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.     

Transverse Orbital Stability of Periodic Traveling Waves for Nonlinear Klein–Gordon Equations

In this paper, we establish the orbital stability of a class of spatially periodic wave train solutions to multidimensional nonlinear Klein–Gordon equations with periodic potential. We show that the orbit generated by the one‐dimensional wave train is stable under the flow of the multidimensional equation under perturbations which are, on one hand, coperiodic with respect to the translation or Galilean variable of propagation, and, on the other hand, periodic (but not necessarily coperiodic) with respect to the transverse directions. That is, we show their transverse orbital stability. The class of periodic wave trains under consideration is the family of subluminal rotational waves, which are periodic in the momentum but unbounded in their position.     

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.     

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.     

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.     

Spectrum created by line defects in periodic structures

We study a Helmholtz‐type spectral problem related to the propagation of electromagnetic waves in photonic crystal waveguides. The waveguide is created by introducing a linear defect into a three‐dimensional periodic medium; the defect is infinitely extended in one direction, but compactly supported in the remaining two. This perturbation introduces guided mode spectrum inside the band gaps of the fully periodic, unperturbed spectral problem. We will show that even small perturbations lead to additional spectrum in the spectral gaps of the unperturbed operator and investigate some properties of the spectrum that is created.     

Three-Dimensional Water Waves

We study the evolution of small-amplitude water waves when the fluid motion is three dimensional. An isotropic pseudodifferential equation that governs the evolution of the free surface of a fluid with arbitrary, uniform depth is derived. It is shown to reduce to the Benney-Luke equation, the Korteweg-de Vries (KdV) equation, the Kadomtsev-Petviashvili (KP) equation, and to the nonlinear shallow water theory in the appropriate limits. We compute, numerically, doubly periodic solutions to this equation. In the weakly two-dimensional long wave limit, the computed patterns and nonlinear dispersion relations agree well with those of the doubly periodic theta function solutions to the KP equation. These solutions correspond to traveling hexagonal wave patterns, and they have been compared with experimental measurements by Hammack, Scheffner, and Segur. In the fully two-dimensional long wave case, the solutions deviate considerably from those of KP, indicating the limitation of that equation. In the finite depth case, both resonant and nonresonant traveling wave patterns are obtained.     

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.     

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.     

The Stability Analysis of the Periodic Traveling Wave Solutions of the mKdV Equation

The stability of periodic solutions of partial differential equations has been an area of increasing interest in the last decade. In this paper, we derive all periodic traveling wave solutions of the focusing and defocusing mKdV equations. We show that in the defocusing case all such solutions are orbitally stable with respect to subharmonic perturbations: perturbations that are periodic with period equal to an integer multiple of the period of the underlying solution. We do this by explicitly computing the spectrum and the corresponding eigenfunctions associated with the linear stability problem. Next, we bring into play different members of the mKdV hierarchy. Combining this with the spectral stability results allows for the construction of a Lyapunov function for the periodic traveling waves. Using the seminal results of Grillakis, Shatah, and Strauss, we are able to conclude orbital stability. In the focusing case, we show how instabilities arise.     

On the resonant reflection of weak,nonlinear sound waves off an entropy wave

We analyze the resonant reflection of very weak, nonlinear sound waves off a weak sawtooth entropy wave for spatially periodic solutions of the one‐dimensional, nonisentropic gas dynamics equations. The case of an entropy wave with a sawtooth profile is of interest because the oscillations of the reflected sound waves are nondispersive with frequency independent of their wavenumber, leading to an unusual type of nonlinear dynamics. On an appropriate long time scale, we show that a complex amplitude function for the spatial profile of the sound waves satisfies a degenerate quasilinear Schrödinger equation. We present some numerical solutions of this equation that illustrate the generation of small spatial scales by a resonant four‐wave cascade and front propagation in compactly supported solutions.     

Blocking and invasion for reaction–diffusion equations in periodic media

We investigate the large time behavior of solutions of reaction–diffusion equations with general reaction terms in periodic media. We first derive some conditions which guarantee that solutions with compactly supported initial data invade the domain. In particular, we relate such solutions with front-like solutions such as pulsating traveling fronts. Next, we focus on the homogeneous bistable equation set in a domain with periodic holes, and specifically on the cases where fronts are not known to exist. We show how the geometry of the domain can block or allow invasion. We finally exhibit a periodic domain on which the propagation takes place in an asymmetric fashion, in the sense that the invasion occurs in a direction but is blocked in the opposite one.     

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.     

Stability of Front Solutions of the Bidomain Equation

The bidomain model is the standard model describing electrical activity of the heart. Here we study the stability of planar front solutions of the bidomain equation with a bistable nonlinearity (the bidomain Allen‐Cahn equation) in two spatial dimensions. In the bidomain Allen‐Cahn equation a Fourier multiplier operator whose symbol is a positive homogeneous rational function of degree two (the bidomain operator) takes the place of the Laplacian in the classical Allen‐Cahn equation. Stability of the planar front may depend on the direction of propagation given the anisotropic nature of the bidomain operator. We establish various criteria for stability and instability of the planar front in each direction of propagation. Our analysis reveals that planar fronts can be unstable in the bidomain Allen‐Cahn equation in striking contrast to the classical or anisotropic Allen‐Cahn equations. We identify two types of instabilities, one with respect to long‐wavelength perturbations, the other with respect to medium‐wavelength perturbations. Interestingly, whether the front is stable or unstable under long‐wavelength perturbations does not depend on the bistable nonlinearity and is fully determined by the convexity properties of a suitably defined Frank diagram. On the other hand, stability under intermediate‐wavelength perturbations does depend on the choice of bistable nonlinearity. Intermediate‐wavelength instabilities can occur even when the Frank diagram is convex, so long as the bidomain operator does not reduce to the Laplacian. We shall also give a remarkable example in which the planar front is unstable in all directions.© 2016 Wiley Periodicals, Inc.     

Instability of cnoidal‐peak for the NLS‐δ equation

We study the existence and stability of standing waves for the periodic cubic nonlinear Schrödinger equation with a point defect determined by the periodic Dirac distribution at the origin. We show that this model admits a smooth curve of periodic‐peak standing wave solutions with a profile determined by the Jacobi elliptic function of cnoidal type. Via a perturbation method and continuation argument, we obtain that in the repulsive defect, the cnoidal‐peak standing wave solutions are unstable in $H^1_{per}$ with respect to perturbations which have the same period as the wave itself. Global well‐posedness is verified for the Cauchy problem in $H^1_{per}$ .     

Stability of periodic traveling flexural‐gravity waves in two dimensions

In this work, we solve the Euler's equations for periodic waves traveling under a sheet of ice. These waves are referred to as flexural‐gravity waves. We compare and contrast two models for the effect of the ice: a linear model and a nonlinear model. The benefit of this reformulation is that it facilitates the asymptotic analysis. We use it to derive the nonlinear Schrödinger equation that describes the modulational instability of periodic traveling waves. We compare this asymptotic result with the numerical computation of stability using the Fourier–Floquet–Hill method to show they agree qualitatively. We show that different models have different stability regimes for large values of the flexural rigidity parameter. Numerical computations are also used to analyze high‐frequency instabilities in addition to the modulational instability. In the regions examined, these are shown to be the same regardless of the model representing ice.     

Moving gap solitons in periodic potentials

We address the existence of moving gap solitons (traveling localized solutions) in the Gross–Pitaevskii equation with a small periodic potential. Moving gap solitons are approximated by the explicit solutions of the coupled‐mode system. We show, however, that exponentially decaying traveling solutions of the Gross–Pitaevskii equation do not generally exist in the presence of a periodic potential due to bounded oscillatory tails ahead and behind the moving solitary waves. The oscillatory tails are not accounted in the coupled‐mode formalism and are estimated by using techniques of spatial dynamics and local center‐stable manifold reductions. Existence of bounded traveling solutions of the Gross–Pitaevskii equation with a single bump surrounded by oscillatory tails on a large interval of the spatial scale is proven by using these techniques. Copyright © 2008 John Wiley & Sons, Ltd.     

Rigorous Justification of the Whitham Modulation Equations for the Generalized Korteweg–de Vries Equation

In this paper, we consider the spectral stability of spatially periodic traveling wave solutions of the generalized Korteweg–de Vries equation to long‐wavelength perturbations. Specifically, we extend the work of Bronski and Johnson by demonstrating that the homogenized system describing the mean behavior of a slow modulation (WKB) approximation of the solution correctly describes the linearized dispersion relation near zero frequency of the linearized equations about the background periodic wave. The latter has been shown by rigorous Evans function techniques to control the spectral stability near the origin, that is, stability to slow modulations of the underlying solution. In particular, through our derivation of the WKB approximation we generalize the modulation expansion of Whitham for the KdV to a more general class of equations which admit periodic waves with nonzero mean. As a consequence, we will show that, assuming a particular nondegeneracy condition, spectral stability near the origin is equivalent with the local well‐posedness of the Whitham system.     

Spots and stripes in nonlinear Schrödinger‐type equations with nearly one‐dimensional potentials

We consider the existence of spots and stripes for a class of nonlinear Schrödinger‐type equations in the presence of nearly one‐dimensional localized potentials. Under suitable assumptions on the potential, we construct various types of waves that are localized in the direction of the potential and have single‐ or multihump, or periodic profile in the perpendicular direction. The analysis relies upon a spatial dynamics formulation of the existence problem, together with a center manifold reduction. This reduction procedure allows these waves to be realized as unipulse or multipulse homoclinic orbits, or periodic orbits in a reduced system of ordinary differential equations. Copyright © 2013 John Wiley & Sons, Ltd.     

Wave Properties of Double One-Dimensional Periodic Sheet Grating

We show that the double one-dimensional periodic sheet gratings always have waveguide properties for acoustic waves. In general, there are two types of pass bands: i.e., the connected sets of frequencies for which there exist harmonic acoustic traveling waves propagating in the direction of periodicity and localized in the neighborhood of the grating. Using numerical-analytical methods, we describe the dispersion relations for these waves, pass bands, and their dependence on the geometric parameters of the problem. The phenomenon is discovered of bifurcation of waveguide frequencies with respect to the parameter of the distance between the gratings that decreases from infinity. Some estimates are obtained for the parameters of frequency splitting or fusion in dependence on the distance between the simple blade gratings forming the double grating. We show that near a double sheet grating there always exist standing waves (in-phase oscillations in the neighboring fundamental cells of the group of translations) localized near the grating. By numerical-analytical methods, the dependences of the standing wave frequencies on the geometric parameters of the grating are determined. The mechanics is described of traveling and standing waves localized in the neighborhood of the double gratings.