Cauchy Problem for Differential Equation with Caputo Derivative

The paper is devoted to the study of the Cauchy problem for a nonlinear differential equation of complex order with the Caputo fractional derivative. The equivalence of this problem and a nonlinear Volterra integral equation in the space of continuously differentiable functions is established. On the basis of this result, the existence and uniqueness of the solution of the considered Cauchy problem is proved. The approximate-iterative method by Dzjadyk is used to obtain the approximate solution of this problem. Two numerical examples are given.

Soliton's eigenvalue based analysis on the generation mechanism of rogue wave phenomenon in optical fibers exhibiting weak third order dispersion

One of the extraordinary aspects of nonlinear wave evolution which has been observed as the spontaneous occurrence of astonishing and statistically extraordinary amplitude wave is called rogue wave. We show that the eigenvalues of the associated equation of nonlinear Schrödinger equation are almost constant in the vicinity of rogue wave and we validate that optical rogue waves are formed by the collision between quasi-solitons in anomalous dispersion fiber exhibiting weak third order dispersion.

Analysis of rogue wave phenomenon in optical fiber based on soliton's eigenvalue

The impact of third-order dispersion (TOD) on optical rogue wave phenomenon is investigated numerically. We validate the TOD coefficient by utilizing the eigenvalue of the associated equation of the nonlinear Schrödinger equation (NLSE). © 2014 OSA.

Multi-threaded parallel numerical modelling of femtosecond pulse propagation in laser machining

Femtosecond laser microfabrication has emerged over the last decade as a 3D flexible technology in photonics. Numerical simulations provide an important insight into spatial and temporal beam and pulse shaping during the course of extremely intricate nonlinear propagation (see e.g. [1,2]). Electromagnetics of such propagation is typically described in the form of the generalized Non-Linear Schrdinger Equation (NLSE) coupled with Drude model for plasma [3]. In this paper we consider a multi-threaded parallel numerical solution for a specific model which describes femtosecond laser pulse propagation in transparent media [4, 5]. However our approach can be extended to similar models. The numerical code is implemented in NVIDIA Graphics Processing Unit (GPU) which provides an effitient hardware platform for multi-threded computing. We compare the performance of the described below parallel code implementated for GPU using CUDA programming interface [3] with a serial CPU version used in our previous papers [4,5]. © 2011 IEEE.

Gain dependent pulse regimes transitions in a dissipative dispersion-managed fibre laser

For the first time, we demonstrate the possibility to switch between three distinct pulse regimes in a dissipative dispersion-managed (DM) fibre laser by solely controlling the gain saturation energy. Nonlinear Schrödinger equation based simulations show the transitions between hyper-Gaussian similaritons, parabolic similaritons, and dissipative solitons in the same laser cavity. It is also shown that such transitions exist in a wide dispersion range from all-normal to slightly net-normal dispersion. This work demonstrates that besides dispersion and filter managements gain saturation energy can be a new degree of freedom to manage pulse regimes in DM fibre lasers, which offers flexibility in designing ultrafast fibre lasers. Also, the result indicates that in contrast to conservative soliton lasers whose intensity profiles are unique, dissipative DM lasers show diversity in pulse shapes. The findings not only give a better understanding of pulse shaping mechanisms in mode-locked lasers, but also provide insight into dissipative systems.

One-dimensional optical wave turbulence:experiment and theory

We present a review of the latest developments in one-dimensional (1D) optical wave turbulence (OWT). Based on an original experimental setup that allows for the implementation of 1D OWT, we are able to show that an inverse cascade occurs through the spontaneous evolution of the nonlinear field up to the point when modulational instability leads to soliton formation. After solitons are formed, further interaction of the solitons among themselves and with incoherent waves leads to a final condensate state dominated by a single strong soliton. Motivated by the observations, we develop a theoretical description, showing that the inverse cascade develops through six-wave interaction, and that this is the basic mechanism of nonlinear wave coupling for 1D OWT. We describe theory, numerics and experimental observations while trying to incorporate all the different aspects into a consistent context. The experimental system is described by two coupled nonlinear equations, which we explore within two wave limits allowing for the expression of the evolution of the complex amplitude in a single dynamical equation. The long-wave limit corresponds to waves with wave numbers smaller than the electrical coherence length of the liquid crystal, and the opposite limit, when wave numbers are larger. We show that both of these systems are of a dual cascade type, analogous to two-dimensional (2D) turbulence, which can be described by wave turbulence (WT) theory, and conclude that the cascades are induced by a six-wave resonant interaction process. WT theory predicts several stationary solutions (non-equilibrium and thermodynamic) to both the long- and short-wave systems, and we investigate the necessary conditions required for their realization. Interestingly, the long-wave system is close to the integrable 1D nonlinear Schrödinger equation (NLSE) (which contains exact nonlinear soliton solutions), and as a result during the inverse cascade, nonlinearity of the system at low wave numbers becomes strong. Subsequently, due to the focusing nature of the nonlinearity, this leads to modulational instability (MI) of the condensate and the formation of solitons. Finally, with the aid of the probability density function (PDF) description of WT theory, we explain the coexistence and mutual interactions between solitons and the weakly nonlinear random wave background in the form of a wave turbulence life cycle (WTLC).

Weak Langmuir optical turbulence in a fiber cavity

We study theoretically and numerically the dynamics of a passive optical fiber ring cavity pumped by a highly incoherent wave: an incoherently injected fiber laser. The theoretical analysis reveals that the turbulent dynamics of the cavity is dominated by the Raman effect. The forced-dissipative nature of the fiber cavity is responsible for a large diversity of turbulent behaviors: Aside from nonequilibrium statistical stationary states, we report the formation of a periodic pattern of spectral incoherent solitons, or the formation of different types of spectral singularities, e.g., dispersive shock waves and incoherent spectral collapse behaviors. We derive a mean-field kinetic equation that describes in detail the different turbulent regimes of the cavity and whose structure is formally analogous to the weak Langmuir turbulence kinetic equation in the presence of forcing and damping. A quantitative agreement is obtained between the simulations of the nonlinear Schrödinger equation with cavity boundary conditions and those of the mean-field kinetic equation and the corresponding singular integrodifferential reduction, without using adjustable parameters. We discuss the possible realization of a fiber cavity experimental setup in which the theoretical predictions can be observed and studied.

Freak waves and modulational dynamics in plasmas with negative ions

The nonlinear dynamics of modulated electrostatic wavepackets propagating in negativeion plasmas is investigated from first principles. A nonlinear Schrödinger equation is derived by adopting a multiscale technique. The stability of breather- like (bright envelope soliton) structures, considered as a precursor to freak wave (rogue wave) formation, is investigated and then tested via numerical simulations.

Atténuation d'effets non linéaires dans les lasers fibrés de haute puissance opérés en régime continu

Les lasers à fibre de haute puissance sont maintenant la solution privilégiée pour les applications de découpe industrielle. Le développement de lasers pour ces applications n’est pas simple en raison des contraintes qu’imposent les normes industrielles. La fabrication de lasers fibrés de plus en plus puissants est limitée par l’utilisation d’une fibre de gain avec une petite surface de mode propice aux effets non linéaires, d’où l’intérêt de développer de nouvelles techniques permettant l’atténuation de ceux-ci. Les expériences et simulations effectuées dans ce mémoire montrent que les modèles décrivant le lien entre la puissance laser et les effets non linéaires dans le cadre de l’analyse de fibres passives ne peuvent pas être utilisés pour l’analyse des effets non linéaires dans les lasers de haute puissance, des modèles plus généraux doivent donc développés. Il est montré que le choix de l’architecture laser influence les effets non linéaires. En utilisant l’équation de Schrödinger non linéaire généralisée, il a aussi été possible de montrer que pour une architecture en co-propagation, la diffusion Raman influence l’élargissement spectral. Finalement, les expériences et les simulations effectuées montrent qu’augmenter la réflectivité nominale et largeur de bande du réseau légèrement réfléchissant de la cavité permet d’atténuer la diffusion Raman, notamment en réduisant le gain Raman effectif.

Lie group symmetry analysis for granular media stress equations

The Airy stress function, although frequently employed in classical linear elasticity, does not receive similar usage for granular media problems. For plane strain quasi-static deformations of a cohesionless Coulomb–Mohr granular solid, a single nonlinear partial differential equation is formulated for the Airy stress function by combining the equilibrium equations with the yield condition. This has certain advantages from the usual approach, in which two stress invariants and a stress angle are introduced, and a system of two partial differential equations is needed to describe the flow. In the present study, the symmetry analysis of differential equations is utilised for our single partial differential equation, and by computing an optimal system of one-dimensional Lie algebras, a complete set of group-invariant solutions is derived. By this it is meant that any group-invariant solution of the governing partial differential equation (provided it can be derived via the classical symmetries method) may be obtained as a member of this set by a suitable group transformation. For general values of the parameters (angle of internal friction and gravity g) it is found there are three distinct classes of solutions which correspond to granular flows considered previously in the literature. For the two limiting cases of high angle of internal friction and zero gravity, the governing partial differential equation admit larger families of Lie point symmetries, and from these symmetries, further solutions are derived, many of which are new. Furthermore, the majority of these solutions are exact, which is rare for granular flow, especially in the case of gravity driven flows.

Parameter estimation for a phenomenological model of the cardiac action potential

The action potential (ap) of a cardiac cell is made up of a complex balance of ionic currents which flow across the cell membrane in response to electrical excitation of the cell. Biophysically detailed mathematical models of the ap have grown larger in terms of the variables and parameters required to model new findings in subcellular ionic mechanisms. The fitting of parameters to such models has seen a large degree of parameter and module re-use from earlier models. An alternative method for modelling electrically exciteable cardiac tissue is a phenomenological model, which reconstructs tissue level ap wave behaviour without subcellular details. A new parameter estimation technique to fit the morphology of the ap in a four variable phenomenological model is presented. An approximation of a nonlinear ordinary differential equation model is established that corresponds to the given phenomenological model of the cardiac ap. The parameter estimation problem is converted into a minimisation problem for the unknown parameters. A modified hybrid Nelder–Mead simplex search and particle swarm optimization is then used to solve the minimisation problem for the unknown parameters. The successful fitting of data generated from a well known biophysically detailed model is demonstrated. A successful fit to an experimental ap recording that contains both noise and experimental artefacts is also produced. The parameter estimation method’s ability to fit a complex morphology to a model with substantially more parameters than previously used is established.

A comparison of stability and bifurcation criteria for inflated spherical elastic shells

The nonlinear stability analysis introduced by Chen and Haughton [1] is employed to study the full nonlinear stability of the non-homogeneous spherically symmetric deformation of an elastic thick-walled sphere. The shell is composed of an arbitrary homogeneous, incompressible elastic material. The stability criterion ultimately requires the solution of a third-order nonlinear ordinary differential equation. Numerical calculations performed for a wide variety of well-known incompressible materials are then compared with existing bifurcation results and are found to be identical. Further analysis and comparison between stability and bifurcation are conducted for the case of thin shells and we prove by direct calculation that the two criteria are identical for all modes and all materials.

Modeling of Channel Potential and Subthreshold Slope of Symmetric Double-Gate Transistor

A new physically based classical continuous potential distribution model, particularly considering the channel center, is proposed for a short-channel undoped body symmetrical double-gate transistor. It involves a novel technique for solving the 2-D nonlinear Poisson's equation in a rectangular coordinate system, which makes the model valid from weak to strong inversion regimes and from the channel center to the surface. We demonstrated, using the proposed model, that the channel potential versus gate voltage characteristics for the devices having equal channel lengths but different thicknesses pass through a single common point (termed ``crossover point''). Based on the potential model, a new compact model for the subthreshold swing is formulated. It is shown that for the devices having very high short-channel effects (SCE), the effective subthreshold slope factor is mainly dictated by the potential close to the channel center rather than the surface. SCEs and drain-induced barrier lowering are also assessed using the proposed model and validated against a professional numerical device simulator.

Unsteady Incompressible Two-Dimensional and Axisymmetric Turbulent Boundary Layer Flows

The unsteady turbulent incompressible boundary-layer flow over two-dimensional and axisymmetric bodies with pressure gradient has been studied. An eddy-viscosity model has been used to model the Reynolds shear stress. The unsteadiness is due to variations in the free stream velocity with time. The nonlinear partial differential equation with three independent variables governing the flow has been solved using Keller's Box method. The results indicate that the free stram velocity distribution exerts strong influence on the boundary-layer characteristics. The point of zero skin friction is found to move upstream as time increases.