Permanent Wave Structures and Resonant Triads in a Layered Fluid

A closed three layer fluid with small density differences between the layers has two closely related modes of gravity wave propagation. The nonlinear interactions between the wave modes are investigated, particularly the nearly resonant or significant interactions. Permanent wave solutions are calculated, and it is shown that a permanent wave of the slower mode can generate resonantly a wave harmonic of the faster mode. The equations governing resonant triads of the two modes are derived, and solutions having a permanent structure are calculated from them. It is found that some resonant triad solutions vanish when the triad is embedded in the set of all harmonics with wavenumbers in its neighborhood     

一类反应扩散方程的行波解

1　引　　言由于反应扩散方程涉及的大量问题来自物理学、化学、生物学和人口动力学中众多的数学模型,因而有广阔的实际背景.其行波解引起了人们的兴趣,行波解是某个常微分方程的解,对某些传播速度,利用几何方法可以建立其解的存在性(见[1][2][3]).在文[4]中J.Canosa讨论了Fisher方程ut=2u2x+u(1-u)(1)行波解的存在性、逼近解和误差估计.所谓方程(1)的行波解是指形为u(x,t)=u(x-ct)=u(z)的解.众所周知,行波解u(x,t)=u(x-ct)=u(z)是方程(1)的行波解的充要条件是d2udz2+cdudz+u(1-u)=0(2)若u(z)是单调有界且不恒为常数,则u(z)叫做(1)的波前…     

The investigation into the exact solutions of the generalized time-delayed Burgers-Fisher equation with positive fractional power terms

The time-delayed Burgers-Fisher equation is very important model to forest fire, population growth, Neolithic transitions, the interaction between the reaction mechanism, convection effect and diffusion transport, etc. In this paper, the solitary wave solutions of the generalized time-delayed Burgers-Fisher equation with positive fractional power terms are derived with the aid of a subsidiary high-order ODE, and the solitary wave solutions of the special type of generalized time-delayed Burgers-Fisher equation are presented. From the expressions of the solitary wave solutions, it is easy to obtain how the time-delayed constant τ works upon soliton velocity and width of the soliton, and these exact solutions are very important to understand the physical mechanism of the phenomena described by the time-delayed Burgers-Fisher equation.     

Strong Interactions between Solitary Waves Belonging to Different Wave Modes

Strong interactions between weakly nonlinear long waves are studied. Strong interactions occur when the linear long wave phase speeds are nearly equal although the waves belong to different modes. Specifically we study this situation in the context of internal wave modes propagating in a density stratified fluid. The interaction is described by two coupled Korteweg-deVries equations, which possess both dispersive and nonlinear coupling terms. It is shown that the coupled equations possess an exact analytical solution involving the characteristic “sech²” profile of the Korteweg-deVries equation. It is also shown that when the coefficients satisfy some special conditions, the coupled equations possess an n-solition solution analogous to the Korteweg-deVries n-solition solution. In general though the coupled equations are found not to be amenable to solution by the inverse scattering transform technique, and thus a numerical method has been employed in order to find solutions. This method is described in detail in Appendix A. Several numerical solutions of the coupled equations are presented.     

Direct Resonance in Double-Diffusive Systems

In thermohaline convection and a number of other systems, both direct and oscillatory modes of instability are possible. Should these modes coalesce at some point in parameter space, nonlinear instabilities are likely to be more dramatic in the neighborhood of such a point than elsewhere. A finite-amplitude evolution equation describing such events is derived here by the method of multiple scales. The results are compared with those obtained previously for model systems and by other methods. The direct resonance point of view is found to provide some new insights. A generalization to allow for slow spatial modulation of the amplitude is given and, among other possibilities, solitary-wave solutions are obtainable.     

The (3,4) Mode Interaction in the GeoFlow Framework

The problem of convection in a self‐gravitating spherical shell of fluid is commonly encountered in sciences like astrophysics and geophysics (earth's liquid core). The GEOFLOW‐experiment is a project of the European Space Agency in order to perform the spherical Rayleigh‐Bénard convection problem on the International Space Station in a micro‐gravity environment: the central force field is simulated by a dielectrophoretic one. Beyond a critical Rayleigh number Ra_c, generically an unique spherical ℓ mode becomes unstable and only stationary or travelling waves solutions are expected near the onset. But, for a critical aspect ratio η_c two consecutive modes (ℓ, ℓ + 1) are unstable. The (1,2) and (2,3) interactions have showed a rich bifurcation diagram, in particular, we have found heteroclinic cycles predicted by the theoretical study. Because of the experiment requirements, only the (3,4) one is possible. So, this paper purposes to analyse this bifurcation in non‐rotating case in the GEOFLOWframework using the theory of bifurcation with the spherical symmetry.     

Free boundary effects on stability of two phase planar fluid motion in annulus. Part I: Migration of the stable mode

We study the linearized stability of a planar dynamical model describing two-phase perfect fluid circulating around a circle with a sufficiently large radius within a central gravitational field. The model is associated with the spatial and temporal structure of the zonally averaged global-scale atmospheric longitudinal circulation around the Earth. Two cases are studied separately; in the first one, the simulations were carried out using the rigid lid approximation at the upper boundary of the outer atmospheric layer. In the second one, the free boundary nonlinear conditions (kinematic and dynamic) were assumed on the outer atmospheric layer. For the both cases, a certain family of steady, explicit solutions which have circular streamlines was considered. The governing equations were linearized at these solutions to find the typical wave numbers of the interfacial wave perturbation to the basic state at which the destabilizing effect of shear, which overcomes the stabilizing effect of stratification, occurs. It is shown that for the both cases, the model always have the same two potentially unstable wave modes while there always exist two wave modes which are stable for any wavelengths. The behavior of the stable and unstable modes were compared for the both cases to investigate the effects of the free boundary on the mixing process at the interface.     

Asymptotic Stability of Traveling Wave Solutions for Perturbations with Algebraic Decay

For a class of scalar partial differential equations that incorporate convection, diffusion, and possibly dispersion in one space and one time dimension, the stability of traveling wave solutions is investigated. If the initial perturbation of the traveling wave profile decays at an algebraic rate, then the solution is shown to converge to a shifted wave profile at a corresponding temporal algebraic rate, and optimal intermediate results that combine temporal and spatial decay are obtained. The proofs are based on a general interpolation principle which says that algebraic decay results of this form always follow if exponential temporal decay holds for perturbation with exponential spatial decay and the wave profile is stable for general perturbations.     

Effect of magnetic field dependent viscosity on ferroconvection in the presence of dust particles

This paper deals with the theoretical investigation of the effect of magnetic field dependent (MFD) viscosity on the thermal convection in a ferromagnetic fluid in the presence of dust particles. For a flat ferromagnetic fluid layer contained between two free boundaries, the exact solution is obtained using a linear stability analysis and a normal mode analysis method. For the case of stationary convection, dust particles always have a destabilizing effect, whereas the MFD viscosity has a stabilizing effect on the onset of convection. In the absence of MFD viscosity, the destabilizing effect of magnetization is depicted but in the presence of MFD viscosity, non-buoyancy magnetization may have a destabilizing or a stabilizing effect on the onset of convection. The critical wave number and critical magnetic thermal Rayleigh number for the onset of stationary convection are also determined numerically for sufficiently large values of buoyancy magnetization parameter M ₁. Graphs have been plotted by giving numerical values to the parameters to depict the stability characteristics. It is observed that the critical magnetic thermal Rayleigh number is reduced solely because the heat capacity of clean fluid is supplemented by that of the dust particles. The principle of exchange of stabilities is found to hold true for the ferromagnetic fluid heated from below in the absence of dust particles. The oscillatory modes are introduced due to the presence of the dust particles, which were non-existent in their absence. A sufficient condition for the non-existence of overstability is also obtained.     

l 1-error estimates on the immersed interface upwind scheme for linear convection equations with piecewise constant coefficients: A simple proof

Abstract A linear convection equation with discontinuous coefcients arises in wave propagation through interfaces.An interface condition is needed at the interface to select a unique solution.An upwind scheme that builds this interface condition into its numerical flux is called the immersed interface upwind scheme.An l1-error estimate of such a scheme was frst established by Wen et al.(2008).In this paper,we provide a simple analysis on the l1-error estimate.The main idea is to formulate the solution to the underline initial-value problem into the sum of solutions to two convection equations with constant coefcients,which can then be estimated using classical methods for the initial or boundary value problems.     

Justification of the Ginzburg–Landau approximation for an instability as it appears for Marangoni convection

The Ginzburg–Landau equation appears as a universal amplitude equation for spatially extended pattern forming systems close to the first instability. It can be derived via multiple scaling analysis for the Marangoni convection problem that is driven by temperature‐dependent surface tension and is the subject of our interest. In this paper, we prove estimates between this formal approximation and true solutions of a scalar pattern forming model problem showing the same spectral picture as the Marangoni convection problem in case of a thin fluid. The new difficulties come from neutral modes touching the imaginary axis for the wave number k = 0 and from identical group velocities at the critical wave number k = k_c and the wave number k = 0. The problem is solved by using the reflection symmetry of the system and by using the fact that the modes concentrate at integer multiples of the critical wave number k = k_c. The paper presents a method that is applicable whenever this kind of instability occurs. Copyright © 2012 John Wiley & Sons, Ltd.     

The null spaces of elliptic partial differential operators in Rn

Overstability in a horizontal layer of a viscoelastic fluid is considered in the presence of a uniform magnetic field. The equations of motion appropriate to hydromagnetics in a Maxwellian fluid have been established and the analysis has been carried out in terms of normal modes. The proper solutions have been obtained for the case of two free boundaries. The dispersion relation obtained is found to be quite complex and involves the Prandtl number p₁, magnetic Prandtl number p₂, a parameter Q characterizing the strength of the magnetic field, and a parameter Γ which characterizes the elasticity of the fluid. Numerical calculations have been performed for different values of the parameters involved and the values of critical Rayleigh numbers, wave numbers, and frequencies for the onset of instability as overstability have been obtained. It is found that the magnetic field has a stabilizing influence on the overstable mode of convection in a viscoelastic fluid. Elasticity is found to have a destabilizing influence as in the absence of a magnetic field. Thus the effect of a magnetic field is the same as that for an ordinary viscous fluid.     

A study on a two‐wave mode Kadomtsev–Petviashvili equation: conditions for multiple soliton solutions to exist

We establish a two‐wave mode equation for the integrable Kadomtsev–Petviashvili equation, which describes the propagation of two different wave modes in the same direction simultaneously. We determine the necessary conditions that make multiple soliton solutions exist for this new equation. The simplified Hirota's method will be used to conduct this work. We also use other techniques to obtain other set of periodic and singular solutions for the two‐mode Kadomtsev‐Petviashvili equation. Copyright © 2017 John Wiley & Sons, Ltd.     

Explicit Traveling Wave Solutions to Nonlinear Evolution Equations

First of all, some technical tools are developed. Then the author studies explicit traveling wave solutions to nonlinear dispersive wave equations, nonlinear dissipative dispersive wave equations, nonlinear convection equations, nonlinear reaction diffusion equations and nonlinear hyperbolic equations, respectively.     

A Note on Resonance and Nonlinear Dispersive Waves

The general theory for the slow dispersion of nonlinear wave trains, first studied by Whitham, is applied to a wave train, which, in the weakly nonlinear limit, exhibits resonant singularities. Numerical and perturbation methods are used to develop singly periodic solutions both away from and near all such critical values. Similarly, the equations governing the slow modulations of such a system are found by asymptotic analysis. The expansions are found to be valid so long as the wave train is sufficiently nonlinear. These ideas should be applicable to other problems where resonant singularities arise, in particular, multiphase modes.     

New solitary wave and periodic wave solutions for general types of KdV and KdV–Burgers equations

In this paper, we first introduced improved projective Riccati method by means of two simplified Riccati equations. Applying the improved method, we consider the general types of KdV and KdV–Burgers equations with nonlinear terms of any order. As a result, many explicit exact solutions, which contain new solitary wave solutions, periodic wave solutions and the combined formal solitary wave solutions are obtained. Some of them are found for the first time.     

A Note on Edge Waves in a Stratified Fluid

Explicit solutions are given for edge waves on a sloping beach in an exponentially stratified fluid. The lowest mode, Stoke's edge wave, is found to be completely insensitive to the density field. The higher modes show that previous restrictions on slope angle etc., no longer apply.     

Arithmetic partial differential equations, II

We continue the study of arithmetic partial differential equations initiated in [7] by classifying “arithmetic convection equations” on modular curves, and by describing their space of solutions. Certain of these solutions involve the Fourier expansions of the Eisenstein modular forms of weight 4 and 6, while others involve the Serre-Tate expansions (Mori, 1995 [13], Buium, 2003 [4]) of the same modular forms; in this sense, our arithmetic convection equations can be seen as “unifying” the two types of expansions. The theory can be generalized to one of “arithmetic heat equations” on modular curves, but we prove that they do not carry “arithmetic wave equations.” Finally, we prove an instability result for families of arithmetic heat equations converging to an arithmetic convection equation.     

On Doubly Periodic Standing Wave Solutions of the Coupled Higgs Field Equation

In this paper, we derive a class of doubly periodic standing wave solutions for a coupled Higgs field equation by employing the Hirota bilinear method and theta function identities. Such solutions are expressed in terms of theta functions with variable separation form. Moreover, it is shown that these solutions can be converted into Jacobi elliptic function representations, and their long‐wave limit yields collision of dark solitons. In comparing with known solutions of the canonical evolution equation, three new aspects will be developed in this paper. First, the periods in the spatial and temporal directions, measured in terms of the theta function parameters τ and τ₁, are independent of each other, quite unlike most similar solutions found earlier in the literature. Second, the doubly periodic wave solutions possess two families of the local maxima, whose locations and types are then examined in detail. Third, we obtain new doubly periodic standing wave solutions for the Davey–Stewartson equation through its similarity transformation to the coupled Higgs field equation.     

Singularly perturbed and nonlocal modulation equations for systems with interacting instability mechanisms

Summary Two related systems of coupled modulation equations are studied and compared in this paper. The modulation equations are derived for a certain class of basic systems which are subject to two distinct, interacting, destabilising mechanisms. We assume that, near criticality, the ratio of the widths of the unstable wavenumber-intervals of the two (weakly) unstable modes is small—as, for instance, can be the case in double-layer convection. Based on these assumptions we first derive a singularly perturbed modulation equation and then a modulation equation with a nonlocal term. The reduction of the singularly perturbed system to the nonlocal system can be interpreted as a limit in which the width of the smallest unstable interval vanishes. We study and compare the behaviour of the stationary solutions of both systems. It is found that spatially periodic stationary solutions of the nonlocal system exist under the same conditions as spatially periodic stationary solutions of the singularly perturbed system. Moreover, these solutions can be interpreted as representing the same quasi-periodic patterns in the underlying basic system. Thus, the ‘Landau reduction’ to the nonlocal system has no significant influence on the stationary quasi-periodic patterns. However, a large variety of intricate heteroclinic and homoclinic connections is found for the singularly perturbed system. These orbits all correspond to so-called ‘localised structures’ in the underlying system: They connect simple periodic patterns atx → ± ∞. None of these patterns can be described by the nonlocal system. So, one may conclude that the reduction to the nonlocal system destroys a rich and important set of patterns.