Nonlinear dynamic analysis of a V-shaped microcantilever of an atomic force microscope

This paper is devoted to investigate the nonlinear behaviors of a V-shaped microcantilever of an atomic force microscope (AFM) operating in its two major modes: amplitude modulation and frequency modulation. The nonlinear behavior of the AFM is due to the nonlinear nature of the AFM tip–sample interaction caused by the Van der Waals attraction/repulsion force. Considering the V-shaped microcantilever as a flexible continuous system, the resonant frequencies, mode shapes, governing nonlinear partial and ordinary differential equations (PDE and ODE) of motion, boundary conditions, frequency and time responses, potential function and phase-plane of the system are obtained analytically. The governing PDE is determined by employing the Hamilton principle. Subsequently, the Galerkin method is utilized to gain the governing nonlinear ODE. Afterward, the resulting ODE is analytically solved by means of some perturbation techniques including the method of multiple scales and the Lindsted–Poincare method. In addition, the effects of different parameters including geometrical one on the frequency response of the system are assessed.     

Soliton solutions via auxiliary function method for a coherently-coupled model in the optical fiber communications

A coherently-coupled nonlinear Schrödinger system in the optical fiber communications, with the mixed self-phase modulation (SPM), cross-phase modulation (XPM) and positive coherent coupling terms, is studied through the bilinear method with an auxiliary function. Solutions for that system are found to be of two types: singular and non-singular ones, and the latter appear as the soliton-typed. Vector bright one- and two-solitons are derived with the corresponding phase-shift parameter constraints. In virtue of computerized symbolic computation and asymptotic behavior analysis, elastic collision mechanisms of such vector solitons are investigated. With the aid of graphical simulation, vector solitons are displayed to be of the single- or double-hump profiles. The formation and collision mechanisms of the vector bright solitons for that system are generated based on the combined effects of SPM, XPM and coherent coupling. Only elastic collisions of the vector solitons occur for that system, which is a distinctive feature amid those of other coherently-coupled nonlinear Schrödinger systems.     

Amplitude Modulation of Nonlinear Waves in Fluid-Filled Tapered Tubes

We study the modulation of nonlinear waves in fluid-filled prestressed tapered tubes. For this, we obtain the nonlinear dynamical equations of motion of a prestressed tapered tube filled with an incompressible inviscid fluid. Assuming that the tapering angle is small and using the reductive perturbation method, we study the amplitude modulation of nonlinear waves and obtain the nonlinear Schrödinger equation with variable coefficients as the evolution equation. A traveling-wave type of solution of such a nonlinear equation with variable coefficients is obtained, and we observe that in contrast to the case of a constant tube radius, the speed of the wave is variable. Namely, the wave speed increases with distance for narrowing tubes and decreases for expanding tubes.     

On the Whitham Equations for the Defocusing Nonlinear Schrodinger Equation with Step Initial Data

The behavior of solutions of the finite-genus Whitham equations for the weak dispersion limit of the defocusing nonlinear Schrodinger equation is investigated analytically and numerically for piecewise-constant initial data. In particular, the dynamics of constant-amplitude initial conditions with one or more frequency jumps (i.e., piecewise linear phase) are considered. It is shown analytically and numerically that, for finite times, regions of arbitrarily high genus can be produced; asymptotically with time, however, the solution can be divided into expanding regions which are either of genus-zero, genus-one, or genus-two type, their precise arrangement depending on the specifics of the initial datum given. This behavior should be compared to that of the Korteweg-de Vries equation, where the solution is divided into regions which are either genus-zero or genus-one asymptotically. Finally, the potential application of these results to the generation of short optical pulses is discussed: The method proposed takes advantage of nonlinear compression via appropriate frequency modulation, and allows control of both the pulse amplitude and its width, as well as the distance along the fiber at which the pulse is produced     

On the Whitham system for the (2+1)-dimensional nonlinear Schrödinger equation

Whitham modulation equations are derived for the nonlinear Schrödinger equation in the plane ((2+1)-dimensional nonlinear Schrödinger [2d NLS]) with small dispersion. The modulation equations are obtained in terms of both physical and Riemann-type variables; the latter yields equations of hydrodynamic type. The complete 2d NLS Whitham system consists of six dynamical equations in evolutionary form and two constraints. As an application, we determine the linear stability of one-dimensional traveling waves. In both the elliptic and hyperbolic cases, the traveling waves are found to be unstable. This result is consistent with previous investigations of stability by other methods and is supported by direct numerical calculations.     

Nonlinear Controller Design for PWM–Controlled Converters

This contribution is concerned with the nonlinear controller design for a certain class of PWM (pulse width modulation) – controlled converter systems. It will be shown that under certain assumptions the SSA (state space averaging) – model of the PWM–controlled converter with the duty ratio as the plant input has a very special mathematical structure. Based on this mathematical model a modified version of the nonlinear H₂–design where an integral term is systematically included in the nonlinear controller will be presented.     

On the validity of the Ginzburg-Landau equation

Summary The famous Ginzburg-Landau equation describes nonlinear amplitude modulations of a wave perturbation of a basic pattern when a control parameterR lies in the unstable regionO(ε ²) away from the critical valueR _c for which the system loses stability. Hereε>0 is a small parameter. G-L's equation is found for a general class of nonlinear evolution problems including several classical problems from hydrodynamics and other fields of physics and chemistry. Up to now, the rigorous derivation of G-L's equation for general situations is not yet completed. This was only demonstrated for special types of solutions (steady, time periodic) or for special problems (the Swift-Hohenberg equation). Here a mathematically rigorous proof of the validity of G-L's equation is given for a general situation of one space variable and a quadratic nonlinearity. Validity is meant in the following sense. For each given initial condition in a suitable Banach space there exists a unique bounded solution of the initial value problem for G-L's equation on a finite interval of theO(1/ε²)-long time scale intrinsic to the modulation. For such a finite time interval of the intrinsic modulation time scale on which the initial value problem for G-L's equation has a bounded solution, the initial value problem for the original evolution equation with corresponding initial conditions, has a unique solutionO(ε²) — close to the approximation induced by the solution of G-L's equation. This property guarantees that, for rather general initial conditions on the intrinsic modulation time scale, the behavior of solutions of G-L's equation is really inherited from solutions of the original problem, and the other way around: to a solution of G-L's equation corresponds a nearby exact solution with a relatively small error.     

阻尼波动方程在模空间上的一些估计

本文研究分数次阻尼波动方程解在模空间上的估计及相应的时空估计, 作为应用, 我们将得到小初值条件下一类Cauchy问题的全局解.     

A generalized nonlinear Schrödinger equation and optical soliton in a gradient index cylindrical media

A generalized form of nonlinear Schrödinger equation is deduced for the propagation of an optical pulse in a fiber with a cylindrical geometry having a gradient in refractive index in the radial direction. The configuration gives a simple model for a fiber with a cladding or multicore fiber. To begin with we have analyzed in detail the modulational instability in terms of Stokes and anti-Stokes side band amplitudes which shows a significant change with respect to the depth parameter L and dispersion constant. Next we have deduced the equations governing the modulation of parameters of a Gaussian pulse as it propagates through it. The moment method is used for the derivation. The gradient of the refractive index leads to the trapping of the pulse, whereas the balance between nonlinearity (Kerr type) and dispersion in the longitudinal direction guides the propagation. Instead of a constant dispersion profile we have considered the standard dispersion map which helps in shaping of the pulse. The numerical simulation of these derived equations shows how the chirp, width, amplitude of the pulse change with type of gradient and the distance travelled.     

Non-integer power estimate in modulation spaces and its applications

We give a method to estimate non-integer power function|u|~ku in modulation space which is an open question in the study of modulation space.As an application,we can study Cauchy problem for the nonlinear Klein-Gordon equation with nonlinear term|u|~ku in modulation space,where k is not an integer.Moreover,we also study the global solution with small initial value for the Klein-Gordon-Hartree equation.The results show some advantages of modulation space both in high and low regularity cases.     

Existence of the positive composite optical vortices

Optical vortices as topological objects exist ubiquitously in nature. In this paper, we use the principle of variational method and mountain pass lemma to develop some existence theorems for the stationary vortex wave solution of a coupled nonlinear Schrödinger equations, which describe the possibility of effective waveguiding of a weak probe beam via the cross‐phase modulation‐type interaction. The main goal is to obtain a positive solution, of minimal action if possible, with all vector components not identically zero. Additionally, as demanded by beam confinement, we prove the exponential decay of the soliton amplitude at infinity. Copyright © 2017 John Wiley & Sons, Ltd.     

Vector breathers in the Manakov system

We study theoretically the nonlinear interactions of vector breathers propagating on an unstable wavefield background. As a model, we use the two-component extension of the one-dimensional focusing nonlinear Schrödinger equation—the Manakov system. With the dressing method, we generate the multibreather solutions to the Manakov model. As shown previously in [D. Kraus, G. Biondini, and G. Kovačič, Nonlinearity 28(9), 3101, (2015)], the class of vector breathers is presented by three fundamental types I, II, and III. Their interactions produce a broad family of the two-component (polarized) nonlinear wave patterns. First, we demonstrate that the type I and the types II and III correspond to two different branches of the dispersion law of the Manakov system in the presence of the unstable background. Then, we investigate the key interaction scenarios, including collisions of standing and moving breathers and resonance breather transformations. Analysis of the two-breather solution allows us to derive general formulas describing phase and space shifts acquired by breathers in mutual collisions. The found expressions enable us to describe the asymptotic states of the breather interactions and interpret the resonance fusion and decay of breathers as a limiting case of infinite space shift in the case of merging breather eigenvalues. Finally, we demonstrate that only type I breathers participate in the development of modulation instability from small-amplitude perturbations withing the superregular scenario, while the breathers of types II and III, belonging to the stable branch of the dispersion law, are not involved in this process.     

Breathers and multi-soliton solutions for the higher-order generalized nonlinear Schrödinger equation

In this paper, the higher-order generalized nonlinear Schrödinger equation, which describes the propagation of ultrashort optical pulse in optical fibers, is analytically investigated. By virtue of the Darboux transformation constructed in this paper, some exact soliton solutions on the continuous wave (cw) background are generated. The following propagation characteristics of those solitons are mainly discussed: (1) Propagation of two types of breathers which delineate modulation instability and bright pulse propagation on a cw background respectively; (2) Two types propagation characteristics of two-solitons: elastic interactions and mutual attractions and repulsions bound solitons. Those results might be useful in the study of ultrashort optical solitons in optical fibers.     

Heat and mass transfer in annular liquid jets: II. g-jitter

The effects of g-jitter on heat and mass transfer in underpressurized, annular liquid jets are analyzed numerically as a function of the amplitude and frequency of the gravitational modulation by means of a mapping technique that transforms the time-dependent geometry of these jets into a unit square and a conservative finite difference method. It is shown that the pressure coefficient, gas concentration at the jet's inner interface, heat fluxes at the jet's inner and outer interfaces and interfacial temperature are periodic functions of time whose amplitudes increase as the amplitude of the g-jitter is increased, but decrease as the jitter frequency is increased. The pressure coefficient is almost in phase with the heat flux at the jet's outer interface, and out of phase with the mass transfer rate at the jet's inner interface. It is also shown that the temperature field adapts itself rapidly to the imposed gravity modulation, and thermal equilibrium is reached quickly. However, mass transfer phenomena are very slow and require a very long time to become periodic.     

Krein signature and Whitham modulation theory: the sign of characteristics and the “sign characteristic”

In classical Whitham modulation theory, the transition of the dispersionless Whitham equations from hyperbolic to elliptic is associated with a pair of nonzero purely imaginary eigenvalues coalescing and becoming a complex quartet, suggesting that a Krein signature is operational. However, there is no natural symplectic structure. Instead, we find that the operational signature is the “sign characteristic” of real eigenvalues of Hermitian matrix pencils. Its role in classical Whitham single‐phase theory is elaborated for illustration. However, the main setting where the sign characteristic becomes important is in multiphase modulation. It is shown that a necessary condition for two coalescing characteristics to become unstable (the generalization of the hyperbolic to elliptic transition) is that the characteristics have opposite sign characteristic. For example the theory is applied to multiphase modulation of the two‐phase traveling wave solutions of coupled nonlinear Schrödinger equation.     

A new scheme for solving nonlinear Stratonovich Volterra integral equations via Bernoulli’s approximation

In this paper, an efficient method for solving nonlinear Stratonovich Volterra integral equations is proposed. By using Bernoulli polynomials and their stochastic operational matrix of integration, these equations can be reduced to the system of nonlinear algebraic equations with unknown Bernoulli coefficient which can be solved by numerical methods such as Newton’s method. Also, an error analysis is valid under fairly restrictive conditions. Furthermore, in order to show the accuracy and reliability of the proposed method, the new approach is compared with the block pulse functions method by some examples. The obtained results reveal that the proposed method is more accurate and efficient than the block pulse functions method.     

Evolution of Optical Pulses in Fiber Lines with Lumped Nonlinear Devices as a Mapping Problem

We analyze the steady-state propagation of optical pulses in fiber transmission systems with lumped nonlinear optical devices (NODs) placed periodically in the line. For the first time to our knowledge, a theoretical model is developed to describe the transmission regime with a quasilinear pulse evolution along the transmission line and the point action of NODs. We formulate the mapping problem for pulse propagation in a unit cell of the line and show that in the particular application to nonlinear optical loop mirrors, the steady-state pulse characteristics predicted by the theory accurately reproduce the results of direct numerical simulations. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 2, pp. 277–289, August, 2005.     

Pulse vaccination on SEIR epidemic model with nonlinear incidence rate

In this paper, we consider an SEIR epidemic model with two time delays and nonlinear incidence rate, and study the dynamical behavior of the model with pulse vaccination. By using the Floquet theorem and comparison theorem, we prove that the infection-free periodic solution is globally attractive when R^*<1, and using a new modelling method, we obtain a sufficient condition for the permanence of the epidemic model with pulse vaccination when R_*>1.