On time-splitting methods for nonlinear Schrödinger equation with highly oscillatory potential

In this work, we consider the numerical solution of the nonlinear Schrödinger equation with a highly oscillatory potential (NLSE-OP). The NLSE-OP is a model problem which frequently occurs in recent studies of some multiscale dynamical systems, where the potential introduces wide temporal oscillations to the solution and causes numerical difficulties. We aim to analyze rigorously the error bounds of the splitting schemes for solving the NLSE-OP to a fixed time. Our theoretical results show that the Lie–Trotter splitting scheme is uniformly and optimally accurate at the first order provided that the oscillatory potential is integrated exactly, while the Strang splitting scheme is not. Our results apply to general dispersive or wave equations with an oscillatory potential. The error estimates are confirmed by numerical results.

Electrical activity in rat retina in a streptozotocin-induced diabetes model

Objectives: Diabetic retinopathy remains the major cause of blindness among the working-age population of developed countries. Considering this, experimental models of diabetes involving laboratory animals are important for assessing clinically significant methods to determine early pathologic alterations of the retina. The early detection of diabetic retinopathy in combination with a search for new pathogenetic targets will enable focusing on new strategies to limit the development of critical changes in the retina and to prolong retinal functioning during the development of diabetes mellitus. Aim: This study aimed to define parameters of electroretinography test that identifies changes due to retinal impairment in diabetes. Methods: Experimental diabetes was induced in Wistar rats by intraperitoneally injecting streptozocin (65 mg/kg; group DM). The control group (CB) received intraperitoneal injections of the vehicle, i.e. citric buffer. On each consecutive day of the experiment, all rats received insulin detemir (2 u/kg). Ophthalmoscopy and electroretinography were conducted before initiating the experiment and after 50, 58 and 66 days of injectin sptreptozocin. Results: Amid 2u\kg insulin injection the glucose level in venous blood in DM group amounted to 30-40 mM. The ophthalmoscopy showed that the optic nerve disk paled by the 50th day, with its line erasing. During electroretinography, wave amplitude in oscillatory potential test tended to decrease. -wave latency of photopic system increased with -wave latency of photopic system and - and -waves latency of scotopic system not altering. In addition, the amplitude of rhythmic stimulation of 8 and 12 Hz decreased. Conclusion: The most apparent parameters of electroretinography for modelling streptozocin-induced diabetes are wave amplitude during the oscillatory potential test, photopic B-wave latency and the amplitude of rhythmic stimulation. These results suggest that in diabetes, ischaemic injury is an important cause of early dysfunction of inner retinal layers.

Faster than the brain’s speed of light: Retinocortical interactions differ in high frequency activity when processing darks and lights

Visual processing of dark visual stimuli has been hypothesized to occur faster relative to bright stimuli. We investigated the timing, processing, and propagation of neural activity in response to darks and lights, operationalized as light offset and onset, in the human visual system by recording electroretinography (ERG) simultaneously with magnetoencephalography (MEG) in humans. We discovered that dark onset resulted in 75–95 Hz retinal activity that we call the dark retinal oscillatory potential, occurring with the same latency as the analogous but more broadband (55–195 Hz) oscillatory potential at light onset. Both retinal oscillations coupled with subsequent cortical activity of corresponding bandwidths, but cortical responses for darks indeed occurred earlier than for lights. Darks therefore propagate from retina to cortex more quickly than lights, potentially resulting from a thalamic advantage. Furthermore, we found that this propagation is effectuated by high frequency retinocortical coupling of narrow bandwidth for darks but wide bandwidth for lights.

Interval Oscillation Criteria for Forced Second-Order Nonlinear Delay Dynamic Equations with Damping and Oscillatory Potential on Time Scales

We are concerned with the interval oscillation of general type of forced second-order nonlinear dynamic equation with oscillatory potential of the formrtg1xt,xΔtΔ+p(t)g2(x(t),xΔ(t))xΔ(t)+q(t)f(x(τ(t)))=e(t), on a time scaleT. We will use a unified approach on time scales and employ the Riccati technique to establish some oscillation criteria for this type of equations. Our results are more general and extend the oscillation criteria of Erbe et al. (2010). Also our results unify the oscillation of the forced second-order nonlinear delay differential equation and the forced second-order nonlinear delay difference equation. Finally, we give some examples to illustrate our results.

Attractive interaction between ions inside a quantum plasma structure

We construct the model of a quantum spherically symmetric plasma structure based on radial oscillations of ions. We suppose that ions are involved in ion-acoustic plasma oscillations. We find the exact solution of the Schrödinger equation for an ion moving in the self-consistent oscillatory potential of an ion-acoustic wave. The system of ions is secondly quantized and its ground state is constructed. Then, we consider the interaction between ions by the exchange of an acoustic wave. It is shown that this interaction can be attractive. We describe the formation of pairs of ions inside a plasma structure and demonstrate that such a plasmoid can exist in a dense astrophysical medium corresponding to the outer core of a neutron star (NS).