Multiscale Expansion Method Research Articles

A multiscale Gaussian expansion method for low-frequency multimodal damping analysis of variable stiffness acoustic black hole beams

To address the issues of slow convergence and poor accuracy in existing dynamics methods caused by non-uniform variable cross-sections and layered material stiffness variations in variable stiffness acoustic black hole structure (VSABH). A multiscale Gaussian expansion method (MSGEM) is proposed in this paper. The structural dimensions are taken into account, and the structural parameters of multiple shape functions are adaptively selected based on stiffness and cross-sectional variation parameters. This results in the formation of shape function groups of various scales, which in turn creates the matrix of multi-scale Gaussian function groups. The feature information of the displacement field is realized to be extracted from multiple scales, and the fitting error of MSGEM to the displacement field of VSABH beam is within 2.02%. The missing solution of eigenfrequency of Gaussian expansion method is avoided, and the modal vibration patterns are basically matched, which verifies the validity of MSGEM. Meanwhile, the advantages of low-frequency multimodal vibration reduction of VSABH beams are highlighted from several aspects, and the trend of the adjustment of the vibration reduction characteristics of VSABH beams by different parameters is clarified, which provides valuable design guidance for ABH metamaterials oriented to low-frequency vibration reduction.

Wavy approach for fluid–structure interaction with high Froude number and undamped structure

This paper addresses the fluid–structure interaction problem, with an interest on the interaction of a gravity wave with a flexible floating structure, anchored to a seabed of constant depth. To achieve this goal, we make use of the model equations, namely, the Navier–Stokes equations and the Navier–Lamé equation, as well as the associated the boundary conditions. Applying the multi-scale expansion method, these set of equations are reduced to a pair of nonlinearly coupled complex cubic Ginzburg–Landau equations (CCGLE). By applying the proposed modified expansion method, the group velocity dispersion and second-order dispersion relation are deduced. In the same vein, modulation instability (MI) is investigated as a mechanism of formation of pulse trains in fluid–structure system using a CCGLE. For the analytical analysis, we made use of the inverse scattering method to find analytical solutions to the coupled nonlinear equations. Through that method, the obtained solutions depict rogue-shaped waves. Our results suggest that uncontrolled MI within the interaction between a flexible body and gravity waves in viscous flow may be considered as the principal source of many structural ruptures, which are the first cause of critical damage due to the great energy and unpredictability of rogue waves. The present work aims to provide tools to model a wide range of physical problems regarding the interaction of surface gravity waves and an offshore-anchored structure, and it aims to be essential to our understanding of the nonlinear characteristics of offshore structures in real-sea states.

A (2+1) -dimensional evolution model of Rossby waves and its resonance Y-type soliton and interaction solutions

In this paper, we derive a Kadomtsev-Petviashvili equation by using the multi-scale expansion and perturbation method, which is a model from the potential vorticity equation in the traditional approximation and describes the Rossby wave propagation properties. The N-soliton solutions, resonance Y-type soliton solutions, resonance X-type soliton solutions and interaction solutions of the equation are obtained with the help of dependent-variable transformation. In addition, the composite graphs are given to view the resonance phenomenon of Rossby waves. The results better enrich the research of Rossby waves in ocean dynamics and atmospheric dynamics.

Nonlinear instability and solitons in a self‐gravitating fluid

We study a spherical, self‐gravitating fluid model, which finds applications in cosmic structure formation. We argue that since the system features nonlinearity and gravity‐induced dispersion, the emergence of solitons becomes possible. We thus employ a multiscale expansion method to study, in the weakly nonlinear regime, the evolution of small‐amplitude perturbations around the equilibrium state. This way, we derive a spherical nonlinear Schrödinger (NLS) equation that governs the envelope of the perturbations. The effective NLS description allows us to predict a “nonlinear instability” (occurring in the nonlinear regime of the system), namely, the modulational instability, which, in turn, may give rise to spherical soliton states. The latter feature a very slow (polynomial) curvature‐induced decay in time. The soliton profiles may be used to describe the shape of dark matter halos at the rims of the galaxies.

The Rayleigh–Bénard problem for water with maximum density effects

Linear stability and weakly nonlinear stability analyses are developed for Rayleigh–Bénard convection in water near 3.98 °C subject to isothermal boundary conditions. The density–temperature relationship (equation of state) is approximated by a cubic polynomial, including linear, quadratic, and cubic terms. The continuity equation, the Navier–Stokes momentum equation, the equation of state, and the energy equation constitute the governing system. Linear stability analysis is used to investigate how the maximum density property of water affects the onset of convective instability and the choice of unstable wave number for four different types of boundary conditions. Then, a weakly nonlinear stability study is done using the spectral Fourier method for isothermal tangential stress-free boundary conditions to quantify the heat transport of the system and demonstrate the transition from regular/periodic convection to chaotic convection. A Stuart-Ginzburg–Landau equation is obtained using the multiscale expansion method. Streamlines and isotherms are presented and analyzed. The influence of maximum density has been shown to delay the onset of instability and is, therefore, a stabilizing mechanism for thermal instability. Due to the maximum density, the onset of chaotic convection is also delayed. Among four different boundaries, the impermeable rigid boundaries require the highest Rayleigh number for instability to begin. Increasing boundary temperatures advance the onset of chaotic convection and improve the heat transport situation.

Propagations of multiple solitons in a deformed ferrite

In this work, we investigate the magnetic soliton profile and propagation properties of ultrashort waves in one-dimensional deformed ferrite. A generalized variable-coefficient (1+1)-dimensional ultrashort wave model is obtained by employing the multi-scale expansion method and short-wave approximation. Using numerical simulations, we examine the impact of multiple inhomogeneities on the transmission of magnetic solitons, which arise from local and periodic deformations. Our results indicate that the localized inhomogeneity alter the velocity of magnetic solitons, allowing for the manipulation of their acceleration or deceleration. Periodic type deformation inhomogeneities, on the other hand, provide the magnetic solitons with a period-like external potential that converts its transmission mode to a controlled oscillation.

Fractional model of blood flow and rogue waves in arterial vessels

More and more researches evidence that hemodynamic factors play an important role in the occurrence and development of cardiovascular diseases. In this paper, a new mathematical model has been proposed to describe rogue waves in arterial vessels. Based on two‐dimensional Navier–Stokes (NS) equation and continuity equation, vorticity equation satisfied by blood flow is given. Further, by employing multiscale analysis and perturbation expansion method, the ( )‐dimensional nonlinear Schrödinger (NLS) equation is derived to describe the envelope solitary waves propagation of blood vessels. Different from the previous model, the ( )‐dimensional NLS model takes into account the propagation of blood flow along the vessel axis and radius. In order to further study, the integer‐order model is generalized to the time‐fractional nonlinear Schrödinger (TF‐NLS) equation by use of the semi‐inverse method and Agrawal's method, and the conservation laws are also investigated. Moreover, the rogue wave solution of the fractional model which can directly describe the behavior of the rogue waves in arterial vessels is obtained. Rogue wave and fractional parameter effect on blood flow volume is analyzed and discussed, which can provide some help for the study of cardiovascular diseases.

A higher-order three-scale reduced homogenization approach for nonlinear mechanical properties of 3D braided composites

A more effective higher-order three-scale reduced homogenization (HTRH) approach is developed for predicting the nonlinear mechanical properties of three-dimensional (3D) 4-directional braided composite. In this work, the 3D braided composites are considered by periodic distributions of different unit cells on microscale and mesoscale configurations. First, the various high-order unit cell functions based on the microscopic and mesoscopic domain are obtained by multiscale expansion methods. Then, two kinds of homogenization coefficients are given, and relative nonlinear homogenized equations are derived on the macroscopic domains. Further, the reduced order systems for the nonlinear multiscale problem are constructed. The significant features of the reduced-order multiscale method are: (i) an asymptotic higher-order homogenized solution evaluated by post-processing that does not need high-order continuities for the homogenization solutions, (ii) an effective model reduction format for investigating the higher-order nonlinear multiscale problems with less computational cost and (iii) an novel three-scale formulas established for analyzing the 3D braided composites. Finally, the validity of the proposed approach in comparison to the direct numerical simulations and classical multiphase method is computed on some damage and elasto-plastic problems. These numerical examples show that the HTRH method is effective to study the nonlinear problems of the 3D braided composites, and provides a potential application for practical engineering calculation.

Universal reductions and solitary waves of weakly nonlocal defocusing nonlinear Schrödinger equations

We study asymptotic reductions and solitary waves of a weakly nonlocal defocusing nonlinear Schrödinger (NLS) model. The hydrodynamic form of the latter is analyzed by means of multiscale expansion methods. To the leading-order of approximation (where only the first of the moments of the response function is present), we show that solitary waves, in the form of dark solitons, are governed by an effective Boussinesq/Benney–Luke (BBL) equation, which describes bidirectional waves in shallow water. Then, for long times, we reduce the BBL equation to a pair of Korteweg–de Vries (KdV) equations for right- and left-going waves, and show that the BBL solitary wave transforms into a KdV soliton. In addition, to the next order of approximation (where both the first and second moment of the response function are present), we find that dark solitons are governed by a higher-order perturbed KdV (pKdV) equation, which has been used to describe ion-acoustic solitons in plasmas and water waves in the presence of higher-order effects. The pKdV equation is approximated by a higher-order integrable system and, as a result, only insubstantial changes in the soliton shape and velocity are found, while no radiation tails (in this effective KdV picture) are produced.

Wave Processes in Rotating Compressible Astrophysical Plasma Flows with Stable Stratification

The wave processes in a rotating layer of compressible astrophysical plasma with a stable stratification and a linear entropy profile are studied theoretically. The compressibility is taken into account in the anelastic approximation. In this approximation the acoustic waves are filtered out, the system contains terms with a potential temperature (entropy), and the continuity equation contains an initial stratified density profile. The Coriolis force in the magnetohydrodynamic equations for a compressible astrophysical plasma is considered in four different approximations: a traditional f-plane, a nontraditional f-plane (given the horizontal component of the Coriolis force), a traditional β-plane, and a nontraditional β-plane. Linear and nonlinear theories of wave processes have been developed for each Coriolis force approximation under consideration. New types of waves have been found, with the rotation, magnetic field, gravity, and compressibility serving as their restoring mechanisms. The compressibility effects are represented in the new dispersion equations by the Brunt–Vaisala frequency for compressible stratified flows dependent on both initial density and pressure profiles. All of the realizable types of three-wave interactions have been revealed through a qualitative analysis of the dispersion curves. A system of equations for the amplitudes of interacting waves and the growth rates of parametric instabilities have been derived by the multiscale expansion method.

The Schamel-Ostrovsky equation in nonlinear wave dynamics of cylindrical shells

On the basis of the smeared stiffener theory, the evolution of nonlinear axisymmetric longitudinal waves in a nonlinear elastic cylindrical shell reinforced by internal stringers is investigated. A non-classical nonlinear physical law is adopted for the shell material, which is characterized by a fractional degree of strain intensity. An analysis of the influence of an external elastic medium on a nonlinear wave process is carried out on the basis of the Winkler and shear-lag model. Using the asymptotic method of multiscale expansions, the Schamel-Ostrovsky quasi-hyperbolic equation for the component of longitudinal displacement was first derived. The Painlevé analysis of the equation showed the impossibility of exact solitary-wave solutions. The dispersion relation is analyzed, the maximum value of the phase velocity of infinitesimal disturbances is determined, above which stationary nonlinear waves can propagate. Based on the finite-difference approach and the Petviashvili method, a numerical simulation of the derived equation is carried out, profiles of stably propagating wave packets and solitary waves are constructed. A special dispersionless case, impossible for an unstiffened shell, has been identified, in which there are exact compactons, peakons and periodic solutions.

Wave Processes in Three-Dimensional Stratified Flows of a Rotating Plasma in the Boussinesq Approximation

Magnetohydrodynamic (MHD) waves in a stratified rotating plasma in a gravitational field are investigated in the Boussinesq approximation. A theory of flows on an f-plane, on a nontraditional f-plane (with regard to the horizontal component of the Coriolis force), on a β-plane, and on a nontraditional β‑plane is developed. In each case, linear solutions to systems of three-dimensional MHD equations in the Boussinesq approximation are obtained that describe magnetic gravito-inertial waves, magnetostrophic waves, and magnetic Rossby waves. All existing types of three-wave interactions are found with the use of dispersion equations. In the case of magnetic Rossby waves in the β-plane approximation, it is shown that the low-frequency mode of a magnetic Rossby wave in the Boussinesq approximation is equivalent to that in the shallow-water MHD approximation. By the multiscale expansion method, a system of amplitude equations for interacting waves and the increments of two types of instability occurring in the system, decay and amplification, are obtained. For each type of three-wave interactions, it is shown that there is a difference in the coefficients and differential operators in the system of three-wave interactions.

Solitary and periodic waves in collisionless plasmas: The Adlam-Allen model revisited.

We consider the Adlam-Allen (AA) system of partial differential equations, which, arguably, is the first model that was introduced to describe solitary waves in the context of propagation of hydrodynamic disturbances in collisionless plasmas. Here, we identify the solitary waves of the model by implementing a dynamical systems approach. The latter suggests that the model also possesses periodic wave solutions-which reduce to the solitary wave in the limiting case of an infinite period-as well as rational solutions that are obtained herein. In addition, employing a long-wave approximation via a relevant multiscale expansion method, we establish the asymptotic reduction of the AA system to the Korteweg-de Vries equation. Such a reduction is not only another justification for the above solitary wave dynamics, but may also offer additional insights for the emergence of other possible plasma waves. Direct numerical simulations are performed for the study of multiple solitary waves and their pairwise interactions. The stability of solitary waves is discussed in terms of potentially relevant criteria, while the robustness of spatially periodic wave solutions is touched upon via numerical experiments.

Soliton pairs in two-dimensional nonlocal media.

We study the interaction of optical beams of different wavelengths, described by a two-component, two-dimensional (2D) nonlocal nonlinear Schrödinger (NLS) model. Using a multiscale expansion method the NLS model is asymptotically reduced to the completely integrable 2D Mel'nikov system, the soliton solutions of which are used to construct approximate dark-bright and antidark-bright soliton solutions of the original NLS model; the latter being unique to the nonlocal NLS system with no relevant counterparts in the local case. Direct numerical simulations show that, for sufficiently small amplitudes, both these types of soliton stripes do exist and propagate undistorted, in excellent agreement with the analytical predictions. Larger amplitude of these soliton stripes, when perturbed along the transverse direction, disintegrate either to filled vortex structures (the dark-bright solitons) or to radiation (the antidark-bright solitons).

A nonlocal variable coefficient KdV equation: Bäcklund transformation and nonlinear waves

A nonlocal two-layer fluid model is constructed through a simple symmetry reduction from the local one. Then, a general variable coefficients nonlocal KdV (VCKdV) equation with shifted space parity and delayed time reversal is derived from it by using multi-scale expansion method, with and without the so-called $$y-$$average method, respectively. A non-auto-Backlund transformation between the VCKdV equation and a constant coefficients KdV (CCKdV) equation is established. By using this transformation, various exact solutions of the VCKdV equation can be obtained from the seed solutions of the CCKdV equation. As some concrete examples, one solitary wave solution and two kinds of periodic wave solutions are given. Due to the inclusion of arbitrary functions in these solutions, they possess abundant dynamical behaviors with some of them analyzed graphically.

Нелокальные солитоны в нелинейной цепочке атомов

A nonlinear 1D chain with nonlocal interaction is considered. Using the multiscale expansion method, a nonlocal equation is obtained that describes the propagation of envelope waves in a medium. The properties of the obtained equation were studied and exact soliton-like solutions were constructed by the Darboux method of transformations.

Numerical Treatment of the Nonconservative Product in a Multiscale Fluid Model for Plasmas in Thermal Nonequilibrium: Application to Solar Physics

This contribution deals with the modeling of collisional multicomponent magnetized plasmas in thermal and chemical nonequilibrium aiming at simulating and predicting magnetic reconnections in the chromosphere of the sun. We focus on the numerical simulation of a simplified fluid model in order to properly investigate the influence on shock solutions of a nonconservative product present in the electron energy equation. Then, we derive jump conditions based on travelling wave solutions and propose an original numerical treatment in order to avoid non-physical shocks for the solution, that remains valid in the case of coarse-resolution simulations. A key element for the numerical scheme proposed is the presence of diffusion in the electron variables, consistent with the physically-sound scaling used in the model developed by Graille et al. following a multiscale Chapman-Enskog expansion method [M3AS, 19 (2009) 527--599]. The numerical strategy is eventually assessed in the framework of a solar physics test case. The computational method is able to capture the travelling wave solutions in both the highly- and coarsely-resolved cases.

A nonlocal nonlinear Schrödinger equation derived from a two-layer fluid model

By applying a simple symmetry reduction on a two-layer liquid model, a nonlocal counterpart of it is obtained. Then, a general form of nonlocal nonlinear Schrodinger (NNLS) equation with shifted parity, charge conjugate and delayed time reversal is obtained by using multi-scale expansion method. Some kinds of elliptic periodic wave solutions of the NNLS equation, which become soliton solutions and kink solutions when the modulus is taken as unity, are obtained by using elliptic function expansion method. Some representative figures of these solutions are given and analyzed in detail. In addition, by carrying out the classical symmetry method on the NNLS equation, not only the Lie symmetry group but also the related symmetry reduction solutions are given.

Envelope solitons in a nonlinear string with mirror nonlocality

The monoatomic nonlinear string in the presence of a mirror-type nonlocal interaction is considered. Using the method of multiscale expansions, a nonlocal equation describing the envelope wave dynamics in such medium is obtained. Properties of derived equation are studied, and exact soliton-like solutions are constructed by using Darboux method.

Modeling and analysis of fractional neutral disturbance waves in arterial vessels

The behavior of neutral disturbance in arterial vessels has attracted more and more attention in recent decades because it carries some important information which can be applied to predict and diagnose related heart disease, such as arteriosclerosis and hypertension, etc. Because of the complexity of blood flow in arteries, it is very necessary to construct accurate mathematical model and analyze the mechanical behavior of neutral disturbance in arterial vessels. In this paper, start from the basic equations of blood flow and the two-dimensional Navier–Stokes equation, the vorticity equation describing the disturbance flow is presented. Then, by use of multi-scale analysis and perturbation expansion method, the ZK equation is put forward which can reflect the behavior of the neutral perturbation flow in arterial vessels. Compared with the traditional KdV model, the model established in the paper can show the propagation of the disturbance flow in the radius direction. Furthermore, the time-fractional ZK equation is derived by semi-inverse method and Agrawal’s method, which is more convenient and accurate for discussing the feature of neutral disturbance in arterial vessels and can provide more information for analyzing some related heart disease. Meanwhile, with the help of the modified extended tanh method, the above mentioned equation is solved. The results show that neutral disturbance exists in arterial vessels and propagates in the form of solitary waves. By calculating, we find the relation of the stroke volume with vascular radius, blood flow velocity as well as the fractional order parameterα, which is very meaningful for preventing and treating related heart disease because the stroke volume is closely linked with heart disease.