Traveling Wave Analysis Research Articles

Travelling wave analysis on high-speed performance of Q-modulated distributed feedback laser

The structure of a Q-modulated distributed feedback laser is designed and simulated. A large reflectivity modulation of the rear reflector is achieved by using an anti-resonant cavity formed by two deep trenches with the one between the modulator and phase section filled by a high index dielectric material. The travelling wave model is presented to analyze the high speed performance of the laser. Due to the effect of the wave propagation in the structure, the modulation extinction ratio decreases with increasing cavity length. It is shown that 40 Gb/s RZ signal modulation can be achieved with an extinction ratio of 7 dB and 10 dB, respectively, for a cavity length of 500 μm and 300 μm.

Green–Naghdi type III viscous fluids

We investigate constitutive theory for a new type of thermoviscous fluid. Equations are derived which reduce to the classical heat conduction equation under Fourier’s law when the new parameters tend to zero. Basic solutions for parallel flows are found and a travelling wave analysis is presented.

Traveling wave analysis for a mathematical model of malignant tumor invasion

This paper is concerned with traveling wave solutions to a malignant tumor invasion model. In 1999, Perumpanani et al. presented a mathematical model of malignant tumor cell invasion into connective tissue without cell diffusion. In their and subsequent researches, traveling waves have been studied mainly by numerical methods. In this paper, the existence of smooth traveling wave solutions to the model is proved rigorously. In order to show the existence of traveling waves, one considers heteroclinic orbits on a phase plane and constructs an invariant region for the orbits. Moreover, their asymptotic behavior is investigated. Finally, a remark on discontinuous traveling waves is given.

Some implications of Scale Relativity theory in avascular stages of growth of solid tumors in the presence of an immune system response

We present a traveling-wave analysis of a reduced mathematical model describing the growth of a solid tumor in the presence of an immune system response in the framework of Scale Relativity theory. Attention is focused upon the attack of tumor cells by tumor-infiltrating cytotoxic lymphocytes (TICLs), in a small multicellular tumor, without necrosis and at some stage prior to (tumor-induced) angiogenesis. For a particular choice of parameters, the underlying system of partial differential equations is able to simulate the well-documented phenomenon of cancer dormancy and propagation of a perturbation in the tumor cell concentration by cnoidal modes, by depicting spatially heterogeneous tumor cell distributions that are characterized by a relatively small total number of tumor cells. This behavior is consistent with several immunomorphological investigations. Moreover, the alteration of certain parameters of the model is enough to induce soliton like modes and soliton packets into the system, which in turn result in tumor invasion in the form of a standard traveling wave. In the same framework of Scale Relativity theory, a very important feature of malignant tumors also results, that even in avascular stages they might propagate and invade healthy tissues, by means of a diffusion on a Newtonian fluid.

Folds, canards and shocks in advection–reaction–diffusion models

Tactically driven cell movement modelled by coupled advection–reaction–diffusion (ARD) equations typically exhibit smooth travelling waves, and less frequently sharp interfaces in the wave form. We study the existence of travelling waves with smooth and sharp interfaces in coupled ARD models by using geometric singular perturbation techniques. In particular, we show that a travelling wave analysis under an appropriate Liénard transformation reveals a generic fold condition to observe shock-like interfaces in the wave form. This geometric approach further explains automatically well-known jump and entropy conditions for shocks in hyperbolic PDE theory (Rankine–Hugoniot and Lax conditions). Our analysis also shows that canards, a special class of solutions within singular perturbation problems, play an important role in the construction of travelling waves with smooth and sharp interfaces.

TRAVELLING WAVE SOLUTIONS FOR DOUBLY DEGENERATE REACTION–DIFFUSION EQUATIONS

AbstractThis paper concerns a nonlinear doubly degenerate reaction–diffusion equation which appears in a bacterial growth model and is also of considerable mathematical interest. A travelling wave analysis for the equation is carried out. In particular, the qualitative behaviour of both sharp and smooth travelling wave solutions is analysed. This travelling wave behaviour is also verified by some numerical computations for a special case.

Transient EMF induced in LV cables due to wind turbine direct lightning strike

This paper presents a novel, easy to use, engineering method for determining the transient electromotive force (EMF) induced in low-voltage (LV) cables, connecting the wind turbine with a near-by transformer, in the event of direct lightning strike into the top of the wind turbine tower. Proposed method is based on the application of the travelling wave analysis onto the system consisted of wind turbine tower, earthing system of wind turbine, earthing system of near-by transformer station and LV cables connecting the wind turbine with associated transformer. Hence, this design gives rise to a complex, mutually connected, earthing system. Direct lightning strike to the wind turbine initiates a travelling wave process in the system consisted of lightning channel, wind turbine tower and earthing system of the wind turbine. Due to the transient nature of the observed phenomenon, current and voltage states at the earthing system as well as in the associated low-voltage cables are formed through the propagation and reflection of the accompanying travelling waves. Transient EMF induced in LV cables could endanger cable main insulation and insulation of the associated transformer LV winding. Developed theory is subsequently applied on the concrete wind turbine example.

An application of traveling wave analysis in economic growth model

A temporal–spatial economic growth model is established in this paper. As a useful tool, traveling wave analysis is used to analyze technological growth and diffusion. Numerical simulation shows that this model has perfect performance.

Travelling-wave analysis of a model describing tissue degradation by bacteria

We study travelling-wave solutions for a reaction-diffusion system arising as a model for host-tissue degradation by bacteria. This system consists of a parabolic equation coupled with an ordinary differential equation. For large values of the ‘degradation-rate parameter’ solutions are well approximated by solutions of a Stefan-like free boundary problem, for which travelling-wave solutions can be found explicitly. Our aim is to prove the existence of travelling waves for all sufficiently large wave speeds for the original reaction-diffusion system and to determine the minimal speed. We prove that for all sufficiently large degradation rates, the minimal speed is identical to the minimal speed of the limit problem. In particular, in this parameter range,non-linearselection of the minimal speed occurs.

Limit cycles in the presence of convection: A traveling wave analysis

We consider a diffusion model with limit cycle reaction functions. In an unbounded domain, diffusion spreads the pattern outwards from the source. Convection adds instability to the reaction-diffusion system. The result of this instability is a readiness to create a pattern. We choose the Lambda-Omega reaction functions for their simple limit cycle. We carry out a transformation of the dependent variables into polar form. From this we consider the initiation of the pattern to approximate a traveling wave. We carry out numerical experiments to test our analysis. These confirm the premise of the analysis, that the initiation can be modeled by a traveling wave. Furthermore, the analysis produces a good estimate of the numerical results. Most significantly, we confirm that the pattern consists of two different types.

Traveling wave solutions of a reaction–diffusion model for bacterial growth

In this paper, we consider a reaction–diffusion model for the bacterial growth. Mathematical analysis on the traveling wave solutions of the model is performed. This includes traveling wave analysis and numerical simulations of wave front propagation for a special case. Specifically, we show that such solutions exist only for wave speeds greater than some minimum speed giving wave with a sharp front. The minimum speed is estimated and the wave profile is calculated and compared with different numerical methods.

Traveling wave solutions of a nonlinear reaction–diffusion–chemotaxis model for bacterial pattern formation

In this paper we consider a nonlinear reaction–diffusion–chemotaxis model for the description of the spatiotemporal evolution of the bacteria of the type Paenibacillus dendritiformis on a thin layer of agar in a Petri dish. We perform a traveling wave analysis for the model equation showing the existence of traveling wave solutions, in particular, the sharp wave front type solutions with minimum speed. Further, we present numerical investigations for a special case. The minimum speed is estimated and the profile of the traveling wave solution is calculated and compared for different numerical methods.

Fundamental mistuning model for determining system properties and predicting vibratory response of bladed disks

A reduced order model called the Fundamental Mistuning Model (FMM) accurately predicts vibratory response of a bladed disk system. The FMM software may describe the normal modes and natural frequencies of a mistuned bladed disk using only its tuned system frequencies and the frequency mistuning of each blade/disk sector (i.e., the sector frequencies). The FMM system identification methods—basic and advanced FMM ID methods—use the normal (i.e., mistuned) modes and natural frequencies of the mistuned bladed disk to determine sector frequencies as well as tuned system frequencies. FMM may predict how much the bladed disk will vibrate under the operating (rotating) conditions. Field calibration and testing of the blades may be performed using traveling wave analysis and FMM ID methods. The FMM model can be generated completely from experimental data. Because of FMM's simplicity, no special interfaces are required for FMM to be compatible with a finite element model. Because of the rules governing abstracts, this abstract should not be used to construe the claims.

Electrooptic Polymer Ring Resonator Modulation up to 165 GHz

Modulation is demonstrated at 84, 111, 139, and 165 GHz resonances of a traveling-wave electrooptic polymer ring-resonator-based modulator. The modulation response is characterized throughout the W-band, illustrating the resonant response at 84 and 111 GHz. A traveling-wave analysis that includes the compound effect of microwave loss and optical/microwave velocity mismatch in a ring-resonator-based modulator is presented and shows a good agreement with experimental results. The ring modulator shows superior performance compared to the Mach-Zehnder modulator in the presence of these limitations when both structures have the same equivalent low-frequency Vpi

Accurate computation of traveling wave solutions of some nonlinear diffusion equations

We consider a class of degenerate diffusion reaction equations where the nonlinearity is assumed to be singular at zero. We carry out a traveling wave analysis for these equations showing the existence of different types of traveling wave solutions, in particular, sharp type solution with minimum speed. We use two different methods for the numerical computation of such wave solutions for a special case of these equations. One of the methods involve the traveling wave equations and leads to accurate computation of wave profiles and speeds. The second method is to solve an initial-boundary-value problem with an adaptive traveling wave condition for the partial differential equation.

Mathematical modelling of the loss of tissue compression responsiveness and its role in solid tumour development

This paper presents a mathematical model of normal and abnormal tissue growth. The modelling focuses on the potential role that stress responsiveness may play in causing proliferative disorders which are at the basis of the development of avascular tumours. In particular, we study how an incorrect sensing of its compression state by a cell population can represent a clonal advantage and can generate hyperplasia and tumour growth with well-known characteristics such as compression of the tissue, structural changes in the extracellular matrix, change in the percentage of cell type (normal or abnormal), extracellular matrix and extracellular liquid. A spatially independent description of the phenomenon is given initially by a system of non-linear ordinary differential equations which is explicitly solved in some cases of biological interest showing a first phase in which some abnormal cells simply replace the normal ones, a second phase in which the hyper-proliferation of the abnormal cells causes a progressive compression within the tissue itself and a third phase in which the tissue reaches a compressed state, which presses on the surrounding environment. A travelling wave analysis is also performed which gives an estimate of the velocity of the growing mass.

Wave interactions in suliciu model for dynamic phase transitions

Elementary waves in Suliciu model for dynamic phase transitions are obtained through traveling wave analysis. For any given initial data with two pieces of constant states, the Riemann solutions are constructed as a combination of elementary waves. When the initial profile contains three pieces of constant states, the solution may be constructed from the Riemann solutions, with each two adjacent states connected by elementary waves. A new Riemann problem forms when these two waves collide. Through the exploration of these Riemann problems, the outcome of wave interactions may be classified in a suitable parametric space.

Lagrangian averaging for the 1D compressible Euler equations

We consider a $1$-dimensional Lagrangian averaged model for an inviscid compressible fluid. As previously introduced in the literature, such equations are designed to model the effect of fluctuations upon the mean flow in compressible fluids. This paper presents a traveling wave analysis and a numerical study for such a model. The discussion is centered around two issues. One relates to the intriguing wave motions supported by this model. The other is the appropriateness of using Lagrangian-averaged models for compressible flow to approximate shock wave solutions of the compressible Euler equations.

Pathogens can Slow Down or Reverse Invasion Fronts of their Hosts

Infectious diseases are often regarded as possible explanations for the sudden collapse of biological invasions. This phenomenon is characterized by a host species, which firstly can successfully establish in a non-native habitat, but then spontaneously disappears again. This study proposes a reaction-diffusion model consisting of a simple SI disease with vital dynamics of Allee effect type. By way of travelling wave analysis, conditions are derived under which the invasion of the host population is slowed down, stopped or reversed as a consequence of a subsequently introduced disease. Hence, pathogens can dramatically control the rate of spread of invasive species.

Travelling-wave analysis of a model of the immune response to cancer

In this paper we present a travelling-wave analysis of a mathematical model describing the growth of a solid tumour in the presence of an immune system response. From a modelling perspective, attention is focused upon the attack of tumour cells by tumour infiltrating cytotoxic lymphocytes (TICLs), in a small multicellular tumour, without necrosis and at some stage prior to (tumour-induced) angiogenesis. As we have shown in previous work, for a particular choice of parameters, the underlying reaction–diffusion–chemotaxis system of partial differential equations is able to simulate the well-documented phenomenon of cancer dormancy by depicting spatially heterogeneous tumour cell distributions that are characterized by a relatively small total number of tumour cells. This behaviour is consistent with several immunomorphological investigations. Moreover, the alteration of certain parameters of the model is enough to induce bifurcations into the system, which in turn result in tumour invasion in the form of a standard travelling wave. The work presented in this paper complements the bifurcation analysis undertaken by Matzavinos et al. [Math. Med. Biol. IMA 21 (2004) 1–34] and establishes the existence of travelling-wave solutions for the system under discussion by promoting the understanding of the geometry of an appropriate phase space. To cite this article: A. Matzavinos, M.A.J. Chaplain, C. R. Biologies 327 (2004).