The purpose of this paper is to present a comparison between the modified nonlinear Schrödinger (MNLS) equation and the focusing and defocusing variants of the (unmodified) nonlinear Schrödinger (NLS) equation in the semiclassical limit. We describe aspects of the limiting dynamics and discuss how the nature of the dynamics is evident theoretically through inverse-scattering and noncommutative steepest descent methods. The main message is that, depending on initial data, the MNLS equation can behave either like the defocusing NLS equation, like the focusing NLS equation (in both cases the analogy is asymptotically accurate in the semiclassical limit when the NLS equation is posed with appropriately modified initial data), or like an interesting mixture of the two. In the latter case, we identify a feature of the dynamics analogous to a sonic line in gas dynamics, a free boundary separating subsonic flow from supersonic flow.

Nonlinear dispersive partial differential equations such as the nonlinear Schrödinger equations can have solutions that blow-up. We numerically study the long time behavior and potential blowup of solutions to the focusing Davey-Stewartson II equation by analyzing perturbations of the lump and the Ozawa exact solutions. It is shown in this way that the lump is unstable to both blowup and dispersion, and that blowup in the Ozawa solution is generic.

Acceleration and sound measurements during granular discharge from silos are used to show that silo music is a sound resonance produced by silo quake. The latter is produced by stick-slip friction between the wall and the granular material in tall narrow silos. For the discharge rates studied, the occurrence and frequency of flow pulsations are determined primarily by the surface properties of the granular material and the silo wall. The measurements show that the pulsating motion of the granular material drives the oscillatory motion of the silo and the occurrence of silo quake does not require a resonant interaction between the silo and the granular material. IntroductionThe discharge of granular materials from silos is often characterized by vibrations or pulsations of the silo, termed 'silo quake', and a loud noise, termed 'silo music' [1][2][3][4][5][6][7][8]. Both of these are undesirable as silo quake may cause structural failure and silo music is a source of noise pollution. Unfortunately, the numerous conflicting studies published in the literature [1][2][3][4][5][6][7][8] do not give the silo designer a simple model to understand the physical processes that cause the pulsations, and to guide silo design or modification that would prevent the pulsations or at least minimize their effect. The purpose of this study is to investigate the cause of the noise and the pulsations, and the interaction between the motion of the granular material and the motion of the structure.Several studies of the discharge of granular material from silos have noted fluctuations in discharge rate and the production of noise and vibration [1][2][3][4][5][6][7][8]. The top of the granular material has been observed to move in discrete steps even though the discharge from the bottom of the silo was continuous [4,6]. For smooth-walled, tall, narrow silos, pulsations occurred during both mass and mixed flow. The pulsations were observed to stop at a critical height of granular material in the silo [1,3]. Methods suggested for preventing pulsations include roughening the walls in the transition zone between the bunker and the orifice [1][2][3] and placement of inserts along the silo walls [4].In an early study, Phillips [6] observed the motion of sand in a tube, which had a glass face, and was closed at the lower end by a flat bottom having a central orifice. When the orifice was opened, the sand in the upper part of the tube moved downward intermittently in jerks. Phillips noted, "when the flow begins, a curious rattling sound is heard which changes to a distinct musical note". He also did experiments in which the tube was first partly filled with mercury and then filled with sand. Once again, the free surface of the sand descended intermittently when the mercury was allowed to flow through the orifice. He observed that the length of the column of sand increased by about 2% during the 'stick' phase. Further, the motion of the granular material caused the wall of the tube to vibrate. Thus both silo music and silo quake occur...

In solving semilinear initial boundary value problems with prescribed non-periodic boundary conditions using implicit-explicit and implicit time stepping schemes, both the function and derivatives of the function may need to be computed accurately at each time step. To determine the best Chebyshev collocation method to do this, the accuracy of the real space Chebyshev differentiation, spectral space preconditioned Chebyshev tau, real space Chebyshev integration and spectral space Chebyshev integration methods are compared in the L 2 and W 2,2 norms when solving linear fourth order boundary value problems; and in the L ∞ ([0, T ]; L 2) and L ∞ ([0, T ]; W 2,2) norms when solving initial boundary value problems. We find that the best Chebyshev method to use for high resolution computations of solutions to initial boundary value problems is the spectral space Chebyshev integration method which uses sparse matrix operations and has a computational cost comparable to Fourier spectral discretization.

This work experimentally investigates the effects of an interstitial fluid on the discharge of granular material within an hourglass. The experiments include observations of the flow patterns, measurements of the discharge rates, and pressure variations for a range of different fluid viscosities, particle densities and diameters, and hourglass geometries. The results are classified into three regimes: (i) granular flows with negligible interstitial fluid effects; (ii) flows affected by the presence of the interstitial fluid; and (iii) a no-flow region in which particles arch across the orifice and do not discharge. Within the fluid-affected region, the flows were visually classified as lubricated and air-coupled flows, oscillatory flows, channeling flows in which the flow preferentially rises along the sidewalls, and fluidized flows in which the upward flow suspends the particles. The discharge rates depends on the Archimedes number, the ratio of the effective hopper diameter to the particle diameter, and hourglass geometry. The hopper-discharge experiments, as well as experiments found in the literature, demonstrate that the presence of the interstitial fluid is important when the nondimensional ratio ͑N͒ of the single-particle terminal velocity to the hopper discharge velocity is less than 10. Flow ceased in all experiments in which the particle diameter was greater than 25% of the effective hopper diameter regardless of the interstitial fluid.