Coupled Evolution Equations Research Articles

Reductions and Solutions of Two Types of Coupled Nonlinear Evolution Equations in Optical Fibers and Fluid Dynamics

Abstract Under investigation in this paper are the coupled nonlinear Schrödinger equations (CNLSEs) and coupled Burgers-type equations (CBEs), which are, respectively, a model for certain birefringent optical fibers Raman-scattering, Kerr and gain/loss effects, and a generalized model in fluid dynamics. Special attention should be paid to the existing claim that the solitons for the CNLSEs do not exist. Through certain dependent-variable transformations, the CNLSEs are reduced to a Manakov system and the CBEs are linearized. In that way, some new solutions of the CNLSEs and CBEs are obtained via symbolic computation. Especially the one-dark-soliton-like solutions for the CNLSEs have been found, against the existing claim.

Theoretical analysis of electric, magnetic and magnetoelectric properties of nano-structured multiferroic composites

Electric, magnetic and magnetoelectric properties of the nano-structured multiferroic composites were studied by using an energy formulation with the consideration of the surface, interface, and size effect. Coupled thermodynamic evolution equations with respect to the spontaneous polarization and magnetization were established, in which the elastic fields in the matrix and inclusions were solved based on the Eshelby's equivalent inclusion concept and the Mori–Tanaka method. Physical properties of the composite, such as the spontaneous order parameters, piezoelectric/piezomagnetic properties, and the magnetoelectric coupling effect are highly dependent on the stress state and the microstructures of the nano-composites. Magnetoelectric coupling voltage coefficient was unstable in the vicinity of the critical size and disappeared below the critical size. The model is versatile enough for various composite structures.

Breakup of an electrified, perfectly conducting, viscous thread in an AC field

We study the axisymmetric breakup and satellite formation of slender jets surrounded by a concentrically placed cylindrical electrode and subjected to time-dependent AC electric fields. The jet is assumed to be a perfectly conducting viscous fluid and surrounded by a dielectric inviscid gas. We use the long-wave approximation to derive coupled evolution equations for the interface position and the axial velocity component, which accounts for electrostatic forcing. The electrostatic force in this case is large and competes with capillary forces near the rupture point, causing the interface to oscillate and the satellite to have shapes that are distinct from the DC case. In particular, our results indicate that it may be possible to use the AC field to control the number of satellites accompanying breakup as well as their size.

Rupture of thin liquid films: Generalization of weakly nonlinear theory

In this paper, we investigate the rupture dynamics of thin liquid films driven by intermolecular forces via weakly nonlinear bifurcation analysis. The dynamic equations governing slow dynamics of the perturbation amplitude of the near-critical mode corresponding to several models describing the evolution of thin liquid films in different physical situations appear to have the same structure. When antagonistic (attractive and repulsive) molecular forces are considered, nonlinear saturation of the instability becomes possible, while the boundary of this supercritical bifurcation is determined solely by the form of the intermolecular potential. The rupture time estimate obtained in closed form shows an excellent agreement with the results of the previously reported numerical simulations of the strongly nonlinear coupled evolution equations upon fitting the amplitude of the small initial perturbation. We further extend the weakly nonlinear analysis of the film dynamics and apply the Galerkin approximation to derive the amplitude equation(s) governing the dynamics of the fastest growing linear mode far from the instability threshold. The comparison of the rupture time derived from this theory with the results of numerical simulations of the original nonlinear evolution equations shows a very good agreement without any adjustable parameters.

Lagrange Equations Coupled to a Thermal Equation: Mechanics as Consequence of Thermodynamics

Following the analytic approach to thermodynamics developed by Stückelberg, we study the evolution equations of a closed thermodynamic system consisting of point particles in a fluid. We obtain a system of coupled differential equations describing the mechanical and the thermal evolution of the system. The coupling between these evolution equations is due to the action of a viscous friction term. Finally, we apply our coupled evolution equations to study the thermodynamics of an isolated system consisting of identical point particles interacting through a harmonic potential.

On surfactant-enhanced spreading and superspreading of liquid drops on solid surfaces

The mechanisms driving the surfactant-enhanced spreading of droplets on the surface of solid substrates, and particularly those underlying the superspreading behaviour sometimes observed, are investigated theoretically. Lubrication theory for the droplet motion, together with advection–diffusion equations and chemical kinetic fluxes for the surfactant transport, leads to coupled evolution equations for the drop thickness, interfacial concentrations of surfactant monomers and bulk concentrations of monomers and micellar, or other, aggregates. The surfactant can be adsorbed on the substrate either directly from the bulk monomer concentrations or from the liquid–air interface through the contact line. An important feature of the spreading model developed here is the surfactant behaviour at the contact line; this is modelled using a constitutive relation, which is dependent on the local surfactant concentration. The evolution equations are solved numerically, using the finite-element method, and we present a thorough parametric analysis for cases of both insoluble and soluble surfactants with concentrations that can, in the latter case, exceed the critical micelle, or aggregate, concentration. The results show that basal adsorption of the surfactant plays a crucial role in the spreading process; the continuous removal of the surfactant that lies upon the liquid–air interface, due to the adsorption at the solid surface, is capable of inducing high Marangoni stresses, close to the droplet edge, driving very fast spreading. The droplet radius grows at a rate proportional to ta with a = 1 or even higher, which is close to the reported experimental values for superspreading. The spreading rates follow a non-monotonic variation with the initial surfactant concentration also in accordance with experimental observations. An accompanying feature is the formation of a rim at the leading edge of the droplet. In some cases, the drop spreads to form a ‘pancake’ or creates a ‘secondary’ front separated from the main droplet.

Analysis of time-dependent nonlinear dynamics of the axisymmetric liquid film on a vertical circular cylinder: Energy integral model

The nonlinear dynamics of an axisymmetric liquid film falling on the outer surface of a vertical cylinder is investigated in the framework of the set of two coupled evolution equations derived recently using the energy integral method (EIM). This set of EIM evolution equations is solved numerically and its solutions are compared with the traveling wave solutions derived from it using AUTO. We find that traveling wave solutions of EIM equations can bifurcate either supercritically or subcritically from the base state. The type of bifurcation depends on the parameter set of the problem. The set of EIM equations studied here admits both traveling wave and nonstationary wave flows. We demonstrate that in the case of subcritical primary bifurcation the film dynamics is sensitive to the choice of the initial condition and coexistence of up to five different flows is possible for the same parameter set in the domain of a given periodicity. The case of supercritical primary bifurcation exhibits much lesser dependence on the initial condition, though coexistence of two different flows for the same parameter set is possible. The synergetic approach based on both direct numerical solution of the governing evolution equations and search of traveling wave solutions using AUTO facilitate a discovery of a large variety of flows and help to conclude about stability of the traveling wave flows found using AUTO.

Exponential Stability of Two Coupled Second-Order Evolution Equations

By using the multiplier technique, we prove that the energy of a system of two coupled second order evolution equations (one is an integro-differential equation) decays exponentially if the convolution kernel decays exponentially. An example is give to illustrate that the result obtained can be applied to concrete partial differential equations.

Nonlinear thermodynamic quantum master equation: Properties and examples

The quantum master equation obtained from two different thermodynamic arguments is seriously nonlinear. We argue that, for quantum systems, nonlinearity occurs naturally in the step from reversible to irreversible equations and we analyze the nature and consequences of the nonlinear contribution. The thermodynamic nonlinearity naturally leads to canonical equilibrium solutions and extends the range of validity to lower temperatures. We discuss the Markovian character of the thermodynamic quantum master equation and introduce a solution strategy based on coupled evolution equations for the eigenstates and eigenvalues of the density matrix. The general ideas are illustrated for the two-level system and for the damped harmonic oscillator. Several conceptual implications of the nonlinearity of the thermodynamic quantum master equation are pointed out, including the absence of a Heisenberg picture and the resulting difficulties with defining multitime correlations.

On diffusion–elastic instabilities in a solid half-space

A model of the diffusion–elastic instability that appears in an ensemble of non-equilibrium atomic defects in unbounded condensed media as well as on the free surface of a half-space is introduced and studied. The dynamical model developed here is based on coupled evolution equations for the elastic displacement of the medium and atomic defect density fields. The idea of an instability model is related to a drift of atomic defects under the influence of elastic fields. It is shown that the development of this instability creates ordered structures of coupled strain and defect-concentration fields. Dispersion relationships for the growth increment of these structures are derived and their characteristic scales are obtained.

On the anomalous large-scale flows in the Universe

Recent combined analyses of the CMB and galaxy cluster data reveal unexpectedly large and anisotropic peculiar velocity fields at large scales. We study cosmic models with included vorticity, acceleration and total angular momentum of the Universe in order to understand the phenomenon. The Zeldovich model is used to mimic the low redshift evolution of the angular momentum. Solving coupled evolution equations of the second kind for density-contrast in corrected Ellis-Bruni covariant and gauge-invariant formalism one can properly normalize and evaluate integrated Sachs-Wolfe effect and peculiar velocity field. The theoretical results compared to the observations favor a much larger matter content of the Universe than that of the concordance model. Large-scale flows appear anisotropic with dominant components placed in the plane perpendicular to the axis of vorticity(rotation). The integrated Sachs-Wolfe term has negative contribution to the CMB fluctuations for the negative cosmological constant and it can explain the observed small power of the CMB TT spectrum at large scales. The rate of the expansion of the Universe can be substantially affected by the angular momentum if its magnitude is large enough.

The tanh–coth method combined with the Riccati equation for solving nonlinear coupled equation in mathematical physics

In this work, we established abundant travelling wave solutions for nonlinear coupled evolution equation. This method was used to construct solitons and travelling wave solutions of nonlinear coupled evolution equation. The tanh–coth method combined with the Riccati equation presents a wider applicability for handling nonlinear wave equations.

Magnetization reversal through flipping solitons under the localized inhomogeneity

We study the nonlinear dynamics of a site-dependent Heisenberg ferromagnetic spin chain with Gilbert damping in the continuum limit and its associated dynamics which is governed by an inhomogeneous generalized higher order nonlinear Schrödinger equation. The effect of inhomogeneity was understood by carrying out a multiple perturbation analysis and the coupled evolution equations for the velocity and amplitude of the soliton were solved using the Jacobi elliptic function method. The evolution of the amplitude and velocity of the soliton leads to magnetization reversal via flipping of solitons in the ferromagnetic medium. Finally, we have also constructed the perturbed soliton solutions.

Concentration–elastic instabilities in a solid half‐space

AbstractThe generation of periodic structures due to concentration–elastic instability that appears in an ensemble of laser‐induced atomic point defects on the surface of a solid is studied. The dynamical model developed here is based on coupled evolution equations for the elastic and atomic defect density fields. The idea of an instability model is related to a reduction in the activation (formation and migration) energy of atomic defects under the influence of elastic fields. It is shown that the development of this instability creates various ordered surface structures of coupled strain and defect‐concentration fields (one‐ and two‐dimensional lattices, concentric rings, radial and radial‐ray structures). Dispersion relationships are derived for the growth increment of these structures and their characteristic scales (grating period, number of rays in a structure) are obtained. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Modelling approaches to the dewetting of evaporating thin films of nanoparticlesuspensions

We review recent experiments on dewetting thin films of evaporating colloidal nanoparticlesuspensions (nanofluids) and discuss several theoretical approaches to describe the ongoingprocesses including coupled transport and phase changes. These approaches range frommicroscopic discrete stochastic theories to mesoscopic continuous deterministicdescriptions. In particular, we describe (i) a microscopic kinetic Monte Carlo model, (ii) adynamical density functional theory and (iii) a hydrodynamic thin film model.Models (i) and (ii) are employed to discuss the formation of polygonal networks, spinodaland branched structures resulting from the dewetting of an ultrathin ‘postcursor film’ thatremains behind a mesoscopic dewetting front. We highlight, in particular, the presence of atransverse instability in the evaporative dewetting front, which results in highly branchedfingering structures. The subtle interplay of decomposition in the film and contact linemotion is discussed.Finally, we discuss a simple thin film model (iii) of the hydrodynamics on the mesoscale.We employ coupled evolution equations for the film thickness profile and mean particleconcentration. The model is used to discuss the self-pinning and depinning of a contact linerelated to the ‘coffee-stain’ effect.In the course of the review we discuss the advantages and limitations of the differenttheories, as well as possible future developments and extensions.

Energy integral method model for the nonlinear dynamics of an axisymmetric thin liquid film falling on a vertical cylinder

The nonlinear dynamics of an axisymmetric liquid film falling on the outer surface of a vertical cylinder is investigated in this paper. Using the energy integral method we derive a set of two coupled evolution equations which constitute a first-order approximation to the original hydrodynamic equations. We carry out the linear stability analysis of the axially uniform axisymmetric flow of a liquid film in the framework of the evolution equations. The results of the linear theory are compared to experimental results and with several other relevant linear stability theories available in literature. Traveling wave solutions of the evolution equations are investigated and favorably compared to experiments and other nonlinear theories describing the dynamics of falling films on a vertical cylinder. A bifurcation structure of traveling wave flows is studied for the cases of fluids with both low and relatively large Kapitza number. Both γ1- and γ2-waves are found and investigated.

Modélisation d'un écoulement coaxial en conduite circulaire de deux fluides visqueux

A simplified model of core-annular flow is derived by using a weighted residual method combined with a long-wavelength expansion developed in 2000 by Ruyer-Quil and Manneville. A set of two coupled evolution equations for two fields, the thickness h ( x , t ) and the flow rate q ( x , t ) is obtained. Linear stability predictions are in good agreement with numerical and experimental results of Preziosi et al. [L. Preziosi, K. Chen, D.D. Joseph, Lubricated pipelining: stability of core-annular flow, J. Fluid Mech. 201 (1989)]. To cite this article: N. Mehidi, N. Amatousse, C. R. Mecanique 337 (2009).

Kinetic Hierarchies and Macroscopic Limits for Crystalline Steps in $1+1$ Dimensions

We apply methods of kinetic theory to study the passage from particle systems to nonlinear partial differential equations (PDEs) in the context of deterministic crystal surface relaxation. Starting with the near-equilibrium motion of N line defects (“steps”) with atomic size a, we derive coupled evolution equations (“kinetic hierarchies”) for correlation functions $F_n^a$, which express correlations of n consecutive steps. We investigate separately the evaporation-condensation and the surface diffusion dynamics in $1+1$ dimensions when each step interacts repulsively with its nearest neighbors. In the limit $a\to0$ with $Na=\mathcal{O}(1)$, where a is appropriately nondimensional, the first equations of the hierarchies reduce to known evolution laws for the surface slope profile. The remaining PDEs take the form of simple continuity equations, which we solve exactly and, thereby, connect continuous limits of $F_n^a$ with the slope profile. In addition, we construct a particular example of $F_n^a$ asymptotically for small but finite a by regularization of measures. In the limit $a\to0$, this construction yields singular correlation functions.

Finite element formulation of a phase field model based on the concept of generalized stresses

A finite element formulation of a phase field model for alloys is proposed within the general framework of continuum thermodynamics in conjunction with the concept of generalized stresses as proposed by Gurtin [1]. Using the principles of the thermodynamics of irreversible processes, balance and constitutive equations are clearly separated in the formulation. Also, boundary conditions for the concentration and order parameter and their dual quantities are clearly stated. The theory is shown to be well-suited for a finite element formulation of the initial boundary value problem. The set of coupled evolution equations, which are the phase field equation and the balance of mass, is solved using an implicit finite element method for space discretization and a finite difference method for time discretization. For an illustrative purpose, the model is used to investigate the growth of an oxide layer at the surface of a pure zirconium slab. Calculations in 1D show a good agreement with an analytical solution for the growth kinetics. Then, 2D calculations of the same process have been undertaken to investigate morphological stability of the oxide layer in order to show the ability of the finite element method to handle arbitrary conditions on complex boundaries.

Modelling film flows down a fibre

Consider the gravity-driven flow of a thin liquid film down a vertical fibre. A model of two coupled evolution equations for the local film thickness h and the local flow rate q is formulated within the framework of the long-wave and boundary-layer approximations. The model accounts for inertia and streamwise viscous diffusion. Evolution equations obtained by previous authors are recovered in the appropriate limit. Comparisons to experimental results show good agreement in both linear and nonlinear regimes. Viscous diffusion effects are found to have a stabilizing dispersive effect on the linear waves. Time-dependent computations of the spatial evolution of the film reveal a strong influence of streamwise viscous diffusion on the dynamics of the flow and the wave selection process.