Rossby Solitary Waves in the Presence of a Critical Layer

This study considers the evolution of weakly nonlinear long Rossby waves in a horizontally sheared zonal current. We consider a stable flow so that the nonlinear time scale is long. These assumptions enable the flow to organize itself into a large‐scale coherent structure in the régime where a competition sets in between weak nonlinearity and weak dispersion. This balance is often described by a Korteweg‐de‐Vries equation. The traditional assumption of a weak amplitude breaks down when the wave speed equals the mean flow velocity at a certain latitude, due to the appearance of a singularity in the leading‐order equation, which strongly modifies the flow in a critical layer. Here, nonlinear effects are invoked to resolve this singularity, because the relevant geophysical flows have high Reynolds numbers. Viscosity is introduced in order to render the nonlinear‐critical‐layer solution unique, but the inviscid limit is eventually taken. By the method of matched asymptotic expansions, this inner flow is matched at the edges of the critical layer with the outer flow. We will show that the critical‐layer–induced flow leads to a strong rearrangement of the related streamlines and consequently of the potential‐vorticity contours, particularly in the neighborhood of the separatrices between the open and closed streamlines. The symmetry of the critical layer vis‐à‐vis the critical level is also broken. This theory is relevant for the phenomenon of Rossby wave breaking and eventual saturation into a nonlinear wave. Spatially localized solutions are described by a Korteweg‐de‐Vries equation, modified by new nonlinear terms; depending on the critical‐layer shape, this leads to depression or elevation waves. The additional terms are made necessary at a certain order of the asymptotic expansion while matching the inner flow on the dividing streamlines. The new evolution equation supports a family of solitary waves. In this paper we describe in detail the case of a depression wave, and postpone for further discussion the more complex case of an elevation wave.     

Asymptotic Analysis of Reacting Materials with Saturated Explosion. II. High-Frequency Waves

The interaction of small-scale material inhomogeneities with high-frequency acoustic waves is known to have a prominent role in accelerating the heat-release rate in liquid and solid explosive materials. In the present paper, simplified asymptotic equations are studied which incorporate the above interaction, and which include reactant depletion at leading order. Because fuel may be completely exhausted, singularities do not always form in the model equations; it is conjectured that when a singularity does form, the material has initiated. The detailed mechanisms by which shock formation and resonant wave interaction can either enhance or retard reaction are explored. In a realistic model for inhomogeneous condensed-phase reaction, with pressure-dependent reaction rate and nonconstant initial fuel concentration, initiation of the material depends on correct placement of the fuel relative to the acoustic waves.     

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.     

Lossless error estimates for the stationary phase method with applications to propagation features for the Schrödinger equation

We consider a version of the stationary phase method in one dimension of A. Erdélyi, allowing the phase to have stationary points of non‐integer order and the amplitude to have integrable singularities. After having completed the original proof and improved the error estimate in the case of regular amplitude, we consider a modification of the method by replacing the smooth cut‐off function employed in the source by a characteristic function, leading to more precise remainder estimates. We exploit this refinement to study the time‐asymptotic behaviour of the solution of the free Schrödinger equation on the line, where the Fourier transform of the initial data is compactly supported and has a singularity. We obtain asymptotic expansions with respect to time in certain space‐time cones as well as uniform and optimal estimates in curved regions, which are asymptotically larger than any space‐time cone. These results show the influence of the frequency band and of the singularity on the propagation and on the decay of the wave packets. Copyright © 2016 John Wiley & Sons, Ltd.     

Interacting Wave Fronts and Rarefaction Waves in a Second Order Model of Nonlinear Thermoviscous Fluids

A wave equation including nonlinear terms up to the second order for a thermoviscous Newtonian fluid is proposed. In the lossless case this equation results from an expansion to third order of the Lagrangian for the fundamental non-dissipative fluid dynamical equations. Thus it preserves the Hamiltonian structure, in contrast to the Kuznetsov equation, a model often used in nonlinear acoustics. An exact traveling wave front solution is derived from a generalized traveling wave assumption for the velocity potential. Numerical studies of the evolution of a number of arbitrary initial conditions as well as head-on colliding and confluent wave fronts exhibit several nonlinear interaction phenomena. These include wave fronts of changed velocity and amplitude along with the emergence of rarefaction waves. An analysis using the continuity of the solutions as well as the boundary conditions is proposed. The dynamics of the rarefaction wave is approximated by a collective coordinate approach in the energy balance equation.     

Asymptotic analysis of contact discontinuities

We study the perturbation of a contact discontinuity by a small amplitude, rapidly oscillating wave train. Under a suitable stability assumption the perturbed solution is still a contact discontinuity, and we give its asymptotic development, as well as that of the contact curve, in terms of the wavelength of the perturbation.     

A Formulation for Water Waves over Topography

Many interesting free-surface flow problems involve a varying bottom. Examples of such flows include ocean waves propagating over topography, the breaking of waves on a beach, and the free surface of a uniform flow over a localized bump. We present here a formulation for such flows that is general and, from the outset, demonstrates the wave character of the free-surface evolution. The evolution of the free surface is governed by a system of equations consisting of a nonlinear wave-like partial differential equation coupled to a time-independent linear integral equation. We assume that the free-surface deformation is weakly nonlinear, but make no a priori assumption about the scale or amplitude of the topography. We also extend the formulation to include the effect of mean flows and surface tension. We show how this formulation gives some of the well-known limits for such problems once assumptions about the amplitude and scale of the topography are made.     

Metrically Regular Vector Field and Iterative Processes for Generalized Equations in Hadamard Manifolds

This paper is focused on the problem of finding a singularity of the sum of two vector fields defined on a Hadamard manifold, or more precisely, the study of a generalized equation in a Riemannian setting. We extend the concept of metric regularity to the Riemannian setting and investigate its relationship with the generalized equation in this new context. In particular, a version of Graves’s theorem is presented and we also define some concepts related to metric regularity, including the Aubin property and the strong metric regularity of set-valued vector fields. A conceptual method for finding a singularity of the sum of two vector fields is also considered. This method has as particular instances: the proximal point method, Newton’s method, and Zincenko’s method on Hadamard manifolds. Under the assumption of metric regularity at the singularity, we establish that the methods are well defined in a suitable neighborhood of the singularity. Moreover, we also show that each sequence generated by these methods converges to this singularity at a superlinear rate.     

Waves in an inhomogeneous fluid in the presence of a dock : PMM vol. 39, n 6, 1975, pp. 1140–1142

We investigate the propagation of waves generated by oscillations of a section of the bottom of a tank through a two-layer fluid, in the presence of a dock. Wave motions in an inhomogeneous fluid generated by displacement of a section of the bottom of a tank were studied in [1] where the upper surface of the fluid was assumed either to be completely free, or completely covered with ice. In the present paper we use the method given in [2] to investigate a similar problem under the assumption that the fluid surface is partly covered with an immovable rigid plate. The expressions obtained for the velocity potential are used to determine the form of the free surface and of the interface. We show that when the fluid is inhomogeneous, the wave amplitude on the free surface increases, while the presence of a plate reduces the amplitude of the surface waves, as well as of the internal waves in the region between the plate and the oscillating section of the bottom.     

Local existence with physical vacuum boundary condition to Euler equations with damping

In this paper, we consider the local existence of solutions to Euler equations with linear damping under the assumption of physical vacuum boundary condition. By using the transformation introduced in Lin and Yang (Methods Appl. Anal. 7 (3) (2000) 495) to capture the singularity of the boundary, we prove a local existence theorem on a perturbation of a planar wave solution by using Littlewood-Paley theory and justifies the transformation introduced in Liu and Yang (2000) in a rigorous setting.     

Solitary wave solutions to the Isobe-Kakinuma model for water waves

We consider the Isobe-Kakinuma model for two-dimensional water waves in the case of a flat bottom. The Isobe-Kakinuma model is a system of Euler-Lagrange equations for a Lagrangian approximating Luke's Lagrangian for water waves. We show theoretically the existence of a family of small amplitude solitary wave solutions to the Isobe-Kakinuma model in the long wave regime. Numerical analysis for large amplitude solitary wave solutions is also provided and suggests the existence of a solitary wave of extreme form with a sharp crest.     

Three-particle Coulomb on-shell vertex functions. Behavior of the amplitude for simultaneous transfer of two charged particles in the vicinity of the nearest singularity in cos θ

The connection between the on-shell vertex function (OSVF) and the asymptotic normalized coefficient of the three-particle wave function of the bound system in the configuration space is found. The explicit form of the leading singular term of the OSVF and its Faddeev component for decay of the three-particle bound state into three charged particles is established. An expression is found for the leading singular term of the exact (in the model of four charged particles) amplitude of sub-Coulomb binary reactions in the vicinity of the nearest singularity.Institute of Nuclear Physics, Uzbek Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 94, No. 3, pp. 448–462, March, 1993.     

Internal waves in an unbounded non-Boussinesq flow

Weakly nonlinear internal waves in an unbounded non-Boussinesq flow with uniform stratification are treated with a Laurent-type expansion. The expansion eliminates the problem encountered with a traditional expansion in wave amplitude where higher harmonics grow exponentially faster with higher order. The results show that the second-order wave correction to the linear estimate of the wave speed of internal waves in an unbounded layer is always negative, meaning that higher amplitude waves travel slower.     

Point source excitation in direct and inverse scattering: The soft and the hard small sphere

A spherical wave emanating from a point source is scatteredby either a soft or a hard body. The incident spherical wavehas a wavelength which is much larger than the characteristicdimension of the scatterer and it is modified in such a wayas to recover the plane wave incidence when the source pointrecedes to infinity. Using low frequency expansions the scatteringproblem is transformed to a sequence of exterior potential problemsin the presence of a monopole singularity located at the sourceof the incident wave field. Complete expansions for the scatteringamplitude are provided. The method is applied to the cases ofa soft and a hard sphere and the first three approximationsfor the near, as well as the far, field are evaluated. It isobserved that every one, after the first, low frequency approximationof the far field, involves one spherical multipole more thanthe corresponding approximation for the case of an incidentplane wave. As the point singularity tends to infinity, therelative results recover all the known expressions for planeincidence. It is shown that for point excitation the Rayleighapproximation of the scattering amplitude for a hard sphereis of the second order, in contrast to the case of plane excitationwhich is of the third order. Simple algorithms that specifythe radius and the position of a soft and a hard sphere areproposed, which are based on the additional dependence of thescattering amplitude represented by the distance from the pointsource to the centre of the scatterer. The inversion algorithmis shown to be stable whenever the source point is not too faraway from the target sphere. A simple way to decide whetherthe sphere is a soft or a hard body is also provided.     

Resolving Singularities in a Ramsey-type Growth Model

The aim of this paper is to discuss a mathematical solution procedure to solve a Ramsay-type growth model that explains the fundamentals of consumption and capital accumula-tion in a dynamic equilibrium setting. The problem is formulated as a system of recursive equations and studied through some numerical experiments for the time path of the different variables of the model under some alternative assumption for the steady-state equilibrium of the labour market conditioning the possible singularity of the model.     

Rossby Elevation Waves in the Presence of a Critical Layer

In a previous paper, we investigated the solitary-wave-like development of small-amplitude Rossby waves propagating in a zonal shear current, for the particular case when the Rossby wave speed equals the mean-flow velocity at a certain latitude in the β-plane. We presented a general theory for the nonlinear critical-layer theory, and illustrated it by explicitly describing the motion of a depression solitary wave (D-wave). Here, we report a continuation of that study and consider the more complex case of an elevation solitary wave (E-wave). The method involves matched asymptotic expansions between the outer flow away from the critical layer and the inner flow inside the latter, both these flows having different scalings. We showed previously that the critical-layer flow expansion diverged in the case of the E-wave on the separatrices bounding the open and closed streamlines, which led us to defer a detailed E-wave study. Thus, in this paper, we examine the motion in the additional layer located along the separatrices where this singularity is removed by using a third scaling and find that the previous undesirable distortions are discarded. The evolution equation is derived and is a Korteveg-de-Vries type-equation modified by new nonlinear terms generated by the nonlinear interactions occuring in the critical layer. This equation supports a family of E-waves provided that the mean flow obeys certain conditions. The energy exchange that occurs between the mean flow and the D or E-wave during the critical-layer formation is evaluated in the quasi-steady régime assumption.     

The p-Factor-Lagrange Methods for Degenerate Nonlinear Programming

The paper presents a new approach to solving nonlinear programming (NLP) problems for which the strict complementarity condition (SCC), a constraint qualification (CQ), and a second-order sufficient condition (SOSC) for optimality are not necessarily satisfied at a solution. Our approach is based on the construction of p-regularity and on reformulating the inequality constraints as equalities. Namely, by introducing the slack variables, we get the equality constrained problem, for which the Lagrange optimality system is singular at the solution of the NLP problem in the case of the violation of the CQs, SCC and/or SOSC. To overcome the difficulty of singularity, we propose the p-factor method for solving the Lagrange system. The method has a superlinear rate of convergence under a mild assumption. We show that our assumption is always satisfied under a standard second-order sufficient condition (SOSC) for optimality. At the same time, we give examples of the problems where the SOSC does not hold, but our assumption is satisfied. Moreover, no estimation of the set of active constraints is required. The proposed approach can be applied to a variety of problems.     

异方差回归中的广义方差比检验

在同方差假设之下,线性模型在回归分析的理论与应用方面起着突出的作用,很受许多研究工作者的青睐.然而,回归模型中同方差性这一标准假设不一定总是成立的.因此我们考虑了用一类基于似残差的方法来检验异方差情形下线性模型拟合观测数据的情况.本文既给出了大量的模拟,又给出了实际数据作为应用的例子.效果都很好.     

Inhibiting Shear Instability Induced by Large Amplitude Internal Solitary Waves in Two-Layer Flows with a Free Surface

We consider a strongly nonlinear long wave model for large amplitude internal waves in two-layer flows with the top free surface. It is shown that the model suffers from the Kelvin–Helmholtz (KH) instability so that any given shear (even if arbitrarily small) between the layers makes short waves unstable. Because a jump in tangential velocity is induced when the interface is deformed, the applicability of the model to describe the dynamics of internal waves is expected to remain rather limited. To overcome this major difficulty, the model is written in terms of the horizontal velocities at the bottom and the interface, instead of the depth-averaged velocities, which makes the system linearly stable for perturbations of arbitrary wavelengths as long as the shear does not exceed a certain critical value.     

Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis

In this paper, we establish the existence and the nonlinear stability of traveling wave solutions to a system of conservation laws which is transformed, by a change of variable, from the well-known Keller-Segel model describing cell (bacteria) movement toward the concentration gradient of the chemical that is consumed by the cells. We prove the existence of traveling fronts by the phase plane analysis and show the asymptotic nonlinear stability of traveling wave solutions without the smallness assumption on the wave strengths by the method of energy estimates.