Multiscale Expansion Method Research Articles

A nonlocal Burgers equation in atmospheric dynamical system and its exact solutions**Project supported by the National Natural Science Foundation of China (Grant Nos. 11405110, 11275129, and 11472177) and the Natural Science Foundationof Zhejiang Province of China (Grant No. LY18A050001).

From a two-vortex interaction model in atmospheric and oceanic systems, a nonlocal counterpart with shifted parity and delayed time reversal is derived by a simple AB reduction. To obtain some approximate analytic solutions of this nonlocal system, the multi-scale expansion method is applied to get an AB-Burgers system. Various exact solutions of the AB-Burgers equation, including elliptic periodic waves, kink waves and solitary waves, are obtained and shown graphically. To show the applications of these solutions in describing correlated events, a simple approximate solution for the two-vortex interaction model is given to show two correlated dipole blocking events at two different places. Furthermore, symmetry reduction solutions of the nonlocal AB-Burgers equation are also given by using the standard Lie symmetry method.

Large-Scale Compressibility in Rotating Flows of Astrophysical Plasma in the Shallow Water Approximation

Two systems of magnetohydrodynamic equations in the shallow water approximation are proposed as a basis for studies in the field of plasma astrophysics: the system of equations with a full allowance for the Coriolis force and the system of equations on the β-plane in which the changes of the Coriolis parameter are linear in coordinate. Both systems of equations take into account such fundamentally important phenomena in plasma astrophysics as the compressibility and external magnetic field effects, increasing significantly the potential for applying these equations to study astrophysical objects. Compressibility in plasma astrophysics is shown to change significantly the dispersion laws for magneto-Poincare, magnetostrophic, and magneto-Rossby waves. The same nonlinear interactions as those in the absence of compressibility have been found to be realized in the case of a compressible rotating plasma. Three-wave equations in the weak nonlinearity approximation, in which the interaction coefficients depend on plasma large-scale compressibility and thermodynamic characteristics, have been derived by the method of multiscale expansions. Expressions for the growth rates of the parametric instabilities of three-wave interactions with large-scale compressibility have been derived.

Solitary waves of the two-dimensional Camassa–Holm—nonlinear Schrödinger equation

In this work, we study solitary waves in a -dimensional variant of the defocusing nonlinear Schrödinger (NLS) equation, the so-called Camassa–Holm-NLS (CH-NLS) equation. We use asymptotic multiscale expansion methods to reduce this model to a Kadomtsev–Petviashvili (KP) equation. The KP model includes both the KP-I and KP-II versions, which possess line and lump soliton solutions. Using KP solitons, we construct approximate solitary wave solutions on top of the stable continuous-wave solution of the original CH-NLS model, which are found to be of both the dark and anti-dark type. We also use direct numerical simulations to investigate the validity of the approximate solutions, study their evolution, as well as their head-on collisions.

Hydrodynamics and two-dimensional dark lump solitons for polariton superfluids.

We study a two-dimensional incoherently pumped exciton-polariton condensate described by an open-dissipative Gross-Pitaevskii equation for the polariton dynamics coupled to a rate equation for the exciton density. Adopting a hydrodynamic approach, we use multiscale expansion methods to derive several models appearing in the context of shallow water waves with viscosity. In particular, we derive a Boussinesq/Benney-Luke-type equation and its far-field expansion in terms of Kadomtsev-Petviashvili-I (KP-I) equations for right- and left-going waves. From the KP-I model, we predict the existence of vorticity-free, weakly (algebraically) localized two-dimensional dark-lump solitons. We find that, in the presence of dissipation, dark lumps exhibit a lifetime three times larger than that of planar dark solitons. Direct numerical simulations show that dark lumps do exist, and their dissipative dynamics is well captured by our analytical approximation. It is also shown that lumplike and vortexlike structures can spontaneously be formed as a result of the transverse "snaking" instability of dark soliton stripes.

Three-Component Soliton States in Spinor F=1 Bose-Einstein Condensates.

Dilute-gas Bose-Einstein condensates are an exceptionally versatile test bed for the investigation of novel solitonic structures. While matter-wave solitons in one- and two-component systems have been the focus of intense research efforts, an extension to three components has never been attempted in experiments. Here, we experimentally demonstrate the existence of robust dark-bright-bright (DBB) and dark-dark-bright solitons in a multicomponent F=1 condensate. We observe lifetimes on the order of hundreds of milliseconds for these structures. Our theoretical analysis, based on a multiscale expansion method, shows that small-amplitude solitons of these types obey universal long-short wave resonant interaction models, namely, Yajima-Oikawa systems. Our experimental and analytical findings are corroborated by direct numerical simulations highlighting the persistence of, e.g., the DBB soliton states, as well as their robust oscillations in the trap.

A Multiscale Perturbation Expansion Approach for Markov State Modeling of Nonstationary Molecular Dynamics

We investigate metastable dynamical systems subject to nonstationary forcing as they appear in molecular dynamics for systems driven by external fields. We show that if the strength of the forcing is inversely proportional to the length of the slow metastable time scales of the unforced system, then the effective behavior of the forced system on slow time scales can be described by a low-dimensional reduced master equation. Our construction is explicit and uses the multiscale perturbation expansion method called two-timing, or method of multiple scales. The reduced master equation---a Markov state model---can be assembled by constructing two equilibrium Markov state models, one for the unforced system and one for a slightly perturbed one.

Asymptotic expansions and solitons of the Camassa–Holm – nonlinear Schrödinger equation

We study a deformation of the defocusing nonlinear Schrödinger (NLS) equation, the defocusing Camassa–Holm NLS, hereafter referred to as CH–NLS equation. We use asymptotic multiscale expansion methods to reduce this model to a Boussinesq-like equation, which is then subsequently approximated by two Korteweg–de Vries (KdV) equations for left- and right-traveling waves. We use the soliton solution of the KdV equation to construct approximate solutions of the CH–NLS system. It is shown that these solutions may have the form of either dark or antidark solitons, namely dips or humps on top of a stable continuous-wave background. We also use numerical simulations to investigate the validity of the asymptotic solutions, study their evolution, and their head-on collisions. It is shown that small-amplitude dark and antidark solitons undergo quasi-elastic collisions.

Rossby waves in the magnetic fluid dynamics of a rotating plasma in the shallow-water approximation

We have studied rotating magnetohydrodynamic flows of a thin layer of astrophysical plasma with a free boundary in the β-plane. Nonlinear interactions of the Rossby waves have been analyzed in the shallow-water approximation based on the averaging of the initial equations of the magnetic fluid dynamics of the plasma over the depth. The shallow-water magnetohydrodynamic equations have been generalized to the case of a plasma layer in an external vertical magnetic field. We have considered two types of the flow, viz., the flow in an external vertical magnetic field and the flow in the presence of a horizontal magnetic field. Qualitative analysis of the dispersion curves shows the presence of three-wave nonlinear interactions of the magnetic Rossby waves in both cases. In the particular case of zero external magnetic field, the wave dynamics in the layer of a plasma is analogous to the wave dynamics in a neutral fluid. The asymptotic method of multiscale expansions has been used for deriving the nonlinear equations of interaction in an external vertical magnetic field for slowly varying amplitudes, which describe three-wave interactions in a vertical external magnetic field as well as three-wave interactions of waves in a horizontal magnetic field. It is shown that decay instabilities and parametric wave amplification mechanisms exist in each case under investigation. The instability increments and the parametric gain coefficients have been determined for the relevant processes.

Vector nematicons: Coupled spatial solitons in nematic liquid crystals

Families of soliton pairs, namely vector solitons, are found within the context of a coupled nonlocal nonlinear Schr\odinger system of equations, as appropriate for modeling beam propagation in nematic liquid crystals. In the focusing case, bright soliton pairs have been found to exist provided their amplitudes satisfy a specific condition. In our analytical approach, focused on the defocusing regime, we rely on a multiscale expansion methods, which reveals the existence of dark-dark and antidark-antidark solitons, obeying an effective Korteweg-de Vries equation, as well as dark-bright solitons, obeying an effective Mel'nikov system. These pairs are discriminated by the sign of a constant that links all physical parameters of the system to the amplitude of the stable continuous wave solutions, and, much like the focusing case, the solitons' amplitudes are linked, leading to mutual guiding.

ZK-Burgers equation for three-dimensional Rossby solitary waves and its solutions as well as chirp effect

Two-dimensional Rossby solitary waves propagating in a line have attracted much attention in the past decade, whereas there is few research on three-dimensional Rossby solitary waves. But as is well known, three-dimensional Rossby solitary waves are more suitable for real ocean and atmosphere conditions. In this paper, using multiscale and perturbation expansion method, a new Zakharov-Kuznetsov (ZK)-Burgers equation is derived to describe three-dimensional Rossby solitary waves that propagate in a plane. By analyzing the equation we obtain the conservation laws of three-dimensional Rossby solitary waves. Based on the sine-cosine method, we give the classical solitary wave solutions of the ZK equation; on the other hand, by the Hirota method we also obtain the rational solutions, which are similar to the solutions of the Benjamin-Ono (BO) equation, the solutions of which can describe the algebraic solitary waves. The rational solutions of the ZK equations are worth of attention. Finally, with the help of the classical solitary wave solutions, similar to the fiber soliton communication, we discuss the dissipation and chirp effect of three-dimensional Rossby solitary waves.

Nonlinear wave interactions in shallow water magnetohydrodynamics of astrophysical plasma

The rotating magnetohydrodynamic flows of a thin layer of astrophysical and space plasmas with a free surface in a vertical external magnetic field are considered in the shallow water approximation. The presence of a vertical external magnetic field changes significantly the dynamics of wave processes in an astrophysical plasma, in contrast to a neutral fluid and a plasma layer in an external toroidal magnetic field. There are three-wave nonlinear interactions in the case under consideration. Using the asymptotic method of multiscale expansions, we have derived nonlinear equations for the interaction of wave packets: three magneto- Poincare waves, three magnetostrophic waves, two magneto-Poincare and one magnetostrophic waves, and two magnetostrophic and one magneto-Poincare waves. The existence of decay instabilities and parametric amplification is predicted. We show that a magneto-Poincare wave decays into two magneto-Poincare waves, a magnetostrophic wave decays into two magnetostrophic waves, a magneto-Poincare wave decays into one magneto-Poincare and one magnetostrophic waves, and a magnetostrophic wave decays into one magnetostrophic and one magneto-Poincare waves. There are the following parametric amplification mechanisms: the parametric amplification of magneto-Poincare waves, the parametric amplification of magnetostrophic waves, the amplification of a magneto-Poincare wave in the field of a magnetostrophic wave, and the amplification of a magnetostrophic wave in the field of a magneto-Poincare wave. The instability growth rates and parametric amplification factors have been found for the corresponding processes.

Phase-field-crystal simulation of edge dislocation climbing and gliding under shear strain

Structural kinetics in crystalline solids is driven heterogeneously at an atomic level by localized defects, which in turn drive mesoscopic and macroscopic phenomena such as structural phase transformation, fracture, and other forms of plastic flows. A complete description of such processes therefore requires a multiscale approach. Existing modeling methods typically operate exclusively either on an atomic scale or on a mesoscopic scale and macroscopic scale. Phase-field-crystal model, on the other hand, provides a framework that combines atomic length scale and mesoacpoic/diffusive time scale, with the potential reaching a mesoacpoic length through systemic multiscale expansion method. In order to study the dislocation movement under shear strain, the free energy density functional including the exerting shear force term is constructed and also the phase field crystal model for system of shear stain is established. The climb and glide of single dislocation in two-grain system are simulated, and the glide velocity of dislocation and the Peierls potential for dislocation gliding are calculated. The results show that the energy curve changing with time are monotonically smooth under a greater shear strain rate, which corresponds to dislocation movement at a constant speed, which is of rigorous characteristic; while under less shear strain rate, the energy change curve of system presents a periodic wave feature and the dislocation movement in the style of periodic “jerky” for gliding with the stick-slip characteristic. There is a critical potential for dislocation starting movement. The Peierls potential wall for climbing movement is many times as high as that for gliding movement. The results in these simulations are in a good agreement with the experimental ones.

Beating dark-dark solitons andZitterbewegungin spin-orbit-coupled Bose-Einstein condensates

We present families of beating dark-dark solitons in spin-orbit (SO) -coupled Bose-Einstein condensates. These families consist of solitons residing simultaneously in the two bands of the energy spectrum. The soliton components are characterized by two different spatial and temporal scales, which are identified by a multiscale expansion method. The solitons are ``beating'' ones, as they perform density oscillations. The characteristic frequency of the latter is relevant to Zitterbewegung (ZB) oscillations, which were recently observed in experiments with SO-coupled condensates [C. Qu et al., Phys. Rev. A 88, 021604(R) (2013); L. J. LeBlanc et al., New J. Phys. 15, 073011 (2013)]. We find that spin oscillations may occur, depending on the parity of each soliton branch, which consequently lead to ZB oscillations of the beating dark solitons. Analytical results are corroborated by numerical simulations, illustrating the robustness of the solitons.

Oscillons and oscillating kinks in the Abelian-Higgs model

We study the classical dynamics of the Abelian Higgs model employing an asymptotic multiscale expansion method, which uses the ratio of the Higgs to the gauge field amplitudes as a small parameter. We derive an effective nonlinear Schrodinger equation for the gauge field, and a linear equation for the scalar field containing the gauge field as a nonlinear source. This equation is used to predict the existence of oscillons and oscillating kinks for certain regimes of the ratio of the Higgs to the gauge field masses. Results of direct numerical simulations are found to be in very good agreement with the analytical findings, and show that the oscillons are robust, while kinks are unstable. It is also demonstrated that oscillons emerge spontaneously as a result of the onset of the modulational instability of plane wave solutions of the model. Connections of the results with the phenomenology of superconductors is discussed.

Matter-Wave Bright Solitons in Spin-Orbit Coupled Bose-Einstein Condensates

We study matter-wave bright solitons in spin-orbit coupled Bose-Einstein condensates with attractive interactions. We use a multiscale expansion method to identify solution families for chemical potentials in the semi-infinite gap of the linear energy spectrum. Depending on the linear and spin-orbit coupling strengths, the solitons may present either a sech2-shaped or a modulated density profile reminiscent of the stripe phase of spin-orbit coupled repulsive Bose-Einstein condensates. Our numerical results are in excellent agreement with our analytical findings and demonstrate the potential robustness of solitons for experimentally relevant conditions.

Existence theory for Rosseland equation and its homogenized equation

The global boundness and existence are presented for the kind of the Rosseland equation with a general growth condition. A linearized map in a closed convex set is defined. The image set is precompact, and thus a fixed point exists. A multi-scale expansion method is used to obtain the homogenized equation. This equation satisfies a similar growth condition.

Asymptotic Bifurcation Solutions for Perturbed Kuramoto—Sivashinsky Equation

Stability and dynamic bifurcation in the perturbed Kuramoto—Sivashinsky (KS) equation with Dirichlet boundary condition are investigated by using central manifold reduction procedure. The result shows, as the bifurcation parameter crosses a critical value, the system undergoes a pitchfork bifurcation to produce two asymptotically stable solutions. Furthermore, when the distance from bifurcation is of comparable order ∊2 (|∊| ≪ 1), the first two terms in ∊-expansions for the new asymptotic bifurcation solutions are derived by multiscale expansion method. Such information is useful to the bifurcation control.

Discrete solitons in electromechanical resonators

We consider a particular type of parametrically driven discrete Klein-Gordon system describing microdevices and nanodevices, with integrated electrical and mechanical functionality. Using a multiscale expansion method we reduce the system to a discrete nonlinear Schrödinger equation. Analytical and numerical calculations are performed to determine the existence and stability of fundamental bright and dark discrete solitons admitted by the Klein-Gordon system through the discrete Schrödinger equation. We show that a parametric driving can not only destabilize onsite bright solitons, but also stabilize intersite bright discrete solitons and onsite and intersite dark solitons. Most importantly, we show that there is a range of values of the driving coefficient for which dark solitons are stable, for any value of the coupling constant, i.e., oscillatory instabilities are totally suppressed. Stability windows of all the fundamental solitons are presented and approximations to the onset of instability are derived using perturbation theory, with accompanying numerical results. Numerical integrations of the Klein-Gordon equation are performed, confirming the relevance of our analysis.

Variable coefficient nonlinear systems derived from an atmospheric dynamical system

Variable coefficient nonlinear systems, the Korteweg de Vries (KdV), the modified KdV (mKdV) and the nonlinear Schrödinger (NLS) type equations, are derived from the nonlinear inviscid barotropic nondivergent vorticity equation in a beta-plane by means of the multi-scale expansion method in two different ways, with and without the so-called y-average trick. The non-auto-Bäcklund transformations are found to transform the derived variable coefficient equations to the corresponding standard KdV, mKdV and NLS equations. Thus, many possible exact solutions can be obtained by taking advantage of the known solutions of these standard equations. Further, many approximate solutions of the original model are ready to be yielded which might be applied to explain some real atmospheric phenomena, such as atmospheric blocking episodes.

Nonlinear effects in the process of propagation of internal waves with regard for the turbulent viscosity and diffusion

By the method of asymptotic multiscale expansions in the Boussinesq approximation, we study nonlinear effects observed in the process of propagation of internal waves with regard for the turbulent viscosity and diffusion. We determine the decrement of attenuation of waves and the boundary-layer solutions at the bottom and on the free surface. The wave-induced mean current is found in the second order of smallness in the wave steepness. The coefficients of the nonlinear Schrödinger equation are obtained for the envelope of the wave packet. It is shown that a weakly nonlinear plane wave is stable under longitudinal modulation in the long-wave limit. If the wavelength is smaller than a certain critical value, then the wave is unstable under modulation.