Stability of Periodic Solutions¶of Conservation Laws with Viscosity:¶Analysis of the Evans Function

Nonclassical conservation laws with viscosity arising in multiphase fluid and solid mechanics exhibit a rich variety of traveling-wave phenomena, including homoclinic (pulse-type) and periodic solutions along with the standard heteroclinic (shock, or front-type) solutions. Here, we investigate stability of periodic traveling waves within the abstract Evans-function framework established by R. A. Gardner. Our main result is to derive a useful stability index analogous to that developed by Gardner and Zumbrun in the traveling-front or -pulse context, giving necessary conditions for stability with respect to initial perturbations that are periodic on the same period T as the traveling wave; moreover, we show that the periodic-stability index has an interpretation analogous to that of the traveling-front or -pulse index in terms of well-posedness of an associated Riemann problem for an inviscid medium, now to be interpreted as allowing a wider class of measure-valued solutionsor, alternatively, in terms of existence and nonsingularity of a local “mass map” from perturbation mass to potential time-asymptotic T-periodic states. A closely related calculation yields also a complementary long-wave stability criterion necessary for stability with respect to periodic perturbations of arbitrarily large period NT, N → ∞. We augment these analytical results with numerical investigations analogous to those carried out by Brin in the traveling-front or -pulse case, approximating the spectrum of the linearized operator about the wave.The stability index and long-wave stability criterion are explicitly evaluable in the same planar, Hamiltonian cases as is the index of Gardner and Zumbrun, and together yield rigorous results of instability similar to those obtained previously for pulse-type solutions; this is established through a novel dichotomy asserting that the two criteria are in certain cases logically exclusive. In particular, we obtain results bearing on the nature and mechanism for formation of highly oscillatory Turing-like patterns observed numerically by Frid and Liu and ?ani? and Peters in models of multiphase flow. Specifically, for the van der Waals model considered by Frid and Liu, we show instability of all periodic waves such that the period increases with amplitude in the one-parameter family of nearby periodic orbits, and in particular of large- and small-amplitude waves; for the standard, double-well potential, this yields instability of all periodic waves.Likewise, for a quadratic-flux model like that considered by ?ani? and Peters, we show instability of large-amplitude waves of the type lying near observed patterns, and of all small-amplitude waves; our numerical results give evidence that intermediate-amplitude waves are unstable as well. These results give support for an alternative mechanism for pattern formation conjectured by Azevedo, Marchesin, Plohr, and Zumbrun, not involving periodic waves.     

Vortex solitons of the (3+1)-dimensional spatially modulated cubic–quintic nonlinear Schrödinger equation with the transverse modulation

Vortex solitons in the spatially modulated cubic–quintic nonlinear media are governed by a (3+1)-dimensional cubic–quintic nonlinear Schrödinger equation with spatially modulated nonlinearity and transverse modulation. Via the variable separation principle with the similarity transformation, we derive two families of vortex soliton solutions in the spatially modulated cubic–quintic nonlinear media. For the disappearing and parabolic transverse modulation, vortex solitons with different configurations are constructed. The similar configurations of vortex solitons exist for the same value of \(l-k\) with the topological charge k and degree number l. Moreover, the number of the inner layer structure of vortex solitons getting rid of the package covering layer is related to \((n-1)/2+1\) with the soliton order number n. For the disappearing transverse modulation, there exist phase azimuthal jumps around their cores of vortex solitons with \(2\pi \) phase change in every jump, and any two jumps one after another realize the change in \(\pi \). For the parabolic transverse modulation, all phases of vortex soliton exist k-jump, and every jump realizes the change in \(2\pi /k\); thus, k-jumps totally realize the azimuthal change in \(2\pi \) around their cores.     

Families of rational solutions of the <Emphasis Type="Italic">y</Emphasis>-nonlocal Davey–Stewartson II equation

The y-nonlocal Davey–Stewartson II equation is an extension of the usual DS II equation involving a partially parity-time-symmetric potential only with respect to the spatial variable y. By using the Hirota bilinear method, families of n-order rational solutions are obtained, which include lumps in the (x, y)-plane and the (y, t)-plane, growing-and-decaying line waves in the (x, t)-plane, and hybrid solutions of interacting line rogue waves and lumps in the (x, y)-plane.     

Wave reflection at a free interface in an anisotropic pyroelectric medium with nonclassical thermoelasticity

In this paper, the well-established two-dimensional mathematical model for linear pyroelectric materials is employed to investigate the reflection of waves at the boundary between a vacuum and an elastic, transversely isotropic, pyroelectric material. A comparative study between the solutions of (a) classical thermoelasticity, (b) Cattaneo–Lord–Shulman theory and (c) Green–Lindsay theory equations, characterised by none, one and two relaxation times, respectively, is presented. Suitable boundary conditions are considered in order to determine the reflection coefficients when incident elasto–electro–thermal waves impinge the free interface. It is established that, in the quasi-electrostatic approximation, three different classes of waves: (1) two principally elastic waves, namely a quasi-longitudinal Primary (qP) wave and a quasi-transverse Secondary (qS) wave; and (2) a mainly thermal (qT) wave. The observed electrical effects are, on the other hand, a direct consequence of mechanical and thermal phenomena due to pyroelectric coupling. The computed reflection coefficients of plane qP waves are found to depend upon the angle of incidence, the elastic, electric and thermal parameters of the medium, as well as the thermal relaxation times. The special cases of normal and grazing incidence are also derived and discussed. Finally, the reflection coefficients are computed for cadmium selenide observing the influence of (1) the anisotropy of the material, (2) the electrical potential and (3) temperature variations and (4) the thermal relaxation times on the reflection coefficients.     

Asymptotic Stability of High-dimensional Zakharov–Kuznetsov Solitons

We prove that solitons (or solitary waves) of the Zakharov–Kuznetsov (ZK) equation, a physically relevant high dimensional generalization of the Korteweg–de Vries (KdV) equation appearing in Plasma Physics, and having mixed KdV and nonlinear Schrödinger (NLS) dynamics, are strongly asymptotically stable in the energy space. We also prove that the sum of well-arranged solitons is stable in the same space. Orbital stability of ZK solitons is well-known since the work of de Bouard [Proc R Soc Edinburgh 126:89–112, 1996]. Our proofs follow the ideas of Martel [SIAM J Math Anal 157:759–781, 2006] and Martel and Merle [Math Ann 341:391–427, 2008], applied for generalized KdV equations in one dimension. In particular, we extend to the high dimensional case several monotonicity properties for suitable half-portions of mass and energy; we also prove a new Liouville type property that characterizes ZK solitons, and a key Virial identity for the linear and nonlinear part of the ZK dynamics, obtained independently of the mixed KdV–NLS dynamics. This last Virial identity relies on a simple sign condition which is numerically tested for the two and three dimensional cases with no additional spectral assumptions required. Possible extensions to higher dimensions and different nonlinearities could be obtained after a suitable local well-posedness theory in the energy space, and the verification of a corresponding sign condition.     

Multipulse Phases in k-Mixtures of Bose–Einstein Condensates

For the system

$-\Delta U_i+ U_i=U_i^3-\beta U_i\sum_{j\neq i}U_j^2,\quad i=1,\dots,k,$

(with k ≧ 3), we prove the existence for β large of positive radial solutions on \({\mathbb R^N}\) . We show that as β →  + ∞, the profile of each component U _i separates, in many pulses, from the others. Moreover, we can prescribe the location of such pulses in terms of the oscillations of the changing-sign solutions of the scalar equation  ? ΔW  +  W  =  W³. Within an Hartree–Fock approximation, this provides a theoretical indication of phase separation into many nodal domains for the k-mixtures of Bose–Einstein condensates.

     

Decay of Almost Periodic Solutions¶of Conservation Laws

We consider the asymptotic behavior of solutions of systems of inviscid or viscous conservation laws in one or several space variables, which are almost periodic in the space variables in a generalized sense introduced by Stepanoff and Wiener, which extends the original one of H. Bohr. We prove that if u(x,t) is such a solution whose inclusion intervals at time t, with respect to ?>0, satisfy l _epsiv;(t)/t→0 as t→∞, and such that the scaling sequence u ^T (x,t)=u(T x,T t) is pre-compact as t→∞ in L _loc ¹(? ^{d +1} ₊, then u(x,t) decays to its mean value \(\), which is independent of t, as t→∞. The decay considered here is in L ¹ _loc of the variable ξ≡x/t, which implies, as we show, that \(\) as t→∞, where M _x denotes taking the mean value with respect to x. In many cases we show that, if the initial data are almost periodic in the generalized sense, then so also are the solutions. We also show, in these cases, how to reduce the condition on the growth of the inclusion intervals l _?(t) with t, as t→∞, for fixed ? > 0, to a condition on the growth of l _?(0) with ?, as ?→ 0, which amounts to imposing restrictions only on the initial data. We show with a simple example the existence of almost periodic (non-periodic) functions whose inclusion intervals satisfy any prescribed growth condition as ?→ 0. The applications given here include inviscid and viscous scalar conservation laws in several space variables, some inviscid systems in chromatography and isentropic gas dynamics, as well as many viscous 2 × 2 systems such as those of nonlinear elasticity and Eulerian isentropic gas dynamics, with artificial viscosity, among others. In the case of the inviscid scalar equations and chromatography systems, the class of initial data for which decay results are proved includes, in particular, the L ^∞ generalized limit periodic functions. Our procedures can be easily adapted to provide similar results for semilinear and kinetic relaxations of systems of conservation laws.     

Application of linear and nonlinear vibration absorbers in micro-milling process in order to suppress regenerative chatter

In this paper, a generalized higher-order variable-coefficient nonlinear Schrödinger equation is studied, which describes the propagation of subpicosecond or femtosecond pulses in an inhomogeneous optical fiber. We derive a set of the integrable constraints on the variable coefficients. Under those constraints, via the symbolic computation and modified Hirota method, bilinear equations, one-, two-,three-soliton solutions and dromion-like structures are obtained. Properties and interactions for the solitons are studied: (a) effects on the solitons resulting from the wave number k, third-order dispersion \(\delta _1(z)\), group velocity dispersion \(\alpha (z)\), gain/loss \(\varGamma _2(z)\) and group-velocity-related \(\gamma (z)\) are discussed analytically and graphically where z is the normalized propagation distance along the fiber; (b) bound state with different values of \(\alpha (z)\), \(\delta _1(z)\), \(\gamma (z)\) and \(\varGamma _2(z)\) are presented where some periodic or quasiperiodic formulae are derived. Interactions between the two solitons and between the bound states and a single soliton are, respectively, discussed; and (c) single, double and triple dromion-like structures with different values of \(\alpha (z)\), \(\delta _1(z)\), \(\gamma (z)\) are also presented, distortions of which are found to be determined by those variable coefficients.     

Travelling waves of density for a fourth-gradient model of fluids

In mean-field theory, the non-local state of fluid molecules can be taken into account using a statistical method. The molecular model combined with a density expansion in Taylor series of the fourth order yields an internal energy value relevant to the fourth-gradient model, and the equation of isothermal motions takes then density’s spatial derivatives into account for waves travelling in both liquid and vapour phases. At equilibrium, the equation of the density profile across interfaces is more precise than the Cahn and Hilliard equation, and near the fluid’s critical point, the density profile verifies an Extended Fisher–Kolmogorov equation, allowing kinks, which converges towards the Cahn–Hillard equation when approaching the critical point. Nonetheless, we also get pulse waves oscillating and generating critical opalescence.     

The mathematical model of reflection and refraction of longitudinal waves in thermo-piezoelectric materials

In this paper, the basic equations of motion, of Gauss and of heat conduction, together with constitutive relations for pyro- and piezoelectric media, are presented. Three thermoelastic theories are considered: classical dynamical coupled theory, the Lord–Shulman theory with one relaxation time and Green and Lindsay theory with two relaxation times. For incident elastic longitudinal, potential electric and thermal waves, referred to as qP, φ-mode and T-mode waves, which impinge upon the interface between two different transversal isotropic media, reflection and refraction coefficients are obtained by solving a set of linear algebraic equations. A case study is investigated: a system formed by two semi-infinite, hexagonal symmetric, pyroelectric–piezoelectric media, namely Cadmium Selenide (CdSe) and Barium Titanate (BaTiO₃). Numerical results for the reflection and refraction coefficients are obtained, and their behavior versus the incidence angle is analyzed. The interaction with the interface give rises to different kinds of reflected and refracted waves: (i) two reflected elastic waves in the first medium, one longitudinal (qP-wave) and the other transversal (qSV-wave), and a similar situation for the refracted waves in the second medium; (ii) two reflected potential electric waves and a similar situation for the refracted waves; (iii) two reflected thermal waves and a similar situation for the refracted waves. The amplitudes of the reflected and refracted waves are functions of the incident angle, of the thermal relaxation times and of the media elastic, electric, thermal constants. This study is relevant to signal processing, sound systems, wireless communications, surface acoustic wave devices and military defense equipment.     

Reversible and Irreversible Tracer Dispersion in an Oscillating Flow Inside a Model Rough Fracture

We study the mixing dynamics of a dyed and a clear miscible fluid by an oscillating flow inside an Hele-Shaw cell with randomly distributed circular obstacles. A transparent setup allows us to analyze the distribution of the two fluids and the reversible and irreversible mixing components. At the lower Péclet numbers Pe (based on the averaged absolute fluid velocity), geometrical dispersion due to the disordered flow field between the obstacles is dominant: the corresponding dispersivity is constant with Pe and, at constant Pe, increases with the amplitude of the oscillations and is negligible at small ones. Compared to echo dispersion with only one injection–suction cycle, oscillating flows are shown to provide additional information when the number of oscillations and, as a result, the distance of transverse mixing are varied. Geometrical dispersion is dominant up to a limiting Pe increasing with the amplitude. At higher \({\textit{Pe}}'{\textit{s}}\), the results are similar to those of Taylor dispersion in cells with smooth walls.     

One-dimensional optical solitons in cubic–quintic–septimal media with $$\varvec{\mathcal {}}$$-symmetric potentials

A (\(1+1\))-dimensional inhomogeneous cubic–quintic–septimal nonlinear Schrödinger equation with \(\mathcal {PT}\)-symmetric potentials is studied, and two families of soliton solutions are obtained. From soliton solutions, the amplitude of soliton is independent of the \(\mathcal {PT}\)-symmetric potential parameter k; however, the phase depends on the parameter k. The phase of soliton alters from negative to positive values at the location of center. Moreover, the evolutional behaviors of these solitons are discussed.     

Traveling Wave Phenomena in a Kermack–McKendrick SIR Model

We study the existence and nonexistence of traveling waves of a general diffusive Kermack–McKendrick SIR model with standard incidence where the total population is not constant. The three classes, susceptible S, infected I and removed R, are all involved in the traveling wave solutions. We show that the minimum wave speed of traveling waves for the three-dimensional non-monotonic system can be derived from its linearizaion at the initial disease-free equilibrium. The proof in this paper is based on Schauder fixed point theorem and Laplace transform. Our study provides a promising method to deal with high dimensional epidemic models.     

On the Existence and Stability of Time Periodic Solution to the Compressible Navier–Stokes Equation on the Whole Space

The existence of a time periodic solution of the compressible Navier–Stokes equation on the whole space is proved for a sufficiently small time periodic external force when the space dimension is greater than or equal to 3. The proof is based on the spectral properties of the time-T-map associated with the linearized problem around the motionless state with constant density in some weighted L ^∞ and Sobolev spaces. The time periodic solution is shown to be asymptotically stable under sufficiently small initial perturbations and the L ^∞ norm of the perturbation decays as time goes to infinity.     

On the Darboux transformation of a generalized inhomogeneous higher-order nonlinear Schrödinger equation

Recently, a paper about the Nth-order rogue waves for an inhomogeneous higher-order nonlinear Schrödinger equation using the generalized Darboux transformation is published. Song et al. (Nonlinear Dyn 82(1):489–500. doi: 10.1007/s11071-015-2170-6, 2015). However, the inhomogeneous equation which admits a nonisospectral linear eigenvalue problem is mistaken for having a constant spectral parameter by the authors. This basic error causes the results to be wrong, especially regarding the Darboux transformation (DT) in Sect. 2 when the inhomogeneous terms are dependent of spatial variable x. In fact, the DT for inhomogeneous equation has an essential difference from the isospectral case, and their results are correct only in the absence of inhomogeneity which was already discussed in detail before. Consequently, we firstly modify the DT based on corresponding nonisospectral linear eigenvalue problem. Then, the nonautonomous solitons are obtained from zero seed solutions. Properties of these solutions in the inhomogeneous media are discussed graphically to illustrate the influences of the variable coefficients. Finally, the failure of finding breather and rogue wave solutions from this modified DT is also discussed.     

Mach Wave Effect on Laminar-Turbulent Transition in Supersonic Flow over a Flat Plate

The effect of a Mach wave (N wave) on laminar-turbulent transition induced by the first instability mode (Tollmien–Schlichting wave) in the flat-plate boundary layer is investigated on the basis of the numerical solution of Navier–Stokes equations at the freestream Mach number of 2.5. In accordance with the experiment, the N wave is generated by a two-dimensional roughness at the computation domain boundary corresponding to the side wall of the test section of a wind tunnel. It is shown that the disturbance induced by the backward front of the N wave in the boundary layer has no effect on the beginning of transition but displaces downstream the nonlinear stage of the first mode development. The disturbance induced by the forward front of the N wave displaces the beginning of transition upstream.     

Multiple Periodic Solutions of an Equation with State-Dependent Delay

Given \({N \in \mathbb N}\) we prove the existence, for parameter values in a certain range, of N distinct periodic solutions of a state-dependent delay equation studied by Walther (Differ Integral Equ 15:923–944, 2002).     

Evaluation of the Spectral Element Dynamic Model for Large-Eddy Simulation on Unstructured,Deformed Meshes

In this paper, the Spectral-Element Dynamic Model (SEDM), suited for Large-Eddy Simulation (LES) using Discontinuous Finite Element Methods (DFEM), is assessed using unstructured meshes. Five test cases of increasing complexity are considered, namely, the Taylor-Green vortex at Re =?5000, the turbulent channel flow at Re_τ =?587, the circular cylinder in cross-flow at Re_D =?3900, the square cylinder in cross-flow at Re_D =?22400 and the channel with periodic constrictions at Re_h =?10595. Various discretization parameters such as the grid spacing, polynomial degree and numerical flux are assessed and very accurate results are reported in all cases. This consistency in the results demonstrates the versatility of the SEDM approach and its ability to gage the actual resolution and quality of the mesh and, accordingly, to introduce an amount of sub-grid dissipation which is adapted to the spatial discretization considered.     

Photographic method to sharpen and double isochromatic fringes

A photographic method is described which gives, upon the superposition of ordinary light- and darkfield isochromatic fringe patterns, a new fringe pattern (mixed field). This mixed-field pattern has fringes at the (N/4 and 3N/4 positions. Use of the mixed-field fringe pattern coupled with the oridinary light- and dark-field patterns permit fringes to be read at theN/4,N/2, 3N/4 andN positions, and thus represents a factor of 2 increase in the number of countable fringes. The method is illustrated on two plane-stress examples, a three-dimensional example, and a case employing birefringent coatings.