Behavior near the focal points of asymptotic solutions to the Cauchy problem for the linearized shallow water equations with initial localized perturbations

We study the behavior of the wave part of asymptotic solutions to the Cauchy problem for linearized shallow water equations with initial perturbations localized near the origin. The global representation for these solutions based on the generalized Maslov canonical operator was given earlier. The asymptotic solutions are also localized in the neighborhood of certain curves (fronts). The simplification of general formulas and the behavior of asymptotic solutions in a neighborhood of the regular part of fronts was also given earlier. Here the behavior of asymptotic solutions in a neighborhood of the focal point of the fronts is discussed in detail and the proof of formulas announced earlier for the wave equation is given. This paper can be regarded as a continuation of the paper in Russiian Journal of Mathematical Physics 15 (2), 192–221 (2008). In memoriam V.A. Borovikov     

Asymptotic Behavior of Soliton Solutions with a Double Spectral Parameter for Principal Chiral Field

The soliton solutions with a double spectral parameter for the principal chiral field are derived by Darboux transformation. The asymptotic behavior of the solutions as time tends to infinity is obtained and the speeds of the peaks in the asymptotic solutions are not constants.     

Asymptotic Behavior of Soliton Solutions with a Double Spectral Parameter for Principal Chiral Field

The soliton solutions with a double spectral parameter for the principal chiral field are derived by Darboux transformation. The asymptotic behavior of the solutions as time tends to infinity is obtained and the speeds of the peaks in the asymptotic solutions are not constants.     

Asymptotic support of localized solutions of the linearized system of magnetohydrodynamics

The asymptotic behavior of solutions of the Cauchy problem for the linearized system of magnetohydrodynamic equations with initial conditions localized near a two-dimensional surface was obtained by the authors earlier. Here, this asymptotic behavior is refined.     

Infinite Kinematic Self-Similarity and Perfect Fluid Spacetimes

Perfect fluid spacetimes admitting a kinematic self-similarity of infinite type are investigated. In the case of plane, spherically or hyperbolically symmetric space-times the field equations reduce to a system of autonomous ordinary differential equations. The qualitative properties of solutions of this system of equations, and in particular their asymptotic behavior, are studied. Special cases, including some of the invariant sets and the geodesic case, are examined in detail and the exact solutions are provided. The class of solutions exhibiting physical self-similarity are found to play an important role in describing the asymptotic behavior of the infinite kinematic self-similar models.     

Asymptotic analysis of Bloch-Floquet waves in a thin bi-material strip with a periodic array of finite-length cracks

We present an asymptotic algorithm to solve a problem of wave propagation in a thin bi-material strip with an array of cracks situated at the interface between two materials. For small frequencies we construct an asymptotic solution which takes into account the singular behavior near the crack tips and the smooth nature of the oscillation far away from them. We construct the boundary layer solutions near the crack tips. The boundary layers are harmonic solutions in scaled domains. Dispersion equations are derived and solved within the frame of the asymptotic model.     

Asymptotic analysis of Bloch–Floquet waves in a thin bi-material strip with a periodic array of finite-length cracks

We present an asymptotic algorithm to solve a problem of wave propagation in a thin bi-material strip with an array of cracks situated at the interface between two materials. For small frequencies we construct an asymptotic solution which takes into account the singular behavior near the crack tips and the smooth nature of the oscillation far away from them. We construct the boundary layer solutions near the crack tips. The boundary layers are harmonic solutions in scaled domains. Dispersion equations are derived and solved within the frame of the asymptotic model.     

Logarithmic asymptotic flatness

We present a general family of asymptotic solutions to Einstein's equation which are asymptotically flat but do not satisfy the peeling theorem. Near scri, the Weyl tensor obeys a logarithmic asymptotic flatness condition and has a partial peeling property. The physical significance of this asymptotic behavior arises from a quasi-Newtonian treatment of the radiation from a collapsing dust cloud. Practically all the scri formalism carries over intact to this new version of asymptotic flatness.     

Scattering for the wave equation on the Schwarzschild Metric

We study the wave equation for the Schwarzschild metric. Wave operators are constructed which yield solutions with given asymptotic behavior either at infinity or on the horizon. We prove asymptotic completeness for these wave operators.Supported by NSF grant No. PHY82-204399.     

New Double Wronskian Solutions of the Whitham-Broer-Kaup System: Asymptotic Analysis and Resonant Soliton Interactions

In this paper, by the Darboux transformation together with the Wronskian technique, we construct new double Wronskian solutions for the Whitham-Broer-Kaup (WBK) system. Some new determinant identities are developed in the verification of the solutions. Based on analyzing the asymptotic behavior of new double Wronskian functions as t → ±∞, we make a complete characterization of asymptotic solitons for the non-singular, non-trivial and irreducible soliton solutions. It turns out that the solutions are the linear superposition of two fully-resonant multi-soliton configurations, in each of which the amplitudes, velocities and numbers of asymptotic solitons are in general not equal as t → ±∞. To illustrate, we present the figures for several examples of soliton interactions occurring in the WBK system.     

Soliton Asymptotics of Rear Part of Non-Localized Solutions of the Kadomtsev-Petviashvili Equation

Abstract

We construct non-localized, real global solutions of the Kadomtsev-Petviashvili-I equation which vanish for x → ?∞ and study their large time asymptotic behavior. We prove that such solutions eject (for t → ∞) a train of curved asymptotic solitons which move behind the basic wave packet.     

Axially symmetric Brans-Dicke-Maxwell solutions

Following a method of John and Goswami new solutions of coupled Brans-Dicke-Maxwell theory are generated from Zipoy's solutions in oblate and prolate spheroidal coordinates for source-free gravitational field. All these solutions become Euclidean at infinity. The asymptotic behavior and the singularity of the solutions are discussed and a comparative study made with the corresponding Einstein-Maxwell solutions. The possibility of a very large red shift from the boundary of the spheroids is also discussed.     

Eigenstates of spherical delta-shells

We study the exact solution of the Schrödinger equation with a δ-potential in the spherical coordinate system. We consider both stationary and nonstationary problems. We obtain solutions for bound states described by functions decreasing rapidly with the radius and solutions characterized by an oscillatory asymptotic behavior.     

Random solution of the Burgers equation in the inviscid case

The structure of solutions of the Burgers equation in the inviscid case is investigated numerically by computing the space-time behavior of the asymptotic solutions expressed as sequences of triangular shock waves. They are sensitively dependent on initial conditions and can display intrinsic randomness, depending on the number of zeros of the initial velocity fields.     

Axially symmetric electrovac solutions

Two classes of electrovac solutions are obtained in oblate spheroidal coordinates, which are the electromagnetic analogs of Zipoy's monopole and dipole solutions. The asymptotic behavior of the solutions is studied to gain some insight into the nature of the source of the gravitational and electromagnetic fields. A similar stationary solution of the pure gravitational field is found to belong to Papapetrou's class.     

Asymmetric self-similar flows of a viscous incompressible fluid along a right-angle corner

Symmetric and asymmetric self-similar flows of a viscous incompressible fluid along a semi-infinite right-angle dihedral corner with a preset streamwise pressure gradient have been considered. Equations describing such flows in the framework of boundary layer approximation have been derived. The asymptotic behavior of solutions of the derived equations far from the corner edge has been theoretically investigated. A new method of computation of these solutions has been developed. Solutions for two types of asymptotic behavior have been obtained.     

Modified Scattering for the Boson Star Equation

We consider the question of scattering for the boson star equation in three space dimensions. This is a semi-relativistic Klein-Gordon equation with a cubic nonlinearity of Hartree type. We combine weighted estimates, obtained by exploiting a special null structure present in the equation, and a refined asymptotic analysis performed in Fourier space, to obtain global solutions evolving from small and localized Cauchy data. We describe the behavior of such solutions at infinity by identifying a suitable nonlinear asymptotic correction to scattering. As a byproduct of the weighted energy estimates alone, we also obtain global existence and (linear) scattering for solutions of semi-relativistic Hartree equations with potentials decaying faster than Coulomb.     

Gradient catastrophes and sawtooth solutions for a generalized Burgers equation on an interval

The asymptotic behavior of solutions of the Burgers equation and its generalizations with initial value-boundary problem on a finite interval with constant boundary conditions is studied. Since it describes a dissipative medium, any initial profile will evolve to a time-invariant solution with the same boundary values. Yet there are three distinctive asymptotic processes: the initial profile may regularly decay to a smooth invariant solution; or a Heaviside-type gap develops through a dispersive shock and multi-oscillations; or an asymptotic limit is a stationary ‘sawtooth’ solution with periodical breaks of derivative.     

Asymptotic Reissner–Nordström black holes

We consider two types of Born–Infeld like nonlinear electromagnetic fields and obtain their interesting black hole solutions. The asymptotic behavior of these solutions is the same as that of a Reissner–Nordström black hole. We investigate the geometric properties of the solutions and find that depending on the value of the nonlinearity parameter, the singularity covered with various horizons.     

Asymptotic analysis of deterministic and stochastic equations with rapidly varying components

The asymptotic character of deterministic and stochastic equations whose solutions have a rapidly varying component is studied. Of particular interest is the class of problems for which the limiting behavior can be described in a contracted and simplified framework.