Infinite-dimensional Evans function theory for elliptic eigenvalue problems in a channel

An infinite-dimensional Evans function theory is developed for the elliptic eigenvalue problem associated with the stability of travelling solitary waves in a channel. Also, a bundle is constructed over the complex domain, so that its first Chern number gives the number of eigenvalues inside the domain.     

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.     

The eigenvalue problem for solitary waves of a boussinesq equation,via a generalization of the rouché theorem

We consider the study of an eigenvalue problem obtained by linearizing about solitary wave solutions of a Boussinesq equation. Instead of using the technique of Evans functions as done by Pego and Weinstein in [R. Pego and M. Weinstein, Convective Linear Stability of Solitary Waves for Boussinesq equation. AMS, 99, 311–375] for this particular problem, we perform Fourier analysis to characterize solutions of the eigenvalue problem in terms of a multiplier operator and use the strong relationship between the eigenvalue problem for the linearized Boussinesq equation and the eigenvalue problem associated with the linearization about solitary wave solutions of a special form of the KdV equation. By using a generalization of the Rouché Theorem and the asymptotic behavior of the Fourier symbol corresponding to the eigenvalues problem for the Boussinesq equation and the Fourier symbol corresponding to the eigenvalues problem for the KdV equation, we show nonexistence of eigenvalues with respect to weighted space in a planar region containing the right-half plane.     

Convective Linear Stability of Solitary Waves for Boussinesq Equations

Boussinesq was the first to explain the existence of Scott Russell's solitary wave mathematically. He employed a variety of asymptotically equivalent equations to describe water waves in the small-amplitude, long-wave regime. We study the linearized stability of solitary waves for three linearly well-posed Boussinesq models. These are problems for which well-developed Lyapunov methods of stability analysis appear to fail. However, we are able to analyze the eigenvalue problem for small-amplitude solitary waves, by comparison to the equation that Boussinesq himself used to describe the solitary wave, which is now called the Korteweg–de Vries equation. With respect to a weighted norm designed to diminish as perturbations convect away from the wave profile, we prove that nonzero eigenvalues are absent in a half-plane of the form R λ>− b for some b >0, for all three Boussinesq models. This result is used to prove the decay of solutions of the evolution equations linearized about the solitary wave, in two of the models. This "convective linear stability" property has played a central role in the proof of nonlinear asymptotic stability of solitary-wave-like solutions in other systems.     

The gap lemma and geometric criteria for instability of viscous shock profiles

An obstacle in the use of Evans function theory for stability analysis of traveling waves occurs when the spectrum of the linearized operator about the wave accumulates at the imaginary axis, since the Evans function has in general been constructed only away from the essential spectrum. A notable case in which this difficulty occurs is in the stability analysis of viscous shock profiles. Here we prove a general theorem, the “gap lemma,” concerning the analytic continuation of the Evans function associated with the point spectrum of a traveling wave into the essential spectrum of the wave. This allows geometric stability theory to be applied in many cases where it could not be applied previously. We demonstrate the power of this method by analyzing the stability of certain undercompressive viscous shock waves. A necessary geometric condition for stability is determined in terms of the sign of a certain Melnikov integral of the associated viscous profile. This sign can easily be evaluated numerically. We also compute it analytically for solutions of several important classes of systems. In particular, we show for a wide class of systems that homoclinic (solitary) waves are linearly unstable, confirming these as the first known examples of unstable viscous shock waves. We also show that (strong) heteroclinic undercompressive waves are sometimes unstable. Similar stability conditions are also derived for Lax and overcompressive shocks and for n × n conservation laws, n ≥ 2. © 1998 John Wiley & Sons, Inc.     

Erratum to Laplacian Instability of Planar Streamer Ionization Fronts—An Example of Pulled Front Analysis

Streamer ionization fronts are pulled fronts that propagate into a linearly unstable state; the spatial decay of the initial condition of a planar front selects dynamically one specific long-time attractor out of a continuous family. A stability analysis for perturbations in the transverse direction has to take these features into account. In this paper we show how to apply the Evans function in a weighted space for this stability analysis. Zeros of the Evans function indicate the intersection of the stable and unstable manifolds; they are used to determine the eigenvalues. Within this Evans function framework, we define a numerical dynamical systems method for the calculation of the dispersion relation as an eigenvalue problem. We also derive dispersion curves for different values of the electron diffusion constant and of the electric field ahead of the front. Numerical solutions of the initial value problem confirm the eigenvalue calculations. The numerical work is complemented with an analysis of the Evans function leading to analytical expressions for the dispersion relation in the limit of small and large wave numbers. The paper concludes with a fit formula for intermediate wave numbers. This empirical fit supports the conjecture that the smallest unstable wave length of the Laplacian instability is proportional to the diffusion length that characterizes the leading edge of the pulled ionization front. G. Derks acknowledges a travel grant of the Royal Society, which initiated this research, and a visitor grant of the Dutch funding agency NWO and the NWO-mathematics cluster NDNS⁺ to finish the work. The work was also supported by a CWI PhD grant for B. Meulenbroek.     

Traveling solitary waves for doubly-resonant media: Computation via simulated annealing

Traveling wave solutions of the Maxwell–Duffing model of doubly-resonant media are studied numerically. The corresponding nonlinear eigenvalue problem is solved via simulated annealing. The model is shown to possess a discrete spectrum of velocities for solitary waves.     

Evans functions for periodic waves on infinite cylindrical domains

Using Galerkin approximations, an Evans function for spatially periodic waves on infinite cylindrical domains is constructed. It is also shown that the Evans function can be used to define a parity index for periodic waves that detects whether the wave admits an odd number of real unstable eigenvalues. This parity index depends only on local information for the existence problem of the wave: in particular, it uses information about the linear dispersion relation near zero and the orientability of the unstable and stable manifolds along the nonlinear wave. The results are applied to small-amplitude wave trains for a scalar equation on an infinite strip.     

Spectral transverse instability of solitary waves in Korteweg fluids

The motion of Korteweg fluids is governed by the Euler-Korteweg model, which admits planar solitary waves for nonmonotone pressure laws such as the van der Waals law below critical temperature. In an earlier work with Danchin, Descombes and Jamet, it was shown by variational arguments and numerical computations that some of these solitary waves are stable in one space dimension. The purpose here is to study their stability with respect to transverse perturbations in several space dimensions. By Evans functions techniques and Rouché's theorem, it is shown that transverse perturbations of large wave length always destabilize solitary waves in the Euler-Korteweg model, whereas energy estimates show that perturbations of short wave length tend to stabilize them.     

Instability of solitons under flexure and twist of an elastic rod

We study the stability of planar soliton solutions of equations describing the dynamics of an infinite inextensible unshearable rod under three-dimensional spatial perturbations. As a result of linearization about the soliton solution, we obtain an inhomogeneous scalar equation. This equation leads to a generalized eigenvalue problem. To establish the instability, we must verify the existence of an unstable eigenvalue (an eigenvalue with a positive real part). The corresponding proof of the instability is done using a local construction of the Evans function depending only on the spectral parameter. This function is analytic in the right half of the complex plane and has at least one zero on the positive real axis coinciding with an unstable eigenvalue of the generalized spectral problem.     

On the Computation of Nonlinear Planetary Waves

A numerical method is developed to solve a class of nonlinear, nonlocal eigenvalue problems defined in an infinite strip, and is applied to compute solitary planetary waves in a sheared zonal current on the beta-plane. This method, an iterative procedure derived from the natural variational structure of these problems, is implemented in the physical case when the ambient parallel flow has a linear or a quadratic velocity profile. The results of the numerical experiments establish rigorous limits on the range of validity of the formal asymptotic theory of weakly nonlinear long waves, and also reveal some new phenomena involving strongly nonlinear waves. The iterative procedure is analyzed in a general setting, and is shown to be globally convergent without restriction on the wave amplitude.     

EVANS FUNCTIONS AND ASYMPTOTIC STABILITY OF TRAVELINGWAVE SOLUTIONS

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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.     

Stability of smooth solitary waves for the generalized Korteweg–de-Vries equation with combined dispersion

The problem of orbital stability of smooth solitary waves in the generalized Korteweg–de-Vries equation with combined dispersion is considered. The results show that the smooth solitary waves are stable for any speed of wave propagation.     

Instability of solitary waves on Euler’s elastica

Stability of solitary waves in a thin inextensible and unshearable rod of infinite length is studied. Solitary-wave profile of the elastica of such a rod without torsion has the form of a planar loop and its speed depends on a tension in the rod. The linear instability of a solitary-wave profile subject to perturbations escaping from the plane of the loop is established for a certain range of solitary-wave speeds. It is done using the properties of the Evans function, an analytic function on the right complex half-plane, that has zeros if and only if there exist the unstable modes of the linearization around a solitary-wave solution. The result follows from comparison of the behaviour of the Evans function in some neighbourhood of the origin with its asymptotic at infinity. The explicit computation of the leading coefficient of the Taylor series of the Evans function near the origin is performed by means of the symbolic computer language. Received: April 6, 2004; revised: December 12, 2004     

Three-dimensional gravity waves in a channel of variable depth

We consider existence of three-dimensional gravity waves traveling along a channel of variable depth. It is well known that the long-wave small-amplitude expansion for such waves results in the stationary Korteweg–de Vries equation, coefficients of which depend on the transverse topography of the channel. This equation has a single-humped solitary wave localized in the direction of the wave propagation. We show, however, that there exists an infinite set of resonant Fourier modes that travel at the same speed as the solitary wave does. This fact suggests that the solitary wave confined in a channel of variable depth is always surrounded by small-amplitude oscillatory disturbances in the far-field profile.     

Solitary wave solutions of the high-order KdV models for bi-directional water waves

An approach, which allows us to construct specific closed-form solitary wave solutions for the KdV-like water-wave models obtained through the Boussinesq perturbation expansion for the two-dimensional water wave problem in the limit of long wavelength/small amplitude waves, is developed. The models are relevant to the case of the bi-directional waves with the amplitude of the left-moving wave of O(ϵ) (ϵ is the amplitude parameter) as compared with that of the right-moving wave. We show that, in such a case, the Boussinesq system can be decomposed into a system of coupled equations for the right- and left-moving waves in which, to any order of the expansion, one of the equations is dependent only on the (main) right-wave elevation and takes the form of the high-order KdV equation with arbitrary coefficients whereas the second equation includes both elevations. Then the explicit solitary wave solutions constructed via our approach may be treated as the exact solutions of the infinite-order perturbed KdV equations for the right-moving wave with the properly specified high-order coefficients. Such solutions include, in a sense, contributions of all orders of the asymptotic expansion and therefore may be considered to a certain degree as modelling the solutions of the original water wave problem under proper initial conditions. Those solitary waves, although stemming from the KdV solitary waves, possess features found neither in the KdV solitons nor in the solutions of the first order perturbed KdV equations.     

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.     

Existence,orbital stability and instability of solitary waves for coupled BBM equations

This paper is concerned with the orbital stability/instability of solitary waves for coupled BBM equations which have Hamiltonian form. The explicit solitary wave solutions will be worked out first. Then by detailed spectral analysis and decaying estimates of solutions for the initial value problem, we obtain the orbital stability/instability of solitary waves.