Triad resonance for internal waves in a uniformly stratified fluid: Rogue waves and breathers.

Three-wave (triad) resonance in a uniformly stratified fluid is investigated as a case study of energy transfer among oscillatory modes. The existence of a degenerate triad is demonstrated explicitly, where two components have identical group velocity. An illuminating example is a resonance involving waves from modes 1, 3, 5 families, but many other combinations are possible. The physical applications and nonlinear dynamics of rogue waves derived analytically in the literature are examined. Exact solutions with four free parameters (two related to the amplitudes of the background plane waves, two related to the frequencies of slowly varying envelopes) describe motions localized in both space and time. The differences between rogue waves of the degenerate versus the nondegenerate cases are highlighted. The phase and profile of the degenerate case rogue waves are correlated. The volume or energy of the rogue wave (defined as the total extent or energy contents of the fluid set in motion for the duration of the rogue wave) may change drastically, if the wave envelope parameters vary. Pulsating modes (breathers) have been studied previously by layered-fluid and modified Korteweg-de Vries models. Here we extend the consideration to stratified fluids but for the simpler case of nondegenerate triads. Instabilities of fission and fusion of breathers are confirmed computationally with Floquet analysis. This knowledge should prove useful for energy transfer processes in the oceans.

Robustness and stability of doubly periodic patterns of the focusing nonlinear Schrödinger equation.

The nonlinear Schrödinger equation possesses doubly periodic solutions expressible in terms of the Jacobi elliptic functions. Such solutions can be realized through doubly periodic patterns observed in experiments in fluid mechanics and optics. Stability and robustness of these doubly periodic wave profiles in the focusing regime are studied computationally by using two approaches. First, linear stability is considered by Floquet theory. Growth will occur if the eigenvalues of the monodromy matrix are of a modulus larger than unity. This is verified by numerical simulations with input patterns of different periods. Initial patterns associated with larger eigenvalues will disintegrate faster due to instability. Second, formation of these doubly periodic patterns from a tranquil background is scrutinized. Doubly periodic profiles are generated by perturbing a continuous wave with one Fourier mode, with or without the additional presence of random noise. Effects of varying phase difference, perturbation amplitude, and randomness are studied. Varying the phase angle has a dramatic influence. Periodic patterns will only emerge if the perturbation amplitude is not too weak. The growth of higher-order harmonics, as well as the formation of breathers and repeating patterns, serve as a manifestation of the classical problem of Fermi-Pasta-Ulam-Tsingou recurrence.

Fermi-Pasta-Ulam-Tsingou recurrence and cascading mechanism for resonant three-wave interactions.

Evolution of resonant three-wave interaction is governed by quadratic nonlinearities. While propagating localized modes and inverse scattering mechanisms have been studied, transient states such as rogue waves and breathers are not fully understood. Modulation instability modes can trigger growth of disturbances and the eventual development of breathers. Here we study computationally the dynamics beyond the first formation of breathers, and demonstrate repeating patterns of breathers as a manifestation of the Fermi-Pasta-Ulam-Tsingou recurrence (FPUT). While nonlinearity governs the actual dynamics, the range of wave numbers for modulation instability remains a useful indicator. Depending on the stability characteristics of the fundamental mode and the higher-order harmonics ("sidebands"), "regular" and "staggered" FPUT patterns can arise. A "cascading mechanism" provides analytical verification, as the fundamental and sideband modes attain the same magnitude at one particular instant, signifying the first occurrence of a breather. A triangular spectrum is also computed, similar to experimental observations of optical pulses. Such spectra can elucidate the spreading of energy among the sidebands and components of the triad resonance. The concept of "effective energy" is examined and the eigenvalues of the inverse scattering mechanism are computed. Both approaches are utilized to correlate with the occurrence of regular or staggered FPUT. These numerical and analytical studies can enhance our understanding of wave interactions in fluid mechanics and optics.

Computational study on the transmission of the SARS-CoV-2 virus through aerosol in an elevator cabin: Effect of the ventilation system.

Aerosol transmission is now well-established as a route in the spread of the SARS-CoV-2 virus. Factors influencing the transport of virus-laden particles in an elevator cabin are investigated computationally and include human respiratory events, locations of the infected person(s), and the ventilation system (ventilation mode, ventilation capacity, and vent schemes). "Breath," "cough," and "sneeze" are defined quantitatively by the fluid jet velocities and particle sizes. For natural ventilation, most particles exhaled by sneezing and coughing tend to deposit on surfaces quickly, but aerosol generated by breathing will remain suspended in the air longer. For forced ventilation, motions of particles under different ventilation capacities are compared. Larger particles otherwise deposited readily on solid surfaces may be slowed down by airflow. Air currents also accelerate the motions of smaller particles, facilitating the subsequent deposition of micrometer or sub-micrometer particles. Locations of the infected person(s) lead to different spreading scenarios due to the distinctive motions of the particles generated by the various respiratory events. Sneeze particles will likely contaminate the person in front of the infected passenger only. Cough particles will increase the risk of all the people around the injector. Breath particles tend to spread throughout the confined environment. An optimized vent scheme is introduced and can reduce particles suspended in the air by up to 80% as compared with commonly used schemes. The purification function of this vent model is robust to various positions of the infected passenger.

Four-wave mixing and coherently coupled Schrödinger equations: Cascading processes and Fermi-Pasta-Ulam-Tsingou recurrence.

Modulation instability, breather formation, and the Fermi-Pasta-Ulam-Tsingou recurrence (FPUT) phenomena are studied in this article. Physically, such nonlinear systems arise when the medium is slightly anisotropic, e.g., optical fibers with weak birefringence where the slowly varying pulse envelopes are governed by these coherently coupled Schrödinger equations. The Darboux transformation is used to calculate a class of breathers where the carrier envelope depends on the transverse coordinate of the Schrödinger equations. A "cascading mechanism" is utilized to elucidate the initial stages of FPUT. More precisely, higher order nonlinear terms that are exponentially small initially can grow rapidly. A breather is formed when the linear mode and higher order ones attain roughly the same magnitude. The conditions for generating various breathers and connections with modulation instability are elucidated. The growth phase then subsides and the cycle is repeated, leading to FPUT. Unequal initial conditions for the two waveguides produce symmetry breaking, with "eye-shaped" breathers in one waveguide and "four-petal" modes in the other. An analytical formula for the time or distance of breather formation for a two-waveguide system is proposed, based on the disturbance amplitude and instability growth rate. Excellent agreement with numerical simulations is achieved. Furthermore, the roles of modulation instability for FPUT are elucidated with illustrative case studies. In particular, depending on whether the second harmonic falls within the unstable band, FPUT patterns with one single or two distinct wavelength(s) are observed. For applications to temporal optical waveguides, the present formulation can predict the distance along a weakly birefringent fiber needed to observe FPUT.

A connection between the maximum displacements of rogue waves and the dynamics of poles in the complex plane.

Rogue waves of evolution systems are displacements which are localized in both space and time. The locations of the points of maximum displacements of the wave profiles may correlate with the trajectories of the poles of the exact solutions from the perspective of complex variables through analytic continuation. More precisely, the location of the maximum height of the rogue wave in laboratory coordinates (real space and time) is conjectured to be equal to the real part of the pole of the exact solution, if the spatial coordinate is allowed to be complex. This feature can be verified readily for the Peregrine breather (lowest order rogue wave) of the nonlinear Schrödinger equation. This connection is further demonstrated numerically here for more complicated scenarios, namely the second order rogue wave of the Boussinesq equation (for bidirectional long waves in shallow water), an asymmetric second order rogue wave for the nonlinear Schrödinger equation (as evolution system for slowly varying wave packets), and a symmetric second order rogue wave of coupled Schrödinger systems. Furthermore, the maximum displacements in physical space occur at a time instant where the trajectories of the poles in the complex plane reverse directions. This property is conjectured to hold for many other systems, and will help to determine the maximum amplitudes of rogue waves.

Haemodynamic Variations of Flow to Renal Arteries in Custom-Made and Pivot Branch Fenestrated Endografting.

OBJECTIVE: This study aimed to investigate variation of blood flow to renal arteries in custom-made and pivot branch (p-branch) fenestrated endografting, using a computational fluid dynamics (CFD) technique. METHODS: Idealised models of custom-made and p-branch fenestrated grafting were constructed on a basis of a 26 mm stent graft. The custom-made fenestration was designed with a 6 mm diameter, while the 5 mm depth renal p-branch was created with a 6 mm inner and 15 mm outer fenestration. Two configurations (option A and option B) were constructed with different locations of p-branches. Option A had both renal p-branches at the same level, whereas option B contained two staggered p-branches at lower positions. The longitudinal stent orientation in both custom-made and p-branch models was represented by a takeoff angle (ToA) between the renal stent and distal stent graft centreline, varying from 55° to 125°. Computational simulations were performed with realistic boundary conditions governing the blood flow. RESULTS: In both custom-made and p-branch fenestrated models, the flow rate and wall shear stress (WSS) were generally higher and recirculation zones were smaller when the renal stent faced caudally. In custom-made models, the highest flow rate (0.390 L/min) was detected at 70° ToA and maximum WSS on vessel segment (16.8 Pa) was attained at 55° ToA. In p-branch models, option A and option B displayed no haemodynamic differences when having the same ToA. The highest flow rate (0.378 L/min) and maximum WSS on vessel segment (16.7 Pa) were both calculated at 55° ToA. The largest and smallest recirculation zones occurred at 90° and 55° ToA respectively in both custom-made and p-branch models. Custom-made fenestrated models exhibited consistently higher flow rate and shear stress and smaller recirculation zones in renal arteries than p-branch models at the same ToA. CONCLUSIONS: Navigating the renal stents towards caudal orientation can achieve better haemodynamic outcomes in both fenestrated devices. Custom-made fenestrated stent grafts are the preferred choice for elective patients. Further clinical evidence is required to validate the computational simulations.

Rogue waves for a system of coupled derivative nonlinear Schrödinger equations.

Rogue waves (RWs) are unexpectedly strong excitations emerging from an otherwise tranquil background. The nonlinear Schrödinger equation (NLSE), a ubiquitous model with wide applications to fluid mechanics, optics, plasmas, etc., exhibits RWs only in the regime of modulation instability (MI) of the background. For a system of multiple waveguides, the governing coupled NLSEs can produce regimes of MI and RWs, even if each component has dispersion and cubic nonlinearity of opposite signs. A similar effect is demonstrated here for a system of coupled derivative NLSEs (DNLSEs) where the special feature is the nonlinear self-steepening of narrow pulses. More precisely, these additional regimes of MI and RWs for coupled DNLSEs depend on the mismatch in group velocities between the components, and the parameters for cubic nonlinearity and self-steepening. RWs considered in this paper differ from those of the NLSEs in terms of the amplification ratio and criteria of existence. Applications to optics and plasma physics are discussed.

A coupled "AB" system: Rogue waves and modulation instabilities.

Rogue waves are unexpectedly large and localized displacements from an equilibrium position or an otherwise calm background. For the nonlinear Schrödinger (NLS) model widely used in fluid mechanics and optics, these waves can occur only when dispersion and nonlinearity are of the same sign, a regime of modulation instability. For coupled NLS equations, rogue waves will arise even if dispersion and nonlinearity are of opposite signs in each component as new regimes of modulation instability will appear in the coupled system. The same phenomenon will be demonstrated here for a coupled "AB" system, a wave-current interaction model describing baroclinic instability processes in geophysical flows. Indeed, the onset of modulation instability correlates precisely with the existence criterion for rogue waves for this system. Transitions from "elevation" rogue waves to "depression" rogue waves are elucidated analytically. The dispersion relation as a polynomial of the fourth order may possess double pairs of complex roots, leading to multiple configurations of rogue waves for a given set of input parameters. For special parameter regimes, the dispersion relation reduces to a cubic polynomial, allowing the existence criterion for rogue waves to be computed explicitly. Numerical tests correlating modulation instability and evolution of rogue waves were conducted.

Pinned modes in two-dimensional lossy lattices with local gain and nonlinearity.

We introduce a system with one or two amplified nonlinear sites ('hot spots', HSs) embedded into a two-dimensional linear lossy lattice. The system describes an array of evanescently coupled optical or plasmonic waveguides, with gain applied to selected HS cores. The subject of the analysis is discrete solitons pinned to the HSs. The shape of the localized modes is found in quasi-analytical and numerical forms, using a truncated lattice for the analytical consideration. Stability eigenvalues are computed numerically, and the results are supplemented by direct numerical simulations. In the case of self-focusing nonlinearity, the modes pinned to a single HS are stable and unstable when the nonlinearity includes the cubic loss and gain, respectively. If the nonlinearity is self-defocusing, the unsaturated cubic gain acting at the HS supports stable modes in a small parametric area, whereas weak cubic loss gives rise to a bistability of the discrete solitons. Symmetric and antisymmetric modes pinned to a symmetric set of two HSs are also considered.

Pinned modes in lossy lattices with local gain and nonlinearity.

We introduce a discrete linear lossy system with an embedded "hot spot" (HS), i.e., a site carrying linear gain and complex cubic nonlinearity. The system can be used to model an array of optical or plasmonic waveguides, where selective excitation of particular cores is possible. Localized modes pinned to the HS are constructed in an implicit analytical form, and their stability is investigated numerically. Stability regions for the modes are obtained in the parameter space of the linear gain and cubic gain or loss. An essential result is that the interaction of the unsaturated cubic gain and self-defocusing nonlinearity can produce stable modes, although they may be destabilized by finite-amplitude perturbations. On the other hand, the interplay of the cubic loss and self-defocusing gives rise to a bistability.

Localized pulses for the quintic derivative nonlinear Schrödinger equation on a continuous-wave background.

Quintic derivative nonlinear Schrödinger equations arise in various physical contexts, notably in the study of hydrodynamic wave packets and media with negative refractive index. A procedure to isolate propagating wave patterns in such nonlinear Schrödinger equations is proposed which is based on two integrals of motion. As an illustration of the method, a "gray" solitary pulse, a "dark" localized mode with nonzero minimum in intensity on a continuous-wave background is identified.

Logarithmic nonlinear Schrödinger equation and irrotational, compressible flows: an exact solution.

A class of irrotational, isentropic, and compressible flows is studied theoretically by formulating the density and the velocity potential in a Madelung transformation. The resulting nonlinear Schrödinger equation is solved in terms of similarity variables. One particular family of exact solutions, valid for any ratio of the specific heat capacities of the gas, permits explicit expressions of the fluid properties and velocities in terms of time and spatial coordinates. Analytically, the density is a Gaussian function of the similarity variable, while the temperature is a function of time only. This method is applicable in one (1D), two, and three dimensional geometries. As a simple example, a 1D gas column, with mass injection on one side and a steadily translating wall on the other, can be formulated exactly. The connection with the evolution of an unsteady velocity potential will also be examined.

University-based peer health education in China: the Shantou experience.

OBJECTIVE: University-based peer health education is a recent development in China. The authors evaluated a newly implemented program in the Guangdong province. PARTICIPANTS AND METHODS: In September 2006, the authors conducted a cross-sectional study using self-administered questionnaires on 30 peer educators and 247 students. RESULTS: All peer educators and the majority of student respondents positively evaluated the program. Although students preferred to seek health information online, approximately one-quarter of the student respondents would contact peer educators. Third-year students were more than twice as likely (29.1%) to contact peer educators than were fourth-year students (13.1%). The peer educators perceived diet, physical activity, safer sex, and mental health as the most relevant student health topics. Peer educators cited acquiring factual information and medical skills, rather than personal development, as the most important things learned from the program. CONCLUSIONS: Despite some promising results, Western-based peer education models may require cultural adaptation for greater effectiveness in China.

Prevalence of albuminuria and cardiovascular risk profile in a referred cohort of patients with type 2 diabetes: an Asian perspective.

BACKGROUND: Microalbuminuria (MA) is a risk marker for diabetic nephropathy and cardiovascular (CV) disease (CVD) in patients with diabetes. This study aimed to describe the prevalence of albuminuria, CV risk factors, and treatments for renal and CV protection in an Asian population with type 2 diabetes. METHODS: This cross-sectional study conducted in eight Asian countries enrolled normotensive/hypertensive adults with type 2 diabetes without known proteinuria and/or non-diabetic kidney disease. Exclusion criteria were type 1 diabetes, menstruation, pregnancy, and acute fever. A single random urinary albumin/creatinine test was carried out in all patients. RESULTS: Of 8,561 patients, 14% had diabetic retinopathy, and 17% and 21% had history of CV disease and smoking, respectively. Normoalbuminuria was seen in 44%, MA in 44%, and macroalbuminuria in 12%. Target glycosylated hemoglobin (HbA1c) (<7%) was reached in only 37% of 3,834 patients with available values. Diabetes was managed by diet alone in 6%, while others received oral hypoglycemic drugs and/or insulin. In total, 75% did not reach target blood pressure (BP) of

A computational study on the biomechanical factors related to stent-graft models in the thoracic aorta.

Endovascular aortic stent-graft is a new, minimally invasive procedure for treating thoracic aortic diseases, and has quickly evolved to be one of the standard treatments subject to anatomic constraints. This procedure involves the placement of a self-expanding stent-graft system in a high-flow thoracic aorta. Stent-graft deployment in the thoracic aorta, especially close to the aortic arch, normally experiences a significant drag force which might lead to the risk of stent-graft failure. A comprehensive investigation on the biomechanical factors affecting the drag force on a stent-graft in the thoracic aorta is thus in order, and the goal is to perform an in-depth study on the contributing biomechanical factors. Three factors affecting the deployed stent-graft are considered, namely, the internal diameter of the vessel, the starting position of the graft and the diameter of curvature of the aortic arch. Computational fluid dynamic techniques are applied to model the blood flow. The inlet velocity and outlet pressure are assumed to be pulsatile. The three-dimensional continuity equation and the time-dependent Navier-Stokes equations for an incompressible fluid were solved numerically. The drag force due to the change of momentum within the stent-graft and the shear stress were calculated and analyzed. The drag force on a stent-graft will depend critically on the internal diameter and the starting position of stent-graft deployment. Larger internal diameter leads to larger drag force and the stent-graft deployed at the more distal position may be associated with significantly diminished drag force. Smaller diameter of curvature of the aortic arch probably results in a decline of the drag force on the stent-graft, even though this factor merely causes only a modest difference. These findings may have important implications for the choice and design of stent-grafts in the future.

Periodic waves in fiber Bragg gratings.

We construct two families of exact periodic solutions to the standard model of fiber Bragg grating (FBG) with Kerr nonlinearity. The solutions are named "sn" and "cn" waves, according to the elliptic functions used in their analytical representation. The sn wave exists only inside the FBG's spectral bandgap, while waves of the cn type may only exist at negative frequencies (omega<0), both inside and outside the bandgap. In the long-wave limit, the sn and cn families recover, respectively, the ordinary gap solitons, and (unstable) antidark and dark solitons. Stability of the periodic solutions is checked by direct numerical simulations and, in the case of the sn family, also through the calculation of instability growth rates for small perturbations. Although, rigorously speaking, all periodic solutions are unstable, a subfamily of practically stable sn waves, with a sufficiently large spatial period and omega>0, is identified. However, the sn waves with omega<0, as well as all cn solutions, are strongly unstable.

On stent-graft models in thoracic aortic endovascular repair: a computational investigation of the hemodynamic factors.

In treating thoracic aortic diseases, endovascular repair involves the placement of a self-expanding stent-graft system across the diseased thoracic aorta. Computational fluid dynamic techniques are applied to model the blood flow by numerically solving the three-dimensional continuity equation and the time-dependent Navier-Stokes equations for an incompressible fluid. From our results, high blood pressure level and high systolic slope of the pressure waveform will significantly increase the drag force on a stent-graft whereas high blood viscosity causes only a mild increase. It indicates that hemodynamic factors might have an important impact on the drag force and thus play a significant role in the risk of stent-graft failure.

A computational investigation on the effect of biomechanical factors related to stent-graft models in the thoracic aorta.

Endovascular aortic stent-graft is a new, minimally invasive procedure for treating thoracic aortic diseases, and has quickly evolved to be one of the standard treatments subject to anatomic constraints. Stent-graft deployment in the thoracic aorta, especially close to the aortic arch, normally experiences a significant drag force which might lead to the risk of stent-graft failure. A comprehensive investigation on the biomechanical factors affecting the drag force on a stent-graft in the thoracic aorta is thus in order, and the goal here is to perform an in-depth study on the contributing biomechanical factors. Three factors affecting the deployed stent-graft are considered, namely, the internal diameter of the vessel, the starting position of the graft and the diameter of curvature of the aortic arch. Computational fluid dynamic techniques are applied to model the blood flow. The three-dimensional continuity equation and the time-dependent Navier-Stokes equations for an incompressible fluid were solved numerically. The drag force due to the change of momentum within the stent-graft and the shear stress were calculated and analyzed. The drag force on a stent-graft will depend critically on the internal diameter and the starting position of stent-graft deployment. Larger internal diameter leads to larger drag force and the stent-graft deployed at the more distal position may be associated with significantly diminished drag force. Smaller diameter of curvature of the aortic arch results in a decrease of the drag force. These findings may have important implications for the choice and design of stent-grafts in the future.