Solitary waves bifurcated from edges of Bloch bands in two-dimensional periodic media are determined
both analytically and numerically in the context of a two-dimensional nonlinear Schrödinger equation with a
periodic potential. Using multiscale perturbation methods, the envelope equations of solitary waves near Bloch
bands are analytically derived. These envelope equations reveal that solitary waves can bifurcate from edges of
Bloch bands under either focusing or defocusing nonlinearity, depending on the signs of the second-order
dispersion coefficients at the edge points. Interestingly, at edge points with two linearly independent Bloch
modes, the envelope equations lead to a host of solitary wave structures, including reduced-symmetry solitons,
dipole-array solitons, vortex-cell solitons, and so on—many of which have not been reported before to our
knowledge. It is also shown analytically that the centers of envelope solutions can be positioned at only four
possible locations at or between potential peaks. Numerically, families of these solitary waves are directly
computed both near and far away from the band edges. Near the band edges, the numerical solutions spread
over many lattice sites, and they fully agree with the analytical solutions obtained from the envelope equations.
Far away from the band edges, solitary waves are strongly localized, with intensity and phase profiles characteristic
of individual families.
Previous work has shown that in a two-dimensional periodic medium under focusing or defocusing cubic
nonlinearities, gap solitons in the form of low-amplitude and slowly modulated single-Bloch-wave packets can
bifurcate out from the edges of Bloch bands. In this paper, linear stability properties of these gap solitons near
band edges are determined both analytically and numerically. Through asymptotic analysis, it is shown that
these gap solitons are linearly unstable if the slope of their power curve at the band edge has the opposite sign
of nonlinearity here focusing nonlinearity is said to have a positive sign, and defocusing nonlinearity to have
a negative sign. An equivalent condition for linear instability is that the power of the gap solitons near the
band edge is lower than the limit power value on the band edge. Through numerical computations of the power
curves, it is found that this condition is always satisfied, thus two-dimensional gap solitons near band edges are
linearly unstable. The analytical formula for the unstable eigenvalue of gap solitons near band edges is also
asymptotically derived. It is shown that this unstable eigenvalue is proportional to the cubic power of the
soliton’s amplitude, and it induces width instabilities of gap solitons. A comparison between this analytical
eigenvalue formula and numerically computed eigenvalues shows excellent agreement.
In this paper, we present a novel pairwise-force smoothed particle hydrodynamics (PF-SPH) model to allow modeling of
various interactions at interfaces in real time. Realistic capture of interactions at interfaces is a challenging problem for SPH-based simulations, especially for scenarios involving multiple interactions at different interfaces. Our PF-SPH model can readily handle multiple kinds of interactions simultaneously in a single simulation; its basis is to use a larger support radius than that used in standard SPH. We adopt a novel anisotropic filtering term to further improve the performance of interaction forces. The proposed model is stable; furthermore, it avoids the particle clustering problem which commonly occurs at the free surface. We show how our model can be used to capture various interactions. We also consider the close connection between droplets and bubbles, and show how to animate bubbles rising in liquid as well as bubbles in air. Our method is versatile, physically plausible and easy-to-implement. Examples are provided to demonstrate the capabilities and effectiveness of our approach.
Surface flow phenomena, such as rain water flowing down a tree trunk and progressive water front in a shower room, are common in real life. However, compared with the 3D spatial fluid flow, these surface flow problems have been much less studied in the graphics community. To tackle this research gap, we present an efficient, robust and high-fidelity simulation approach based on the shallow-water equations. Specifically, the standard shallow-water flow model is extended to general triangle meshes with a feature-based bottom friction model, and a series of coherent mathematical formulations are derived to represent the full range of physical effects that are important for real-world surface flow phenomena. In addition, by achieving compatibility with existing 3D fluid simulators and by supporting physically realistic interactions with multiple fluids and solid surfaces, the new model is flexible and readily extensible for coupled phenomena. A wide range of simulation examples are presented to demonstrate the performance of the new approach.
We apply our model of quantum gravity to a Kerr-AdS spacetime of dimension $2 m+1$, $m\ge2$, where all rotational parameters are equal, resulting in a wave equation in a quantum spacetime which has a sequence of solutions that can be expressed as a product of stationary and temporal eigenfunctions. The stationary eigenfunctions can be interpreted as radiation and the temporal as gravitational waves. The event horizon corresponds in the quantum model to a Cauchy hypersurface that can be crossed by causal curves in both directions such that the information paradox does not occur. We also prove that the Kerr-AdS spacetime can be maximally extended by replacing in a generalized Boyer-Lindquist coordinate system the $r$ variable by $\rho=r^2$ such that the extended spacetime has a timelike curvature singularity in $\rho=-a^2$.