The Influence of Modulational Instability on Energy Exchange in Coupled Sine-Gordon Equations

We consider a two-component system of coupled sine-Gordon equations, particular solutions of which represent a continuum generalization of periodic energy exchange in a system of coupled pendulums. Weakly nonlinear solutions describing periodic energy exchange between waves traveling in the two components are governed, depending on the length scale of the amplitude variation, either by two nonlocally coupled nonlinear Schrödinger equations, with different transport terms due to the group velocity, or by a model that is nondispersive to the leading order. Using both asymptotic analysis and numerical simulations, we show that the effects of dispersion significantly influence the structure of these solutions, causing modulational instability and the formation of localized structures but preserving the pattern of energy exchange between the components.     

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.     

Modulation theory solution for nonlinearly resonant,fifth‐order Korteweg–de Vries,nonclassical, traveling dispersive shock waves

A new class of resonant dispersive shock waves was recently identified as solutions of the Kawahara equation— a Korteweg–de Vries (KdV) type nonlinear wave equation with third‐ and fifth‐order spatial derivatives— in the regime of nonconvex, linear dispersion. Linear resonance resulting from the third‐ and fifth‐order terms in the Kawahara equation was identified as the key ingredient for nonclassical dispersive shock wave solutions. Here, nonlinear wave (Whitham) modulation theory is used to construct approximate nonclassical traveling dispersive shock wave (TDSW) solutions of the fifth‐ order KdV equation without the third derivative term, hence without any linear resonance. A self‐similar, simple wave modulation solution of the fifth order, weakly nonlinear KdV–Whitham equations is obtained that matches a constant to a heteroclinic traveling wave via a partial dispersive shock wave so that the TDSW is interpreted as a nonlinear resonance. The modulation solution is compared with full numerical solutions, exhibiting excellent agreement. The TDSW is shown to be modulationally stable in the presence of sufficiently small third‐order dispersion. The Kawahara–Whitham modulation equations transition from hyperbolic to elliptic type for sufficiently large third‐order dispersion, which provides a possible route for the TDSW to exhibit modulational instability.     

Three Dimensional Flexural–Gravity Waves

Waves propagating on the surface of a three–dimensional ideal fluid of arbitrary depth bounded above by an elastic sheet that resists flexing are considered in the small amplitude modulational asymptotic limit. A Benney–Roskes–Davey–Stewartson model is derived, and we find that fully localized wavepacket solitary waves (or lumps) may bifurcate from the trivial state at the minimum of the phase speed of the problem for a range of depths. Results using a linear and two nonlinear elastic models are compared. The stability of these solitary wave solutions and the application of the BRDS equation to unsteady wave packets is also considered. The results presented may have applications to the dynamics of continuous ice sheets and their breakup.     

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.     

The Transverse Instability of Periodic Waves in Zakharov–Kuznetsov Type Equations

In this paper, we investigate the instability of one‐dimensionally stable periodic traveling wave solutions of the generalized Korteweg‐de Vries equation to long wavelength transverse perturbations in the generalized Zakharov–Kuznetsov equation in two space dimensions. By deriving appropriate asymptotic expansions of the periodic Evans function, we derive an index which yields sufficient conditions for transverse instabilities to occur. This index is geometric in nature, and applies to any periodic traveling wave profile under some minor smoothness assumptions on the nonlinearity. We also describe the analogous theory for periodic traveling waves of the generalized Benjamin–Bona–Mahony equation to long wavelength transverse perturbations in the gBBM–Zakharov–Kuznetsov equation.     

Modulational Instability in the Whitham Equation for Water Waves

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.     

Slow Modulations of Periodic Waves in Hamiltonian PDEs,with Application to Capillary Fluids

Since its elaboration by Whitham almost 50 years ago, modulation theory has been known to be closely related to the stability of periodic traveling waves. However, it is only recently that this relationship has been elucidated and that fully nonlinear results have been obtained. These only concern dissipative systems though: reaction–diffusion systems were first considered by Doelman et al. (Mem Am Math Soc 199(934):viii+105, 2009), and viscous systems of conservation laws have been addressed by Johnson et al. (Invent Math, 2013). Here, only nondissipative models are considered, and a most basic question is investigated, namely, the expected link between the hyperbolicity of modulated equations and the spectral stability of periodic traveling waves to sideband perturbations. This is done first in an abstract Hamiltonian framework, which encompasses a number of dispersive models, in particular the well-known (generalized) Korteweg–de Vries equation and the less known Euler–Korteweg system, in both Eulerian coordinates and Lagrangian coordinates. The latter is itself an abstract framework for several models arising in water wave theory, superfluidity, and quantum hydrodynamics. As regards its application to compressible capillary fluids, attention is paid here to untangle the interplay between traveling waves/modulation equations in Eulerian coordinates and those in Lagrangian coordinates. In the most general setting, it is proved that the hyperbolicity of modulated equations is indeed necessary for the spectral stability of periodic traveling waves. This extends earlier results by Serre (Commun Partial Differ Equ 30(1–3):259–282, 2005), Oh and Zumbrun (Arch Ration Mech Anal 166(2):99–166, 2003), and Johnson et al. (Phys D 239(23–24):2057–2065, 2010). In addition, reduced necessary conditions are obtained in the small-amplitude limit. Then numerical investigations are carried out for the modulated equations of the Euler–Korteweg system with two types of “pressure” laws, namely, the quadratic law of shallow-water equations and the nonmonotone van der Waals pressure law. Both the evolutionarity and the hyperbolicity of the modulated equations are tested, and regions of modulational instability are thus exhibited.     

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.     

Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction-diffusion equations

Using spatial domain techniques developed by the authors and Myunghyun Oh in the context of parabolic conservation laws, we establish under a natural set of spectral stability conditions nonlinear asymptotic stability with decay at Gaussian rate of spatially periodic traveling waves of systems of reaction-diffusion equations. In the case that wave-speed is identically zero for all periodic solutions, we recover and slightly sharpen a well-known result of Schneider obtained by renormalization/Bloch transform techniques; by the same arguments, we are able to treat the open case of nonzero wave-speeds to which Schneider?s renormalization techniques do not appear to apply.     

Transverse nonlinear instability for two-dimensional dispersive models

We present a method to prove nonlinear instability of solitary waves in dispersive models. Two examples are analyzed: we prove the nonlinear long time instability of the KdV solitary wave (with respect to periodic transverse perturbations) under a KP-I flow and the transverse nonlinear instability of solitary waves for the cubic nonlinear Schrödinger equation.     

Transverse spectral instabilities in Konopelchenko–Dubrovsky equation

We study the transverse spectral stability of the one-dimensional small-amplitude periodic traveling wave solutions of the (2+1)-dimensional Konopelchenko–Dubrovsky (KD) equation. We show that these waves are transversely unstable with respect to two-dimensional perturbations that are periodic in both directions with long wavelength in the transverse direction. We also show that these waves are transversely stable with respect to perturbations which are either mean-zero periodic or square-integrable in the direction of the propagation of the wave and periodic in the transverse direction with finite or short wavelength. We discuss the implications of these results for special cases of the KD equation—namely, KP-II and mKP-II equations.     

Stability of small periodic waves for the nonlinear Schrödinger equation

The nonlinear Schrödinger equation possesses three distinct six-parameter families of complex-valued quasiperiodic traveling waves, one in the defocusing case and two in the focusing case. All these solutions have the property that their modulus is a periodic function of x−ct for some c∈R. In this paper we investigate the stability of the small amplitude traveling waves, both in the defocusing and the focusing case. Our first result shows that these waves are orbitally stable within the class of solutions which have the same period and the same Floquet exponent as the original wave. Next, we consider general bounded perturbations and focus on spectral stability. We show that the small amplitude traveling waves are stable in the defocusing case, but unstable in the focusing case. The instability is of side-band type, and therefore cannot be detected in the periodic set-up used for the analysis of orbital stability.     

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.     

Stability of periodic Kuramoto–Sivashinsky waves

In this note, we announce a general result resolving the long-standing question of nonlinear modulational stability, or stability with respect to localized perturbations, of periodic traveling-wave solutions of the generalized Kuramoto–Sivashinsky equation, establishing that spectral modulational stability, defined in the standard way, implies nonlinear modulational stability with sharp rates of decay. The approach extends readily to other second- and higher-order parabolic equations, for example, the Cahn Hilliard equation or more general thin film models.     

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.     

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.     

Spatiotemporal propagation of a time-periodic reaction–diffusion SI epidemic model with treatment

This work is concerned with the spatiotemporal propagation phenomena for a time-periodic reaction-diffusion susceptible-infectious (SI) epidemic model with treatment in terms of the asymptotic speed of spread and periodic traveling waves. First, the asymptotic speed of spread $c^{\ast }$ is characterized and the spreading properties of the model are analyzed by combining the periodic principal eigenvalue problem, comparison method, and the uniform persistence idea for a dynamical system. Second, by constructing suitable super- and subsolutions for truncation problems corresponding to the traveling wave system, the existence of periodic traveling waves is established via the fixed point theorem twice. It turned out that the asymptotic speed of spread coincides with the minimum wave speed of periodic traveling waves. Finally, via numerical simulation, the effects of some important parameters (such as diffusion rate, treatment rate, etc.) on the spreading speed are discussed, and the asymptotic properties of the periodic traveling waves are explored.     

Whitham modulation theory for the Zakharov–Kuznetsov equation and stability analysis of its periodic traveling wave solutions

We derive the Whitham modulation equations for the Zakharov–Kuznetsov equation via a multiple scales expansion and averaging two conservation laws over one oscillation period of its periodic traveling wave solutions. We then use the Whitham modulation equations to study the transverse stability of the periodic traveling wave solutions. We find that all periodic solutions traveling along the first spatial coordinate are linearly unstable with respect to purely transversal perturbations, and we obtain an explicit expression for the growth rate of perturbations in the long wave limit. We validate these predictions by linearizing the equation around its periodic solutions and solving the resulting eigenvalue problem numerically. We also calculate the growth rate of the solitary waves analytically. The predictions of Whitham modulation theory are in excellent agreement with both of these approaches. Finally, we generalize the stability analysis to periodic waves traveling in arbitrary directions and to perturbations that are not purely transversal, and we determine the resulting domains of stability and instability.     

Instability of nonlinear dispersive solitary waves

We consider linear instability of solitary waves of several classes of dispersive long wave models. They include generalizations of KDV, BBM, regularized Boussinesq equations, with general dispersive operators and nonlinear terms. We obtain criteria for the existence of exponentially growing solutions to the linearized problem. The novelty is that we dealt with models with nonlocal dispersive terms, for which the spectra problem is out of reach by the Evans function technique. For the proof, we reduce the linearized problem to study a family of nonlocal operators, which are closely related to properties of solitary waves. A continuation argument with a moving kernel formula is used to find the instability criteria. These techniques have also been extended to study instability of periodic waves and of the full water wave problem.