Kwok Wing Chow

A class of doubly periodic waves for nonlinear evolution equations

Abstract A class of doubly periodic waves for several nonlinear evolution equations is studied by the Hirota bilinear method. Analytically these waves can be expressed as rational functions of elliptic functions with different moduli, and may correspond to standing as well as propagating waves. The two moduli are related by a condition determined as part of the solution process, and the condition translates into constraints on the wavenumbers allowed. Such solutions for the nonlinear Schrodinger equation agree with results derived earlier in the literature by a different method. The present method of combining the Hirota method, elliptic and theta functions is applicable to a wider class of equations, e.g., the Davey–Stewartson, the sinh-Poisson and the higher dimensional sine-Gordon equations. A long wave limit is studied for these special doubly periodic solutions of the Davey–Stewartson and Kadomtsev–Petviashvili equations, and the results are the generation of new solutions and the emergence of the component solitons as the fundamental building blocks, respectively. The validity of these doubly periodic solutions is verified by MATHEMATICA.

Darboux covariant Lax pairs and infinite conservation laws of the (2+1)-dimensional breaking soliton equation

In this paper, the binary Bell polynomials are applied to succinctly construct bilinear formulism, bilinear Backlund transformations, Lax pairs, and Darboux covariant Lax pairs for the (2+1)-dimensional breaking soliton equation. An extra auxiliary variable is introduced to get the bilinear formulism. The infinitely local conservation laws of the equation are found by virtue of its Lax equation and a generalized Miura transformation. All conserved densities and fluxes are given with explicit recursion formulas.

Ermakov–Ray–Reid systems in nonlinear optics

A hydrodynamics-type system incorporating a Madelung–Bohm-type quantum potential, as derived by Wagner et al via Maxwells equations and the paraxial approximation in nonlinear optics, is reduced to a nonlinear Schrodinger canonical form. A two-parameter nonlinear Ermakov–Ray–Reid system that arises from this model, and which governs the evolution of beam radii in an elliptically polarised medium is shown to be reducible to a classical Poschl–Teller equation. A class of exact solutions to the Ermakov-type system is constructed in terms of elliptic dn functions. It is established that integrable two-component Ermakov–Ray–Reid subsystems likewise arise in a coupled (2+1)-dimensional nonlinear optics model descriptive of the two-pulse interaction in a Kerr medium. The Hamiltonian structure of these subsystems allows their complete integration.

A computational fluid dynamic study of stent graft remodeling after endovascular repair of thoracic aortic dissections

OBJECTIVES Significant stent graft remodeling commonly occurs after endovascular repair of thoracic aortic dissections because of continuing expansion of the true lumen. A suboptimal proximal landing zone, minimal oversizing, and lack of a healthy distal attachment site are unique factors affecting long-term stent graft stability. We used computational fluid dynamic techniques to analyze the biomechanical factors associated with stent graft remodeling in these patients. PATIENTS AND METHODS A series of computational fluid dynamic models were constructed to investigate the biomechanical factors affecting the drag force on a thoracic stent graft. The resultant drag force as a net change of fluid momentum was calculated on the basis of varying three-dimensional geometry and deployment positions. A series of 12 patients with type B aortic dissections treated by thoracic stent graft and followed up for more than 12 months were then studied. Computed tomography transaxial images of each patient shortly after stent graft deployment and on subsequent follow-up were used to generate three-dimensional geometric models that were then fitted with a surface mesh. Computational fluid dynamic simulations were then performed on each stent graft model according to its geometric parameters to determine the actual change in drag force experienced by the stent graft as it remodels over time. RESULTS The drag force on the stent graft model increases linearly with its internal diameter and becomes highest when the deployment position is closer to the proximal arch. Aortic curvature is not a significant factor. Serial computed tomography scans of patients showed an increase in mean inlet area from 1030 mm(2) to 1140 mm(2), and mean outlet area from 586 mm(2) to 884 mm(2) (increase of 11% and 58%, respectively; P = .05, .01). These increases are associated with a change in resultant drag force on the stent graft from 21.0 N to 24.8 N (mean increase, 19.5%; range, 0%-63.2%; P = .002). There is a positive relationship between increase in drag force and increase in stent-graft area. CONCLUSION The drag force on thoracic stent grafts is high. A significant change in stent-graft diameter occurs after endovascular repair for type B dissections, which is associated with an increase in hemodynamic drag force. These stent grafts may be subjected to a higher risk of distal migration, and continuing surveillance is mandatory.

High Sensitivity, Wearable, Piezoresistive Pressure Sensors Based on Irregular Microhump Structures and Its Applications in Body Motion Sensing.

UNLABELLED A pressure sensor based on irregular microhump patterns has been proposed and developed. The devices show high sensitivity and broad operating pressure regime while comparing with regular micropattern devices. Finite element analysis (FEA) is utilized to confirm the sensing mechanism and predict the performance of the pressure sensor based on the microhump structures. Silicon carbide sandpaper is employed as the mold to develop polydimethylsiloxane (PDMS) microhump patterns with various sizes. The active layer of the piezoresistive pressure sensor is developed by spin coating PEDOT PSS on top of the patterned PDMS. The devices show an averaged sensitivity as high as 851 kPa(-1) , broad operating pressure range (20 kPa), low operating power (100 nW), and fast response speed (6.7 kHz). Owing to their flexible properties, the devices are applied to human body motion sensing and radial artery pulse. These flexible high sensitivity devices show great potential in the next generation of smart sensors for robotics, real-time health monitoring, and biomedical applications.

Rogue wave modes for the long wave-short wave resonance model

The long wave–short wave resonance model arises physically when the phase velocity of a long wave matches the group velocity of a short wave. It is a system of nonlinear evolution equations solvable by the Hirota bilinear method and also possesses a Lax pair formulation. ‘‘Rogue wave’’ modes, algebraically localized entities in both space and time, are constructed from the breathers by a singular limit involving a ‘‘coalescence’’ of wavenumbers in the long

Transcritical flow of a stratified fluid: The forced extended Korteweg–de Vries model

Transcritical, or resonant, flow of a stratified fluid over an obstacle is studied using a forced extended Korteweg–de Vries model. This model is particularly relevant for a two-layer fluid when the layer depths are near critical, but can also be useful in other similar circumstances. Both quadratic and cubic nonlinearities are present and they are balanced by third-order dispersion. We consider both possible signs for the cubic nonlinear term but emphasize the less-studied case when the cubic nonlinear term and the dispersion term have the same-signed coefficients. In this case, our numerical computations show that two kinds of solitary waves are found in certain parameter regimes. One kind is similar to those of the well-known forced Korteweg–de Vries model and occurs when the cubic nonlinear term is rather small, while the other kind is irregularly generated waves of variable amplitude, which may continually interact. To explain this phenomenon, we develop a hydraulic theory in which the dispersion ter...

Transmission and stability of solitary pulses in complex Ginzburg-Landau equations with variable coefficients

A class of complex Ginzburg–Landau (CGL) equations with variable coefficients is solved exactly by means of the Hirota bilinear method. Two novel features, elaborated in recent works on the bilinear method, are incorporated. One is a modified definition of the bilinear operator, which has been used to construct pulse, hole and front solutions for equations with constant coefficients. The other is the usage of time- or space-dependent wave numbers, which was employed to handle nonlinear Schrodinger (NLS) equations with variable coefficient. One-soliton solutions of the CGL equations with variable coefficients are obtained in an analytical form. A restriction imposed by the method is that the coefficient of the second-order dispersion must be real. However, nonlinear, loss (or gain) is permitted. A simple example of an exponentially modulated dispersion profile is worked out in detail to illustrate the principle. The competition between the linear gain and nonlinear loss, and vice versa , is investigated. T...

Generation of solitary waves by transcritical flow over a step

It is well-known that transcritical flow over a localized obstacle generates upstream and downstream nonlinear wavetrains. The flow has been successfully modelled in the framework of the forced Korteweg-de Vries equation, where numerical and asymptotic analytical solutions have shown that the upstream and downstream nonlinear wavetrains have the structure of unsteady undular bores, connected by a locally steady solution over the obstacle, which is elevated on the upstream side and depressed on the downstream side. In this paper we consider the analogous transcritical flow over a step, primarily in the context of water waves. We use numerical and asymptotic analytical solutions of the forced Korteweg-de Vries equation, together with numerical solutions of the full Euler equations, to demonstrate that a positive step generates only an upstream-propagating undular bore, and a negative step generates only a downstream-propagating undular bore.

Periodic waves in bimodal optical fibers

We consider coupled non-linear Schrodinger equations (CNLSE) which govern the propagation of non-linear waves in bimodal optical fibers. The non-linear transform of a dual-frequency signal is used to generate an ultra-short-pulse train. To predict the energy and width of pulses in the train, we derive three new types of travelling periodic-wave solutions, using the Hirota bilinear method. We also show that all the previously reported periodic wave solutions of CNLSE can be derived in a systematic way, using the Hirota method.