arXiv subjects

Mathew A. Johnson

Publications and source records attributed to Mathew A. Johnson.

At least 19 recordsLinked to original sources

arXiv2026

Linear Asymptotic Stability of the Smooth 1-Solitons for the Degasperis-Procesi Equation

In this paper, we study the asymptotic stability of smooth 1-solitons in the Degasperis-Procesi (DP) equation. Such solutions necessarily exist on a non-zero background, and their spectral and orbital stability has previously been verified by Li, Liu & Wu and by Lafortune & Pelinovsky. Using the complete integrability of the DP equation to establish the strong spectral stability of smooth solitary waves, namely that the origin is the only eigenvalue of the associated linearized operator acting on $L^2(\mathbb{R})$ and that, moreover, in appropriate exponentially weighted spaces the non-zero spectrum for the linearized operator admits a spectral gap away from the imaginary axis. This spectral gap result {{is then}} upgraded to an exponential decay estimate on the semigroup associated with the linearized operator, establishing a linear asymptotic stability result in exponentially weighted spaces. Finally, we outline analytical challenges with extending our result to the nonlinear level.

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arXiv2025

On the Modulation of Wave Trains in the Ostrovsky Equation

We consider the nonlinear wave modulation of arbitrary amplitude periodic traveling wave solutions of the Ostrovsky equation, which arises as a model for the unidirectional propagation of small-amplitude, weakly nonlinear surface and internal gravity waves in a rotating fluid of finite depth. While the modulation of such waves with asymptotically small amplitudes of oscillation (the so-called Stokes waves) has been studied in several works, our goal is to understand the modulational dynamics of general amplitude wave trains. To this end, we first use Whitham's theory of modulations to derive a dispersionless system of quasilinear partial differential equations that is expected to model the slow evolution of the fundamental characteristics of a given wave train. In practice, the modulational stability or instability of a given wave train is considered to be determined by the hyperbolicity or ellipticity, respectively, of the resulting system of Whitham modulation equations. Using rigorous spectral perturbation theory we then study the spectral (linearized) stability problem for a given wave train solution of the Ostrovsky equation, directly connecting the hyperbolicity or ellipticity of the associated Whitham system to the rigorous spectral stability problem for the underlying wave. Specifically, we prove that strict hyperbolicity of the Whitham system implies spectral stability near the origin in the spectral plane, i.e. so-called spectral modulational stability, while ellipticity implies spectral instability of the underlying wave train.

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arXiv2024

Modulational Stability of Wave Trains in the Camassa-Holm Equation

In this paper, we study the nonlinear wave modulation of arbitrary amplitude periodic traveling wave solutions of the Camassa-Holm (CH) equation. Slow modulations of wave trains is often described through Whitham's theory of modulations, which at leading order models the slow evolution of the fundamental wave characteristics (such as the wave's frequency, mass and momentum) through a disperionless system of quasi-linear partial differential equations. The modulational stability or instability of such a slowly modulated wave is considered to be determined by the hyperbolicity or ellipticity of this Whitham modulation system of equations. In work by Abenda \& Grava, the Whitham modulation system for the CH equation was derived through averaged Lagrangian methods and was further shown to always be hyperbolic (although strict hyperbolicity may fail). In this work, we provide an independent derivation of the Whitham modulation system for the CH equation through nonlinear WKB / multiple scales expansions. We further provide a rigorous connection between the Whitham modulation equations for the CH equation and the spectral stability of the underlying periodic wave train to localized (i.e. integrable on the line) perturbations. In particular, we prove that the strict hyperbolicity of the Whitham system implies spectral stability in a neighborhood of the origin in the spectral plane, i.e. spectral modulational stability. As an illustration of our theory, we examine the Whitham modulation system for wave trains with asymptotically small oscillations about their total mass.

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arXiv2024

Modulational Instability of Small Amplitude Periodic Traveling Waves in the Novikov Equation

We study the spectral stability of smooth, small-amplitude periodic traveling wave solutions of the Novikov equation, which is a Camassa-Holm type equation with cubic nonlinearities. Specifically, we investigate the $L^2(\mathbb{R})$-spectrum of the associated linearized operator, which in this case is an integro-differential operator with periodic coefficients, in a neighborhood of the origin in the spectral plane. Our analysis shows that such small-amplitude periodic solutions are spectrally unstable to long-wavelength perturbations if the wave number if greater than a critical value, bearing out the famous Benmajin-Feir instability for the Novikov equation. On the other hand, such waves with wave number less than the critical value are shown to be spectrally stable. Our methods are based on applying spectral perturbation theory to the associated linearization.

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arXiv2024

Orbital Stability of Smooth Solitary Waves for the Novikov Equation

We study the orbital stability of smooth solitary wave solutions of the Novikov equation, which is a Camassa-Holm type equation with cubic nonlinearities. These solitary waves are shown to exist as a one-parameter family (up to spatial translations) parameterized by their asymptotic endstate, and are encoded as critical points of a particular action functional. As an important step in our analysis we must study the spectrum the Hessian of this action functional, which turns out to be a nonlocal integro-differential operator acting on $L^2(\mathbb{R})$. We provide a combination of analytical and numerical evidence that the necessary spectral hypotheses always holds for the Novikov equation. Together with a detailed study of the associated Vakhitov-Kolokolov condition, our analysis indicates that all smooth solitary wave solutions of the Novikov equation are nonlinearly orbitally stable.

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arXiv2024

Orbital Stability of Periodic Traveling Waves in the $b$-Camassa-Holm Equation

In this paper, we identify criteria that guarantees the nonlinear orbital stability of a given periodic traveling wave solution within the b-family Camassa-Holm equation. These periodic waves exist as 3-parameter families (up to spatial translations) of smooth traveling wave solutions, and their stability criteria are expressed in terms of Jacobians of the conserved quantities with respect to these parameters. The stability criteria utilizes a general Hamiltonian structure which exists for every $b>1$, and hence applies outside of the completely integrable cases ($b=2$ and $b=3$).

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arXiv2023

Nonlinear Subharmonic Dynamics of Spectrally Stable Lugiato-Lefever Periodic Waves

We study the nonlinear dynamics of perturbed, spectrally stable $T$-periodic stationary solutions of the Lugiato-Lefever equation (LLE), a damped nonlinear Schr\"odinger equation with forcing that arises in nonlinear optics. It is known that for each $N\in\mathbb{N}$, such a $T$-periodic wave train is (orbitally) asymptotically stable against $NT$-periodic, i.e. subharmonic, perturbations. Unfortunately, in such results both the allowable size of initial perturbations as well as the exponential decay rates of perturbations depend on $N$ and, in fact, tend to zero as $N\to\infty$, leading to a lack of uniformity in the period of the perturbation. In recent work, the authors performed a delicate decomposition of the associated linearized solution operator and obtained linear estimates which are uniform in $N$. The dynamical description suggested by this uniform linear theory indicates that the corresponding nonlinear iteration can only be closed if one allows for a spatio-temporal phase modulation of the underlying wave. However, such a modulated perturbation is readily seen to satisfy a quasilinear equation, yielding an inherent loss of regularity. We regain regularity by transferring a nonlinear damping estimate, which has recently been obtained for the LLE in the case of localized perturbations to the case of subharmonic perturbations. Thus, we obtain a nonlinear, subharmonic stability result for periodic stationary solutions of the LLE that is uniform in $N$. This in turn yields an improved nonuniform subharmonic stability result providing an $N$-independent ball of initial perturbations which eventually exhibit exponential decay at an $N$-dependent rate. Finally, we argue that our results connect in the limit $N \to \infty$ to previously established stability results against localized perturbations, thereby unifying existing theories.

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arXiv2023

Modulational Instability in the Ostrovsky Equation and Related Models

We study the modulational instability of small-amplitude periodic traveling wave solutions in a dispersion generalized Ostrovsky equation. Specifically, we investigate the invertibility of the associated linearized operator in the vicinity of the origin and derive a modulational instability index that depends on the dispersion and nonlinearity. For the classical Ostrovsky equation, we recover the well-known Lighthill condition for modulational instability of small-amplitude periodic traveling waves, and further provide a rigorous connection of the Lighthill condition to the spectral instability of the underlying wave. Our results and methodologies further apply to a wide-class of Ostrovsky type models that incorporate various dispersive effects. As such, we present new results illuminating the effects of rotation on various full-dispersion models arising in the study of weakly nonlinear surface water waves.

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arXiv2021

Solitary waves in a Whitham equation with small surface tension

Using a nonlocal version of the center manifold theorem and a normal form reduction, we prove the existence of small-amplitude generalized solitary-wave solutions and modulated solitary-wave solutions to the steady gravity-capillary Whitham equation with weak surface tension. Through the application of the center manifold theorem, the nonlocal equation for the solitary wave profiles is reduced to a four-dimensional system of ODEs inheriting reversibility. Along particular parameter curves, relating directly to the classical gravity-capillary water wave problem, the associated linear operator is seen to undergo either a reversible $0^{2+}(i k_0)$ bifurcation or a reversible $(i s)^2$ bifurcation. Through a normal form transformation, the reduced system of ODEs along each relevant parameter curve is seen to be well approximated by a truncated system retaining only second-order or third-order terms. These truncated systems relate directly to systems obtained in the study of the full gravity-capillary water wave equation and, as such, the existence of generalized and modulated solitary waves for the truncated systems is guaranteed by classical works, and they are readily seen to persist as solutions of the gravity-capillary Whitham equation due to reversibility. Consequently, this work illuminates further connections between the gravity-capillary Whitham equation and the full two-dimensional gravity-capillary water wave problem.

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arXiv2021

Subharmonic Dynamics of Wave Trains in the Korteweg-de Vries / Kuramoto-Sivashinsky Equation

We study the stability and nonlinear local dynamics of spectrally stable periodic wave trains of the Korteweg-de Vries / Kuramoto-Sivashinsky equation when subjected to classes of periodic perturbations. It is known that for each $N\in\mathbb{N}$, such a $T$-periodic wave train is asymptotically stable to $NT$-periodic, i.e., subharmonic, perturbations, in the sense that initially nearby data will converge asymptotically to a small Galilean boost of the underlying wave, with exponential rates of decay. However, both the allowable size of initial perturbations and the exponential rates of decay depend on $N$ and, in fact, tend to zero as $N\to\infty$, leading to a lack of uniformity in such subharmonic stability results. Our goal here is to build upon a recent methodology introduced by the authors in the reaction-diffusion setting and achieve a subharmonic stability result which is uniform in $N$. This work is motivated by the dynamics of such wave trains when subjected to perturbations which are localized (i.e., integrable on the line).

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arXiv2021

Nonlinear Modulational Dynamics of Spectrally Stable Lugiato-Lefever Periodic Waves

We consider the nonlinear stability of spectrally stable periodic waves in the Lugiato-Lefever equation (LLE), a damped nonlinear Schr\"odinger equation with forcing that arises in nonlinear optics. So far, nonlinear stability of such solutions has only been established against co-periodic perturbations by exploiting the existence of a spectral gap. In this paper, we consider perturbations which are localized, i.e., integrable on the line. Such localized perturbations naturally yield the absence of a spectral gap, so we must rely on a substantially different method with origins in the stability analysis of periodic waves in reaction-diffusion systems. The relevant linear estimates have been obtained in recent work by the first three authors through a delicate decomposition of the associated linearized solution operator. Since its most critical part just decays diffusively, the nonlinear iteration can only be closed if one allows for a spatio-temporal phase modulation. However, the modulated perturbation satisfies a quasilinear equation yielding an apparent loss of regularity. To overcome this obstacle, we incorporate tame estimates on the unmodulated perturbation, which satisfies a semilinear equation in which no derivatives are lost, yet where decay is too slow to close an independent iteration scheme. We obtain nonlinear stability of periodic steady waves in the LLE against localized perturbations with precisely the same decay rates as predicted by the linear theory.

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arXiv2021

Subharmonic Dynamics of Wave Trains in Reaction Diffusion Systems

We investigate the stability and nonlinear local dynamics of spectrally stable wave trains in reaction-diffusion systems. For each $N\in\mathbb{N}$, such $T$-periodic traveling waves are easily seen to be nonlinearly asymptotically stable (with asymptotic phase) with exponential rates of decay when subject to $NT$-periodic, i.e., subharmonic, perturbations. However, both the allowable size of perturbations and the exponential rates of decay depend on $N$, and, in particular, they tend to zero as $N\to\infty$, leading to a lack of uniformity in such subharmonic stability results. In this work, we build on recent work by the authors and introduce a methodology that allows us to achieve a stability result for subharmonic perturbations which is uniform in $N$. Our work is motivated by the dynamics of such waves when subject to perturbations which are localized (i.e. integrable on the line), which has recently received considerable attention by many authors.

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arXiv2021

Linear Modulational and Subharmonic Dynamics of Spectrally Stable Lugiato-Lefever Periodic Waves

We study the linear dynamics of spectrally stable $T$-periodic stationary solutions of the Lugiato-Lefever equation (LLE), a damped nonlinear Schrödinger equation with forcing that arises in nonlinear optics. Such $T$-periodic solutions are nonlinearly stable to $NT$-periodic, i.e. subharmonic, perturbations for each $N\in\mathbb{N}$ with exponential decay rates of perturbations of the form $e^{-δ_N t}$. However, both the exponential rates of decay $δ_N$ and the allowable size of the initial perturbations tend to $0$ as $N\to\infty$, so that this result is non-uniform in $N$ and, in fact, empty in the limit $N=\infty$. The primary goal of this paper is to introduce a methodology, in the context of the LLE, by which a uniform stability result for subharmonic perturbations may be achieved, at least at the linear level. The obtained uniform decay rates are shown to agree precisely with the polynomial decay rates of localized, i.e. integrable on the real line, perturbations of such spectrally stable periodic solutions of the LLE. This work both unifies and expands on several existing works in the literature concerning the stability and dynamics of such waves, and sets forth a general methodology for studying such problems in other contexts.

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arXiv2019

Modulational Instability of Viscous Fluid Conduit Periodic Waves

In this paper, we are interested in studying the modulational dynamics of interfacial waves rising buoyantly along a conduit of a viscous liquid. Formally, the behavior of modulated periodic waves on large space and time scales may be described through the use of Whitham modulation theory. The application of Whitham theory, however, is based on formal asymptotic (WKB) methods, thus removing a layer of rigor that would otherwise support their predictions. In this study, we aim at rigorously verifying the predictions of the Whitham theory, as it pertains to the modulational stability of periodic waves, in the context of the so-called conduit equation, a nonlinear dispersive PDE governing the evolution of the circular interface separating a light, viscous fluid rising buoyantly through a heavy, more viscous, miscible fluid at small Reynolds numbers. In particular, using rigorous spectral perturbation theory, we connect the predictions of Whitham theory to the rigorous spectral (in particular, modulational) stability of the underlying wave trains. This makes rigorous recent formal results on the conduit equation obtained by Maiden and Hoefer.

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arXiv2019

On the Dynamics of Traveling Fronts Arising in Nanoscale Pattern Formation

We study the stability and dynamics of traveling-front solutions of a modified Kuramoto--Sivashinsky equation arising in the modeling of nanoscale ripple patterns that form when a nominally flat solid surface is bombarded with a broad ion beam at an oblique angle of incidence. Structurally, the linearized operators associated with these fronts have unstable essential spectrum---corresponding to instability of the spatially asymptotic states---and stable point spectrum---corresponding to stability of the transition profile of the front. We show that these waves are linearly orbitally asymptotically stable in appropriate exponentially weighted spaces. While the technical device of exponential weights allows us to accommodate the unstable essential spectrum of individual waves in our linear analysis, it does not shed light on the long-time pattern formation that is observed experimentally and in numerical simulations. To begin to address this issue, we consider a periodic array of unstable front and back solutions. While not an exact solution of the governing equation, this periodic pattern mimics experimentally observed phenomena. Our numerical experiments suggest that the convecting instabilities associated with each individual wave are damped as they pass through transition layers and that this stabilization mechanism underlies the pattern formation seen in experiments.

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arXiv2019

On the bifurcation diagram of the capillary-gravity Whitham equation

We study the bifurcation of periodic travelling waves of the capillary-gravity Whitham equation. This is a nonlinear pseudo-differential equation that combines the canonical shallow water nonlinearity with the exact (unidirectional) dispersion for finite-depth capillary-gravity waves. Starting from the line of zero solutions, we give a complete description of all small periodic solutions, unimodal as well bimodal, using simple and double bifurcation via Lyapunov--Schmidt reductions. Included in this study is the resonant case when one wavenumber divides another. Some bifurcation formulas are studied, enabling us, in almost all cases, to continue the unimodal bifurcation curves into global curves. By characterizing the range of the surface tension parameter for which the integral kernel corresponding to the linear dispersion operator is completely monotone (and therefore positive and convex; the threshold value for this to happen turns out to be $T = \frac{4}{π^2}$, not the critical Bond number $\frac{1}{3}$), we are able to say something about the nodal properties of solutions, even in the presence of surface tension. Finally, we present a few general results for the equation and discuss, in detail, the complete bifurcation diagram as far as it is known from analytical and numerical evidence. Interestingly, we find, analytically, secondary bifurcation curves connecting different branches of solutions; and, numerically, that all supercritical waves preserve their basic nodal structure, converging asymptotically in $L^2(\SM)$ (but not in $L^\infty$) towards one of the two constant solution curves.

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arXiv2018

Nondegeneracy and Stability of Antiperiodic Bound States for Fractional Nonlinear Schrödinger Equations

We consider the existence and stability of real-valued, spatially antiperiodic standing wave solutions to a family of nonlinear Schrödinger equations with fractional dispersion and power-law nonlinearity. As a key technical result, we demonstrate that the associated linearized operator is nondegenerate when restricted to antiperiodic perturbations, i.e. that its kernel is generated by the translational and gauge symmetries of the governing evolution equation. In the process, we provide a characterization of the antiperiodic ground state eigenfunctions for linear fractional Schrödinger operators on $\mathbb{R}$ with real-valued, periodic potentials as well as a Sturm-Liouville type oscillation theory for the higher antiperiodic eigenfunctions.

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arXiv2018

Generalized Solitary Waves in the Gravity-Capillary Whitham Equation

We study the existence of traveling wave solutions to a unidirectional shallow water model which incorporates the full linear dispersion relation for both gravitational and capillary restoring forces. Using functional analytic techniques, we show that for small surface tension (corresponding to Bond numbers between $0$ and ${1}/{3}$) there exists small amplitude solitary waves that decay to asymptotically small periodic waves at spatial infinity. The size of the oscillations in the far field are shown to be small beyond all algebraic orders in the amplitude of the wave. We also present numerical evidence, based on the recent analytical work of Hur \& Johnson, that the asymptotic end states are modulationally stable for all Bond numbers between $0$ and $1/3$.

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