K. W. Chow

Periodic solutions for a system of four coupled nonlinear Schrödinger equations

Abstract Exact, periodic solutions for a system of four coupled nonlinear Schrodinger equations are obtained by the Hirota bilinear method and theta functions identities. The solutions involve products of up to four elliptic functions. The validity is verified independently by a computer algebra software. The long wave limit is studied.

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.

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.

Embedded fibre Bragg grating sensors for non-uniform strain sensing in composite structures

A methodology for evaluating the response of embedded fibre Bragg grating (FBG) sensors in composite structures based on the strain in a host material is introduced. In applications of embedded FBG sensors as strain sensing devices, it is generally assumed that the strain experienced in a fibre core is the same as the one measured in the host material. The FBG sensor is usually calibrated by a strain gauge through a tensile test, centred on obtaining the relationship between the surface strain in the host material and the corresponding Bragg wavelength shift obtained from the FBG sensor. However, such a calibration result can only be valid for uniform strain measurement. When the strain distribution along a grating is non-uniform, average strain measured by the strain gauge cannot truly reflect the in-fibre strain of the FBG sensor. Indeed, the peak in the reflection spectrum becomes broad, may even split into multiple peaks, in sharp contrast with a single sharp peak found in the case of the uniform strain measurement. In this paper, a strain transfer mechanism of optical fibre embedded composite structure is employed to estimate the non-uniform strain distribution in the fibre core. This in-fibre strain distribution is then utilized to simulate the response of the FBG sensor based on a transfer-matrix formulation. Validation of the proposed method is preceded by comparing the reflection spectra obtained from the simulations with those obtained from experiments.

Breathers and rogue waves for a third order nonlocal partial differential equation by a bilinear transformation

Abstract Breathers and rogue waves as exact solutions of a nonlocal partial differential equation of the third order are derived by a bilinear transformation. Breathers denote families of pulsating modes and can occur for both continuous and discrete systems. Rogue waves are localized in both space and time, and are obtained theoretically as a limiting case of breathers with indefinitely large periods. Both entities are demonstrated analytically to exist for special classes of nonlocal equations relevant to optical waveguides.

Another exact solution for two-dimensional, inviscid sinh Poisson vortex arrays

Arrays of vortices are considered for two-dimensional, inviscid flows when the functional relationship between the stream function and the vorticity is a hyperbolic sine. An exact solution as a doubly periodic array of vortices is expressed in terms of the Jacobi elliptic functions. There is a threshold value of the period parameter such that a transition from globally smooth distributions of vorticity to singular distributions occurs.

Vortex arrays for sinh-Poisson equation of two-dimensional fluids: Equilibria and stability

The sinh-Poisson equation describes a stream function configuration of a stationary two-dimensional (2D) Euler flow. We study two classes of its exact solutions for doubly periodic domains (or doubly periodic vortex arrays in the plane). Both types contain vortex dipoles of different configurations, an elongated “cat-eye” pattern, and a “diagonal” (symmetric) configuration. We derive two new solutions, one for each class. The first one is a generalization of the Mallier–Maslowe vortices, while the second one consists of two corotating vortices in a square cell. Next, we examine the dynamic stability of such vortex dipoles to initial perturbations, by numerical simulations of the 2D Euler flows on periodic domains. One typical member from each class is chosen for analysis. The diagonally symmetric equilibrium maintains stability for all (even strong) perturbations, whereas the cat-eye pattern relaxes to a more stable dipole of the diagonal type.

Inviscid two dimensional vortex dynamics and a soliton expansion of the sinh-Poisson equation

The dynamics of inviscid, steady, two dimensional flows is examined for the case of a hyperbolic sine functional relation between the vorticity and the stream function. The 2-soliton solution of the sinh-Poisson equation with complex wavenumbers will reproduce the Mallier-Maslowe pattern, a row of counter-rotating vortices. A special 4-soliton solution is derived and the corresponding flow configuration is studied. By choosing special wavenumbers complex flows bounded by two rigid walls can result. A conjecture regarding the number of recirculation regions and the wavenumber of the soliton expansion is offered. The validity of the new solution is verified independently by direct differentiation with a computer algebra software. The circulation and the vorticity of these novel flow patterns are finite and are expressed in terms of well defined integrals. The questions of the linear stability and the nonlinear evolution of a finite amplitude disturbance of these steady vortices are left for future studies.

Analytic doubly periodic wave patterns for the integrable discrete nonlinear Schrödinger (Ablowitz–Ladik) model

We derive two new solutions in terms of elliptic functions, one for the dark and one for the bright soliton regime, for the semi-discrete cubic nonlinear Schrodinger equation of Ablowitz and Ladik. When considered in the complex plane, these two solutions are identical. In the continuum limit, they reduce to known elliptic function solutions. In the long wave limit, the dark one reduces to the collision of two discrete dark solitons, and the bright one to a discrete breather.

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.