Modulational instability in a full‐dispersion shallow water model

We propose a shallow water model that combines the dispersion relation of water waves and Boussinesq equations, and that extends the Whitham equation to permit bidirectional propagation. We show that its sufficiently small and periodic traveling wave is spectrally unstable to long wavelength perturbations if the wave number is greater than a critical value, like the Benjamin‐Feir instability of a Stokes wave. We verify that the associated linear operator possesses infinitely many collisions of purely imaginary eigenvalues, but they do not contribute to instability to the leading order in the amplitude parameter. We discuss the effects of surface tension. The results agree with those from a formal asymptotic expansion and a numerical computation for the physical problem.     

Transverse Spectral Stability of Small Periodic Traveling Waves for the KP Equation

The Kadomtsev–Petviashvili (KP) equation possesses a four‐parameter family of one‐dimensional periodic traveling waves. We study the spectral stability of the waves with small amplitude with respect to two‐dimensional perturbations which are either periodic in the direction of propagation, with the same period as the one‐dimensional traveling wave, or nonperiodic (localized or bounded). We focus on the so‐called KP‐I equation (positive dispersion case), for which we show that these periodic waves are unstable with respect to both types of perturbations. Finally, we briefly discuss the KP‐II equation, for which we show that these periodic waves are spectrally stable with respect to perturbations that are periodic in the direction of propagation, and have long wavelengths in the transverse direction.     

On the linear stability of simple and semi-simple periodic waves for a system of cubic Klein–Gordon equations

We study the linear stability of traveling wave solutions for the nonlinear wave equation and coupled nonlinear wave equations. It is shown that periodic waves of the dnoidal type are spectrally unstable with respect to co-periodic perturbations. Our arguments rely on a careful spectral analysis of various self-adjoint operators, both scalar and matrix and on instability index count theory for Hamiltonian systems.     

A Numerical Study of the Stability of Solitary Waves of the Bona–Smith Family of Boussinesq Systems

In this paper we study, from a numerical point of view, some aspects of stability of solitary-wave solutions of the Bona–Smith systems of equations. These systems are a family of Boussinesq-type equations and were originally proposed for modelling the two-way propagation of one-dimensional long waves of small amplitude in an open channel of water of constant depth. We study numerically the behavior of solitary waves of these systems under small and large perturbations with the aim of illuminating their long-time asymptotic stability properties and, in the case of large perturbations, examining, among other, phenomena of possible blow-up of the perturbed solutions in finite time.      

Transverse Instability of Line Solitary Waves in Massive Dirac Equations

Working in the context of localized modes in periodic potentials, we consider two systems of the massive Dirac equations in two spatial dimensions. The first system, a generalized massive Thirring model, is derived for the periodic stripe potentials. The second one, a generalized massive Gross–Neveu equation, is derived for the hexagonal potentials. In both cases, we prove analytically that the line solitary waves are spectrally unstable with respect to periodic transverse perturbations of large periods. The spectral instability is induced by the spatial translation for the generalized massive Thirring model and by the gauge rotation for the generalized massive Gross–Neveu model. We also observe numerically that the spectral instability holds for the transverse perturbations of any period in the generalized massive Thirring model and exhibits a finite threshold on the period of the transverse perturbations in the generalized massive Gross–Neveu model.     

Non-linear waves on shallow water under an ice cover. Higher order expansions

Non-linear wave processes on the surface of shallow water under a layer of ice are considered taking bending deformations and tension compression into account. A closed system of equations in the water level perturbations and the velocity potential is derived to describe them. From the consistancy conditions for this system, using the method of multiple scales and perturbation theory, a ninth-order non-linear evolution equation is obtained for describing the perturbations of the water level, taking into account higher order corrections in the small parameters. A periodic solution of the equation obtained is constructed, expressed in terms of Weierstrass elliptic functions. Solutions are obtained in the form of solitary waves, expressed in terms of hyperbolic functions, using a modification of the simplest equations method. It is shown that, for periodic and solitary waves, two forms of wave profiles exist depending on the parameters of the mathematical model.     

Spectrally Stable Encapsulated Vortices for Nonlinear Schr?dinger Equations

Summary. A large class of multidimensional nonlinear Schrodinger equations admit localized nonradial standing-wave solutions that carry nonzero intrinsic angular momentum. Here we provide evidence that certain of these spinning excitations are spectrally stable. We find such waves for equations in two space dimensions with focusing-defocusing nonlinearities, such as cubic-quintic. Spectrally stable waves resemble a vortex (nonlocalized solution with asymptotically constant amplitude) cut off at large radius by a kink layer that exponentially localizes the solution. For the evolution equations linearized about a localized spinning wave, we prove that unstable eigenvalues are zeroes of Evans functions for a finite set of ordinary differential equations. Numerical computations indicate that there exist spectrally stable standing waves having central vortex of any degree.     

Envelope Solitary Waves and Periodic Waves in the AB Equations

This article deals with the envelope solitary waves and periodic waves in the AB equations that serve as model equations describing marginally unstable baroclinic wave packets in geophysical fluids and also ultra‐short pulses in nonlinear optics. An envelope solitary wave has a width proportional to its velocity and inversely proportional to its amplitude. The velocity of the envelope solitary wave is partially dependent on its amplitude in the sense that the amplitude determines the upper or lower limit of the velocity. When two envelope solitary waves collide, they survive the collision and retain their identities except for a shift in the positions of both the envelopes and the carrier waves. The periodic wave solutions in sine wave form may be stable or unstable depending upon the wave parameters. When the sine wave is destabilized by small perturbations, its long‐time evolution shows a Fermi–Pasta–Ulam‐type oscillation.     

Linear Stability of Viscous Roll Waves

The purpose of this paper is to study the linear stability of “viscous” roll waves. These are periodic continuous traveling waves solutions of viscous perturbations of inhomogeneous hyperbolic systems. We first study the scalar case for the Burgers equation and for an inhomogeneous hyperbolic equation. Then we analyze the stability of roll waves, solutions of the shallow water equations with a real viscosity. In both cases, we first analyze the Evans function and compute an asymptotic expansion in the low frequency regime. Under a strong spectral stability condition, we prove the linear stability of viscous roll waves, solutions of the Saint Venant equations, with pointwise estimates on the Green functions.     

Can Parity‐Time‐Symmetric Potentials Support Families of Non‐Parity‐Time‐Symmetric Solitons?

For the one‐dimensional nonlinear Schrödinger equations with parity‐time (PT) symmetric potentials, it is shown that when a real symmetric potential is perturbed by weak PT‐symmetric perturbations, continuous families of asymmetric solitary waves in the real potential are destroyed. It is also shown that in the same model with a general PT‐symmetric potential, symmetry breaking of PT‐symmetric solitary waves does not occur. Based on these findings, it is conjectured that one‐dimensional PT‐symmetric potentials cannot support continuous families of non‐PT‐symmetric solitary waves.     

Modulational Instability and Variational Structure

We study the modulational instability of periodic traveling waves for a class of Hamiltonian systems in one spatial dimension. We examine how the Jordan block structure of the associated linearized operator bifurcates for small values of the Floquet exponent to derive a criterion governing instability to long wavelengths perturbations in terms of the kinetic and potential energies, the momentum, the mass of the underlying wave, and their derivatives. The dispersion operator of the equation is allowed to be nonlocal, for which Evans function techniques may not be applicable. We illustrate the results by discussing analytically and numerically equations of Korteweg‐de Vries type.     

A spectrally accurate algorithm for electromagnetic scattering in three dimensions

In this work we develop, implement and analyze a high-order spectrally accurate algorithm for computation of the echo area, and monostatic and bistatic radar cross-section (RCS) of a three dimensional perfectly conducting obstacle through simulation of the time-harmonic electromagnetic waves scattered by the conductor. Our scheme is based on a modified boundary integral formulation (of the Maxwell equations) that is tolerant to basis functions that are not tangential on the conductor surface. We test our algorithm with extensive computational experiments using a variety of three dimensional perfect conductors described in spherical coordinates, including benchmark radar targets such as the metallic NASA almond and ogive. The monostatic RCS measurements for non-convex conductors require hundreds of incident waves (boundary conditions). We demonstrate that the monostatic RCS of small (to medium) sized conductors can be computed using over one thousand incident waves within a few minutes (to a few hours) of CPU time. We compare our results with those obtained using method of moments based industrial standard three dimensional electromagnetic codes CARLOS, CICERO, FE-IE, FERM, and FISC. Finally, we prove the spectrally accurate convergence of our algorithm for computing the surface current, far-field, and RCS values of a class of conductors described globally in spherical coordinates.     

The Modulated Phase Shift for Weakly Dissipated Nonlinear Oscillatory Waves of the Korteweg-deVries Type

Nonlinear dispersive oscillatory waves are analyzed for Korteweg-deVries type partial differential equations with slowly varying coefficients and arbitrary small perturbations. Spatial and temporal evolution of the amplitude parameters are determined in the usual way by the possible dissipation of the wave actions for both momentum and energy. For dissipative perturbations, both wave actions are shown to be valid to a higher order. Thus, the first variation of the wave action equations is used to derive equations for the slow modulations of the phase shift. It is shown that the phase shift satisfies a set of two coupled linear equations.     

Pointwise stability estimates for periodic traveling wave solutions of systems of viscous conservation laws

In the previous paper [9], we showed time asymptotic behavior with detailed decaying rates of perturbations of periodic traveling reaction–diffusion waves under small initial perturbations with a Gaussian rate and an algebraic rate. Here, we establish pointwise nonlinear stability up to an appropriate modulation of periodic traveling waves of systems of viscous conservation laws under small algebraic decaying initial data. Similar to the reaction–diffusion equations, by using Bloch decomposition, we start with pointwise bounds on the Green function of the linearized operator about underlying solutions.     

Stability of peakons for the Degasperis‐Procesi equation

The Degasperis‐Procesi equation can be derived as a member of a one‐parameter family of asymptotic shallow‐water approximations to the Euler equations with the same asymptotic accuracy as that of the Camassa‐Holm equation. In this paper, we study the orbital stability problem of the peaked solitons to the Degasperis‐Procesi equation on the line. By constructing a Lyapunov function, we prove that the shapes of these peakon solitons are stable under small perturbations. © 2007 Wiley Periodicals, Inc.     

Numerical Bifurcation and Spectral Stability of Wavetrains in Bidirectional Whitham Models

We consider several different bidirectional Whitham equations that have recently appeared in the literature. Each of these models combines the full two‐way dispersion relation from the incompressible Euler equations with a canonical shallow water nonlinearity, providing nonlocal model equations that may be expected to exhibit some of the interesting high‐frequency phenomena present in the Euler equations that standard “long‐wave” theories fail to capture. Of particular interest here is the existence and stability of periodic traveling wave solutions in such models. Using numerical bifurcation techniques, we construct global bifurcation diagrams for each system and compare the global structure of branches, together with the possibility of bifurcation branches terminating in a “highest” singular (peaked/cusped) wave. We also numerically approximate the stability spectrum along these bifurcation branches and compare the stability predictions of these models. Our results confirm a number of analytical results concerning the stability of asymptotically small waves in these models and provide new insights into the existence and stability of large amplitude waves.     

Interactions of delta shock waves for the relativistic Chaplygin Euler equations with split delta functions

In this article, we are concerned with the interactions of delta shock waves with contact discontinuities for the relativistic Euler equations for Chaplygin gas by using split delta functions method. The solutions are obtained constructively and globally when the initial data consists of three piecewise constant states. The global structure and large time‐asymptotic behaviors of the solutions are analyzed case by case. During the process of the interaction, the strengths of delta shock waves are computed completely. Moreover, it can be found that the Riemann solutions are stable for such small perturbations with special initial data by letting perturbed parameter ε tends to zero. Copyright © 2014 John Wiley & Sons, Ltd.     

Nonlinear Hamiltonian waves with constant frequency and surface waves on vorticity discontinuities

Waves with constant, nonzero linearized frequency form an interesting class of nondispersive waves whose properties differ from those of nondispersive hyperbolic waves. We propose an inviscid Burgers‐Hilbert equation as a model equation for such waves and give a dimensional argument to show that it models Hamiltonian surface waves with constant frequency. Using the method of multiple scales, we derive a cubically nonlinear, quasi‐linear, nonlocal asymptotic equation for weakly nonlinear solutions. We show that the same asymptotic equation describes surface waves on a planar discontinuity in vorticity in two‐dimensional inviscid, incompressible fluid flows. Thus, the Burgers‐Hilbert equation provides an effective equation for these waves. We describe the Hamiltonian structure of the Burgers‐Hilbert and asymptotic equations, and show that the asymptotic equation can also be derived by means of a near‐identity transformation. We derive a semiclassical approximation of the asymptotic equation and show that spatially periodic, harmonic traveling waves are linearly and modulationally stable. Numerical solutions of the Burgers‐Hilbert and asymptotic equations are in excellent agreement in the appropriate regime. In particular, the lifespan of small‐amplitude smooth solutions of the Burgers‐Hilbert equation is given by the cubically nonlinear timescale predicted by the asymptotic equation. © 2009 Wiley Periodicals, Inc.     

Birkhoff Normal Form and Long Time Existence for Periodic Gravity Water Waves

We consider the gravity water waves system with a periodic one-dimensional interface in infinite depth and give a rigorous proof of a conjecture of Dyachenko-Zakharov [16] concerning the approximate integrability of these equations. More precisely, we prove a rigorous reduction of the water waves equations to its integrable Birkhoff normal form up to order 4. As a consequence, we also obtain a long-time stability result: periodic perturbations of a flat interface that are initially of size ε remain regular and small up to times of order ${\epsilon }^{-3}$. This time scale is expected to be optimal. © 2022 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.     

Global Existence for Capillary Water Waves

Consider the capillary water waves equations, set in the whole space with infinite depth, and consider small data (i.e., sufficiently close to zero velocity, and constant height of the water). We prove global existence and scattering. The proof combines in a novel way the energy method with a cascade of energy estimates, the space‐time resonance method and commuting vector fields. © 2015 Wiley Periodicals, Inc.