Amplitude Modulation of Nonlinear Waves in Fluid-Filled Tapered Tubes

We study the modulation of nonlinear waves in fluid-filled prestressed tapered tubes. For this, we obtain the nonlinear dynamical equations of motion of a prestressed tapered tube filled with an incompressible inviscid fluid. Assuming that the tapering angle is small and using the reductive perturbation method, we study the amplitude modulation of nonlinear waves and obtain the nonlinear Schrödinger equation with variable coefficients as the evolution equation. A traveling-wave type of solution of such a nonlinear equation with variable coefficients is obtained, and we observe that in contrast to the case of a constant tube radius, the speed of the wave is variable. Namely, the wave speed increases with distance for narrowing tubes and decreases for expanding tubes.     

Analysis of ray tube area in geometrical shock dynamics

We provide a rigorous analysis of structure of a ray tube in geometrical shock dynamics. Our aim is to shed more lights on the cross-sectional area function of a ray tube. We have shown that for a given ray tube and a given initial value of cross-sectional area, then its cross-sectional area can be determined uniquely everywhere. We give a definition of cross-sectional area of a ray tube in precise mathematical terms and from that we derive a set of relations, each describing the cross-sectional area for an arbitrary ray tube in geometrical shock dynamics. We have shown that from our results one can deduce Whitham’s area function relation as a partial differential equation from our general formulations. Some applications are discussed.     

Energy Dissipated Across Shocks in Weak Solutions of Conservation Laws

Manipulation of conservation laws through multiplication by mappings of the unknown function result in singular terms that are densities supported on the discontinuity locus. We derive an expression for these singular terms which have been interpreted as the energy dissipated at the shocks. As an application, the expression for these singular terms is obtained from the transport equation governing the propagation of small amplitude high frequency waves in hyperbolic conservation laws. Analogous formulas are obtained for rich systems in several space dimensions.     

An asymptotic derivation of weakly nonlinear ray theory

Using a method of expansion similar to Chapman-Enskog expansion, a new formal perturbation scheme based on high frequency approximation has been constructed. The scheme leads to an eikonal equation in which the leading order amplitude appears. The transport equation for the amplitude has been deduced with an errorO(ε²) where ε is the small parameter appearing in the high frequency approximation. On a length scale over which Choquet-Bruhat’s theory is valid, this theory reduces to the former. The theory is valid on a much larger length scale and the leading order terms give the weakly nonlinear ray theory (WNLRT) of Prasad, which has been very successful in giving physically realistic results and also in showing that the caustic of a linear theory is resolved when nonlinear effects are included. The weak shock ray theory with infinite system of compatibility conditions also follows from this theory.     

Algebra of symmetry operators for an invariant nonlinear-transport equation

The invariant form is obtained for a nonlinear-transport equation, and extended systems are found for nonlinear-transport equations in terms of the variables (x, t), (e, t), and (e, x), where x is the Euler coordinate, t is time, and e is the energy space variable. The algebra of point-symmetry operators is calculated for the invariant nonlinear-transport equation and this algebra is shown to be admitted by the extended systems of nonlinear transport.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 105. No. 3. pp. 364–370, December, 1995.     

Justification of modulation equations for hyperbolic systems via normal forms

The justification problem for the Nonlinear Schr?dinger equation as a modulation equation for almost spatial periodic wavetrains of small amplitude is considered. We show exact estimates between solutions of the original system and their approximations which are obtained by the solutions of the Nonlinear Schr?dinger equation. By a normal form transform the a priori dangerous quadratic terms of the considered hyperbolic systems are eliminated. Then the transformed systems start with cubic terms. This allows to justify the Nonlinear Schr?dinger equation by a simple application of Gronwall's inequality. Moreover, the influence of resonances is estimated. Received September 16, 1996     

Stability of semi-autonomous difference equations

For a linear difference equation with constant coefficients and several bounded variable delays we obtain criteria for the uniform and uniformly exponential stability expressed in terms of parameters of the initial problem. We adduce examples that prove the exactness of the boundaries of the obtained stability domain.     

Dispersion and Strichartz estimates for the Liouville equation

We consider the Liouville equation associated to a metric g and we prove dispersion and Strichartz estimates for the solution of this equation in terms of the geometry of the trajectories associated to g. In particular, we obtain global Strichartz estimates in time for metrics where dispersion estimate is false even locally in time. We also study the analogy between Strichartz estimates obtained for the Liouville equation and the Schrödinger equation with variable coefficients. To cite this article: D. Salort, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Traveling waves in the complex Ginzburg-Landau equation

Summary In this paper we consider a modulation (or amplitude) equation that appears in the nonlinear stability analysis of reversible or nearly reversible systems. This equation is the complex Ginzburg-Landau equation with coefficients with small imaginary parts. We regard this equation as a perturbation of the real Ginzburg-Landau equation and study the persistence of the properties of the stationary solutions of the real equation under this perturbation. First we show that it is necessary to consider a two-parameter family of traveling solutions with wave speedυ and (temporal) frequencyθ; these solutions are the natural continuations of the stationary solutions of the real equation. We show that there exists a two-parameter family of traveling quasiperiodic solutions that can be regarded as a direct continuation of the two-parameter family of spatially quasi-periodic solutions of the integrable stationary real Ginzburg-Landau equation. We explicitly determine a region in the (wave speedυ, frequencyθ)-parameter space in which the weakly complex Ginzburg-Landau equation has traveling quasi-periodic solutions. There are two different one-parameter families of heteroclinic solutions in the weakly complex case. One of them consists of slowly varying plane waves; the other is directly related to the analytical solutions due to Bekki & Nozaki [3]. These solutions correspond to traveling localized structures that connect two different periodic patterns. The connections correspond to a one-parameter family of heteroclinic cycles in an o.d.e. reduction. This family of cycles is obtained by determining the limit behaviour of the traveling quasi-periodic solutions as the period of the amplitude goes to ∞. Therefore, the heteroclinic cycles merge into the stationary homoclinic solution of the real Ginzburg-Landau equation in the limit in which the imaginary terms disappear.     

A UNIFORM FIRST-ORDER METHOD FOR THE DISCRETE ORDINATE TRANSPORT EQUATION WITH INTERFACES IN X,Y-GEOMETRY

A uniformly first-order convergent numerical method for the discrete-ordinate transport equation in the rectangle geometry is proposed in this paper. Firstly we approximate the scattering coefficients and source terms by piecewise constants determined by their cell averages. Then for each cell, following the work of De Barros and Larsen [1, 19], the solution at the cell edge is approximated by its average along the edge. As a result, the solution of the system of equations for the cell edge averages in each cell can be obtained analytically. Finally, we piece together the numerical solution with the neighboring cells using the interface conditions. When there is no interface or boundary layer, this method is asymptotic-preserving, which implies that coarse meshes （meshes that do not resolve the mean free path） can be used to obtain good numerical approximations. Moreover, the uniform first-order convergence with respect to the mean free path is shown numerically and the rigorous proof is provided.     

A representation of solution of stochastic differential equations

We prove that the logarithm of the formal power series, obtained from a stochastic differential equation, is an element in the closure of the Lie algebra generated by vector fields being coefficients of equations. By using this result, we obtain a representation of the solution of stochastic differential equations in terms of Lie brackets and iterated Stratonovich integrals in the algebra of formal power series.     

A phenomenological and kinetic description of diffusion and heat transport in multicomponent gas mixtures and plasma

Different forms of expressing diffusion and heat fluxes in multicomponent mixtures, obtained by methods of non-equilibrium thermodynamics and the kinetic theory of gas mixtures, are analysed and compared. It is shown that an alternative representation of the linear relations of non-equilibrium thermodynamics is possible, which enables them to be written in a form similar to that of the well-known Stefan–Maxwell equations. A relation between the phenomenological coefficients of non-equilibrium thermodynamics and the corresponding transport coefficients obtained in kinetic theory is established, with a confirmation that the Onsager reciprocity relations are satisfied. It is shown that there is an advantage in writing the transport relations on the basis of the “forces in terms of fluxes” representation, compared with the classical “fluxes in terms of forces” representation, used in standard schemes of phenomenological non-equilibrium thermodynamics and the Chapman–Enskog method, traditional for kinetic theory. A generalization of the Stefan-Maxwell equations and the equation for the heat flux is considered, which takes into account the contribution to these equations of the time and space derivatives of the fluxes. The relaxation form of the equations obtained enable one to approach the analysis of the propagation of small heat and concentration perturbations in gas mixtures to be justified, which, within the framework of classical transport relations, propagate with infinitely high velocity. The results presented in this review enable one to determine the areas of effective application of different methods of describing diffusion and heat transfer in multicomponent gas mixtures when solving specific gas-dynamic problems.     

Spatial behavior for solutions in heat conduction with two delays

In this note, we investigate the spatial behavior of the solutions of the equation proposed to describe a theory for the heat conduction with two delay terms. We obtain an alternative of the Phragmén-Lindelöf type, which means that the solutions either decay or blow-up at infinity, both options in an exponential way. We also describe how to obtain an upper bound for the amplitude term. This is the first contribution on spatial behavior for partial differential equations involving two delay terms. We use energy arguments. The main point of the contribution is the use of an exponentially weighted energy function.     

Bright soliton solution to a generalized Burgers–KdV equation with time-dependent coefficients

We consider a generalized Burgers–KdV type equation with time-dependent coefficients incorporating a generalized evolution term, the effects of third-order dispersion, dissipation, nonlinearity, nonlinear diffusion and reaction. The exact bright soliton solution for the considered model is obtained by using a solitary wave ansatz in the form of sech^s function. The physical parameters in the soliton solution are obtained as functions of the time varying coefficients and the dependent exponents. The dependent exponents and the temporal variations of the model coefficients satisfy certain parametric conditions as shown by the obtained soliton solution. This solution may be useful to explain some physical phenomena in genuinely nonlinear dynamical systems that are described by Burgers–KdV type models.     

Oscillatory asymptotic expansion for semilinear wave equation

We study oscillatory properties of the solution to semilinear wave equation, assuming oscillatory terms in initial data have sufficiently small amplitude. The main result gives an a priori estimate of the remainder in the approximation by means of the method of geometric optics. The method of establishing this estimate is based on a combination between energy type estimates for transport equation and Sobolev embedding. Copyright © 2001 John Wiley & Sons, Ltd.     

Stability of solutions to delay differential equations with periodic coefficients of linear terms

Under study are the systems of quasilinear delay differential equations with periodic coefficients of linear terms. We establish sufficient conditions for the asymptotic stability of the zero solution, obtain estimates for solutions which characterize the decay rate at infinity, and find the attractor of the zero solution. Similar results are obtained for systems with parameters.     

Painlev\'{e} Analysis and Auto-B\"{a}cklund Transformation for a General Variable Coefficient Burgers Equation with Linear Damping Term

This paper investigates a general variable coefficient (gVC) Burgers equation with linear damping term. We derive the Painlev\''{e} property of the equation under certain constraint condition of the coefficients. Then we obtain an auto-B\"{a}cklund transformation of this equation in terms of the Painlev\''{e} property. Finally, we find a large number of new explicit exact solutions of the equation. Especially, infinite explicit exact singular wave solutions are obtained for the first time. It is worth noting that these singular wave solutions will blow up on some lines or curves in the $(x,t)$ plane. These facts reflect the complexity of the structure of the solution of the gVC Burgers equation with linear damping term. It also reflects the complexity of nonlinear wave propagation in fluid from one aspect.     

Stationary Schr?dinger equation in nonrelativistic quantum mechanics and the functional integral

We formulate a method for representing solutions of homogeneous second-order equations in the form of a functional integral or path integral. As an example, we derive solutions of second-order equations with constant coefficients and a linear potential. The method can be used to find general solutions of the stationary Schr?dinger equation. We show how to find the spectrum and eigenfunctions of the quantum oscillator equation. We obtain a solution of the stationary Schr?dinger equation in the semiclassical approximation, without a singularity at the turning point. In that approximation, we find the coefficient of transmission through a potential barrier. We obtain a representation for the elastic potential scattering amplitude in the form of a functional integral.     

Harnack inequality and smoothness for quasilinear degenerate elliptic equations

We prove local and global regularity for the positive solutions of a quasilinear variational degenerate equation, assuming minimal hypothesis on the coefficients of the lower order terms. As an application we obtain Hölder continuity for the gradient of solutions to nonvariational quasilinear equations.