Asymptotic behavior of dispersive waves in a spiral structure at large times

In the framework of the sine-Gordon integrable model for spiral magnetic structures, we investigate the behavior at large times of a weakly nonlinear dispersive wave field generated by a spatially local initial excitation of the structure. The method used is based on a direct asymptotic analysis of the corresponding matrix of the Riemann problem on the torus.     

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.     

Asymptotic behavior of nonlinear sound waves in inviscid media with thermal and molecular relaxation

Nonlinear sound propagation through media with thermal and molecular relaxation can be modeled by third-order in time wave-like equations with memory. We investigate the asymptotic behavior of a Cauchy problem for such a model, the nonlocal Jordan–Moore–Gibson–Thompson equation, in the so-called critical case, which corresponds to propagation through inviscid fluids or gases. The memory has an exponentially fading character and type I, meaning that involves only the acoustic velocity potential. A major challenge in studying global behavior is that the linearized equation’s decay estimates are of regularity-loss type. As a result, the classical energy methods fail to work for the nonlinear problem. To overcome this difficulty, we construct appropriate time-weighted norms, where weights can have negative exponents. These problem-tailored norms create artificial damping terms that help control the nonlinearity and the loss of derivatives, and ultimately allow us to discover the model’s asymptotic behavior.     

Asymptotic behavior toward nonlinear waves for radially symmetric solutions of the multi-dimensional Burgers equation

The present paper is concerned with the asymptotic behaviors of radially symmetric solutions for the multi-dimensional Burgers equation on the exterior domain in ${\mathbb{R}}^n,n\ge 3$, where the boundary and far field conditions are prescribed. We show that in some case where the corresponding 1-D Riemann problem for the non-viscous part admits a shock wave, the solution tends toward a linear superposition of stationary and rarefaction waves as time goes to infinity, and also show the decay rate estimates. Furthermore, we improve the results on the asymptotic stability of the stationary waves which are treated in the previous papers [2], [3]. Finally, for the case of $n=3$, we give the complete classification of the asymptotic behaviors, which includes even a linear superposition of stationary and viscous shock waves.     

Asymptotic models for internal waves

Derived here in a systematic way, and for a large class of scaling regimes are asymptotic models for the propagation of internal waves at the interface between two layers of immiscible fluids of different densities, under the rigid lid assumption and with a flat bottom. The full (Euler) model for this situation is reduced to a system of evolution equations posed spatially on , d=1,2, which involve two nonlocal operators. The different asymptotic models are obtained by expanding the nonlocal operators with respect to suitable small parameters that depend variously on the amplitude, wave-lengths and depth ratio of the two layers. We rigorously derive classical models and also some model systems that appear to be new. Furthermore, the consistency of these asymptotic systems with the full Euler equations is established.     

Asymptotic behavior of grafting rays

In this paper we study the convergence behavior of grafting rays to the Thurston boundary of Teichmüller space. When the grafting is done along a weighted system of simple closed curves or along a maximal uniquely ergodic lamination this behavior is the same as for Teichmüller geodesics and lines of minima. We also show that the rays grafted along a weighted system of simple closed curves are at bounded distance from Teichmüller geodesics.     

Asymptotic behavior of random heaps

We consider a random walk W_n on the locally free group (or equivalently a signed random heap) with m generators subject to periodic boundary conditions. Let #T(W_n) denote the number of removable elements, which determines the heap's growth rate. We prove that lim_n→∞??(#T(W_n))/m ≤ 0.32893 for m ≥ 4. This result disproves a conjecture (due to Vershik, Nechaev and Bikbov [Comm Math Phys 212 (2000), 469–501]) that the limit tends to 1/3 as m → ∞. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2005     

Asymptotic behavior of generalized convolutions

We study the behavior of a class of convolution-type nonlinear transformations. Under some smallness conditions we prove the existence of fixed points and analyze the spectrum of the associated linearized operator.      

Asymptotic behavior of Brownian polymers

Summary We consider a system that models the shape of a growing polymer. Our basic problem concerns the asymptotic behavior ofX _t , the location of the end of the polymer at timet. We obtain bounds onX _t in the (physically uninteresting) case thatd=1 and the interaction functionf(x)0. If, in addition,f(x) behaves for largex likeCx ^– with <1 we obtain a strong law that gives the exact growth rate.Partially supported by the National Science Foundation and the Army Research Office through the Mathematical Sciences Institute (MSI) at Cornell UniversityPartially supported by SERC grant GR/G02307 and MSI, this work was done while the second author was visiting Cornell, April–July, 1990     

Asymptotic behavior of charge-exchange amplitude

The eikonal approximation in the distorted wave formalism is modified for application to three-particle scattering with rearrangement in the region in which neither the Born nor the adiabatic approximation is valid. The long-range effects due to Coulomb interaction between the particles in both the entrance and exit reaction channels are investigated. The leading asymptotic term (in the large impact parameter) of the eikonal charge exchange amplitude is calculated for transitions betweens states of atoms. It is shown that the general expression for the amplitude correctly describes both limiting casesvv ₀ andvv ₀.State University, Uzhgorod; P. N. Lebedev Physics Institute, USSR Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 91, No. 1, pp. 66–82, April, 1992.     

Asymptotic behavior of permutation records

We study the asymptotic behavior of two statistics defined on the symmetric group Sn when n tends to infinity: the number of elements of Sn having k records, and the number of elements of Sn for which the sum of the positions of their records is k. We use a probabilistic argument to show that the scaled asymptotic behavior of these statistics can be described by remarkably simple functions.     

Asymptotic behavior of reduction numbers

It is shown that the reduction number and the big reduction number of are linear functions of for all large . Here is a homogeneous ideal of a polynomial ring .

     

Asymptotic waves and collapse of self-gravitating gaseous systems

The asymptotic wave approach has been used to study the gravitational collapse of self-gravitating gaseous systems. In the case of sonic waves it is shown that macroscopic phenomena, such as shock-waves formation, can arise even if initial perturbations are confined in layers of linear dimensions very small compared to the dimensions of the medium. The occurring of critical times in a model of spherically symmetric self-gravitating interstellar gas cloud, suggests a mechanism in which the strong shock waves so generated might explain the formation of a cluster of protostars in the framework of the modern theory of Star Formation. A procedure is also described which let to overcome the solving of the wave front equation which turns out to be very useful in numerical applications.     

Asymptotic representation of surface waves in the form of two traveling burgers waves

Chair of General Mathematics, Department of Computational Mathematics and Cybernetics, M. V. Lomonosov Moscow State University. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 29, No. 3, pp. 25–40, July–September, 1995.