High-Frequency Instabilities of a Boussinesq–Whitham System: A Perturbative Approach

Abstract

We analyze the spectral stability of small-amplitude, periodic, traveling-wave solutions of a Boussinesq–Whitham system. These solutions are shown numerically to exhibit high-frequency instabilities when subject to bounded perturbations on the real line. We use a formal perturbation method to estimate the asymptotic behavior of these instabilities in the small-amplitude regime. We compare these asymptotic results with direct numerical computations.

1. Introduction

In 1967, Whitham [1] proposed a nonlinear, integro-differential equation to model shallow water waves with a linearized dispersion relation that matches one full branch of the 1D Euler water wave problem (WWP) [2]. Because its dispersion relation excludes the other branch, however, Whitham’s equation does not capture all the instabilities of traveling-wave solutions of the WWP [3].

In this manuscript, we investigate the spectral stability of small-amplitude, $2\pi /\kappa$-periodic, traveling waves of a Boussinesq–Whitham system proposed by Hur and Pandey [4] and Hur and Tao [5]:

$$\begin{array}{cc}\hfill {\eta }_t& =-h_0{u}_x-{\left(\eta u\right)}_x,\hfill \\ \hfill u_t& =-g\mathcal{K}\left[{\eta }_x\right]-u{u}_x.\hfill \end{array}$$

(1)

In this model, $\eta (x,t)$ represents the displacement of a wave profile from its equilibrium depth $h_0$, $u(x,t)$ is the horizontal velocity along $\eta$, and $\mathcal{K}$ is a Fourier multiplier operator defined so that the linearized dispersion relation of (1) matches both branches of the WWP. For functions $f\in {\mathrm{L}}_{\mathrm{per}}^1(-\pi /\kappa ,\pi /\kappa )$, $\mathcal{K}$ is defined as

$$\widehat{\mathcal{K}\left[f\right]}\left(\kappa n\right)=\frac{tanh\left(\kappa n{h}_0\right)}{\kappa n{h}_0}\widehat{f}\left(\kappa n\right),n\in \mathbb{Z},$$

(2)

where $\widehat{·}$ denotes the Fourier transform of f:

$$\widehat{f}\left(k\right)=\frac{\kappa }{2\pi }{\int }_{-\pi /\kappa }^{\pi /\kappa }f\left(x\right)e^{-ikx}dx.$$

(3)

Alternatively, $\mathcal{K}$ can be defined in physical variables as the pseudo-differential operator

$$\mathcal{K}\left[f\right]=\left(\frac{tanh\left(h_{0}D\right)}{h_{0}D}\right)f,$$

(4)

where $D=-i{\partial }_x$. For the remainder of this manuscript, we refer to (1) as the Hur-Pandey-Tao–Boussinesq–Whitham system, or HPT–BW for short.

The HPT–BW system is Hamiltonian [4], with

$$\begin{array}{c}\hfill \mathcal{H}=\frac{1}{2}{\int }_{-\pi /\kappa }^{\pi /\kappa }\left(h_0{u}^2+g\eta \mathcal{K}\left[\eta \right]+\eta u^2\right)dx,\end{array}$$

(5)

and non-canonical Poisson structure

$$\begin{array}{c}\hfill J=-\left(\begin{array}{cc}0& {\partial }_x\\ {\partial }_x& 0\end{array}\right).\end{array}$$

(6)

The system has a one-parameter family of small-amplitude, $2\pi /\kappa$-periodic, traveling-wave solutions for each $\kappa >0$. We call these solutions the Stokes waves of HPT–BW by analogy with solutions of the WWP of the same name [6,7,8]. In Section 2, we derive a power series expansion for HPT–BW Stokes waves in a small parameter $\epsilon$ that scales with the amplitude of the waves.

Perturbing Stokes waves yield a spectral problem after linearizing the governing equations of the perturbations. The spectral elements of this problem define the stability spectrum of Stokes waves (Section 3). The stability spectrum is purely continuous [9,10,11], but Floquet theory decomposes the spectrum into an uncountable union of point spectra. Each of these point spectra is indexed by the Floquet exponent [12,13,14].

The stability spectrum inherits quadrafold symmetry from the Hamiltonian structure of (1); i.e., the spectrum is invariant under conjugation and negation [10,13]. Because of quadrafold symmetry, all elements of the stability spectrum have non-positive real component only if the stability spectrum is a subset of the imaginary axis. Therefore, HPT–BW Stokes waves are spectrally stable only if the spectrum lies on the imaginary axis. Otherwise, the Stokes waves are spectrally unstable.

If the aspect ratio $\kappa h_0$ is sufficiently large, both HPT–BW [4] and WWP Stokes waves [15,16,17] have stability spectra near the origin that leave the imaginary axis for $0<\left|\epsilon \right|\ll 1$, resulting in modulational instability. Using the Floquet–Fourier–Hill (FFH) method [12], recent numerical work by [18] and [19,20] shows, respectively, that HPT–BW and WWP Stokes waves also have stability spectra away from the origin that leave the imaginary axis, regardless of $\kappa h_0$. These spectra give rise to the so-called high-frequency instabilities [3], shown schematically in Figure 1.

High-frequency instabilities arise from the collision of nonzero elements of the zero-amplitude ($\epsilon =0$) stability spectrum. At these collided spectral elements, a Hamiltonian-Hopf bifurcation occurs, resulting in a locus of spectral elements bounded away from the origin that leave the imaginary axis as $\left|\epsilon \right|$ increases. We refer to this locus of spectral elements as a high-frequency isola.

High-frequency isolas are difficult to find using numerical methods like FFH. To capture the isola closest to the origin, for example, the corresponding interval of Floquet exponents has width $\mathcal{O}\left({\epsilon }^2\right)$ (Figure 2). In general, the jth isola from the origin appears to have width $O\left({\epsilon }^{j+1}\right)$. The isolas also drift from their initial collision sites (Figure 2), meaning that the Floquet exponent that gives rise to the collided spectral elements at $\epsilon =0$ is not contained in the interval parameterizing the corresponding isola for sufficiently large $\left|\epsilon \right|$. To circumvent these difficulties, one must supply the numerical method with asymptotic expressions for the interval of Floquet exponents corresponding to the desired isolas. We discover these expressions for high-frequency isolas of HPT–BW.

Our motivation for studying the HPT–BW system, apart from its inherent interest, is that it retains the full dispersion relation (both branches) of the more complicated WWP. Our goal is the application of the perturbation method developed herein to the WWP. The first step towards this goal was the investigation of the stability spectra of Stokes waves of the Kawahara Equation [21]. The investigations presented here constitute our second step, before we proceed to the WWP.

For a given high-frequency isola of an HPT–BW Stokes wave, we obtain (i) an asymptotic range of Floquet exponents that parameterizes the isola, (ii) an asymptotic estimate for the most unstable spectral element of the isola, (iii) expressions of curves that are asymptotic to the isola, and (iv) wavenumbers for which the given isola is not present. Our approach is inspired by a perturbation method outlined in [22], but modified appropriately for higher-order calculations. We compare all asymptotic results with numerical results computed by the FFH method.

2. Small-Amplitude Stokes Waves

In a traveling frame moving with velocity c, $x\to x-ct$ and (1) becomes

$$\begin{array}{cc}\hfill {\eta }_t& =c{\eta }_x-h_0{u}_x-{\left(\eta u\right)}_x,\hfill \\ \hfill u_t& =c{u}_x-g\mathcal{K}\left({\eta }_x\right)-u{u}_x.\hfill \end{array}$$

(7)

Non-dimensionalizing (7) according to $\eta \to h_0\eta$, $u\to u\sqrt{g{h}_0}$, $x\to {\alpha }^{-1}{h}_{0}x$, $t\to t\sqrt{h_0/g}$, and $c\to c\sqrt{g{h}_0}$ yields the following system:

$$\begin{array}{cc}\hfill {\alpha }^{-1}{\eta }_t& =c{\eta }_x-u_x-{\left(\eta u\right)}_x,\hfill \\ \hfill {\alpha }^{-1}{u}_t& =c{u}_x-{\mathcal{K}}_{\alpha }\left({\eta }_x\right)-u{u}_x.\hfill \end{array}$$

(8)

The parameter $\alpha$ is chosen to map $2\pi /\kappa$-periodic solutions of (7) to $2\pi$-periodic solutions of (8). Consequently, $\alpha =\kappa h_0>0$ and

$$\begin{array}{c}\hfill \widehat{{\mathcal{K}}_{\alpha }\left[f\right]}\left(n\right)=\frac{tanh\left(\alpha n\right)}{\alpha n}\widehat{f}\left(n\right),n\in \mathbb{Z},\end{array}$$

(9)

or, alternatively,

$$\begin{array}{c}\hfill {\mathcal{K}}_{\alpha }\left[f\right]=\left(\frac{tanh\left(\alpha D\right)}{\alpha D}\right)f,\end{array}$$

(10)

for $f\in {\mathrm{L}}_{\mathrm{per}}^1(-\pi ,\pi )$ with the Fourier transform (3) redefined over $(-\pi ,\pi )$.

Stokes wave solutions of (8) are independent of time. Equating time derivatives in (8) to zero and integrating in x, we find

$$\begin{array}{cc}\hfill c\eta & =u+\eta u+{\mathcal{I}}_1\hfill \\ \hfill cu& ={\mathcal{K}}_{\alpha }\left(\eta \right)+\frac{1}{2}{u}^2+{\mathcal{I}}_2,\hfill \end{array}$$

(11)

where ${\mathcal{I}}_j$ are integration constants. For each $\alpha >0$, there exists a three-parameter family of infinitely differentiable, even, small-amplitude, $2\pi$-periodic solutions of (11), provided that ${\mathcal{I}}_j$ are sufficiently small [4]. We call these solutions the HPT–BW Stokes waves, denoted ${({\eta }_S(x;\epsilon ,{\mathcal{I}}_j),u_S(x;\epsilon ,{\mathcal{I}}_j))}^T$, where $\epsilon$ is a small-amplitude parameter defined implicitly in terms of the first Fourier mode of ${\eta }_S(x;\epsilon ,{\mathcal{I}}_j)$:

$$\begin{array}{c}\hfill \epsilon :=2\widehat{{\eta }_S}\left(1\right)=\frac{1}{\pi }{\int }_{-\pi }^{\pi }{\eta }_S(x;\epsilon ,{\mathcal{I}}_j)e^{-ix}dx.\end{array}$$

(12)

Remark 1.

Redefining $c\to c-{\mathcal{I}}_1$ and $u\to u-{\mathcal{I}}_1$ in (11) implies ${\mathcal{I}}_1=0$ without loss of generality. If we also equate ${\mathcal{I}}_2=0$, our Stokes waves reduce to a one-parameter family of solutions to (11) such that (12) ensures that ${\eta }_S(x;\epsilon )\sim \epsilon cos\left(x\right)$ as $\epsilon \to 0$. We restrict to this case for simplicitly, but the methodology in Section 4 and Section 5 are unchanged if ${\mathcal{I}}_2\ne 0$. For series representations of Stokes waves that include ${\mathcal{I}}_j$, see [4].

The Stokes waves and their velocity may be expanded as power series in $\epsilon$:

$$\begin{array}{cc}\hfill {\eta }_S& ={\eta }_S(x;\epsilon )=\epsilon cos\left(x\right)+\sum _{j=2}^{\infty }{\eta }_j\left(x\right){\epsilon }^j,\hfill \end{array}$$

(13a)

$$\begin{array}{cc}\hfill u_S& =u_S(x;\epsilon )=c_0\epsilon cos\left(x\right)+\sum _{j=2}^{\infty }{u}_j\left(x\right){\epsilon }^j,\hfill \end{array}$$

(13b)

$$\begin{array}{cc}\hfill c& =c\left(\epsilon \right)=c_0+\sum _{j=1}^{\infty }{c}_{2j}{\epsilon }^{2j},c_0^2=\frac{tanh\left(\alpha \right)}{\alpha },\hfill \end{array}$$

(13c)

where ${\eta }_j\left(x\right)$ and $u_j\left(x\right)$ are analytic, even, and $2\pi$-periodic for each j. Substituting these expansions into (11) (with ${\mathcal{I}}_j=0$) and following a Poincaré-Lindstedt perturbation method [2], one determines ${\eta }_j\left(x\right)$, $u_j\left(x\right)$, and $c_{2j}$ order by order. In Appendix A, we report expansions of ${\eta }_S$, $u_S$, and c up to the fourth order in $\epsilon$; this is sufficient for our asymptotic calculations of high-frequency isolas discussed in Section 4 and Section 5.

3. The Stability Spectrum of Stokes Waves

We consider perturbations to ${({\eta }_S,u_S)}^T$ of the form

$$\left(\begin{array}{c}\eta (x,t;\epsilon ,\rho )\\ u(x,t;\epsilon ,\rho )\end{array}\right)=\left(\begin{array}{c}{\eta }_S(x;\epsilon )\\ u_S(x;\epsilon )\end{array}\right)+\rho \left(\begin{array}{c}H(x,t;\epsilon )\\ U(x,t;\epsilon )\end{array}\right)+\mathcal{O}\left({\rho }^2\right),$$

(14)

where $0<\left|\rho \right|\ll 1$ is a parameter independent of $\epsilon$ and H, and U are sufficiently smooth, bounded functions of x on $\mathbb{R}$ for all $t\ge 0$. When (14) is substituted into (8), terms of $\mathcal{O}\left({\rho }^0\right)$ cancel by (11) (with ${\mathcal{I}}_j=0$). Equating terms of $\mathcal{O}\left(\rho \right)$, the perturbation ${(H,U)}^T$ solves the linear system

$$\frac{\partial }{\partial t}\left(\begin{array}{c}H\\ U\end{array}\right)=\alpha \left(\begin{array}{cc}-u_S^{\prime }+(c-u_S){\partial }_x& -{\eta }_S^{\prime }-(1+{\eta }_S){\partial }_x\\ -\frac{itanh\left(\alpha D\right)}{\alpha }& -u_S^{\prime }+(c-u_S){\partial }_x\end{array}\right)\left(\begin{array}{c}H\\ U\end{array}\right),$$

(15)

where primes denote differentiation with respect to x. Formally separating variables,

$$\begin{array}{c}\hfill \left(\begin{array}{c}H(x,t;\epsilon )\\ U(x,t;\epsilon )\end{array}\right)=e^{\lambda t}\left(\begin{array}{c}\mathcal{H}(x,t;\epsilon )\\ \mathcal{U}(x,t;\epsilon )\end{array}\right),\end{array}$$

(16)

where ${(\mathcal{H},\mathcal{U})}^T$ solves the spectral problem

$$\begin{array}{c}\hfill \lambda \left(\begin{array}{c}\mathcal{H}\\ \mathcal{U}\end{array}\right)=\alpha \left(\begin{array}{cc}-u_S^{\prime }+(c-u_S){\partial }_x& -{\eta }_S^{\prime }-(1+{\eta }_S){\partial }_x\\ -\frac{itanh\left(\alpha D\right)}{\alpha }& -u_S^{\prime }+(c-u_S){\partial }_x\end{array}\right)\left(\begin{array}{c}\mathcal{H}\\ \mathcal{U}\end{array}\right).\end{array}$$

(17)

Since the entries of the matrix operator above are $2\pi$-periodic, one can use Floquet theory to solve (17) for ${(\mathcal{H},\mathcal{U})}^T$. (Strictly speaking, Floquet theory applies only to linear, local operators. Work by [23] extends this theory to nonlocal operators.) These solutions take the form

$$\begin{array}{c}\hfill \left(\begin{array}{c}\mathcal{H}(x;\epsilon )\\ \mathcal{U}(x;\epsilon )\end{array}\right)=e^{i\mu x}\left(\begin{array}{c}\mathfrak{h}(x;\epsilon )\\ \mathfrak{u}(x;\epsilon )\end{array}\right),\end{array}$$

(18)

where $\mu \in [-1/2,1/2]$ is called the Floquet exponent and $\mathfrak{h},\mathfrak{u}\in {\mathrm{H}}_{\mathrm{per}}^1(-\pi ,\pi )$. Substituting (18) into (17) results in a spectral problem for $\mathbf{w}={(\mathfrak{h},\mathfrak{u})}^{\mathbf{T}}$:

$$\begin{array}{c}\hfill {\lambda }_{\epsilon ,\mu }\mathbf{w}={\mathcal{L}}_{\epsilon ,\mu }\mathbf{w},\end{array}$$

(19)

with

$$\begin{array}{c}\hfill {\mathcal{L}}_{\epsilon ,\mu }=\alpha \left(\begin{array}{cc}-u_S^{\prime }+(c-u_S)(i\mu +{\partial }_x)& -{\eta }_S^{\prime }-(1+{\eta }_S)(i\mu +{\partial }_x)\\ -\frac{itanh\left(\alpha \right(\mu +D\left)\right)}{\alpha }& -u_S^{\prime }+(c-u_S)(i\mu +{\partial }_x)\end{array}\right).\end{array}$$

(20)

For sufficiently small $\epsilon$, (19) has a countable collection of eigenvalues ${\lambda }_{\epsilon ,\mu }$ for each Floquet exponent $\mu$ [14]. The union of these eigenvalues over $\mu \in [-1/2,1/2]$ recovers the purely continuous spectrum of (17) for fixed $\epsilon$; this is the stability spectrum of HPT–BW Stokes waves. We use the Floquet–Fourier–Hill method [12] to compute the stability spectrum numerically.

If there exists a $\mu$ such that there is a ${\lambda }_{\epsilon ,\mu }$ with $\mathrm{Re}\left({\lambda }_{\epsilon ,\mu }\right)>0$, then there exists a perturbation (16) that grows exponentially in time, and Stokes waves of amplitude $\epsilon$ are spectrally unstable. If no such $\mu$ is found, then the Stokes waves are spectrally stable. Because of the quadrafold symmetry mentioned in the introduction, Stokes waves are spectrally stable if and only if their stability spectrum is a subset of the imaginary axis.

When $\epsilon =0$, ${\mathcal{L}}_{0,\mu }$ has constant coefficients, and its spectral elements are given exactly by

$$\begin{array}{c}\hfill {\lambda }_{0,\mu ,n}^{\left(\sigma \right)}=-i{\mathsf{\Omega }}_{\sigma }(n+\mu ),n\in \mathbb{Z},\sigma =\pm 1,\end{array}$$

(21)

where ${\mathsf{\Omega }}_{\sigma }$ are the two branches of the linear dispersion relation of (8) with $c\to c_0$ ($c_0$ is given in (13)). Explicitly,

$$\begin{array}{c}\hfill {\mathsf{\Omega }}_{\sigma }\left(k\right)=-\alpha c_{0}k+\sigma {\omega }_{\alpha }\left(k\right),\end{array}$$

(22)

where

$$\begin{array}{c}\hfill {\omega }_{\alpha }\left(k\right)=\mathrm{sgn}\left(k\right)\sqrt{\alpha ktanh\left(\alpha k\right)}.\end{array}$$

(23)

As expected, ${\lambda }_{0,\mu ,n}^{\left(\sigma \right)}$ is a countable collection of eigenvalues for each $\mu$, and the resulting stability spectrum has quadrafold symmetry. In addition, the stability spectrum coincides with the imaginary axis, implying that zero-amplitude Stokes waves are spectrally stable.

For some $\mu ={\mu }_0$, nonzero eigenvalues of ${\mathcal{L}}_{0,{\mu }_0}$ with double multiplicity may give rise to Hamiltonian–Hopf bifurcations and, thus, to high-frequency instabilities for $0<\left|\epsilon \right|\ll 1$. These eigenvalues exist, provided there exists ${\mu }_0$, m, and n, such that

$$\begin{array}{c}\hfill {\lambda }_{0,{\mu }_0,n}^{\left({\sigma }_1\right)}={\lambda }_{0,{\mu }_0,m}^{\left({\sigma }_2\right)}\ne 0.\end{array}$$

(24)

We view (24) as a collision of two simple, nonzero eigenvalues. It can be shown that such a collision occurs only if ${\sigma }_1\ne {\sigma }_2$ [3,4,24]. Theorem A1 in Appendix B shows that, for any $p\in \mathbb{Z}\backslash \{0,\pm 1\}$, there exist unique ${\mu }_0$, m, and n that satisfy (24) with $m-n=p$. Thus, there are a countably infinite number of nonzero eigenvalue collisions in the zero-amplitude stability spectrum; each of which has potential to develop a high-frequency instability in the small-amplitude stability spectrum.

Remark 2.

Using results in Appendix B, it can be shown that the Krein signatures [25] of the colliding eigenvalues have opposite signs. This is a second necessary criterion for the occurance of high-frequency instabilities [3,26].

Remark 3.

The WWP shares the same collided eigenvalues with HPT–BW, since (22) is also the dispersion relation of the WWP.

4. High-Frequency Instabilities: p = 2

We use perturbation methods to investigate the high-frequency instability that develops from the collision of ${\lambda }_{0,{\mu }_0,n}^{\left(1\right)}$ and ${\lambda }_{0,{\mu }_0,m}^{(-1)}$, where ${\mu }_0\in [-1/2,1/2]$ is the unique Floquet exponent for which (24) is satisfied and $m-n=2$. (Because the spectrum (22) has the symmetry ${\overline{\lambda }}_{0,-{\mu }_0,-n}^{\left(\sigma \right)}={\lambda }_{0,{\mu }_0,n}^{\left(\sigma \right)}$, where the overbar denotes complex conjugation, choosing $p=-2$ gives the isola conjugate to that for $p=2$. Thus, we may choose $p=2$ without loss of generality.) This instability corresponds to the high-frequency isola closest to the origin; see Theorem A1 in Appendix B. For sufficiently small $\epsilon$, this is also the isola with largest real component.

4.1. The $\mathcal{O}\left({\epsilon }^0\right)$ Problem

The $p=2$ isola develops from the spectral data

$$\begin{array}{cc}\hfill {\lambda }_0& ={\lambda }_{0,{\mu }_0,n}^{\left(1\right)}=-i{\mathsf{\Omega }}_1({\mu }_0+n)=-i{\mathsf{\Omega }}_{-1}({\mu }_0+m)={\lambda }_{0,{\mu }_0,m}^{(-1)}\ne 0,\hfill \end{array}$$

(25a)

$$\begin{array}{cc}\hfill {\mathbf{w}}_{\mathbf{0}}\left(x\right)& =\left(\begin{array}{c}{\mathfrak{h}}_0\left(x\right)\\ {\mathfrak{u}}_0\left(x\right)\end{array}\right)={\gamma }_0\left(\begin{array}{c}1\\ -\frac{{\omega }_{\alpha }(m+{\mu }_0)}{\alpha (m+{\mu }_0)}\end{array}\right)e^{imx}+{\gamma }_1\left(\begin{array}{c}1\\ \frac{{\omega }_{\alpha }(n+{\mu }_0)}{\alpha (n+{\mu }_0)}\end{array}\right)e^{inx},\hfill \end{array}$$

(25b)

where ${\gamma }_j$ are arbitrary, nonzero constants. As $\left|\epsilon \right|$ increases, we assume the spectral data vary analytically [24] with $\epsilon$:

$$\begin{array}{cc}\hfill \lambda & ={\lambda }_0+\epsilon {\lambda }_1+{\epsilon }^2{\lambda }_2+\mathcal{O}\left({\epsilon }^3\right),\hfill \end{array}$$

(26a)

$$\begin{array}{cc}\hfill \mathbf{w}& ={\mathbf{w}}_{\mathbf{0}}+\epsilon {\mathbf{w}}_{\mathbf{1}}+{\epsilon }^2{\mathbf{w}}_{\mathbf{2}}+\mathcal{O}\left({\epsilon }^3\right)\hfill \end{array}$$

(26b)

$$\begin{array}{cc}\hfill & =\left(\begin{array}{c}{\mathfrak{h}}_0\\ {\mathfrak{u}}_0\end{array}\right)+\epsilon \left(\begin{array}{c}{\mathfrak{h}}_1\\ {\mathfrak{u}}_1\end{array}\right)+{\epsilon }^2\left(\begin{array}{c}{\mathfrak{h}}_2\\ {\mathfrak{u}}_2\end{array}\right)+\mathcal{O}\left({\epsilon }^3\right),\hfill \end{array}$$

(26c)

where we suppress functional dependencies for ease of notation. We normalize $\mathbf{w}$ so that

$$\begin{array}{c}\hfill \widehat{\mathfrak{h}}\left(n\right)=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }\mathfrak{h}{e}^{-inx}dx=1,\end{array}$$

(27)

or, alternatively, so that

$$\begin{array}{cc}\hfill \widehat{{\mathfrak{h}}_0}\left(n\right)& =1,\hfill \\ \hfill \widehat{{\mathfrak{h}}_j}\left(n\right)& =0,\forall j\in \mathbb{N}.\hfill \end{array}$$

(28)

This normalization ensures that ${\mathfrak{h}}_0$ fully resolves the nth Fourier mode of $\mathfrak{h}$, a convenient choice for the perturbation calculations that follow. With this normalization,

$$\begin{array}{c}\hfill {\mathbf{w}}_{\mathbf{0}}\left(x\right)=\left(\begin{array}{c}{\mathfrak{h}}_0\left(x\right)\\ {\mathfrak{u}}_0\left(x\right)\end{array}\right)={\gamma }_0\left(\begin{array}{c}1\\ -\frac{{\omega }_{\alpha }(m+{\mu }_0)}{\alpha (m+{\mu }_0)}\end{array}\right)e^{imx}+\left(\begin{array}{c}1\\ \frac{{\omega }_{\alpha }(n+{\mu }_0)}{\alpha (n+{\mu }_0)}\end{array}\right)e^{inx}.\end{array}$$

(29)

The arbitrary constant ${\gamma }_0$ will be determined at higher order, leading to a unique expression for ${\mathbf{w}}_{\mathbf{0}}$.

Remark 4.

The eigenvalue corrections ${\lambda }_j$ derived below are independent of the normalization chosen for $\mathbf{w}$.

If ${\lambda }_0$ is a semi-simple, isolated eigenvalue of ${\mathcal{L}}_{0,{\mu }_0}$, we may justify (26) using analytic perturbation theory [27], provided the Floquet exponent is fixed. For sufficiently small $\epsilon$, this method of proof gives two spectral elements on the isola. Numerical and asymptotic calculations show that these spectral elements quickly leave the isola as a result of the Floquet parameterization’s dependence on $\epsilon$ (Figure 2). To account for this dependence, we formally assume that the Floquet exponent depends analytically on $\epsilon$ as well:

$$\begin{array}{c}\hfill \mu ={\mu }_0+\epsilon {\mu }_1+{\epsilon }^2{\mu }_2+\mathcal{O}\left({\epsilon }^3\right).\end{array}$$

(30)

Remark 5.

In the calculations that follow, explicit expressions of select quantities are suppressed for ease of readability. The interested reader may consult the file in Data Availability Statement hptbw_isolap2.nb for these expressions.

4.2. The $\mathcal{O}\left(\epsilon \right)$ Problem

Substituting the expansions of the Stokes wave (13), spectral data (26), and Floquet exponent (30) into the spectral problem (19) and collecting terms of $\mathcal{O}\left(\epsilon \right)$, we find

$$\begin{array}{c}\hfill \left({\mathcal{L}}_{0,{\mu }_0}-{\lambda }_0\right){\mathbf{w}}_{\mathbf{1}}={\lambda }_1{\mathbf{w}}_{\mathbf{0}}-{\mathcal{L}}_1{\mathbf{w}}_{\mathbf{0}},\end{array}$$

(31)

with

$$\begin{array}{c}\hfill {\mathcal{L}}_1=\alpha \left(\begin{array}{cc}-u_1^{\prime }+i{c}_0{\mu }_1-u_1(i{\mu }_0+{\partial }_x)& -{\eta }_1^{\prime }-i{\mu }_1-{\eta }_1(i{\mu }_0+{\partial }_x)\\ -i{\mu }_1{\mathrm{sech}}^2\left(\alpha ({\mu }_0+D)\right)& -u_1^{\prime }+i{c}_0{\mu }_1-u_1(i{\mu }_0+{\partial }_x)\end{array}\right).\end{array}$$

(32)

The inhomogeneous terms on the RHS of (31) can be evaluated using expressions for ${\eta }_1$, $u_1$, and ${\mathbf{w}}_{\mathbf{0}}$. Each of these quantities are finite linear combinations of $2\pi$-periodic sinusoids. As a result, the inhomogeneous terms can be rewritten as a finite Fourier series, and (31) becomes

$$\begin{array}{c}\hfill \left({\mathcal{L}}_{0,{\mu }_0}-{\lambda }_0\right){\mathbf{w}}_{\mathbf{1}}=\sum _{j=n-1}^{m+1}{\mathbf{T}}_{\mathbf{1},\mathbf{j}}{e}^{\mathit{ijx}},\end{array}$$

(33)

where ${\mathbf{T}}_{\mathbf{1},\mathbf{j}}$ depend on ${\mu }_0$, $\alpha$, and ${\gamma }_0$; see the file in Data Availability Statement for details.

Remark 6.

Since $m-n=2$, the index $j\in \{n-1,n,1+n,m,m+1\}$. When evaluating the inhomogeneous terms, one finds vector multiples of exp$\left(i\right(1+n\left)x\right)$ and exp$\left(i\right(m-1\left)x\right)$. These vectors are combined to give ${\mathbf{T}}_{\mathbf{1},\mathbf{1}+\mathbf{n}}$.

For (33) to have a solution ${\mathbf{w}}_{\mathbf{1}}$, the inhomogeneous terms must be orthogonal (in the L${}_{\mathrm{per}}^2(-\pi ,\pi )\times {\mathrm{L}}_{\mathrm{per}}^2(-\pi ,\pi )$ sense) to the nullspace of the hermitian adjoint of ${\mathcal{L}}_{0,{\mu }_0}-{\lambda }_0$ by the Fredholm alternative. The hermitian adjoint of ${\mathcal{L}}_{0,{\mu }_0}-{\lambda }_0$ is

$$\begin{array}{c}\hfill {\left({\mathcal{L}}_{0,{\mu }_0}-{\lambda }_0\right)}^{†}=\left(\begin{array}{cc}-\alpha c_0(i{\mu }_0+{\partial }_x)-\overline{{\lambda }_0}& itanh\left(\alpha ({\mu }_0+D)\right)\\ \alpha (i{\mu }_0+{\partial }_x)& -\alpha c_0(i{\mu }_0+{\partial }_x)-\overline{{\lambda }_0}\end{array}\right),\end{array}$$

(34)

where overbars denote complex conjugation. Its nullspace is

$$\begin{array}{c}\hfill \mathrm{Null}\left[{\left({\mathcal{L}}_{0,{\mu }_0}-{\lambda }_0\right)}^{†}\right]=\mathrm{Span}\left[\left(\begin{array}{c}1\\ \frac{\alpha ({\mu }_0+n)}{{\omega }_{\alpha }({\mu }_0+n)}\end{array}\right)e^{inx},\left(\begin{array}{c}1\\ -\frac{\alpha ({\mu }_0+m)}{{\omega }_{\alpha }({\mu }_0+m)}\end{array}\right)e^{imx}\right].\end{array}$$

(35)

Thus, according to the Fredholm alternative, there exists a solution ${\mathbf{w}}_{\mathbf{1}}$ to (33) if

$$\begin{array}{c}\hfill 〈\left(\begin{array}{c}1\\ \frac{\alpha ({\mu }_0+n)}{{\omega }_{\alpha }({\mu }_0+n)}\end{array}\right)e^{inx},{\mathbf{T}}_{\mathbf{1},\mathbf{n}}{e}^{\mathit{inx}}〉=0,〈\left(\begin{array}{c}1\\ -\frac{\alpha ({\mu }_0+m)}{{\omega }_{\alpha }({\mu }_0+m)}\end{array}\right)e^{imx},{\mathbf{T}}_{\mathbf{1},\mathbf{m}}{e}^{\mathit{imx}}〉=0,\end{array}$$

(36)

where $〈·,·〉$ is the standard inner product on L${}_{\mathrm{per}}^2(-\pi ,\pi )\times {\mathrm{L}}_{\mathrm{per}}^2(-\pi ,\pi )$. Substituting expressions for ${\mathbf{T}}_{\mathbf{1},\mathbf{n}}$ and ${\mathbf{T}}_{\mathbf{1},\mathbf{m}}$ gives solvability conditions

$$\begin{array}{cc}\hfill {\lambda }_1+i{\mu }_1{c}_{g_1}({\mu }_0+n)& =0,\hfill \end{array}$$

(37a)

$$\begin{array}{cc}\hfill {\gamma }_0\left({\lambda }_1+i{\mu }_1{c}_{g_{-1}}({\mu }_0+m)\right)& =0,\hfill \end{array}$$

(37b)

where $c_{g_{\sigma }}\left(k\right)={\mathsf{\Omega }}_{\sigma }^{\prime }\left(k\right)$ is the group velocity of ${\mathsf{\Omega }}_{\sigma }$. Lemma A3 in Appendix B shows that $c_{g_1}({\mu }_0+n)\ne c_{g_{-1}}({\mu }_0+m)$. Since ${\gamma }_0$ is nonzero,

$$\begin{array}{c}\hfill {\lambda }_1=0={\mu }_1.\end{array}$$

(38)

Consequently, ${\mathbf{T}}_{\mathbf{1},\mathbf{n}}=\mathbf{0}={\mathbf{T}}_{\mathbf{1},\mathbf{m}}$, simplifying the inhomogeneous terms in (33).

With the solvability conditions satisfied, we solve for the particular solution of ${\mathbf{w}}_{\mathbf{1}}$ in (33). Combining with the nullspace of ${\mathcal{L}}_{0,{\mu }_0}-{\lambda }_0$,

$$\begin{array}{c}\hfill {\mathbf{w}}_{\mathbf{1}}={\displaystyle \sum _{\begin{array}{c}j=n-1\\ j\ne n,m\end{array}}^{m+1}}{\mathcal{W}}_{\mathbf{1},\mathbf{j}}{e}^{\mathit{ijx}}+{\beta }_{1,m}\left(\begin{array}{c}1\\ -\frac{{\omega }_{\alpha }(m+{\mu }_0)}{\alpha (m+{\mu }_0)}\end{array}\right)e^{\mathit{imx}}+{\beta }_{1,n}\left(\begin{array}{c}1\\ \frac{{\omega }_{\alpha }(n+{\mu }_0)}{\alpha (n+{\mu }_0)}\end{array}\right)e^{\mathit{inx}},\end{array}$$

(39)

where ${\beta }_{1,j}$ are arbitrary constants and ${\mathcal{W}}_{\mathbf{1},\mathbf{j}}$ are found in the file in Data Availability Statement. Enforcing the normalization condition (28), one finds ${\beta }_{1,n}=0$. For ease of notation, let ${\beta }_{1,m}\to {\gamma }_1$ so that

$$\begin{array}{c}\hfill {\mathbf{w}}_{\mathbf{1}}={\displaystyle \sum _{\begin{array}{c}j=n-1\\ j\ne n,m\end{array}}^{m+1}}{\mathcal{W}}_{\mathbf{1},\mathbf{j}}{e}^{\mathit{ijx}}+{\gamma }_1\left(\begin{array}{c}1\\ -\frac{{\omega }_{\alpha }(m+{\mu }_0)}{\alpha (m+{\mu }_0)}\end{array}\right)e^{\mathit{imx}}.\end{array}$$

(40)

4.3. The $\mathcal{O}\left({\epsilon }^2\right)$ Problem

Using (38), the spectral problem (19) at $\mathcal{O}\left({\epsilon }^2\right)$ is

$$\begin{array}{c}\hfill \left({\mathcal{L}}_{0,{\mu }_0}-{\lambda }_0\right){\mathbf{w}}_{\mathbf{2}}={\lambda }_2{\mathbf{w}}_{\mathbf{0}}-{\mathcal{L}}_2{{|}_{{\mu }_1=0}{\mathbf{w}}_{\mathbf{0}}-{\mathcal{L}}_1|}_{{\mu }_1=0}{\mathbf{w}}_{\mathbf{1}},\end{array}$$

(41)

where ${\mathcal{L}}_1{|}_{{\mu }_1=0}$ is the same as above, but evaluated at ${\mu }_1=0$, and

$$\begin{array}{c}\hfill {\mathcal{L}}_2{|}_{{\mu }_1=0}=\alpha \left(\begin{array}{cc}-u_2^{\prime }+(c_2-u_2)(i{\mu }_0+{\partial }_x)+i{\mu }_2{c}_0& -{\eta }_2^{\prime }-i{\mu }_2-{\eta }_2(i{\mu }_0+{\partial }_x)\\ -i{\mu }_2{\mathrm{sech}}^2\left(\alpha ({\mu }_0+D)\right)& -u_2^{\prime }+(c_2-u_2)(i{\mu }_0+{\partial }_x)+i{\mu }_2{c}_0\end{array}\right).\end{array}$$

(42)

One can evaluate the inhomogeneous terms of (42) using ${\eta }_j$, $u_j$, and ${\mathbf{w}}_{\mathbf{j}-\mathbf{1}}$ for $j\in \{1,2\}$. These inhomogeneous terms can be expressed as a finite Fourier series, giving

$$\begin{array}{c}\hfill \left({\mathcal{L}}_{0,{\mu }_0}-{\lambda }_0\right){\mathbf{w}}_{\mathbf{2}}=\sum _{\begin{array}{c}j=n-2\\ j\ne n-1\end{array}}^{m+2}{\mathbf{T}}_{\mathbf{2},\mathbf{j}}{e}^{ijx}.\end{array}$$

(43)

It can be shown that ${\mathbf{T}}_{\mathbf{2},\mathbf{n}-\mathbf{1}}=\mathbf{0}$.

Proceeding similarly to the previous order, solvability conditions for (43) are

$$\begin{array}{cc}\hfill 2\left({\lambda }_2+i{\mathcal{C}}_{1,n}\right)+i{\gamma }_0{\mathcal{S}}_{2,n}& =0,\hfill \end{array}$$

(44a)

$$\begin{array}{cc}\hfill 2{\gamma }_0\left({\lambda }_2+i{\mathcal{C}}_{-1,m}\right)+i{\mathcal{S}}_{2,m}& =0,\hfill \end{array}$$

(44b)

where

$$\begin{array}{cc}\hfill {\mathcal{C}}_{1,n}& ={\mu }_2{c}_{g_1}({\mu }_0+n)-{\mathcal{P}}_{2,n},\hfill \end{array}$$

(45a)

$$\begin{array}{cc}\hfill {\mathcal{C}}_{-1,m}& ={\mu }_2{c}_{g_{-1}}({\mu }_0+m)-{\mathcal{P}}_{2,m}.\hfill \end{array}$$

(45b)

Expressions for ${\mathcal{S}}_{2,j}$ and ${\mathcal{P}}_{2,j}$ have no dependence on ${\gamma }_0$, ${\gamma }_1$, ${\mu }_2$, or ${\lambda }_2$; see the file in Data Availability Statement for details.

Conditions (44a) and (44b) form a nonlinear system for ${\gamma }_0$ and ${\lambda }_2$. Solving for ${\lambda }_2$ yields

$$\begin{array}{c}\hfill {\lambda }_2=-i\left(\frac{{\mathcal{C}}_{-1,m}+{\mathcal{C}}_{1,n}}{2}\right)\pm \sqrt{-{\left(\frac{{\mathcal{C}}_{-1,m}-{\mathcal{C}}_{1,n}}{2}\right)}^2+\frac{{\mathcal{S}}_2^2}{4{\omega }_{\alpha }({\mu }_0+m){\omega }_{\alpha }({\mu }_0+n)}},\end{array}$$

(46)

where

$$\begin{array}{c}\hfill {\mathcal{S}}_{2,n}{\mathcal{S}}_{2,m}=-\frac{{\mathcal{S}}_2^2}{{\omega }_{\alpha }({\mu }_0+m){\omega }_{\alpha }({\mu }_0+n)}.\end{array}$$

(47)

A corollary of Lemma A3 in Appendix B shows that ${\omega }_{\alpha }({\mu }_0+m){\omega }_{\alpha }({\mu }_0+n)$ is positive. (This corollary is equivalent to satisfying the Krein signature condition mentioned in Section 3) Provided ${\mathcal{S}}_2\ne 0$ and $c_{g_{-1}}({\mu }_0+m)\ne c_{g_1}({\mu }_0+n)$, ${\lambda }_2$ has nonzero real part for ${\mu }_2\in (M_{2,-},M_{2,+})$, where

$$\begin{array}{cc}\hfill M_{2,\pm }=& {\mu }_{2,\ast }\pm \frac{|{\mathcal{S}}_2|}{|c_{g_{-1}}({\mu }_0+m)-c_{g_1}({\mu }_0+n)|\sqrt{{\omega }_{\alpha }({\mu }_0+m){\omega }_{\alpha }({\mu }_0+n)}},\hfill \end{array}$$

(48)

and

$${\mu }_{2,\ast }=\frac{{\mathcal{P}}_{2,m}-{\mathcal{P}}_{2,n}}{c_{g_{-1}}({\mu }_0+m)-c_{g_1}({\mu }_0+n)}.$$

(49)

That $c_{g_{-1}}({\mu }_0+m)\ne c_{g_1}({\mu }_0+n)$ follows from Lemma A2 in Appendix B. A plot of ${\mathcal{S}}_2$ as a function of $\alpha$ suggests that ${\mathcal{S}}_2>0$ for all values of $\alpha >0$ (Figure 3). We conjecture that HPT–BW Stokes waves of any wavenumer experience a $p=2$ high-frequency instability at $\mathcal{O}\left({\epsilon }^2\right)$.

For ${\mu }_2\in (M_{2,-},M_{2,+})$, a quick calculation shows that (46) parameterizes an ellipse asymptotic to the numerically observed $p=2$ high-frequency isola (Figure 4). The ellipse has semi-major and -minor axes that scale like ${\epsilon }^2$, and the center of the ellipse drifts along the imaginary axis from ${\lambda }_0$ like ${\epsilon }^2$.

The midpoint of $(M_{2,-},M_{2,+})$ maximizes the real part of ${\lambda }_2$. Thus, the most unstable spectral element of the isola has Floquet exponent

$$\begin{array}{c}\hfill {\mu }_{\ast }={\mu }_0+{\mu }_{2,\ast }{\epsilon }^2+\mathcal{O}\left({\epsilon }^3\right),\end{array}$$

(50)

and its real and imaginary components are

$$\begin{array}{cc}\hfill {\lambda }_{r,\ast }& =\left(\frac{|{\mathcal{S}}_2|}{2\sqrt{{\omega }_{\alpha }({\mu }_0+m){\omega }_{\alpha }({\mu }_0+n)}}\right){\epsilon }^2+\mathcal{O}\left({\epsilon }^3\right),\hfill \end{array}$$

(51a)

$$\begin{array}{cc}\hfill {\lambda }_{i,\ast }& =-{\mathsf{\Omega }}_1({\mu }_0+n)-{\mathcal{C}}_{1,n}{\epsilon }^2+\mathcal{O}\left({\epsilon }^3\right),\hfill \end{array}$$

(51b)

respectively. These expansions agree well with the FFH results (Figure 5).

5. High-Frequency Instabilities: p = 3

According to Theorem A1 in Appendix B, the $p=3$ high-frequency instability is the second-closest to the origin. This instability arises at $\mathcal{O}\left({\epsilon }^3\right)$. Let ${\mu }_0$ correspond to the unique Floquet exponent in $[-1/2,1/2]$ that satisfies the collision condition (24) with $m-n=3$. Then, the spectral data (25) give rise to the $p=3$ high-frequency instability. We assume these data and the Floquet exponent vary analytically with $\epsilon$. For uniqueness, we normalize the eigenfunction $\mathbf{w}$ according to (28) so that ${\mathbf{w}}_{\mathbf{0}}$ is given by (29).

The calculations proceed similarly as in the $p=2$ case, with two major exceptions. (i) At $\mathcal{O}\left({\epsilon }^2\right)$, the solvability conditions are linear in ${\lambda }_2$ and ${\mu }_2$ and independent of ${\gamma }_0$, leading only to a purely imaginary correction to the isola at this order. (ii) At $\mathcal{O}\left({\epsilon }^3\right)$, the solvability conditions depend nonlinearly on ${\lambda }_3$ and ${\gamma }_0$, but also include a term proportional to ${\gamma }_1$. One can show that this term vanishes due to the solvability conditions at $\mathcal{O}\left({\epsilon }^2\right)$. For more details, see Appendix C.

Remark 7.

In Appendix C and what follows, explicit expressions of select quantities are suppressed for ease of readability. Some quantities borrow notation from the $p=2$ case (but evaluate differently). The interested reader may consult the file in Data Availability Statement hptbw_isolap3.nb for these expressions.

Solving for ${\lambda }_3$ in the solvability conditions at $\mathcal{O}\left({\epsilon }^3\right)$, one finds

$$\begin{array}{cc}\hfill {\lambda }_3=& -i{\mu }_3\left(\frac{c_{g_{-1}}({\mu }_0+m)+c_{g_1}({\mu }_0+n)}{2}\right)\hfill \\ \hfill & \pm \sqrt{-{\mu }_3^2{\left(\frac{c_{g_{-1}}({\mu }_0+m)-c_{g_1}({\mu }_0+n)}{2}\right)}^2+\frac{{\mathcal{S}}_3^2}{4{\omega }_{\alpha }({\mu }_0+m){\omega }_{\alpha }({\mu }_0+n)}},\hfill \end{array}$$

(52)

where ${\mathcal{S}}_3$ is given in the file in Data Availability Statement. As in the $p=2$ case, ${\omega }_{\alpha }({\mu }_0+m){\omega }_{\alpha }({\mu }_0+n)>0$ and $c_{g_{-1}}({\mu }_0+m)\ne c_{g_1}({\mu }_0+n)$. Provided ${\mathcal{S}}_3\ne 0$, ${\lambda }_3$ has nonzero real part if ${\mu }_3\in (-M_3,M_3)$, where

$$\begin{array}{c}\hfill M_3=\frac{|{\mathcal{S}}_3|}{|c_{g_{-1}}({\mu }_0+m)-c_{g_1}({\mu }_0+n)|\sqrt{{\omega }_{\alpha }({\mu }_0+m){\omega }_{\alpha }({\mu }_0+n)}}.\end{array}$$

(53)

A plot of ${\mathcal{S}}_3$ vs. $\alpha$ reveals that ${\mathcal{S}}_3=0$ only at $\alpha =1.1862...$ (Figure 6). For this aspect ratio, the $p=3$ instability does not occur at $\mathcal{O}\left({\epsilon }^3\right)$. In fact, Figure 6 shows that, if $\alpha$ approaches 1.1862.... for fixed $\epsilon$, the numerically computed $p=3$ isola shrinks to a point on the imaginary axis. We conjecture that HPT–BW Stokes waves with $\alpha =1.1862...$ do not have a $p=3$ instability, even beyond $\mathcal{O}\left({\epsilon }^3\right)$. In the next subsection, we find that ${\lambda }_4$ is purely imaginary, so Stokes waves with aspect ratio $\alpha =1.1862...$ do not exhibit $p=3$ instabilities to $\mathcal{O}\left({\epsilon }^4\right)$.

Assuming $\alpha \ne 1.1862...$, ${\mu }_3\in (-M_3,M_3)$ parameterizes an ellipse asymptotic to the $p=3$ high-frequeny isola; see Figure 7. The ellipse has semi-major and -minor axes that scale like ${\epsilon }^3$. The center of this ellipse drifts along the imaginary axis like ${\epsilon }^2$ due to the purely imaginary correction found at $\mathcal{O}\left({\epsilon }^2\right)$.

The interval of Floquet exponents that parameterizes the $p=3$ isola is

$$\begin{array}{c}\hfill \mu \in \left({\mu }_0+{\mu }_2{\epsilon }^2-M_3{\epsilon }^3,{\mu }_0+{\mu }_2{\epsilon }^2+M_3{\epsilon }^3\right)+\mathcal{O}\left({\epsilon }^4\right),\end{array}$$

(54)

where ${\mu }_2$ is given by (A23b). The width of this interval is an order of magnitude smaller than that of the $p=2$ isola. Consequently, the $p=3$ isola is more challenging to find numerically than the $p=2$ isola, at least for methods similar to FFH (

Table 1. Intervals of Floquet exponents that parameterize the $p=2$ and $p=3$ high-frequency isolas with $\epsilon ={10}^{-3}$ and $\alpha =1/2,1$, and 2. The first digit for which the boundary values disagree is underlined and colored red. If a uniform mesh of Floquet exponents in $[-1/2,1/2]$ is used for numerical methods like FFH, the spacing of the mesh must be finer than ${\epsilon }^2$ to capture the $p=2$ instability and ${\epsilon }^3$ to capture the $p=3$ instability. The intervals vary with $\alpha$ as well, making it difficult to adapt and refine a uniform mesh to find high-frequency isolas.

	$\mathit{p}=2$
$\alpha =\frac{1}{2}$	(−0.106478813547533, −0.106478633575956)
$\alpha =1$	(−0.260909131823605, −0.260908917941151)
$\alpha =2$	(−0.330352196060556, −0.330352275321770)
	$\mathbf{p}=\mathbf{3}$
$\alpha =\frac{1}{2}$	(−0.375448877009085, −0.375448875412116)
$\alpha =1$	(0.257196721100572, 0.257196721343587)
$\alpha =2$	(0.044058331346416, 0.044058331384758)

).

For $\alpha =1$ and $\left|\epsilon \right|<5\times {10}^{-4}$, (54) provides an excellent approximation to the numerically computed interval of Floquet exponents (Figure 8). Fourth-order corrections are necessary to improve agreement between (54) and numerical computations for larger $\epsilon$, as is done below.

Choosing ${\mu }_3=0$ maximizes the real part of ${\lambda }_3$. Thus, the most unstable spectral element of the $p=3$ isola has Floquet exponent

$$\begin{array}{c}\hfill {\mu }_{\ast }={\mu }_0+{\mu }_2{\epsilon }^2+\mathcal{O}\left({\epsilon }^4\right),\end{array}$$

(55)

where ${\mu }_2$ is as in (A23b), and its real and imaginary components are

$$\begin{array}{cc}\hfill {\lambda }_{r,\ast }& =\left(\frac{|{\mathcal{S}}_3|}{2\sqrt{{\omega }_{\alpha }({\mu }_0+m){\omega }_{\alpha }({\mu }_0+n)}}\right){\epsilon }^3+\mathcal{O}\left({\epsilon }^4\right),\hfill \end{array}$$

(56a)

$$\begin{array}{cc}\hfill {\lambda }_{i,\ast }& =-{\mathsf{\Omega }}_1({\mu }_0+n)-\left(\frac{{\mathcal{P}}_{2,m}{c}_{g_1}({\mu }_0+n)-{\mathcal{P}}_{2,n}{c}_{g_{-1}}({\mu }_0+m)}{c_{g_{-1}}({\mu }_0+m)-c_{g_1}({\mu }_0+n)}\right){\epsilon }^2+\mathcal{O}\left({\epsilon }^4\right),\hfill \end{array}$$

(56b)

respectively. Expressions for ${\mathcal{P}}_{2,n}$ and ${\mathcal{P}}_{2,m}$ are found in the file in Data Availability Statement. The expansion for ${\lambda }_{r,\ast }$ is in excellent agreement with numerical results using the FFH method (Figure 8). As with (54), corrections to ${\mu }_{\ast }$ and ${\lambda }_{i,\ast }$ at $\mathcal{O}\left({\epsilon }^4\right)$ improve the agreement between numerical and asymptotic results for these quantities.

Before proceeding to $\mathcal{O}\left({\epsilon }^4\right)$, we solve the $\mathcal{O}\left({\epsilon }^3\right)$ problem (Equation (A27) in Appendix C), subject to the solvability and normalization conditions. We find

$$\begin{array}{c}\hfill {\mathbf{w}}_{\mathbf{3}}=\sum _{\begin{array}{c}j=n-3\\ j\ne n-2\end{array}}^{m+3}{\mathcal{W}}_{\mathbf{3},\mathbf{j}}{e}^{ijx}+{\gamma }_3\left(\begin{array}{c}1\\ -\frac{{\omega }_{\alpha }(m+{\mu }_0)}{\alpha (m+{\mu }_0)}{e}^{imx}\end{array}\right),\end{array}$$

(57)

where ${\gamma }_3$ is arbitrary, ${\mathcal{W}}_{\mathbf{3},\mathbf{n}-\mathbf{2}}=\mathbf{0}$, and the remaining ${\mathcal{W}}_{\mathbf{3},\mathbf{j}}$ are in the file in Data Availability Statement.

The $\mathcal{O}\left({\epsilon }^4\right)$ Problem

In this section, we derive corrections to the $p=3$ isola beyond its leading-order behavior. As will be seen, a new condition must be introduced to close the solvability conditions at this order. This extends the perturbation method introduced in [22]. We anticipate that our extension generalizes for corrections beyond the leading-order behavior of any isola.

At $\mathcal{O}\left({\epsilon }^4\right)$,

$$\begin{array}{c}\hfill \left({\mathcal{L}}_{0,{\mu }_0}-{\lambda }_0\right){\mathbf{w}}_{\mathbf{4}}=\sum _{j=2}^4{\lambda }_j{\mathbf{w}}_{\mathbf{3}-\mathbf{j}}-\sum _{j=1}^4{\mathcal{L}}_j{|}_{{\mu }_1=0}{\mathbf{w}}_{\mathbf{3}-\mathbf{j}},\end{array}$$

(58)

where ${\mathcal{L}}_j{|}_{{\mu }_1=0}$ are as before and

$$\begin{array}{c}\hfill {\mathcal{L}}_4{|}_{{\mu }_1=0}=\alpha \left(\begin{array}{cc}{\mathcal{L}}_4^{(1,1)}& {\mathcal{L}}_4^{(1,2)}\\ {\mathcal{L}}_4^{(2,1)}& {\mathcal{L}}_4^{(1,1)}\end{array}\right),\end{array}$$

(59)

with

$$\begin{array}{cc}\hfill {\mathcal{L}}_4^{(1,1)}& =i{c}_0{\mu }_4-i{\mu }_3{u}_1+i{\mu }_2(c_2-u_2)+(c_4-u_4)(i{\mu }_0+{\partial }_x)-u_4^{\prime },\hfill \end{array}$$

(60a)

$$\begin{array}{cc}\hfill {\mathcal{L}}_4^{(1,2)}& =-i{\mu }_4-i{\mu }_3{\eta }_1-i{\mu }_2{\eta }_2-{\eta }_4(i{\mu }_0+{\partial }_x)-{\eta }_4^{\prime },\hfill \end{array}$$

(60b)

$$\begin{array}{cc}\hfill {\mathcal{L}}_4^{(2,1)}& =-i{\mu }_4{\mathrm{sech}}^2\left(\alpha ({\mu }_0+D)\right)+i\alpha {\mu }_2{\mathrm{sech}}^2\left(\alpha ({\mu }_0+D)\right)tanh\left(\alpha ({\mu }_0+D)\right).\hfill \end{array}$$

(60c)

Substituting ${\eta }_j$, $u_j$, and ${\mathbf{w}}_{\mathbf{j}-\mathbf{1}}$ for $j\in \{1,2,3\}$ into (58), we find

$$\begin{array}{c}\hfill \left({\mathcal{L}}_{0,{\mu }_0}-{\lambda }_0\right){\mathbf{w}}_{\mathbf{4}}=\sum _{\begin{array}{c}j=n-4\\ j\ne n-3\end{array}}^{m+4}{\mathbf{T}}_{\mathbf{4},\mathbf{j}}{e}^{ijx},\end{array}$$

(61)

where ${\mathbf{T}}_{\mathbf{4},\mathbf{n}-\mathbf{3}}=\mathbf{0}$ (since ${\mathcal{W}}_{\mathbf{3},\mathbf{n}-\mathbf{2}}=\mathbf{0}$).

The solvability conditions for (61) can be expressed as

$$\begin{array}{c}\hfill \left(\begin{array}{cc}2& i{\mathcal{S}}_{3,n}\\ 2{\gamma }_0& 2({\lambda }_3+i{\mu }_3{c}_{g_{-1}}({\mu }_0+m))\end{array}\right)\left(\begin{array}{c}{\lambda }_4\\ {\gamma }_1\end{array}\right)+i{\gamma }_2\left(\begin{array}{c}0\\ {\mathcal{T}}_{4,m}\end{array}\right)=-2i\left(\begin{array}{c}{\mu }_4{c}_{g_1}({\mu }_0+n)-{\mathcal{P}}_{4,n}\\ {\gamma }_0\left({\mu }_4{c}_{g_{-1}}({\mu }_0+m)-{\mathcal{P}}_{4,m}\right)\end{array}\right).\end{array}$$

(62)

Expressions for ${\mathcal{P}}_{4,j}$ are in the file in Data Availability Statement. Using the solvability condition (A22b) together with the collision condition (24) shows that ${\mathcal{T}}_{4,m}\equiv 0$. What remains is a linear system for ${\lambda }_4$ and ${\gamma }_1$.

If $\alpha \ne 1.1862...$, then an application of the third-order solvability condition (A28a) shows that, for ${\mu }_3\in (-M_3,M_3)$,

$$\begin{array}{c}\hfill \det \left(\begin{array}{cc}2& i{\mathcal{S}}_{3,n}\\ 2{\gamma }_0& 2({\lambda }_3+i{\mu }_3{c}_{g_{-1}}({\mu }_0+m))\end{array}\right)=8{\lambda }_{3,r},\end{array}$$

(63)

where ${\lambda }_{3,r}=\mathrm{Re}\left({\lambda }_3\right)$. For ${\mu }_3$ in this interval, ${\lambda }_{3,r}\ne 0$ by construction; thus, (62) is an invertible linear system.

We solve (62) for ${\lambda }_4$ by Cramer’s rule, using (A28a) to eliminate the dependence on ${\gamma }_0$ and ${\mathcal{S}}_{3,n}$. Then,

$$\begin{array}{cc}\hfill {\lambda }_4& =-i[\frac{({\lambda }_3+i{\mu }_3{c}_{g_{-1}}({\mu }_0+m))({\mu }_4{c}_{g_1}({\mu }_0+n)-{\mathcal{P}}_{4,n})}{2{\lambda }_{3,r}}\hfill \\ \hfill & +\frac{({\lambda }_3+i{\mu }_3{c}_{g_1}({\mu }_0+n))({\mu }_4{c}_{g_{-1}}({\mu }_0+m)-{\mathcal{P}}_{4,m})}{2{\lambda }_{3,r}}].\hfill \end{array}$$

(64)

We decompose (64) into its real and imaginay components, noting ${\mathcal{P}}_{4,n},{\mathcal{P}}_{4,m},$ and ${\mu }_4\in \mathbb{R}$. We find ${\lambda }_4={\lambda }_{4,r}+i{\lambda }_{4,i}$, where

$$\begin{array}{cc}\hfill {\lambda }_{4,r}& =\frac{{\mu }_3\left(c_{g_{-1}}({\mu }_0+m)-c_{g_1}({\mu }_0+n)\right)\left[-{\mu }_4(c_{g_{-1}}({\mu }_0+m)-c_{g_1}({\mu }_0+n))+{\mathcal{P}}_{4,m}-{\mathcal{P}}_{4,n}\right]}{4{\lambda }_{3,r}},\hfill \end{array}$$

(65a)

$$\begin{array}{cc}\hfill {\lambda }_{4,i}& =-\frac{1}{2}\left[{\mu }_4(c_{g_{-1}}({\mu }_0+m)+c_{g_1}({\mu }_0+n))-({\mathcal{P}}_{4,m}+{\mathcal{P}}_{4,n})\right].\hfill \end{array}$$

(65b)

As $|{\mu }_3|\to M_3$, ${\lambda }_{3,r}\to 0$. If ${\lambda }_{4,r}$ is to remain bounded, the numerator of (65a) must vanish in this limit. Since $c_{g_{-1}}({\mu }_0+m)\ne c_{g_1}({\mu }_0+n)$, we must have

$$\begin{array}{c}\hfill {\mu }_4=\frac{{\mathcal{P}}_{4,m}-{\mathcal{P}}_{4,n}}{c_{g_{-1}}({\mu }_0+m)-c_{g_1}({\mu }_0+n)}.\end{array}$$

(66)

We refer to this equality as the regular curve condition: it ensures that the curve asymptotic to the $p=3$ isola remains continuous near its intersections with the imaginary axis. From the regular curve condition, we obtain

$$\begin{array}{c}\hfill {\lambda }_4=-i\left(\frac{{\mathcal{P}}_{4,m}{c}_{g_1}({\mu }_0+n)-{\mathcal{P}}_{4,n}{c}_{g_{-1}}({\mu }_0+m)}{c_{g_{-1}}({\mu }_0+m)-c_{g_1}({\mu }_0+n)}\right).\end{array}$$

(67)

As expected, the Floquet parameterization and imaginary component of the $p=3$ isola have a nonzero correction at $\mathcal{O}\left({\epsilon }^4\right)$. These corrections improve the agreement between numerical and asymptotic results observed at the previous order (Figure 9 and Figure 10). No corrections to the real component of the isola are found at fourth order.

Remark 8.

If $\alpha =1.1862\dots$, one can show that ${\lambda }_3=0={\mu }_3$ and ${\mathcal{S}}_{3,n}=0$. Applying the Fredholm alternative to (62) gives (66). Then, ${\lambda }_4$ is given by (67), and ${\gamma }_0=1$. The constant ${\gamma }_1$ remains arbitrary at this order for this value of α only.

6. Conclusions

We have extended a formal perturbation method, introduced in [22], to obtain asymptotic behavior of the largest ($p=2,3$) high-frequency instabilities of small-amplitude, HPT–BW Stokes waves. In particular, we have computed explicit expressions for (i) the interval of Floquet exponents that asymptotically parameterize the pth isola, (ii) the leading-order behavior of its most unstable spectral elements, (iii) the leading-order curve asymptotic to the isola, and (iv) wavenumbers that do not have a pth isola. Items (i)–(iii) can be extended to higher-order if necessary using the regular curve condition. In all instances, our perturbation calculations are in excellent agreement with numerical results computed by the FFH method [12].

We restrict to $p=2$ and $p=3$ in this work, but our method can provide asymptotic expressions for $p>3$ isolas. We conjecture that this method yields the first real-component correction of the isola at $\mathcal{O}\left({\epsilon }^p\right)$, similar to the cases $p=2$ and $p=3$. If correct, this conjecture highlights the main difficulty of computing higher-order high-frequency instabilities, both numerically and perturbatively.

The asymptotic expressions derived in this paper are intimidating and cumbersome. Although it is satisfying to have asymptotic expressions for the results previously obtained only numerically, this is not the main point of our work. Rather, (i) the perturbation method demonstrated allows one to approximate an entire isola at once, going beyond standard eigenvalue perturbation theory [27], (ii) the results obtained constitute a first step toward a proof of the presence of the high-frequency instabilities, and (iii) the asymptotic expressions for the range of Floquet exponents allow for a far more efficient numerical computation of the high-frequency isolas, which are difficult to track numerically as the amplitude of the solution increases.