Decaying states of perturbed wave equations

We study the solutions of perturbed wave equations that represent free wave motion outside some ball. When there are no trapped rays, it is shown that every solution whose total energy decays to zero must be smooth. This extends results of Rauch to the even-dimensional case and to systems having more than one sound speed. In these results, obstacles are not considered. We show that, even allowing obstacles, waves with compact spatial support cannot decay, assuming a unique continuation hypothesis. An example with obstacle is given where nonsmooth, compactly supported, decaying waves exist.     

General soliton solutions to a reverse-time nonlocal nonlinear Schrödinger equation

General soliton solutions to a reverse-time nonlocal nonlinear Schrödinger (NLS) equation are discussed via a matrix version of binary Darboux transformation. With this technique, searching for solutions of the Lax pair is transferred to find vector solutions of the associated linear differential equation system. From vanishing and nonvanishing seed solutions, general vector solutions of such linear differential equation system in terms of the canonical forms of the spectral matrix can be constructed by means of triangular Toeplitz matrices. Several explicit one-soliton solutions and two-soliton solutions are provided corresponding to different forms of the spectral matrix. Furthermore, dynamics and interactions of bright solitons, degenerate solitons, breathers, rogue waves, and dark solitons are also explored graphically.     

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.     

An expansion theorem for two‐dimensional elastic waves and its application

We prove an Atkinson–Wilcox‐type expansion for two‐dimensional elastic waves in this paper. The approach developed on the two‐dimensional Helmholtz equation will be applied in the proof. When the elastic fields are involved, the situation becomes much harder due to two wave solutions propagating at different phase velocities. In the last section, we give an application about the reconstruction of an obstacle from the scattering amplitude. Copyright © 2006 John Wiley & Sons, Ltd.     

An improved time domain linear sampling method for Robin and Neumann obstacles

We consider inverse obstacle scattering problems for the wave equation with Robin or Neumann boundary conditions. The problem of reconstructing the geometry of such obstacles from measurements of scattered waves in the time domain is tackled using a time domain linear sampling method. This imaging technique yields a picture of the scatterer by solving a linear operator equation involving the measured data for many right-hand sides given by singular solutions to the wave equation. We analyse this algorithm for causal and smooth impulse shapes, we discuss the effect of different choices of the singular solutions used in the algorithm, and finally we propose a fast FFT-based implementation.     

Resonant Generation of Finite-Amplitude Waves by Flow Past Topography on a β-Plane

The forced Korteweg-de Vries (fKdV) equation is the generic equation for resonant flow past an obstacle. However, for flow past topography on a β-plane, the case when the upstream flow is uniform is anomalous in that there is no quadratic nonlinear term in the fKdV equation. Here we show that in this important case an alternative theory is required and obtain a new evolution equation, which has some similarities to the fKdV equation with two significant differences. These are that a small-amplitude topography now produces finite-amplitude waves and the flow response is limited by a wave breakdown characterized by an incipient flow reversal. Various numerical solutions are described.     

Solitons of the resonant nonlinear Schrödinger equation with nontrivial boundary conditions: Hirota bilinear method

We use the Hirota bilinear approach to consider physically relevant soliton solutions of the resonant nonlinear Schrödinger equation with nontrivial boundary conditions, recently proposed for describing uniaxial waves in a cold collisionless plasma. By the Madelung representation, the model transforms into the reaction-diffusion analogue of the nonlinear Schrödinger equation, for which we study the bilinear representation, the soliton solutions, and their mutual interactions.     

Resonances and Particle Stochastization in Nonhomogeneous Electromagnetic Fields

In the present paper we investigate the resonant interaction between monochromatic electromagnetic waves and charged particles in configurations with magnetic field reversals (e.g., in the earth magnetotail). The smallness of certain physical parameters allows us to solve this problem using perturbation theory, reducing the problem of resonant wave–particle interaction to the analysis of slow passages of a particle through a resonance. We discuss in detail two of the most important resonant phenomena: capture into resonance and scattering on resonance. We show that these processes result in destruction of the adiabatic invariants and chaotization of particles; they also may lead to significant (almost free) acceleration of particles and may govern transport in the phase space. We calculate the characteristic times of mixing due to resonant effects and separatrix crossings, and discuss the relative importance of these phenomena.     

On resonant interactions of gravity‐capillary waves without energy exchange

We consider resonant triad interactions of gravity‐capillary waves and investigate in detail special resonant triads that exchange no energy during their interactions so that the wave amplitudes remain constant in time. After writing the resonance conditions in terms of two parameters (or two angles of wave propagation), we first identify a region in the two‐dimensional parameter space, where resonant triads can be always found, and then describe the variations of resonant wavenumbers and wave frequencies over the resonance region. Using the amplitude equations recovered from a Hamiltonian formulation for water waves, it is shown that any resonant triad inside the resonance region can interact without energy exchange if the initial wave amplitudes and relative phase satisfy the two conditions for fixed point solutions of the amplitude equations. Furthermore, it is shown that the symmetric resonant triad exchanging no energy forms a transversely modulated traveling wave field, which can be considered a two‐dimensional generalization of Wilton ripples.     

Convex backscattering support in electric impedance tomography

This paper reinvestigates a recently introduced notion of backscattering for the inverse obstacle problem in impedance tomography. Under mild restrictions on the topological properties of the obstacles, it is shown that the corresponding backscatter data are the boundary values of a function that is holomorphic in the exterior of the obstacle(s), which allows to reformulate the obstacle problem as an inverse source problem for the Laplace equation. For general obstacles, the convex backscattering support is then defined to be the smallest convex set that carries an admissible source, i.e., a source that yields the given (backscatter) data as the trace of the associated potential. The convex backscattering support can be computed numerically; numerical reconstructions are included to illustrate the viability of the method.     

双障碍问题中梯度的局部和全局可积性

本文研究形如△·Ap(x,u,△u)=0的二阶拟线性椭圆方程的双障碍问题,获得了双障碍问题中解的梯度的局部和全局可积性,这些结果可用于证明双障碍问题解的稳定性.     

Resonant frequencies for a system of time-harmonic elastic wave

In this work, we consider the propagation of elastic waves outside a (compact) obstacle with smooth boundary. Existence, uniqueness results for the solution are established in a simple way. We study the meromorphic continuation to the whole complex plane of the solution. A proof alternative of the existence of resonant frequencies is given.     

Solitary-wave families of the Ostrovsky equation: An approach via reversible systems theory and normal forms

The Ostrovsky equation is an important canonical model for the unidirectional propagation of weakly nonlinear long surface and internal waves in a rotating, inviscid and incompressible fluid. Limited functional analytic results exist for the occurrence of one family of solitary-wave solutions of this equation, as well as their approach to the well-known solitons of the famous Korteweg–de Vries equation in the limit as the rotation becomes vanishingly small. Since solitary-wave solutions often play a central role in the long-time evolution of an initial disturbance, we consider such solutions here (via the normal form approach) within the framework of reversible systems theory. Besides confirming the existence of the known family of solitary waves and its reduction to the KdV limit, we find a second family of multihumped (or N-pulse) solutions, as well as a continuum of delocalized solitary waves (or homoclinics to small-amplitude periodic orbits). On isolated curves in the relevant parameter region, the delocalized waves reduce to genuine embedded solitons. The second and third families of solutions occur in regions of parameter space distinct from the known solitary-wave solutions and are thus entirely new. Directions for future work are also mentioned.     

Diffraction of reaction-diffusion waves: the conformal-mapped eikonal equation

The interaction of reaction-diffusion (RD) waves with obstacles is considered. The conformal map transformation of the eikonal equation is used for investigating the behavior of waves in the presence of movable boundaries. It is shown that using the conformal map, the complicated boundary conditions become simple Neumann boundary conditions and can be easily dealt with numerically. The process is applied to diffraction of RD waves by two disks in two dimensions. The approach is extended to three-dimensions and the obstacles considered are 2 two-tori. A stable stationary solution, in the form of an unduloid, trapped between 2 two-tori, is obtained. It is shown that, if the obstacles are located a distance apart, the wave moves away from its stationary position giving rise to regular and irregular motions, depending on the choice of initial solutions.     

Asymptotic behavior for the singularly perturbed damped Boussinesq equation

This work is focused on the long‐time behavior of solutions to the singularly perturbed damped Boussinesq equation in a 3D case where ε > 0 is small enough. Without any growth restrictions on the nonlinearity f(u), we establish in an appropriate bounded phase space a finite dimensional global attractor as well as an exponential attractor of optimal regularity. The key step is the estimate of the difference between the solutions of the damped Boussinesq equation and the corresponding pseudo‐parabolic equation.Copyright © 2014 John Wiley & Sons, Ltd.     

Undercompressive shock waves in Hall‐magnetohydrodynamics

We study the propagation of nonlinear waves in a Hall‐magnetohydrodynamic model. An asymptotic method is used to derive the Gardner‐Burgers equation for fast magnetosonic waves; here, the flux function is nonconvex with both quadratic and cubic nonlinearities, and the evolution equation involves both second‐ and third‐order derivatives representing diffusion and dispersion terms, respectively. Effects of Hall parameter are discussed on the evolution of waves and their interaction by solving a pair of Riemann problems both analytically and numerically. It is shown that the Hall parameter is responsible for shock splitting—a phenomenon that is completely absent in ideal magnetohydrodynamic; indeed, the Hall parameter plays a significant role in deciding about the structure of the solution that involves undercompressive shocks and their interaction with refracted waves and the Lax shocks. It is found that increasing Hall parameter means increasing dispersion that triggers the physical mechanism causing speed and strength of an undercompressive shock to increase and the wave‐fan width to decrease; numerical solutions substantiate these features predicted by the analytical solution.     

Subordinacy theory for extended CMV matrices

We develop subordinacy theory for extended Cantero-Moral-Velázquez(CMV) matrices,i.e.,we provide explicit supports for the singular and absolutely continuous parts of the canonical spectral measure associated with a given extended CMV matrix in terms of the presence or absence of subordinate solutions to the generalized eigenvalue equation.Some corollaries and applications of this result are described as well.     

Lower semicontinuous obstacles for the porous medium equation

We deal with the obstacle problem for the porous medium equation in the slow diffusion regime $m>1$. Our main interest is to treat fairly irregular obstacles assuming only boundedness and lower semicontinuity. In particular, the considered obstacles are not regular enough to work with the classical notion of variational solutions, and a different approach is needed. We prove the existence of a solution in the sense of the minimal supersolution lying above the obstacle. As a consequence, we can show that non-negative weak supersolutions to the porous medium equation can be approximated by a sequence of supersolutions which are bounded away from zero.