Showing papers in "Studies in Applied Mathematics in 2015"
TL;DR: In this paper, a consistent Riccati expansion (CRE) is proposed for solving nonlinear systems with the help of a CRE equation, and a system having a CRE is then defined to be CRE solvable.
Abstract: A consistent Riccati expansion (CRE) is proposed for solving nonlinear systems with the help of a Riccati equation. A system having a CRE is then defined to be CRE solvable. The CRE solvability is demonstrated quite universal for various integrable systems including the Korteweg–de Vries, Kadomtsev–Petviashvili, Ablowitz–Kaup–Newell–Segur (and then nonlinear Schrodinger), sine-Gordon, Sawada–Kotera, Kaup–Kupershmidt, modified asymmetric Nizhnik–Novikov–Veselov, Broer–Kaup, dispersive water wave, and Burgers systems. In addition, it is revealed that many CRE solvable systems share a similar determining equation describing the interactions between a soliton and a cnoidal wave. They have a common nonlocal symmetry expression and they possess a formally universal once Backlund transformation.
142 citations
TL;DR: In this paper, it was shown that periodic traveling waves with sufficiently small amplitudes of the Whitham equation, which incorporates the dispersion relation of surface water waves and the nonlinearity of the shallow water equations, are spectrally unstable to long-wavelengths perturbations if the wave number is greater than a critical value, bearing out the Benjamin-Feir instability of Stokes waves.
Abstract: We show that periodic traveling waves with sufficiently small amplitudes of the Whitham equation, which incorporates the dispersion relation of surface water waves and the nonlinearity of the shallow water equations, are spectrally unstable to long-wavelengths perturbations if the wave number is greater than a critical value, bearing out the Benjamin–Feir instability of Stokes waves; they are spectrally stable to square integrable perturbations otherwise. The proof involves a spectral perturbation of the associated linearized operator with respect to the Floquet exponent and the small-amplitude parameter. We extend the result to related, nonlinear dispersive equations.
87 citations
TL;DR: In this paper, it is shown that the solution of the coupled nonlinear Schrodinger equation on the half-line can be expressed in terms of the unique solution of a matrix Riemann-Hilbert problem formulated in the complex k-plane.
Abstract: Initial-boundary value problems for the coupled nonlinear Schrodinger equation on the half-line are investigated via the Fokas method. It is shown that the solution can be expressed in terms of the unique solution of a matrix Riemann–Hilbert problem formulated in the complex k-plane, whose jump matrix is defined in terms of the matrix spectral functions and that depend on the initial data and all boundary values, respectively. If there exist spectral functions satisfying the global relation, it can be proved that the function defined by the above Riemann–Hilbert problem solves the coupled nonlinear Schrodinger equation and agrees with the prescribed initial and boundary values. The most challenging problem in the implementation of this method is to characterize the unknown boundary values that appear in the spectral function . For a particular class of boundary conditions so-called linearizable boundary conditions, it is possible to compute the spectral function in terms of and given boundary conditions by using the algebraic manipulation of the global relation. For the general case of boundary conditions, an effective characterization of the unknown boundary values can be obtained by employing perturbation expansion.
76 citations
TL;DR: In this article, the boundary value problem is converted into a system of three Fredholm type integral equations and various integrals associated with the integral equations are evaluated using appropriate Gauss quadrature formulae.
Abstract: In the present study, oblique surface wave scattering by a submerged vertical flexible porous plate is investigated in both the cases of water of finite and infinite depths. Using Green's function technique, the boundary value problem is converted into a system of three Fredholm type integral equations. Various integrals associated with the integral equations are evaluated using appropriate Gauss quadrature formulae and the system of integral equations are converted into a system of algebraic equations. Further, using Green's second identity, expressions for the reflection and transmission coefficients are obtained in terms of the velocity potential and its normal derivative. Energy balance relations for wave scattering by flexible porous plates and permeable membrane barriers are derived using Green's identity and used to check the correctness of the computational results. From the general formulation of the submerged plate, wave scattering by partial plates such as (i) surface-piercing and (ii) bottom-standing plates are studied as special cases. Further, oblique wave scattering by bottom-standing and surface-piercing porous membrane barriers are studied in finite water depth as particular cases of the flexible plate problem. Various numerical results are presented to study the effect of structural rigidity, angle of incidence, membrane tension, structural length, porosity and water depth on wave scattering. It is found that wave reflection is more for a surface-piercing flexible porous plate in infinite water depth compared to finite water depth and opposite trend is observed for a submerged flexible porous plate. For a surface-piercing nonpermeable membrane, zeros in transmission coefficient are observed for waves of intermediate water depth which disappear with the inclusion of porosity. The study reveals that porosity has small influence on the wave-induced excitation of the structure with higher flexibility but it tends to reduce the deflection of a stiffer structure. In case of partial flexible plates and membrane barriers, irrespective of the gap length, full transmission occurs due to wave diffraction through the gap in the very long wave regime while, full reflection occurs by complete flexible impermeable barriers for similar wave condition.
47 citations
TL;DR: A generalized two-component model with peakon solutions is proposed in this paper, which allows an arbitrary function to be involved in as well as including some existing integrable peakon equations as special reductions.
Abstract: A generalized two-component model with peakon solutions is proposed in this paper. It allows an arbitrary function to be involved in as well as including some existing integrable peakon equations as special reductions. The generalized two-component system is shown to possess Lax pair and infinitely many conservation laws. Bi-Hamiltonian structures and peakon interactions are discussed in detail for typical representative equations of the generalized system. In particular, a new type of N-peakon solution, which is not in the traveling wave type, is obtained from the generalized system.
47 citations
TL;DR: In this article, the well-posedness theory for Landau-Lifshitz equation with the frame of inverse scattering method is given, and the generalized Darboux transformation is rigorous considered in the frame for inverse scattering transformation.
Abstract: The Landau–Lifshitz equation is analyzed via the inverse scattering method. First, we give the well-posedness theory for Landau–Lifshitz equation with the frame of inverse scattering method. The generalized Darboux transformation is rigorous considered in the frame of inverse scattering transformation. Finally, we give the high-order soliton solution formula of Landau–Lifshitz equation and vortex filament equation.
34 citations
TL;DR: In this paper, the effects of quadratic singular curves in integrable wave equations are studied by using the bifurcation theory of dynamical system, and an intrinsic connection is constructed between the singular traveling waves in the phase plane of a traveling wave system, where the new singular periodic waves, pseudo-cuspons and compactons emerge if corresponding periodic orbits or homoclinic orbits are tangent to a hyperbola, ellipse, and parabola.
Abstract: In this paper, the effects of quadratic singular curves in integrable wave equations are studied by using the bifurcation theory of dynamical system. Some new singular solitary waves (pseudo-cuspons) and periodic waves are found more weak than regular singular traveling waves such as peaked soliton (peakon), cusp soliton (cuspon), cusp periodic wave, etc. We show that while the first-order derivatives of the new singular solitary wave and periodic waves exist, their second-order derivatives are discontinuous at finite number of points for the solitary waves or at infinitely countable points for the periodic wave. Moreover, an intrinsic connection is constructed between the singular traveling waves and quadratic singular curves in the phase plane of traveling wave system. The new singular periodic waves, pseudo-cuspons, and compactons emerge if corresponding periodic orbits or homoclinic orbits are tangent to a hyperbola, ellipse, and parabola. In particular, pseudo-cuspon is proposed for the first time. Finally, we study the qualitative behavior of the new singular solitary wave and periodic wave solutions through theoretical analysis and numerical simulation.
26 citations
TL;DR: In this paper, the interface problem for the linear Schrodinger equations in one-dimensional piecewise homogeneous domains is examined by providing an explicit solution in each domain, where the location of the interfaces is known and the continuity of the wave function and a jump in their derivative at the interface are the only conditions imposed.
Abstract: The interface problem for the linear Schrodinger equations in one-dimensional piecewise homogeneous domains is examined by providing an explicit solution in each domain. The location of the interfaces is known and the continuity of the wave function and a jump in their derivative at the interface are the only conditions imposed. The problem of two semi-infinite domains and that of two finite-sized domains are examined in detail. The problem and the method considered here extend that of an earlier paper by Deconinck et al. (2014) [1]. The dispersive nature of the problem presents additional difficulties that are addressed here.
25 citations
TL;DR: In this paper, two elementary and a binary Darboux transformations are constructed for the supersymmetric modified Korteweg-de Vries, sinh-Gordon, and nonlinear Schrodinger equations.
Abstract: In this paper, we consider a supersymmetric AKNS spectral problem. Two elementary and a binary Darboux transformations are constructed. By means of reductions, Darboux and Backlund transformations are given for the supersymmetric modified Korteweg-de Vries, sinh-Gordon, and nonlinear Schrodinger equations. These Darboux and Backlund transformations are adopted for the constructions of integrable discrete super systems, and both semidiscrete and fully discrete systems are presented. Also, the continuum limits of the relevant discrete systems are worked out.
23 citations
TL;DR: In this article, the authors consider the Caudrey-Beals-Coifman linear problem and the theory of the Recursion Operators (Generating Operators) related to it in the presence of Zh-reductions of Mikhailov type defined by Coxeter automorphism of the underlying algebra.
Abstract: We consider the Caudrey–Beals–Coifman linear problem (CBC problem) and the theory of the Recursion Operators (Generating Operators) related to it in the presence of Zh-reductions of Mikhailov type defined by Coxeter automorphism of the underlying algebra. We discuss mainly the spectral aspects of the theory of the Recursion Operators related to the expansions over the adjoint solutions of the CBC problem but pay attention also to the algebraic issues related to the recursion relations through which the hierarchies of the nonlinear evolution equations associated with the CBC auxiliary problem are obtained.
21 citations
TL;DR: In this paper, a reduction algorithm was proposed to reduce a large system to a smaller one consisting of m differential equations, where m is the number of variables in the system and n is the dimension of the system.
Abstract: We consider systems of differential equations with quadratic nonlinearities having applications for biochemistry and population dynamics, which may have a large dimension n. Due to the complexity of these systems, reduction algorithms play a crucial role in study of their large time behavior. Our approach aims to reduce a large system to a smaller one consisting of m differential equations, where . Under some restrictions (that allow us to separate slow and fast variables in the system) we obtain a new system of differential equations, involving slow variables only. This reduction is feasible from a computational point of view for large n that allows us to investigate sensitivity of dynamics with respect to random variations of parameters. We show that the quadratic systems are capable to generate all kinds of structurally stable dynamics including chaos.
TL;DR: In this paper, the double scaling limit of the Hankel determinants was studied in terms of the Jimbo-Miwa-Okamoto �-function and asymptotic approximations of the determinants were obtained.
Abstract: We study the Hankel determinants associated with the weight w(x;t) = (1 − x 2 ) � (t 2 − x 2 ) � h(x), x ∈ (−1,1), where � > −1, � + � > −1, t > 1, h(x) is analytic in a domain containing [−1,1] and h(x) > 0 for x ∈ [−1,1]. In this paper, based on the Deift-Zhou nonlinear steepest descent analysis, we study the double scaling limit of the Hankel determinants as n → ∞ and t → 1. We obtain the asymptotic approximations of the Hankel determinants, evaluated in terms of the Jimbo-Miwa-Okamoto �-function
TL;DR: In this article, a straightforward procedure for using Renormalization Group methods to solve a significant variety of perturbation problems, including some that result from applying a nonlinear version of variation of parameters, is presented.
Abstract: This paper presents a straightforward procedure for using Renormalization Group methods to solve a significant variety of perturbation problems, including some that result from applying a nonlinear version of variation of parameters A regular perturbation procedure typically provides asymptotic solutions valid for bounded t values as a positive parameter e tends to zero One can eliminate secular terms by introducing a slowly-varying amplitude obtained as a solution of an amplitude equation on intervals where is bounded With sufficient stability hypotheses, the results may even hold for all These ideas are illustrated for a number of nontrivial problems involving ordinary differential equations
TL;DR: In this article, it was shown that the singular case can be expressed in terms of the solution to the Blasius equation, which is the case in this paper, and that the slope Chebyshev series has only a fourth-order convergence until a simple change of coordinate restores a geometric rate of convergence, empirically proportional to the square root singularity.
Abstract: The Kidder problem is with and where . This looks challenging because of the square root singularity. We prove, however, that for all . Other very simple but very accurate curve fits and bounds are given in the text; . Maple code for a rational Chebyshev pseudospectral method is given as a table. Convergence is geometric until the coefficients are when the coefficients . An initial-value problem is obtained if is known; the slope Chebyshev series has only a fourth-order rate of convergence until a simple change-of-coordinate restores a geometric rate of convergence, empirically proportional to . Kidder's perturbation theory (in powers of α) is much inferior to a delta-expansion given here for the first time. A quadratic-over-quadratic Pade approximant in the exponentially mapped coordinate predicts the slope at the origin very accurately up to about . Finally, it is shown that the singular case can be expressed in terms of the solution to the Blasius equation.
TL;DR: In this article, the authors theoretically show the existence of Cartesian vector solutions for general N-dimensional compressible Euler equations, which can be explicitly expressed by an appropriate formulae.
Abstract: In this paper, based on matrix and curve integration theory, we theoretically show the existence of Cartesian vector solutions for the general N-dimensional compressible Euler equations. Such solutions are global and can be explicitly expressed by an appropriate formulae. One merit of this approach is to transform analytically solving the Euler equations into algebraically constructing an appropriate matrix . Once the required matrix is chosen, the solution is directly obtained. Especially, we find an important solvable relation between the dimension of equations and pressure parameter, which avoid additional independent constraints on the dimension N in existing literatures. Special cases of our results also include some interesting conclusions: (1) If the velocity field is a linear transformation on , then the pressure p is a relevant quadratic form. (2) The compressible Euler equations admit the Cartesian solutions if is an antisymmetric matrix. (3) The pressure p possesses radial symmetric form if is an antisymmetrically orthogonal matrix.
TL;DR: In this article, the integrable discretization of a negative order AKNS equation via Hirota's bilinear method is investigated and two semidiscrete and one fully discrete versions of the system are given.
Abstract: The paper investigates the integrable discretization of a negative order AKNS equation. Two semidiscrete and one fully discrete versions of the system are given via Hirota's bilinear method. Soliton solutions for the derived discrete systems are also presented. Dynamics of one-soliton and two-soliton solutions of spacial-discrete system are characterized.
TL;DR: In this paper, the authors considered a discrete version of the first Painleve equation (qPI), whose phase space (space of initial values) is a rational surface of type.
Abstract: In this paper, we present new, unstable solutions, which we call quicksilver solutions, of a q-difference Painleve equation in the limit as the independent variable approaches infinity. The specific equation we consider in this paper is a discrete version of the first Painleve equation (qPI), whose phase space (space of initial values) is a rational surface of type . We describe four families of almost stationary behaviors, but focus on the most complicated case, which is the vanishing solution. We derive this solution's formal power series expansion, describe the growth of its coefficients, and show that, while the series is divergent, there exist true analytic solutions asymptotic to such a series in a certain q-domain. The method, while demonstrated for qPI, is also applicable to other q-difference Painleve equations.
TL;DR: In this article, the initial-boundary value problem for the Boussinesq equation on the half-line with various initial and boundary conditions is considered. And the authors present a solution for the linearized version of the problem.
Abstract: This work is concerned with the initial-boundary value problem for the Boussinesq equation. By employing the unified transform method of Fokas, novel solution formulae for the linearized “good” Boussinesq equation on the half-line with various initial and boundary conditions are obtained. Moreover, these solution formulae are numerically illustrated in the case of concrete data. Finally, Boussinesq's original physical derivation of the so-called “bad” Boussinesq equation is provided.
TL;DR: In this article, the authors proposed an explicit approximation for the nonlinear Dirichlet-to-Neumann map supplied by the defocusing nonlinear Schrodinger equation and use it to provide approximate solutions of general nonhomogeneous boundary value problems for this equation posed as an initial-boundary value problem on the half-line.
Abstract: Initial-boundary value problems for integrable nonlinear partial differential equations have become tractable in recent years due to the development of so-called unified transform techniques. The main obstruction to applying these methods in practice is that calculation of the spectral transforms of the initial and boundary data requires knowledge of too many boundary conditions, more than are required to make the problem well-posed. The elimination of the unknown boundary values is frequently addressed in the spectral domain via the so-called global relation, and types of boundary conditions for which the global relation can be solved are called linearizable. For the defocusing nonlinear Schrodinger equation, the global relation is only known to be explicitly solvable in rather restrictive situations, namely homogeneous boundary conditions of Dirichlet, Neumann, and Robin (mixed) type. General nonhomogeneous boundary conditions are not known to be linearizable. In this paper, we propose an explicit approximation for the nonlinear Dirichlet-to-Neumann map supplied by the defocusing nonlinear Schrodinger equation and use it to provide approximate solutions of general nonhomogeneous boundary value problems for this equation posed as an initial-boundary value problem on the half-line. Our method sidesteps entirely the solution of the global relation. The accuracy of our method is proven in the semiclassical limit, and we provide explicit asymptotics for the solution in the interior of the quarter-plane space-time domain.
TL;DR: In this article, the variable-coefficient Kortweg-de Vries equation was examined for the situation when the coefficient of the quadratic nonlinear term changes sign at a certain critical point.
Abstract: We examine the variable-coefficient Kortweg-de Vries equation for the situation when the coefficient of the quadratic nonlinear term changes sign at a certain critical point. This case has been widely studied for a solitary wave, which is extinguished at the critical point and replaced by a train of solitary waves of the opposite polarity to the incident wave, riding on a pedestal of the original polarity. Here, we examine the same case but for a modulated periodic wave train. Using an asymptotic analysis, we show that in contrast a periodic wave is preserved with a finite amplitude as it passes through the critical point, but a phase change is generated causing the wave to reverse its polarity.
TL;DR: In this article, a numerical method for computing time-periodic and relative timeperiodic solutions in dissipative wave systems is proposed, where the unknown parameters are expressed in terms of the solution through quasi-Rayleigh quotients.
Abstract: A numerical method is proposed for computing time-periodic and relative time-periodic solutions in dissipative wave systems. In such solutions, the temporal period, and possibly other additional internal parameters such as the propagation constant, are unknown priori and need to be determined along with the solution itself. The main idea of the method is to first express those unknown parameters in terms of the solution through quasi-Rayleigh quotients, so that the resulting integrodifferential equation is for the time-periodic solution only. Then this equation is computed in the combined spatiotemporal domain as a boundary value problem by Newton-conjugate-gradient iterations. The proposed method applies to both stable and unstable time-periodic solutions; its numerical accuracy is spectral; it is fast-converging; its memory use is minimal; and its coding is short and simple. As numerical examples, this method is applied to the Kuramoto–Sivashinsky equation and the cubic-quintic Ginzburg–Landau equation, whose time-periodic or relative time-periodic solutions with spatially periodic or spatially localized profiles are computed. This method also applies to systems of ordinary differential equations, as is illustrated by its simple computation of periodic orbits in the Lorenz equations. MATLAB codes for all numerical examples are provided in the Appendices to illustrate the simple implementation of the proposed method.
TL;DR: In this article, an integral representation of the solution of the biharmonic equation in the interior of a convex polygon is presented, which is more general than the one presented in [1].
Abstract: A new method for analyzing linear elliptic partial differential equations in the interior of a convex polygon was developed in the late 1990s. This method does not rely on the classical approach of separation of variables and on the use of classical integral transforms and therefore is well suited for the investigation of the biharmonic equation. Here, we present a novel integral representation of the solution of the biharmonic equation in the interior of a convex polygon. This representation contains certain free parameters and therefore is more general than the one presented in [1]. For a given boundary value problem, by choosing these free parameters appropriately, one can obtain the simplest possible representation for the solution. This representation still involves certain unknown boundary values, thus for this formula to become effective it is necessary to characterize the unknown boundary values in terms of the given boundary conditions. This requires the investigation of certain relations refereed to as the global relations. A general approach for analyzing these relations is illustrated by solving several problems formulated in the interior of a semistrip. In addition, for completeness, similar results are presented for the Poisson equation by employing an integral representation for the Laplace equation which is more general than the one derived in the late 1990s.
TL;DR: In this paper, the authors derived the Bernoulli integral for 3D flow with two-component vorticity field and compared it with the known integrals for potential flows or 2D stationary vortical flows of inviscid fluid.
Abstract: It has been shown in our previous paper that there is a wide class of 3D motions of incompressible viscous fluid which can be described by one scalar function dabbed the quasi-potential. This class of fluid flows is characterized by three-component velocity field having two-component vorticity field; both these fields can depend of all three spatial variables and time, in general. Governing equations for the quasi-potential have been derived and simple illustrative example of 3D flow has been presented. Here, we derive the Bernoulli integral for that class of flows and compare it against the known Bernoulli integrals for the potential flows or 2D stationary vortical flows of inviscid fluid. We show that the Bernoulli integral for this class of fluid motion possesses unusual features: it is valid for the vortical nonstationary motions of a viscous incompressible fluid. We present a new very nontrivial analytical example of 3D flow with two-component vorticity which hardly can be obtained by any of known methods. In the last section, we suggest a generalization of the developed concept which allows one to describe a certain class of 3D flows with the 3D vorticity.
TL;DR: In this article, the impact of small parity-time symmetric perturbations on nonlinear optical honeycomb lattices in the tight-binding limit was examined, and the existence of semilocalized gap solitons was shown.
Abstract: We examine the impact of small parity-time () symmetric perturbations on nonlinear optical honeycomb lattices in the tight-binding limit. We show for strained lattices that complex dispersion relationships do not form under perturbation, and we find a variety of nonlinear wave equations which describe the effective dynamics in this regime. The existence of semilocalized gap solitons in this case is also shown, though we numerically demonstrate these solitons are likely unstable. We show for unstrained lattices under the effect of a restricted class of perturbations, which prevent complex dispersion relationships from appearing, that nontrivial phase dynamics emerge as a result of the perturbation. This phase can be understood as momentum imparted to optical beams by the lattice, thus showing perturbations offer potentially novel means for the control of light in honeycomb lattices.
TL;DR: In this article, the evolution of barotropic vortex Rossby waves in a cyclonic vortex in a two-dimensional configuration where the variation of the Coriolis force with latitude is ignored is described.
Abstract: Analyses of observational data on hurricanes in the tropical atmosphere indicate the existence of spiral rainbands which propagate outward from the eye and affect the structure and intensity of the hurricane. These disturbances may be described as vortex Rossby waves. This paper describes the evolution of barotropic vortex Rossby waves in a cyclonic vortex in a two-dimensional configuration where the variation of the Coriolis force with latitude is ignored. The waves are forced by a constant-amplitude boundary condition at a fixed radius from the center of the vortex and propagate outward. The mean flow angular velocity profile is taken to be a quadratic function of the radial distance from the center of the vortex and there is a critical radius at which it is equal to the phase speed of the waves. For the case of waves with steady amplitude, an exact solution is derived for the steady linearized equations in terms of hypergeometric functions; this solution is valid in the outer region away from the critical radius. For the case of waves with time-dependent amplitude, asymptotic solutions of the linearized equations, valid for late time, are obtained in the outer and inner regions. It is found that there are strong qualitative similarities between the conclusions on the evolution of the vortex waves in this configuration and those obtained in the case of Rossby waves in a rectangular configuration where the latitudinal gradient of the Coriolis parameter is taken into account. In particular, the amplitude of the steady-state outer solution is greatly attenuated and there is a phase change of -π across the critical radius, and in the linear time-dependent configuration, the outer solution approaches a steady state in the limit of infinite time, while the amplitude of the inner solution grows on a logarithmic time scale and the width of the critical layer approaches zero.
TL;DR: In this article, the degeneracy of the Whitham equations is considered where one of the characteristic speeds is zero and it is shown that conservation of wave action morphs into the Korteweg-de Vries (KdV) equation on a longer time scale.
Abstract: A Wiley Company. The Whitham modulation theory for periodic traveling waves of PDEs generated by a Lagrangian produces first-order dispersionless PDEs that are, generically, either hyperbolic or elliptic. In this paper, degeneracy of the Whitham equations is considered where one of the characteristic speeds is zero. In this case, the Whitham equations are no longer valid. Reformulation and rescaling show that conservation of wave action morphs into the Korteweg-de Vries (KdV) equation on a longer time scale thereby generating dispersion in the Whitham modulation equations even for finite amplitude waves.
TL;DR: In this article, the authors derive a new representation for the incomplete gamma function, exploiting the reformulation of the method of steepest descents by Howls 1992, and obtain numerically computable bounds for the remainder term of the asymptotic expansion.
Abstract: In this paper, we derive a new representation for the incomplete gamma function, exploiting the reformulation of the method of steepest descents by Howls 1992. Using this representation, we obtain numerically computable bounds for the remainder term of the asymptotic expansion of the incomplete gamma function Γ−a,λa with large a and fixed positive λ, and an asymptotic expansion for its late coefficients. We also give a rigorous proof of Dingle's formal result regarding the exponentially improved version of the asymptotic series of Γ−a,λa.
TL;DR: In this article, the linear stability of the solitary waves for the one-dimensional Benney-Luke equation in the case of strong surface tension is investigated rigorously and the critical wave speeds are computed explicitly.
Abstract: The linear stability of the solitary waves for the one-dimensional Benney–Luke equation in the case of strong surface tension is investigated rigorously and the critical wave speeds are computed explicitly. For the Klein–Gordon equation, the stability of the traveling standing waves is considered and the exact ranges of the wave speeds and the frequencies needed for stability are derived. This is achieved via the abstract stability criteria recently developed by Stanislavova and Stefanov.
TL;DR: In this article, the authors present a two-dimensional configuration that represents the propagation of forced Rossby waves in a cyclonic vortex and use it to explore mechanisms by which critical layer interactions could contribute to the evolution of the secondary eyewall location.
Abstract: Vortex Rossby waves in cyclones in the tropical atmosphere are believed to play a role in the observed eyewall replacement cycle, a phenomenon in which concentric rings of intense rainbands develop outside the wall of the cyclone eye, strengthen and then contract inward to replace the original eyewall. In this paper, we present a two-dimensional configuration that represents the propagation of forced Rossby waves in a cyclonic vortex and use it to explore mechanisms by which critical layer interactions could contribute to the evolution of the secondary eyewall location. The equations studied include the nonlinear terms that describe wave-mean-flow interactions, as well as the terms arising from the latitudinal gradient of the Coriolis parameter. Asymptotic methods based on perturbation theory and weakly nonlinear analysis are used to obtain the solution as an expansion in powers of two small parameters that represent nonlinearity and the Coriolis effects. The asymptotic solutions obtained give us insight into the temporal evolution of the forced waves and their effects on the mean vortex. In particular, there is an inward displacement of the location of the critical radius with time which can be interpreted as part of the secondary eyewall cycle.
TL;DR: In this paper, a forced Benney-Luke (fBL) equation is derived from the Euler equations for the propagation of three-dimensional surface waves in water with an ambient current over a varying bathymetry.
Abstract: This paper studies the propagation of three-dimensional surface waves in water with an ambient current over a varying bathymetry. When the ambient flow is near the critical speed, under the shallow water assumptions, a forced Benney–Luke (fBL) equation is derived from the Euler equations. An asymptotic approximation of the water's reaction force over the varying bathymetry is derived in terms of topographic stress. Numerical simulations of the fBL equation over a trough are compared to those using a forced Kadomtsev–Petviashvilli equation. For larger variations in the bathymetry that upstream-radiating three-dimensional solitons are observed, which are different from the upstream-radiating solitons simulated by the forced Kadomtsev–Petviashvilli equation. In this case, we show the fBL equation is a singular perturbation of the forced Kadomtsev–Petviashvilli equation which explains the significant differences between the two flows.