclination angle , the
dimensionless noncommutative parameter is also found
to affect the shape of the black hole shadow. The result shows that the size of
the shadow slightly decreases with the parameter ,
while the distortion increases with it. Compared to the Kerr black hole, the
parameter increases the deformation of the shadow.
This may offer a way to distinguish noncommutative geometry inspired black hole
from Kerr one via astronomical instruments in the near future.Comment: 18 pages, 6 figures, 3 table
Interaction-induced exotic vortex states in an optical lattice clock with spin-orbit coupling
Motivated by a recent experiment [L. F. Livi, et al., Phys. Rev. Lett. 117,
220401(2016)], we study the ground-state properties of interacting fermions in
a one-dimensional optical lattice clock with spin-orbit coupling. As the
electronic and the hyperfine-spin states in the clock-state manifolds can be
treated as effective sites along distinct synthetic dimensions, the system can
be considered as multiple two-leg ladders with uniform magnetic flux
penetrating the plaquettes of each ladder. As the inter-orbital spin-exchange
interactions in the clock-state manifolds couple individual ladders together,
we show that exotic interaction-induced vortex states emerge in the
coupled-ladder system, which compete with existing phases of decoupled ladders
and lead to a rich phase diagram. Adopting the density matrix renormalization
group approach, we map out the phase diagram, and investigate in detail the
currents and the density-density correlations of the various phases. Our
results reveal the impact of interactions on spin-orbit coupled systems, and
are particularly relevant to the on-going exploration of spin-orbit coupled
optical lattice clocks
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