Shock Waves in Dispersive Eulerian Fluids

The long-time behavior of an initial step resulting in a dispersive shock wave (DSW) for the one-dimensional isentropic Euler equations regularized by generic, third-order dispersion is considered by use of Whitham averaging. Under modest assumptions, the jump conditions (DSW locus and speeds) for admissible, weak DSWs are characterized and found to depend only upon the sign of dispersion (convexity or concavity) and a general pressure law. Two mechanisms leading to the breakdown of this simple wave DSW theory for sufficiently large jumps are identified: a change in the sign of dispersion, leading to gradient catastrophe in the modulation equations, and the loss of genuine nonlinearity in the modulation equations. Large amplitude DSWs are constructed for several particular dispersive fluids with differing pressure laws modeled by the generalized nonlinear Schrödinger equation. These include superfluids (Bose–Einstein condensates and ultracold fermions) and “optical fluids.” Estimates of breaking times for smooth initial data and the long-time behavior of the shock tube problem are presented. Detailed numerical simulations compare favorably with the asymptotic results in the weak to moderate amplitude regimes. Deviations in the large amplitude regime are identified with breakdown of the simple wave DSW theory.     

Inaccessible States in Time-Dependent Reaction Diffusion

Using asymptotic methods we show that the long-time dynamic behavior in certain systems of nonlinear parabolic differential equations is described by a time-dependent, spatially inhomogeneous nonlinear evolution equation. For problems with multiple stable states, the solution develops sharp fronts separating slowly varying regions. By studying the basins of attraction of Abel's nonlinear differential equation, we demonstrate that the presence of explicit time dependence in the asymptotic evolution equation creates “forbidden regions” where the existence of interfaces is excluded. Consequently, certain configurations of stable states in the nonlinear system become inaccessible and cannot be achieved from any set of real initial conditions.     

On the Cauchy problem for nonlinear dissipative systems

The large-time asymptotic behavior of solutions of the Cauchy problem for a system of nonlinear evolutionary equations with dissipation is studied. The approach used in the case of small initial data is based on the construction of solutions by the method of contracting mappings. In the case of large initial data, we will obtain the large-time asymptotics of solutions with a certain symmmetry of a nonlinear term taken into account. In the critical case, it is proved that if the initial data has a nonzero total mass, then the principal term of the large-time asymptotics of a solution is given by the self-similar solution uniquely determined by the total mass of the initial data. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 29, Voronezh Conference-1, 2005.     

Multiple-scale Homogenization for Weakly Nonlinear Conservation Laws with Rapid Spatial Fluctuations

We consider hyperbolic conservation laws with rapid periodic spatial fluctuations and study initial value problems that correspond to small perturbations about a steady state. Weakly nonlinear solutions are computed asymptotically using multiple spatial and temporal scales to capture the homogenized solution as well as its long-term behavior. We show that the linear problem may be destabilized through interactions between two solution modes and the periodic structure. We also show that a discontinuity, either in the initial data or due to shock formation, introduces rapid spatial and temporal fluctuations to leading order in its zone of influence. The evolution equations we derive for the homogenized leading-order solution are more general than their counterparts for conservation laws having no rapid spatial variations. In particular, these equations may be diffusive for certain general flux vectors. Selected examples are solved numerically to substantiate the asymptotic results.     

Asymptotic behavior of nonlinear waves in elastic media with dispersion and dissipation

In the case of nonlinear elastic quasitransverse waves in composite media described by nonlinear hyperbolic equations, we study the nonuniqueness problem for solutions of a standard self-similar problem such as the problem of the decay of an arbitrary discontinuity. The system of equations is supplemented with terms describing dissipation and dispersion whose influence is manifested in small-scale processes. We construct solutions numerically and consider self-similar asymptotic approximations of the obtained solution of the equations with the initial data in the form of a “spreading” discontinuity for large times. We find the regularities for realizing various self-similar asymptotic approximations depending on the choice of the initial conditions including the dependence on the form of the functions determining the small-scale smoothing of the original discontinuity. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 2, pp. 240–256, May, 2006.     

APPROXIMATION TO THE GLOBAL ATTRACTOR FOR A NONLINEAR SCHRODINGER EQUATION

Abstract. In the present paper, we deal with the long-time behavior of dissipative partial differenttial equations, and we construct the approximate inertial mardfolds for the nonlbaear Stringer equation with a zero order dlssipation. The order of approximation of these manlfolde to the global attractor is derived.     

一类耗散型电流体动力学方程组自相似解的渐近稳定性

主要考虑一类来源于电流体动力学中的由非线性非局部方程组耦合而成的耗散型系统的初值问题.利用Lorentz空间中广义L~p-L~q热半群估计和广义Hardy-Littlewood-Sobolev不等式,首先证明了该系统在Lorentz空间中自相似解的整体存在性和唯一性,然后建立了自相似解当时间趋于无穷时的渐近稳定性.因为Lorentz空间包含了具有奇性的齐次函数,因次上述结果保证了具有奇性的初值所对应的自相似解的整体存在性和渐近稳定性.     

Asymptotic behavior of solution to nonlinear evolution equations with damping

In this paper, we consider the global existence and the asymptotic behavior of solutions to the Cauchy problem for the following nonlinear evolution equations with ellipticity and dissipative effects

(E)     

Self-similar asymptotics of wave problems and the structures of non-classical discontinuities in non-linearly elastic media with dispersion and dissipation

Solutions of the non-linear hyperbolic equations describing quasi-transverse waves in composite elastic media are investigated within the framework of a previously proposed model, which takes into account small dissipative and dispersion processes. It is well known for this model that if a solution of the problem of the decay of an arbitrary discontinuity is constructed using Riemann waves and discontinuities having a structure, the solution turns out to be non-unique. In order to study the problem of non-uniqueness, solutions of non-self-similar problems are constructed numerically within the framework of the proposed model with initial data in the form of a “smooth” step. With time passing the solutions acquire a self-similar asymptotic form, corresponding to a certain solution of the problem of the decay of an arbitrary discontinuity. It is shown that, by changing the method of smoothing the step, one can construct any of the self-similar asymptotic forms, as was done previously in Ref. [Chugainova AP. The asymptotic behaviour of non-linear waves in elastic media with dispersion and dissipation. Teor Mat Fiz 2006;147(2):240–56] for media with terms of opposite sign, responsible for the non-linearity, although the set of admissible discontinuities and the structure of the solutions of the problems in these cases turn out to be different.     

Global existence, asymptotic behavior and blowup of solutions for a class of nonlinear wave equations with dissipative term

The paper studies the global existence, asymptotic behavior and blowup of solutions to the initial boundary value problem for a class of nonlinear wave equations with dissipative term. It proves that under rather mild conditions on nonlinear terms and initial data the above-mentioned problem admits a global weak solution and the solution decays exponentially to zero as t→+∞, respectively, in the states of large initial data and small initial energy. In particular, in the case of space dimension N=1, the weak solution is regularized to be a unique generalized solution. And if the conditions guaranteeing the global existence of weak solutions are not valid, then under the opposite conditions, the solutions of above-mentioned problem blow up in finite time. And an example is given.     

On the Whitham Equations for the Defocusing Nonlinear Schrodinger Equation with Step Initial Data

The behavior of solutions of the finite-genus Whitham equations for the weak dispersion limit of the defocusing nonlinear Schrodinger equation is investigated analytically and numerically for piecewise-constant initial data. In particular, the dynamics of constant-amplitude initial conditions with one or more frequency jumps (i.e., piecewise linear phase) are considered. It is shown analytically and numerically that, for finite times, regions of arbitrarily high genus can be produced; asymptotically with time, however, the solution can be divided into expanding regions which are either of genus-zero, genus-one, or genus-two type, their precise arrangement depending on the specifics of the initial datum given. This behavior should be compared to that of the Korteweg-de Vries equation, where the solution is divided into regions which are either genus-zero or genus-one asymptotically. Finally, the potential application of these results to the generation of short optical pulses is discussed: The method proposed takes advantage of nonlinear compression via appropriate frequency modulation, and allows control of both the pulse amplitude and its width, as well as the distance along the fiber at which the pulse is produced     

Cauchy problem for quasi-linear wave equations with viscous damping

The paper studies the global existence and asymptotic behavior of weak solutions to the Cauchy problem for quasi-linear wave equations with viscous damping. It proves that when pmax{m,α}, where m+1, α+1 and p+1 are, respectively, the growth orders of the nonlinear strain terms, the nonlinear damping term and the source term, the Cauchy problem admits a global weak solution, which decays to zero according to the rate of polynomial as t→∞, as long as the initial data are taken in a certain potential well and the initial energy satisfies a bounded condition. Especially in the case of space dimension N=1, the solutions are regularized and so generalized and classical solution both prove to be unique. Comparison of the results with previous ones shows that there exist clear boundaries similar to thresholds among the growth orders of the nonlinear terms, the states of the initial energy and the existence, asymptotic behavior and nonexistence of global solutions of the Cauchy problem.     

Nonlinear Amplitude Evolution of Baroclinic Wave Trains and Wave Packets

The weakly nonlinear theory of baroclinic wave trains and wave packets is examined by the use of systematic expansion procedures in appropriate powers of a small parameter measuring the supercriticality according to linear theory; well-known multiple scaling techniques are employed. Crucial importance is ascribed to the magnitude of parameters measuring dissipation and dispersion relative to each other and to the supercriticality, and equations describing the slow evolution in space and time of the wave amplitude are established for a range of parameter values. For vanishingly small dissipation the wave train equations have straightforward oscillatory solutions, dependent on initial conditions, and for large dissipation steady equilibration, independent of initial conditions, is predicted. For moderately small dissipation, however, a wide variety of behaviors is possible—including steady equilibration, single and multiple periodicity, and aperiodicity—in the solutions of the equations, which are recognizable as generalisations of the well-known Lorenz attractor equations. Equations describing the evolution of wave packets take a variety of forms; for vanishingly small dissipation or for large dissipation, they are essentially parabolic and of nonlinear Schrödinger type, whilst for moderate dissipation they are of Lorenz type, modified by spatial variations. Solutions of a number of these equations are discussed and compared, where appropriate, with experimental results.     

Ellipsoidal BGK model for polyatomic molecules near Maxwellians: A dichotomy in the dissipation estimate

We consider the global existence and asymptotic behavior of classical solutions to the ellipsoidal BGK model for polyatomic molecules when the initial data starts sufficiently close to a global polyatomic Maxwellian. We observe that the linearized relaxation operator is decomposed into a truly polyatomic part and an essentially monatomic part, leading to a dichotomy in the dissipative property in the sense that the degeneracy of the dissipation shows an abrupt jump as the relaxation parameter θ reaches zero. Accordingly, we employ two different sets of micro–macro system to derive the full coercivity and close the energy estimate.     

Estimating the number of asymptotic degrees of Freedom for Nonlinear Dissipative Systems

We show that the long-time behavior of the projection of the exact solutions to the Navier-Stokes equations and other dissipative evolution equations on the finite-dimensional space of interpolant polynomials determines the long-time behavior of the solution itself provided that the spatial mesh is fine enough. We also provide an explicit estimate on the size of the mesh. Moreover, we show that if the evolution equation has an inertial manifold, then the dynamics of the evolution equation is equivalent to the dynamics of the projection of the solutions on the finite-dimensional space spanned by the approximating polynomials. Our results suggest that certain numerical schemes may capture the essential dynamics of the underlying evolution equation.

     

Self-similar asymptotics describing nonlinear waves in elastic media with dispersion and dissipation

Solutions of problems for the system of equations describing weakly nonlinear quasi-transverse waves in an elastic weakly anisotropic medium are studied analytically and numerically. It is assumed that dissipation and dispersion are important for small-scale processes. Dispersion is taken into account by terms involving the third derivatives of the shear strains with respect to the coordinate, in contrast to the previously considered case when dispersion was determined by terms with second derivatives. In large-scale processes, dispersion and dissipation can be neglected and the system of equations is hyperbolic. The indicated small-scale processes determine the structure of discontinuities and a set of admissible discontinuities (with a steady-state structure). This set is such that the solution of a self-similar Riemann problem constructed using solutions of hyperbolic equations and admissible discontinuities is not unique. Asymptotics of non-self-similar problems for equations with dissipation and dispersion were numerically found, and it appeared that they correspond to self-similar solutions of the Riemann problem. In the case of nonunique self-similar solutions, it is shown that the initial conditions specified as a smoothed step lead to a certain self-similar solution implemented as the asymptotics of the unsteady problem depending on the smoothing method.     

On the asymptotic behavior of solutions of nonlinear second-order parabolic equations

We study the asymptotic behavior as t → +∞ of solutions to a semilinear second-order parabolic equation in a cylindrical domain bounded in the spatial variable. We find the leading term of the asymptotic expansion of a solution as t → +∞ and show that each solution of the problem under consideration is asymptotically equivalent to a solution of some nonlinear ordinary differential equation.     

The asymptotic manifold of a perturbed linear differential equation as determined by the comparison principle

The asymptotic properties of the solutions of a linear homogeneous system of differential equations determine, under suitable restrictions, the asymptotic properties of a set of solutions of a nonlinear perturbation of this linear equation. The comparison principle is used here to generate an asymptotic manifold of the perturbed equation. The majorant function that is used in connection with the comparison technique is usually assumed to be nondecreasing in the dependent variable. However, properties of the asymptotic manifold are discussed here under the opposite monotonicity assumption, namely, that the majorant function is nonincreasing in the dependent variable. This type of majorant, function arises, for example, in certain gravitation problems. The main result on the structure of asymptotic manifolds which have an asymptotic uniformity is that solutions close to the manifold are either in the manifold or do not exist in the future. This research was supported in part by the National Science Foundation under grant GP-11543. Entrata in Redazione il 6 giugno 1970.     

CONVERGENCE OF SOLUTIONS FOR RLC-NONLINEAR NETWORKS WITH TIME-VARYING ELEMENTS

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Self-similar Asymptotics for Linear and Nonlinear Diffusion Equations

The long-time asymptotic solutions of initial value problems for the heat equation and the nonlinear porous medium equation are self-similar spreading solutions. The symmetries of the governing equations yield three-parameter families of these solutions given in terms of their mass, center of mass, and variance. Unlike the mass and center of mass, the variance, or “time-shift,” of a solution is not a conserved quantity for the nonlinear problem. We derive an optimal linear estimate of the long-time variance. Newman's Lyapunov functional is used to produce a maximum entropy time-shift estimate. Results are applied to nonlinear merging and time-dependent, inhomogeneously forced diffusion problems.