Multiple-scale Expansion Method Research Articles

Nonlinear Wave Interaction of Nanorods Embedded in a Viscoelastic Medium

Abstract Purpose Nonlinear interactions between two acoustic waves in nanorods traveling at various wave numbers, group velocities, and frequencies are examined in this study. Methods The nonlinear equation of the nanorod in a viscoelastic medium is obtained using the theory of nonlocal elasticity. Furthermore, the multiple-scale expansion method is applied to study strongly dispersive, weakly nonlinear waves in a nonlocal viscoelastic medium. Using this expansion technique, we can derive the coupled nonlinear Schrödinger equations as the governing equations, which we solve as differential equations of some parameters by expanding the field quantities into an asymptotic series of the smallness parameter. Results We give the nonlinear plane wave solutions to these equations in several special cases. The plane wave solutions show how the wave amplitude affects the frequencies of nonlinear plane waves. Additionally, we show numerically how the real and imaginary parts of the group velocities and natural frequency of the system for a carbon nanotube in a viscoelastic medium are affected by the nonlocal, damping, and stiffness parameters.

Two-dimensional vector solitons in Bose-Einstein-condensate mixtures

We derive two decoupled KP-I equations from the system of two-dimensional (2D) Gross-Pitaevskii equations for a two-component Bose-Einstein condensate (BEC), using the multiple-scale expansion method. We produce asymptotic analytical vector-soliton solutions, viz., dark-dark (DD) and dark-antidark (DAD) one-soliton and two-soliton states, by tuning coupling constants and norms of species, and address their evolution numerically under the action of the harmonic-oscillator (HO) trap, in the local-density approximation. We find that shallow single-line DD and DAD solitons are stable, while single-lump DAD solutions (weakly localized truly-2D states) split and lead to nucleation of two half-vortices. We also find that the BEC mixture placed in the HO trap admits stable asymptotic multi-soliton solutions, e.g., two-line DD and DAD solitons and two-lump DD ones, which were not reported before. In particular, the two-lump solutions describe inelastic collisions between the lumps. The analysis developed in this work may also be applied to systems with spin-orbit coupling and gauge fields, which have been realized in atomic BEC.

Theoretical study on the downshift of class II Bragg resonance

In this paper, a new solution for class II Bragg resonance was derived based on the multiple-scale expansion method. The proposed solution can accurately predict the reflection coefficient and effectively characterize the downshift behavior of the Bragg resonance. Subsequently, a formula that quantifies the magnitude of the wave resonance frequency is proposed. The formula's predictive ability of the frequency downshift is demonstrated through comparisons against existing analytical and numerical solutions. Furthermore, investigation of the interactions between the waves and ripples reveals the underlying mechanism of the downshift behavior, highlighting the significant influence of wave re-reflection by the doubly sinusoidal ripples. Particularly, the process where waves are initially reflected by one of the ripple components and subsequently re-interacts with their conjugates results in the downward shift from the Bragg resonance condition. Moreover, the study of the effect of ripple length on Bragg resonance demonstrated that the downshift magnitude increases up to a certain limit as the ripple length tends to infinity, and this limit is substantially affected by the ripple amplitudes. In addition, the effects of the two ripple components are examined, which reveals the existence of a threshold ratio of the second wavenumber (larger) to the first ripple wavenumber (smaller) for a specific water depth. When the ratio is smaller than the threshold value, the downshift behavior of Bragg resonance is primarily influenced by the second ripple component. Conversely, as the ratio exceeds this value, the downshift behavior is dominated by the first ripple component.

A study of the acoustic effect inside cylindrical bubble produced by underwater electrical discharge

The evolution of a cylindrical gaseous bubble produced by an underwater electrical discharge is considered in the present study. Both the gas flow inside and the water flow around the bubble are theoretically analyzed in a cylindrical coordinate system. By using the potential flow theory and multiple scale expansion method, governing equations of both flows and the bubble surface are formulated. The radial oscillation of the bubble surface is composed of a slow-changing equilibrium part and a fast-changing displacement. The former corresponds to a quiescent water domain and a uniform gas column, and the latter corresponds to acoustic waves in gas and water flows. The axial gas wave can evolve into a stable standing wave if the bubble length is multiples of half a wavelength. The internal acoustic standing wave then causes a synchronous small-amplitude oscillation of the bubble surface when the frequency of the acoustic wave is close to the natural frequency of the bubble surface. An underwater discharge experiment is implemented to validate our theory. Finally, a novel method to estimate the plasma pressure is proposed based on our theory.

On the large-Weissenberg-number scaling laws in viscoelastic pipe flows

This work explains a scaling law of the first Landau coefficient of the derived Ginzburg–Landau equation in the weakly nonlinear analysis of axisymmetric viscoelastic pipe flows in the large-Weissenberg-number ($Wi$) limit, recently reported in Wanet al.(J. Fluid Mech., vol. 929, 2021, A16). Using an asymptotic method, we derive a reduced system, which captures the characteristics of the linear centre-mode instability near the critical condition in the large-$Wi$limit. Based on the reduced system we then conduct a weakly nonlinear analysis using a multiple-scale expansion method, which readily explains the aforementioned scaling law of the Landau coefficient and some other scaling laws. Particularly, the equilibrium amplitude of disturbance near linear critical conditions is found to scale as$Wi^{-1/2}$, which may be of interest to experimentalists. The current analysis reduces the numbers of parameters and unknowns and exemplifies an approach to studying the viscoelastic flow at large$Wi$, which could shed new light on the understanding of its nonlinear dynamics.

Nonlinear free vibration of functionally graded fiber-reinforced composite hexagon honeycomb sandwich cylindrical shells

In the present study, the nonlinear free vibration of functionally graded fiber-reinforced composite hexagon honeycomb sandwich cylindrical shells is investigated with consideration of the large-amplitude effect. First, an analytical model of such shell structures is proposed, which considers the functionally graded fiber-reinforced composite skins and a hexagon honeycomb core with their respective effective material properties being evaluated through the micromechanical model of Halpin-Tsai and the mixture law, as well as the modified Gibson's formula. Meanwhile, based on the first-order shear deformation theory and von Kármán geometrical nonlinear relations, the nonlinear partial differential governing equations are deduced via Hamilton’s principle, which is further discretized into several ordinary differential equations by the Galerkin approach. Subsequently, the static condensation technique is adopted to decrease the degrees of freedom, and the frequency-amplitude relationships are obtained by the multiple scale expansion method. Following the validation of the developed model via comparing the predicted results to literature ones, a detailed parameter study of critical geometric and material parameters on the nonlinear vibration characteristics of the structure is conducted, with some important conclusions being provided to weaken the nonlinear hardening-spring behavior of the structure.

Information pattern stability in memristive Izhikevich neural networks

In this paper, modulational instability (MI) of information via membrane potential is studied analytically and numerically in an improved Izhikevich neural network under electromagnetic induction. By applying the powerful discrete multiple scale expansion method, a spatiotemporal nonlinear amplitude differential-difference equation governing the information dynamics is derived from the generic model. Linear stability of plane impulse wave solution is then performed on the latter and the impact of electromagnetic induction feedback through the memristor couplings is portrayed on the growth rate diagram. From the diagram, it is found that negative memristor coupling parameter decreases the critical amplitude while positive parameter increases the critical amplitude. To support our analytical predictions, numerical simulations are performed and data selected from the unstable zone of MI lead to the formation of localized solitonic energy patterns, related to the energy coding patterns in the nervous system. Furthermore, the sampled time series for membrane potential under the influence of memristor coupling revealed the breakdown of action potential into multiple impulse-wave trains for high parameter values thus confirming an analytical prediction. Our results provide a potential way to manipulate information coding in the brain.

Nonlinear spatiotemporal instabilities in two-dimensional electroconvective flows

This work studies nonlinear spatiotemporal stability of two-dimensional electroconvection between two flat plates subjected to a through-flow, using numerical simulations and weakly nonlinear analyses. We found that the traveling speeds of the leading and trailing edges of the wave packet in the nonlinear regime are consistent with the linear ones. We derived for the first time the Ginzburg-Landau equation (GLE) using an amplitude expansion method extending earlier work of Pham and Suslov. This GLE can predict the absolute growth rate even when the parameters are away from the linear critical conditions, outperforming the GLE derived using a multiple-scale expansion method.

Dynamics of modulated waves and localized energy in a Burridge and Knopoff model of earthquake with velocity-dependant and hydrodynamics friction forces

We investigate both analytically and numerically the dynamics of modulated waves in a modified Burridge and Knopoff mechanical model of earthquake. We take into account the velocity-dependent and hydrodynamics friction forces between the rock blocks of the system. The theoretical framework for the analysis is presented using the multiple scale expansion method. We show that the system can be governed by the complex Ginzburg–Landau equation, which admits soliton solutions. The modulational instability of the system is performed, showing that the bandwidth of instability increases with the friction forces parameter. The planar wave solution used as the initial condition leads to more unstable waves in the system with chaotic-like behaviour, allowing energy localization. It is also found that the wave parameters, namely, the frequency, magnitude and velocity (as well as the energy of the system) strongly depend on η. In the non-friction limit, the obtained waves propagate with a coherent behaviour for a very short time, before disappearing. Our results suggest that the damages caused by an earthquake can be important as the friction forces increase and may excite the possibility of relative predictions for the dynamics of waves propagating in the earth in the field of nonlinear science.

Cylindrical and spherical ion acoustic shock waves with two temperature superthermal electrons in dusty plasma

In the present study, the cylindrical/spherical ion acoustic shock waves have been investigated in unmagnetized dusty plasma consisting of positive ions, immobile dust particles and kappa distributed cold and hot electrons. A multiple scale expansion method is employed to derive Burgers equation (BE) and modified Burgers equation (MBE) by including higher order nonlinearity. The basic characeristics of the shock waves have been analyzed numerically and graphically for different physical parameters relevant to space and laboratory dusty plasma environments through 2D figures. We show that the amplitude of the wave decreases faster as one departs away from the axis of the cylinder or centre of the sphere. Such decaying behaviour continues as time progresses. Furthermore, the parametric dependence of wave properties (amplitude, width) on kappa index, density and temperature of cold and hot electrons, concentration of dust particles, thermal effects and kinematic viscosity of ions has been studied in detail and findings obtained here will be beneficial to further astrophysical investigations.

Nonplanar ion acoustic waves in dusty plasma with two temperature electrons: Application to Saturn's E ring

In the present paper, we have studied the cylindrical/spherical ion acoustic waves in dusty plasma consisting of positive ions, immobile dust particles, and q nonextensive distributed cold and hot electrons. A multiple scale expansion method is employed to derive a modified Korteweg–de Vries (mKdV) equation. Exact analytical solution for the mKdV equation in nonplanar geometry is obtained using the weighted residual method. This solution is, then, plotted for different physical parameters relevant to Saturn's E ring through 2D figures. We show that the amplitude of the wave decreases faster as one departs away from the axis of the cylinder or the center of the sphere. Such a decaying behavior continues as time progresses. This is expected from the obtained solution where the cylindrical (spherical) amplitude component (τ∞/τ)m decreases as the time progresses. We have also studied the role of the nonadiabatic dust charging fluctuating dissipation process and analyzed the resulting damped mKdV equation. Furthermore, the parametric dependence of wave properties (amplitude and width) on the q nonextensive parameter, density, and temperature of cold and hot electrons, concentration of dust particles, and thermal effects of ions has been studied in detail, and findings obtained here will be beneficial to further astrophysical investigations.

Base pairs opening and bubble transport in damped DNA dynamics with transport memory effects

Transport memory effects on nonlinear wave propagation are addressed in a damped Peyrard-Bishop-Dauxois model of DNA dynamics. Under the continuum and overdamped limits, the multiple-scale expansion method is employed to show that an open-state configuration of the DNA molecule is described by a complex nonlinear Schrödinger equation. For the latter, solutions are proposed as bright solitons, which suitably represent the open-state configuration that takes place along the DNA molecule in the form of bubbles. A good agreement between numerical experiments and analytical predictions on the impact of memory effects on the angular frequency, velocity, width, and amplitude of the moving bubble is obtained. It also appears that memory effects can modify qualitatively and quantitatively the nonlinear dynamics of DNA, including the energy brought by enzymes for the initiation of the processes of replication and transcription.

DNA Breathers and Cell Dynamics

Molecular-morphological signs of oncogenesis can be linked to multiscale collective effects in molecular and cell ensembles. It was shown that nonlinear behavior of biological systems can be associated with the generation of characteristic collective modes representing the open states in molecular and cell organization as the mechanism of the coherent expression dynamics. The mechanical DNA model is developed to study the nonlinear dynamics of the helicoidal geometry DNA molecule. To construct the model of DNA the Peyrard–Bishop–Barbi approach has been applied. The analytical small localized solutions as the discrete breather and the antikink have been obtained by multiple scale expansion method for multicomponent lattices. The set of collective open states (breathers) in the molecular ensembles provides the collective expression dynamics to attract cells toward a few preferred global states. This result allows the formulation of the experimental strategy to analyze the qualitative changes in cell dynamics induced by mentioned collective modes. The biomechanical changes have been shown experimentally using the original data of Coherent Phase Microscopy analyzing the time series of phase thickness fluctuations. Study of the mechanical aspects of the behavior of single cells is a prerequisite for the understanding of cell functions in the case of qualitative changes in diseases affecting the properties of cells and tissues morphology to develop diagnostic and treatment design methodology.

Dissipative electron-acoustic solitons in a cold electron beam plasma with superthermal trapped electrons

The linear and nonlinear properties of high-frequency electron-acoustic (EA) solitons are investigated in multi-component dissipative plasma with a cold beam electron fluid, the Schamel-kappa distributed hot trapped electrons, and stationary ions. The linear phase speed is found to be modified significantly due to variations in dissipative, superthermality, and beam speed parameters. The impact of dissipation (cold electron-to-neutrals collisions) and superthermality on the characteristics of electron acoustics waves (EAWs) is elaborated. The multiple scale expansion method is employed to derive time-varying Schamel equation for small-amplitude electrostatic potential disturbances, carrying dissipative processes as well. The variations in superthermality, and beam speed parameters have been found to strongly impact the profiles of dissipative solitons obtained through numerical time evolution of Schamel equation. The significance of the work lies in the fact that positive potential electron acoustic solitons are sustained, which correspond to holes (or humps) in the cold (hot) electron number density. As trapping is a nonlinear phenomenon, so its impact on these solitary dissipative structures have also been highlighted with suitable application parameters. This study should be beneficial for understanding nonlinear structures reported in the dayside auroral zone and other regions of the magnetosphere.

Effects of thin plaque on blood hammer—An asymptotic theory

We describe a theory of blood hammer in a long and stiffened artery affected by the presence of plaque. Based on the model of a viscous fluid in laminar flow, we derive explicit expressions of oscillatory pressure and wall shear stress when part of the blood vessel is lined with thin plaque. The asymptotic method of multiple-scale expansions is utilized to obtain information on wave reflection and frictional damping analytically. The effects of plaque thickness and length on blood pressure and wall shear stress are studied.

Nonlinear dynamics of DNA with topological constraints

The mechanical DNA model is developed to study the nonlinear dynamics, topological constrains of the helicoidal geometry DNA molecule, dissipation effects and influence of the external force. To construct the model of DNA the Peyrard-Bishop-Barbi approach has been applied. We have derived equations of motion for the DNA base pairs in the presence of a damping term and external driving force to follow the Caldirola-Kanai method. The analytical small localized solutions as the discrete breather and the antikink have been obtained by multiple scale expansion method for multicomponent lattices. The impact of the damping effect and external forces on the breather and the antikink propagation has been investigated. The prospects of using the helicoidal model of DNA with the energy dissipation and external force are discussed. The dissipative term and the external force term in the DNA motion equations follow from corresponding Hamiltonian energy representation. The advantage of this approach is the promising way to study both the thermodynamic properties and nonlinear dynamics of DNA system related to the types of the collective modes (breathers, antikinks) as new thermodynamic degrees of freedom. This result allows the formulation of the experimental strategy to analyze the qualitative changes in cell dynamics induced by mentioned collective modes.

Pushing amplitude equations far from threshold: application to the supercritical Hopf bifurcation in the cylinder wake

The purpose of this review article is to push amplitude equations as far as possible from threshold. We focus on the Stuart–Landau amplitude equation describing the supercritical Hopf bifurcation of the flow in the wake of a cylinder for critical Reynolds number . After having reviewed Stuart's weakly nonlinear multiple-scale expansion method, we first demonstrate the crucial importance of the choice of the critical parameter. For the wake behind a cylinder considered in this paper, choosing instead of considerably improves the prediction of the Landau equation. Although Sipp and Lebedev (2007 J. Fluid Mech 593 333–58) correctly identified the adequate bifurcation parameter ϵ, they have plotted their results adding an additional linearization, which amounts to using as approximation to ϵ. We then illustrate the risks of calculating ‘running’ Landau constants by projection formulas at arbitrary values of the control parameter. For the cylinder wake case, this scheme breaks down and diverges close to . We propose an interpretation based on the progressive loss of the non-resonant compatibility condition, which is the cornerstone of Stuart's multiple-scale expansion method. We then briefly review a self-consistent model recently introduced in the literature and demonstrate a link between its properties and the above-mentioned failure.

Initial–boundary layer associated with the nonlinear Darcy–Brinkman–Oberbeck–Boussinesq system

In this paper, we study the vanishing Darcy number limit of the nonlinear Darcy–Brinkman–Oberbeck–Boussinesq system (DBOB). This singular perturbation problem involves singular structures both in time and in space giving rise to initial layers, boundary layers and initial–boundary layers. We construct an approximate solution to the DBOB system by the method of multiple scale expansions. The convergence with optimal convergence rates in certain Sobolev norms is established rigorously via the energy method.

Multiple scales approach to the gas-piston non-equilibrium themodynamics

The non-equilibrium thermodynamics of a gas inside a piston is a conceptually simple problem where analytic results are rare. For example, it is hard to find in the literature analytic formulas that describe the heat exchanged with the reservoir when the system either relaxes to equilibrium or is compressed over a finite time. In this paper we derive this kind of analytic formula. To achieve this result, we take the equations derived by Cerino et al (2015 Phys. Rev. E 91 032128) describing the dynamic evolution of a gas-piston system, we cast them in a dimensionless form, and we solve the dimensionless equations with the multiple scales expansion method. With the approximated solutions we obtained, we express in a closed form the heat exchanged by the gas-piston system with the reservoir for a large class of relevant non-equilibrium situations.

Weakly nonlinear stability analysis of subcritical electrohydrodynamic flow subject to strong unipolar injection

We analyse in detail the weakly nonlinear stability of electrohydrodynamic (EHD) flow of insulating fluids subject to strong unipolar injection, with and without cross-flow. We first consider the hydrostatic electroconvetion induced by a Coulomb force confined between two infinite flat electrodes, taking into account the charge diffusion effect. The effects of various non-dimensionalized parameters are examined in order to depict in detail and to understand better the subcritical bifurcation of hydrostatic electroconvetion. In addition, electrohydrodynamics with low- or high-$Re$cross-flow is also considered for investigating the combined effect of inertia and the electric field. It is found that the base cross-flow is modified by the electric effect and that, even when the inertia is dominating, the electric field can still strengthen effectively the subcritical characteristics of canonical channel flow. In this process, however, the electric field does not contribute directly to the subcriticality of the resultant flow and the intensified subcritical feature of such flow is thus entirely due to the modified hydrodynamic field as a result of the imposed electric field. This finding might be important for flow control strategies involving an electric field. Theoretically, the above results are obtained from a multiple-scale expansion method, which gives rise to the Ginzburg–Landau equation governing the amplitude of the first-order perturbation. The conclusions are deduced by probing the changes of value of the coefficients in this equation. In particular, the sign of the first Landau coefficient indicates the type of bifurcation, being subcritical or supercritical. Moreover, as a quintic-order Ginzburg–Landau equation is derived, the effects of higher-order nonlinear terms in EHD flow are also discussed.