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 $n$ 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 $a_{ij}(x)=\delta_{ij}, when |x|\geq R. The exponents$p$satisfies$ 1<p<p_{1}(n)$in (0.1), and$p \leq p_{2}(n)$in (0.2), where$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 $n\geq 2$ space dimensions, there exists two indexes $p$ and $q$ 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 $q=4$, 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 $\exp (c\varepsilon^{-2})$ for initial data of size $\varepsilon$. 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 $\varepsilon^{-18}$ for the initial data of size $\varepsilon$, 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 ss- and pp-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 $s$- and $p$-wave pairings in an ultracold spin-1/2 Fermi gas. Its occurrence is facilitated by the co-existence of comparable $s$- and $p$-wave interactions, which is realizable in a two-component $^{40}$K Fermi gas with close-by $s$- and $p$-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 $p$-wave pairing in the fermion species that has no $p$-wave interactions. The hybridized nature of this novel superfluid can also be confirmed by measuring the $s$-wave and $p$-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-$\frac{1}{2}$ 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 $a/M_{0}$ with $M_{0}$ black hole mass and inclination angle $i$, the dimensionless noncommutative parameter $\sqrt{\vartheta}/M_{0}$ 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 $\sqrt{\vartheta}/M_{0}$, while the distortion increases with it. Compared to the Kerr black hole, the parameter $\sqrt{\vartheta}/M_{0}$ 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