Modulational instability in the full-dispersion Camassa–Holm equation

Abstract

We determine the stability and instability of a sufficiently small and periodic travelling wave to long-wavelength perturbations, for a nonlinear dispersive equation which extends a Camassa–Holm equation to include all the dispersion of water waves and the Whitham equation to include nonlinearities of medium-amplitude waves. In the absence of the effects of surface tension, the result qualitatively agrees with the Benjamin–Feir instability of a Stokes wave. In the presence of the effects of surface tension, it qualitatively agrees with those from formal asymptotic expansions of the physical problem and improves upon that for the Whitham equation, predicting the critical wave number at the strong surface tension limit. We discuss the modulational stability and instability in the Camassa–Holm equation and other related models.

1. Introduction

In the 1960s, Whitham (see [1], for instance) proposed

$${\eta }_t+c_{\mathrm{ww}}\left(\sqrt{\beta }|{\mathrm{\partial }}_x|\right){\eta }_x+(3\sqrt{(1+\alpha \eta )}-3){\eta }_x=0,$$ 1.1

to argue for wave breaking in shallow water. That is, the solution remains bounded but its slope becomes unbounded in finite time. Here, $t\in \mathbb{R}$ is proportional to elapsed time, $x\in \mathbb{R}$ is the spatial variable in the primary direction of wave propagation and η=η(x,t) is the fluid surface displacement from the undisturbed depth=1; α is the dimensionless amplitude parameter and β is the long-wavelength parameter. Moreover, c_ww(|∂_x|) is a Fourier multiplier operator, defined as

$$\widehat{c_{\mathrm{ww}}(|{\mathrm{\partial }}_x|)f}(\kappa )=\sqrt{\frac{\tanh \kappa }{\kappa }}\hat{f}(\kappa ).$$ 1.2

Note that c_ww(κ) is the phase speed in the linear theory of water waves. For small-amplitude waves satisfying α≪1, we may expand the nonlinearity of (1.1) up to terms of order α to arrive at

$${\eta }_t+c_{\mathrm{ww}}\left(\sqrt{\beta }|{\mathrm{\partial }}_x|\right){\eta }_x+\frac{3}{2}\alpha \eta {\eta }_x=0.$$ 1.3

For relatively shallow water or, equivalently, relatively long waves satisfying β≪1, we may expand the right-hand side of (1.2) up to terms of order β to find

$$c_{\mathrm{ww}}\left(\sqrt{\beta }k\right)=1-\frac{1}{6}\beta k^2+O({\beta }^2).$$

Therefore, for small-amplitude and long waves satisfying α=O(β) and β≪1, (1.1) becomes the famous Korteweg–de Vries equation

$${\eta }_t+{\eta }_x+\frac{1}{6}\beta {\eta }_{xxx}+\frac{3}{2}\alpha \eta {\eta }_x=0.$$ 1.4

As a matter of fact, for physically relevant initial data, the solutions of the Whitham equation and the Korteweg–de Vries equation differ from those of the water wave problem merely by higher-order terms during the relevant time scale; see [2], for instance, for details. But (1.3) and (1.2) may offer improvements over (1.4) for short waves. Whitham conjectured wave breaking in (1.3) and (1.2), and Hur [3] (see also [4]) recently solved it. By contrast, no solutions of (1.4) break.

Moreover, Johnson & Hur [5] showed that a sufficiently small and 2π/κ periodic travelling wave of the Whitham equation is spectrally unstable to long-wavelength perturbations, provided that κ>1.146… . In other words, (1.3) (or (1.1)) and (1.2) predict the Benjamin–Feir instability of a Stokes wave (see [6–9], for instance). By contrast, periodic travelling waves of the Korteweg–de Vries equation are all modulationally stable. By the way, under the assumption that η_t+η_x is small, one may modify (1.4) to arrive at the Benjamin–Bona–Mahony equation

$${\eta }_t+{\eta }_x-\frac{1}{6}\beta {\eta }_{xxt}+\frac{3}{2}\alpha \eta {\eta }_x=0.$$ 1.5

Hur & Pandey [10] showed that a sufficiently small and 2π/κ periodic travelling wave of (1.5) is modulationally unstable if $\kappa >\sqrt{3}$. Hence, the Benjamin–Bona–Mahony equation seems to predict the Benjamin–Feir instability. But the instability mechanism is different from that in the Whitham equation, or the water wave problem; see [10] for details.

Furthermore, in the presence of the effects of surface tension, Johnson & V.M.H. [11] determined the modulational stability and instability of a sufficiently small and periodic travelling wave of (1.3) and

$$\widehat{c_{\mathrm{ww}}(|{\mathrm{\partial }}_x|;T)f}(\kappa )=\sqrt{(1+T{\kappa }^2)\frac{\tanh \kappa }{\kappa }}\hat{f}(\kappa ),$$ 1.6

where T is the coefficient of surface tension. The result agrees by and large with those in [12,13], for instance, from formal asymptotic expansions of the physical problem. But it fails to predict the critical wave number at the ‘strong surface tension’ limit. Perhaps, this is not surprising because (1.3) neglects higher-order nonlinearities of the water wave problem. It is interesting to find a model which predicts the modulational stability and instability of a gravity capillary wave.

By the way, V.M.H & A.K.P. [14] recently extended the Whitham equation to include bidirectional propagation and showed that the ‘full-dispersion shallow water equations’ correctly predict the capillary effects on the Benjamin–Feir instability. Here, we seek higher-order nonlinearities in a unidirectional propagation model.

As a matter of fact, for medium-amplitude and long waves satisfying $\alpha =O\left(\sqrt{\beta }\right)$ and β≪1, the Camassa–Holm equations for the fluid surface displacement

$${\eta }_t+{\eta }_x+\beta (a{\eta }_{xxx}+b{\eta }_{xxt})+\frac{3}{2}\alpha \eta {\eta }_x-\frac{3}{8}{\alpha }^2{\eta }^2{\eta }_x+\frac{3}{16}{\alpha }^3{\eta }^3{\eta }_x=-\alpha \beta (c\eta {\eta }_{xxx}+d{\eta }_x{\eta }_{xx})$$ 1.7

and for the average horizontal velocity

$$u_t+u_x+\beta (a{u}_{xxx}+b{u}_{xxt})+\frac{3}{2}\alpha u{u}_x=-\alpha \beta (cu{u}_{xxx}+d{u}_x{u}_{xx}),$$ 1.8

where

$$0\le a\le \frac{1}{6},b=a-\frac{1}{6},c=\frac{3}{2}a+\frac{1}{6}\text{and}d=\frac{9}{2}a+\frac{5}{24},$$

extend the Korteweg–de Vries equation to include higher-order nonlinearities; see [2], for instance, for details. In the case of $a=\frac{1}{12}$, (1.7) becomes

$${\eta }_t+{\eta }_x+\frac{1}{12}\beta ({\eta }_{xxx}-{\eta }_{xxt})+\frac{3}{2}\alpha \eta {\eta }_x-\frac{3}{8}{\alpha }^2{\eta }^2{\eta }_x+\frac{3}{16}{\alpha }^3{\eta }^3{\eta }_x=-\frac{7}{24}\alpha \beta (\eta {\eta }_{xxx}+2{\eta }_x{\eta }_{xx}),$$

which is particularly interesting because it predicts wave breaking; see [2] and references therein. Note that

$$\frac{3\alpha \eta }{1+\sqrt{1+\alpha \eta }}=\frac{3}{2}\alpha \eta -\frac{3}{8}{\alpha }^2{\eta }^2+\frac{3}{16}{\alpha }^3{\eta }^3+O({\alpha }^4).$$

Lannes [2] combined the dispersion relation in the linear theory of water waves and nonlinearities of a Camassa–Holm equation, to propose the full-dispersion Camassa–Holm (FDCH) equation for the fluid surface displacement

$${\eta }_t+c_{\mathrm{ww}}\left(\sqrt{\beta }|{\mathrm{\partial }}_x|\right){\eta }_x+\frac{3\alpha \eta }{1+\sqrt{1+\alpha \eta }}{\eta }_x=-\alpha \beta (\frac{5}{12}\eta {\eta }_{xxx}+\frac{23}{24}{\eta }_x{\eta }_{xx}),$$ 1.9

where c_ww(|∂_x|) is in (1.2), or (1.6) in the presence of the effects of surface tension. For long waves satisfying β≪1, (1.9) and (1.2) agree with (1.7), where $a=\frac{1}{6}$, up to terms of order β. But, including all the dispersion of water waves, (1.9) and (1.2) may offer improvements over (1.7) for short waves. For small-amplitude waves satisfying α≪1, (1.9) agrees with (1.3) up to terms of order α. But, including higher-order nonlinearities, (1.9) may offer improvements over (1.3) for medium-amplitude waves. For the average horizontal velocity, we may combine (1.2), or (1.6) in the presence of the effects of surface tension, and (1.8) to propose

$$u_t+c_{\mathrm{ww}}\left(\sqrt{\beta }|{\mathrm{\partial }}_x|\right)u_x+\frac{3}{2}\alpha u{u}_x=-\alpha \beta (\frac{5}{12}u{u}_{xxx}+\frac{23}{24}{u}_x{u}_{xx}).$$ 1.10

Note that it differs from (1.9) by higher power nonlinearities.

We follow along the same line as the arguments in [5,10,11] (see also [15]) and investigate the modulational stability and instability in the FDCH equation. A main difference lies in that the nonlinearities of (1.9) include higher-order derivatives than that of the Whitham equation. For instance, a periodic travelling wave of (1.9) is not a priori smooth. We examine the mapping properties of various operators to construct a smooth solution.

In the absence of the effects of surface tension, we show that a sufficiently small and 2π/κ periodic travelling wave of (1.9) and (1.2) is spectrally unstable to long-wavelength perturbations, provided that

$$\kappa >1.420\dots$$

and stable to square integrable perturbations otherwise. The result qualitatively agrees with the Benjamin–Feir instability (see [7,8], for instance) and that for the Whitham equation [5]. The critical wave number compares reasonably well with 1.363… in [7,8]. Including the effects of surface tension, in the κ and $\kappa \sqrt{T}$ plane, we determine the regions of modulational stability and instability for a sufficiently small and periodic travelling wave of (1.9) and (1.6); see figure 3 for details. The result qualitatively agrees with those in [12,13], for instance, from formal asymptotic expansions of the physical problem, and it improves upon that in [11] for the Whitham equation. In particular, $\kappa (T)\sqrt{T}\to 1.283\dots$ as $T\to \mathrm{\infty }$, where κ(T) is a critical wave number, whereas the limit is unbounded for the Whitham equation [11].

Figure 3.

Stability diagram for sufficiently small and periodic travelling wave of (2.1) and (1.6). To interpret, for any T>0, one must envisage a line through the origin with slope T. ‘S’ and ‘U’ denote stable and unstable regions, respectively. Solid curves labelled 1 through 4 represent the roots of i₁ through i₄, respectively. (Online version in colour.)

Moreover, we show that a sufficiently small and 2π/κ periodic travelling wave of (1.7) is modulationally unstable if κ>6. To the best of the authors’ knowledge, this is new. Hence, the Camassa–Holm equation seems to predict the Benjamin–Feir instability. But the instability mechanism is different from that in (1.9) and (1.2), or the water wave problem, similarly to the Benjamin–Bona–Mahony equation [10]. It is interesting to use the Evans function and other ODE methods to determine the modulational stability and instability in (1.7) for all amplitudes. Our result indicates that the stability result depends on the carrier wave, unlike for the Korteweg–de Vries equation.

In the absence of the effects of surface tension, we show that a sufficiently small and 2π/κ periodic travelling wave of (1.10) and (1.2) is modulationally unstable if κ is greater than a critical value, similarly to the Benjamin–Feir instability. But, in the presence of the effects of surface tension, the modulational stability and instability in (1.10) and (1.6) qualitatively agree with that in the Whitham equation [11]. In particular, it fails to predict the critical wave number at the strong surface tension limit. Therefore, we learn that the higher-power nonlinearities of (1.9) capture the capillary effects on the Benjamin–Feir instability, not the higher-derivative nonlinearities.

It is interesting to explore breaking, peaking and other phenomena of water waves in (1.9) and (1.2) (or (1.6)). By the way, (1.9) and (1.10), where $a=\frac{1}{12}$, both predict wave breaking, but $\underset{x\in \mathbb{R}}{max}{\eta }_x(x,\cdot )\to \mathrm{\infty }$ at the breaking time for (1.9), while $\underset{x\in \mathbb{R}}{min}{u}_x(x,\cdot )\to -\mathrm{\infty }$ for (1.10); see [2] and references therein.

Notation. Let $\mathbb{T}$ denote the unit circle in $\mathbb{C}$. We identify functions over $\mathbb{T}$ with 2π periodic functions over $\mathbb{R}$ via f(e^iz)=F(z) and, for simplicity of notation, we write f(z) rather than f(e^iz). For p in the range $[1,\mathrm{\infty }]$, let $L^p(\mathbb{T})$ consist of real- or complex-valued, Lebesgue measurable, and 2π periodic functions over $\mathbb{R}$ such that

$$\parallel \hspace{0.17em}f{\parallel }_{L^p(\mathbb{T})}:={(\frac{1}{2\pi }{\int }_{-\pi }^{\pi }|f(z){|}^p\hspace{0.17em}\mathrm{d}z)}^{1/p}<\mathrm{\infty }\text{if}\hspace{0.17em}p<\mathrm{\infty }$$

and $\parallel \hspace{0.17em}f{\parallel }_{L^{\mathrm{\infty }}(\mathbb{T})}:={\text{ess sup}}_{-\pi <z\le \pi }|f(z)|<\mathrm{\infty }$ if $p=\mathrm{\infty }$. Let $H^1(\mathbb{T})$ consist of $L^2(\mathbb{T})$ functions whose derivative is in $L^2(\mathbb{T})$. Let $H^{\mathrm{\infty }}(\mathbb{T})=\bigcap _{k=0}^{\mathrm{\infty }}{H}^k(\mathbb{T})$.

For $f\in L^1(\mathbb{T})$, the Fourier series of f is defined by

$$\sum _{n\in \mathbb{Z}}\hspace{0.17em}\hat{f}(n)\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}nz},\text{where}\hspace{0.17em}\hat{f}(n)=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }f(z)\hspace{0.17em}{\mathrm{e}}^{-\mathrm{i}nz}\hspace{0.17em}\mathrm{d}z.$$

If $f\in L^2(\mathbb{T}),$ then its Fourier series converges to f pointwise almost everywhere. We define the $L^2(\mathbb{T})$ inner product as

$${⟨\hspace{0.17em}{f}_1,f_2⟩}_{L^2(\mathbb{T})}=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }{f}_1(z)f_2^{\ast }(z)\hspace{0.17em}\mathrm{d}z=\sum _{n\in \mathbb{Z}}\hspace{0.17em}{\hat{f}}_1(n){\hat{f}}_2^{\ast }(n).$$ 1.11

2. Sufficiently small and periodic travelling waves

We determine periodic travelling waves of the FDCH equation, after normalization of parameters,

$${\eta }_t+c_{\mathrm{ww}}(|{\mathrm{\partial }}_x|;T){\eta }_x+\frac{3\eta }{1+\sqrt{1+\eta }}{\eta }_x=-(\frac{5}{12}\eta {\eta }_{xxx}+\frac{23}{24}{\eta }_x{\eta }_{xx}),$$ 2.1

where c_ww(|∂_x|;T) is in (1.6), and we calculate their small-amplitude expansion.

For any T≥0, c_ww(⋅;T) is even and real analytic, and c_ww(0;T)=1. Note that $c_{\mathrm{ww}}(|{\mathrm{\partial }}_x|;0):H^s(\mathbb{R})\to H^{s+1/2}(\mathbb{R})$ for any $s\in \mathbb{R}$, and for T>0, $c_{\mathrm{ww}}(|{\mathrm{\partial }}_x|;T):H^{s+1/2}(\mathbb{R})\to H^s(\mathbb{R})$; see [5,11], for instance, for details.

Note that c_ww(⋅;0) decreases to zero monotonically away from the origin. For $T\ge \frac{1}{3}$, c_ww(⋅ ;T) increases monotonically and unboundedly away from the origin. For $0<T<\frac{1}{3}$, on the other hand, c_ww′(0;T)=0, c_ww′′(0;T)<0 and $c_{\mathrm{ww}}(\kappa ;T)\to \mathrm{\infty }$ as $\kappa \to \mathrm{\infty }$. Hence, c_ww(⋅;T) possesses a unique minimum over the interval $(0,\mathrm{\infty })$ (figure 1).

Figure 1.

Schematic plots of c_ww(⋅;T) for (a) T=0, (b) $T\ge \frac{1}{3}$ and (c) $0<T<\frac{1}{3}$. (Online version in colour.)

By a travelling wave of (2.1) and (1.6), we mean a solution of the form η(x,t)=η(x−ct) for some c>0, the wave speed, where η satisfies by quadrature

$$(c_{\mathrm{ww}}(|{\mathrm{\partial }}_x|;T)-c-3)\eta +2{(1+\eta )}^{3/2}-2+\frac{5}{12}\eta {\eta }_{xx}+\frac{13}{48}{\eta }_x^2={(1-c)}^{2}b$$

for some $b\in \mathbb{R}$. We seek a periodic travelling wave of (2.1) and (1.6). That is, η is a 2π periodic function of z:=κx for some κ>0, the wave number, and it satisfies

$$(c_{\mathrm{ww}}(\kappa |{\mathrm{\partial }}_z|;T)-c-3)\eta +2{(1+\eta )}^{3/2}-2+\frac{5}{12}{\kappa }^2\eta {\eta }_{zz}+\frac{13}{48}{\kappa }^2{\eta }_z^2={(1-c)}^{2}b.$$ 2.2

Note that

$$c_{\mathrm{ww}}(\kappa |{\mathrm{\partial }}_z|;0):H^s(\mathbb{T})\to H^{s+1/2}(\mathbb{T})\text{and}{c}_{\mathrm{ww}}(\kappa |{\mathrm{\partial }}_z|;T):H^{s+1/2}(\mathbb{T})\to H^s(\mathbb{T})$$ 2.3

for T>0, for any κ>0 and $s\in \mathbb{R}$. Note that

$$c_{\mathrm{ww}}(\kappa |{\mathrm{\partial }}_z|;T)\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}nz}=c_{\mathrm{ww}}(n\kappa ;T)\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}nz}\text{for }n\in \mathbb{Z}.$$ 2.4

Note that (2.2) remains invariant under

$$z\mapsto z+z_0\text{and}z\mapsto -z$$ 2.5

for any $z_0\in \mathbb{R}$. Hence, we may assume that η is even. But (2.2) does not possess scaling invariance. Hence, we may not a priori assume that κ=1. Moreover, (2.2) does not possess Galilean invariance. Hence, we may not a priori assume that b=0. By contrast, the Whitham equation for periodic travelling waves possesses Galilean invariance; see [11], for instance.

We follow along the same line as the arguments in [5,10,11], for instance, to construct periodic travelling waves of (2.1) and (1.6). But a difference lies in the lack of a priori smoothness of solutions of (2.2).

For any T≥0 and an integer k≥0, let

$$F:H^{k+2}(\mathbb{T})\times {\mathbb{R}}_{+}\times \mathbb{R}\times {\mathbb{R}}_{+}\to H^k(\mathbb{T})$$

denote

$$F(\eta ,c;b,\kappa ,T)=(c_{\mathrm{ww}}(\kappa |{\mathrm{\partial }}_z|;T)-c-3)\eta +2{(1+\eta )}^{3/2}-2+\frac{5}{12}{\kappa }^2\eta {\eta }_{zz}+\frac{13}{48}{\kappa }^2{\eta }_z^2-{(1-c)}^{2}b.$$ 2.6

It is well defined by (2.3) and a Sobolev inequality. We seek a solution $\eta \in H^{k+2}(\mathbb{T})$, c>0 and $b\in \mathbb{R}$ of

$$F(\eta ,c;b,\kappa ,T)=0.$$ 2.7

As k is arbitrary, $\eta \in H^{\mathrm{\infty }}(\mathbb{T})$. Note that F is invariant under (2.5). Hence, we may assume that η is even.

For any T≥0, and c>0, $b\in \mathbb{R}$, κ>0, note that

$$\begin{array}{rl}{F}_{\eta }(\eta ,c;b,\kappa ,T)\zeta & =(c_{\mathrm{ww}}(\kappa |{\mathrm{\partial }}_z|;T)-c-3+3{(1+\eta )}^{1/2}\\ & + {\kappa }^2(\frac{5}{12}({\eta }_{zz}+\eta {\mathrm{\partial }}_z^2)+\frac{13}{24}{\eta }_z{\mathrm{\partial }}_z))\zeta :H^{k+2}(\mathbb{T})\to H^k(\mathbb{T})\end{array}$$

is continuous by (2.3) and a Sobolev inequality. Here, a subscript means Fréchet differentiation. Moreover, for any T≥0, and $\eta \in H^{k+2}(\mathbb{T})$, κ>0, $b\in \mathbb{R}$, note that $F_c(\eta ;\kappa ,c,b)=-\eta +2(1-c)b:\mathbb{R}\to H^k(\mathbb{T})$ is continuous. As F_b(η;κ,c,b)=−(1−c)² and

$$F_{\kappa }(\eta ;\kappa ,c,b):=c_{\mathrm{ww}}^{\prime }(\kappa |{\mathrm{\partial }}_z|;T)\eta +\frac{5}{6}\kappa \eta {\eta }_{zz}+\frac{13}{24}\kappa {\eta }_z^2$$

are continuous likewise, F depends continuously differentiably on its arguments. Furthermore, as the Fréchet derivatives of F with respect to η, and c, b of all orders ≥3 are zero everywhere by brutal force, and as c_ww is a real analytic function, F is a real analytic operator.

(a). Bifurcation condition

For any T≥0, κ>0, for any c>0, $b\in \mathbb{R}$ and |b| sufficiently small, note that

$${\eta }_0(c;b,\kappa ,T)=b(1-c)+O(b^2),$$ 2.8

makes a constant solution of (2.6)–(2.7) and, hence, (2.2). It follows from the implicit function theorem that if non-constant solutions of (2.6)–(2.7) and, hence, (2.2) bifurcate from η=η₀ for some c=c₀ then, necessarily,

$$L_0:=F_{\eta }({\eta }_0,c_0;b,\kappa ,T):H^{k+2}(\mathbb{T})\to H^k(\mathbb{T}),$$

where

$$L_0=c_{\mathrm{ww}}(\kappa |{\mathrm{\partial }}_z|;T)-c_0-3+3{(1+{\eta }_0)}^{1/2}+\frac{5}{12}{\kappa }^2{\eta }_0{\mathrm{\partial }}_z^2,$$ 2.9

is not an isomorphism. Here, η₀ depends on c₀, but we suppress it for simplicity of notation. A straightforward calculation reveals that L₀ e^inz=0, $n\in \mathbb{Z}$, if and only if

$$c_0=c_{\mathrm{ww}}(n\kappa ;T)-3+3{(1+{\eta }_0)}^{1/2}-\frac{5}{12}{\kappa }^2{n}^2{\eta }_0.$$ 2.10

For b=0 and, hence, η₀=0 by (2.8), it simplifies to c₀=c_ww(nκ;T). Without loss of generality, we restrict the attention to n=1. For |b| sufficiently small, (2.10) and (2.8) become, respectively,

$$c_0(b,\kappa ,T)=c_{\mathrm{ww}}(\kappa ;T)+b(\frac{3}{2}-\frac{5}{12}{\kappa }^2)(1-c_{\mathrm{ww}}(\kappa ;T))+O(b^2)$$ 2.11

and

$${\eta }_0(b,\kappa ,T)=b(1-c_{\mathrm{ww}}(\kappa ;T))+O(b^2).$$ 2.12

For T=0, as c_ww(κ;0)>c_ww(nκ;0) for n=2,3,… everywhere in $\mathbb{R}$ (figure 1a), it is straightforward to verify that, for any κ>0, $b\in \mathbb{R}$ and |b| sufficiently small, the kernel of $L_0:H^{k+2}(\mathbb{T})\to H^k(\mathbb{T})$ is two-dimensional and spanned by e^±iz. Moreover, the co-kernel of L₀ is two-dimensional. Therefore, L₀ is a Fredholm operator of index zero.

Similarly, for $T\ge \frac{1}{3}$, since c_ww(κ;T)<c_ww(nκ;T) for n=2,3,… everywhere in $\mathbb{R}$ (figure 1b), for any κ>0, $b\in \mathbb{R}$ and |b| sufficiently small, $L_0:H^{k+2}(\mathbb{T})\to H^k(\mathbb{T})$ is a Fredholm operator of index zero, whose kernel is two-dimensional and spanned by e^±iz.

For $0<T<\frac{1}{3}$, on the other hand, for any integer n≥2, it is possible to find some κ such that c_ww(κ;T)=c_ww(nκ;T) (figure 1c). If

$$c_{\mathrm{ww}}(\kappa ;T)\ne c_{\mathrm{ww}}(n\kappa ;T)\text{for any }n=2,3,\dots ,$$ 2.13

then $L_0:H^{k+2}(\mathbb{T})\to H^k(\mathbb{T})$ is likewise a Fredholm operator of index zero, whose kernel is two-dimensional and spanned by e^±iz. But if c_ww(κ;T)=c_ww(nκ;T) for some integer n≥2, resulting in the resonance of the fundamental mode and the nth harmonic, then the kernel is four-dimensional. We do not discuss this here.

(b). Lyapunov–Schmidt procedure

For any T≥0, κ>0 satisfying (2.13), $b\in \mathbb{R}$ and |b| sufficiently small, we employ a Lyapunov–Schmidt procedure to construct non-constant solutions of (2.6)–(2.7) and, hence, of (2.2) bifurcating from η=η₀ and c=c₀, where η₀ and c₀ are in (2.12) and (2.11). Throughout the proof, T, κ and b are fixed and suppressed for simplicity of notation.

Recall that F(η₀,c₀)=0, where F is in (2.6), and L₀ e^±iz=0, where L₀ is in (2.9). We write that

$$\eta (z)={\eta }_0+\frac{1}{2}(a\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}z}+a^{\ast }\hspace{0.17em}{\mathrm{e}}^{-\mathrm{i}z})+{\eta }_r(z)\text{and}c=c_0+c_r,$$ 2.14

and we require that $a\in \mathbb{C}$, ${\eta }_r\in H^{k+2}(\mathbb{T})$ be even and

$${⟨{\eta }_r,{\mathrm{e}}^{\pm \mathrm{i}z}⟩}_{L^2(\mathbb{T})}=0,$$ 2.15

and $c_r\in \mathbb{R}$. Substituting (2.14) into (2.6)–(2.7), we use F(η₀,c₀)=0, L₀ e^±iz=0 and we make an explicit calculation to arrive at

$$\begin{array}{rl}{L}_0{\eta }_r& =(3{(1+{\eta }_0)}^{1/2}+c_r)(\frac{1}{2}(a\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}z}+a^{\ast }\hspace{0.17em}{\mathrm{e}}^{-\mathrm{i}z})+{\eta }_r)\\ & -2{(1+{\eta }_0+\frac{1}{2}(a\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}z}+a^{\ast }\hspace{0.17em}{\mathrm{e}}^{-\mathrm{i}z})+{\eta }_r)}^{3/2}\\ & -\frac{13}{48}{\kappa }^2{(\frac{\mathrm{i}}{2}(a\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}z}-a^{\ast }\hspace{0.17em}{\mathrm{e}}^{-\mathrm{i}z})+{\eta }_r^{\prime })}^2\\ & -\frac{5}{12}{\kappa }^2(\frac{1}{2}(a\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}z}+a^{\ast }\hspace{0.17em}{\mathrm{e}}^{-\mathrm{i}z})+{\eta }_r)(-\frac{1}{2}(a\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}z}+a^{\ast }\hspace{0.17em}{\mathrm{e}}^{-\mathrm{i}z})+{\eta }_r^{″})\\ & =:g({\eta }_r;a,a^{\ast },c_r).\end{array}$$ 2.16

Here and elsewhere, the prime means ordinary differentiation. Note that

$$g:H^{k+2}(\mathbb{T})\times \mathbb{C}\times \mathbb{C}\times \mathbb{R}\to H^k(\mathbb{T}).$$

Recall that F is a real analytic operator. Hence, g depends analytically on its arguments. Clearly, g(0;0,0,c_r)=0 for all $c_r\in \mathbb{R}$.

Let $\mathrm{\Pi }:L^2(\mathbb{T})\to \ker L_0$ denote the spectral projection, defined as

$$\mathrm{\Pi }f(z)=\hat{f}(1)\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}z}+\hat{f}(-1)\hspace{0.17em}{\mathrm{e}}^{-\mathrm{i}z}.$$

As Πη_r=0 by (2.15), we may rewrite (2.16) as

$$L_0{\eta }_r=(1-\mathrm{\Pi })g({\eta }_r;a,a^{\ast },c_r)\text{and}0=\mathrm{\Pi }g({\eta }_r;a,a^{\ast },c_r).$$ 2.17

Moreover, for any T≥0, and κ>0 satisfying (2.13), note that L₀ is invertible on $(1-\mathrm{\Pi })H^k(\mathbb{T})$. Specifically,

$${L_0}^{-1}f(z)=\sum _{n\ne \pm 1}\frac{\hat{f}(n)}{c_{\mathrm{ww}}(\kappa n;T)-c_{\mathrm{ww}}(\kappa ;T)+(5/12){\kappa }^2{\eta }_0(1-n^2)}\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}nz}.$$

Hence, we may rewrite (2.17) as

$${\eta }_r=L_0^{-1}(I-\mathrm{\Pi })g({\eta }_r;a,a^{\ast },c_r)\text{and}0=\mathrm{\Pi }g({\eta }_r;a,a^{\ast },c_r).$$ 2.18

Note that $L_0^{-1}:(1-\mathrm{\Pi })H^k(\mathbb{T})\to H^k(\mathbb{T})$ is bounded. We claim that

$$L_0^{-1}:(1-\mathrm{\Pi })H^k(\mathbb{T})\to H^{k+2}(\mathbb{T})$$

is bounded. As a matter of fact,

$$\left|\frac{n^2\hat{f}(n)}{c_{\mathrm{ww}}(\kappa n;T)-c_{\mathrm{ww}}(\kappa ;T)+\left(\frac{5}{12}\right){\kappa }^2{\eta }_0(1-n^2)}\right|\le C|\hat{f}(n)|$$

for some constant C>0 for $n\in \mathbb{Z}$ and |n| sufficiently large. Therefore, for any $a,a^{\ast }\in \mathbb{C}$ and $c_r\in \mathbb{R}$,

$$L_0^{-1}(1-\mathrm{\Pi })g:H^{k+2}(\mathbb{T})\to H^{k+2}(\mathbb{T})$$

is bounded. Note that it depends analytically on its argument. As g(0;0,0,c_r)=0 for any $c_r\in \mathbb{R}$, it follows from the implicit function theorem that a unique solution η₂=η_r(a,a*,c_r) exists for the former equation of (2.18) near η_r=0 for $a\in \mathbb{C}$ and |a| sufficiently small for any $c_r\in \mathbb{R}$. Note that η₂ depends analytically on its arguments and it satisfies (2.15) for |a| sufficiently small for any $c_r\in \mathbb{R}$. The uniqueness implies

$${\eta }_2(0,0,c_r)=0\text{for any }{c}_r\in \mathbb{R}.$$ 2.19

Moreover, as (2.6)–(2.7) and, hence, (2.18) are invariant under (2.5) for any $z_0\in \mathbb{R}$, it follows that

$${\eta }_2(a,a^{\ast },c_r)(z+z_0)={\eta }_2(a\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{z}_0},a^{\ast }\hspace{0.17em}{\mathrm{e}}^{-\mathrm{i}{z}_0},c_r)\text{and}{\eta }_2(a,a^{\ast },c_r)(-z)={\eta }_2(a,a^{\ast },c_r)(z)$$ 2.20

for any $z_0\in \mathbb{R}$ for any $a\in \mathbb{C}$ and |a| sufficiently small, and $c_r\in \mathbb{R}$.

To proceed, we rewrite the latter equation in (2.18) as

$$\mathrm{\Pi }g({\eta }_2(a,a^{\ast },c_r);a,a^{\ast },c_r)=0$$

for $a\in \mathbb{C}$ and |a| sufficiently small for $c_r\in \mathbb{R}$. This is solvable, provided that

$${\pi }_{\pm }(a,a^{\ast },c_r):={⟨g({\eta }_2(a,a^{\ast },c_r);a,a^{\ast },c_r),a\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}z}\pm a^{\ast }\hspace{0.17em}{\mathrm{e}}^{-\mathrm{i}z}⟩}_{L^2(\mathbb{T})}=0.$$ 2.21

We use (2.20), where $z_0=-2\arg (a)$, and (2.21) to show that

$${\pi }_{-}(a^{\ast },a,c_r)={\pi }_{-}(a,a^{\ast },c_r)=-{\pi }_{-}(a^{\ast },a,c_r).$$

Hence, π₋(a,a*,c_r)=0 holds for any $a\in \mathbb{C}$ and |a| sufficiently small for any $c_r\in \mathbb{R}$. Moreover, we use (2.20), where $z_0=-\arg (a)$, and (2.21) to show that

$${\pi }_{+}(a,a^{\ast },c_r)={\pi }_{+}(|a|,|a|,c_r).$$

Hence, it suffices to solve π₊(a,a,c_r)=0 for any $a,c_r\in \mathbb{R}$ and |a| sufficiently small.

Substituting (2.16) into (2.21), where η_r=η₂(a,a,c_r), we make an explicit calculation to arrive at

$${\pi }_{+}(a,a,c_r)=a^2(\pi c_r+{\pi }_r(a,c_r)),$$

where

$$\begin{array}{rl}{\pi }_r(a,c_r)& =-2a^{-1}⟨(1+{\eta }_0+a\cos z+{\eta }_2(a,a,c_r){(z))}^{3/2},\cos z⟩\\ & -\frac{5}{12}{\kappa }^2(⟨{\eta }_2^{″}(a,a,c_r)(z)-{\eta }_2(a,a,c_r)(z),{\cos }^{2}z⟩-a^{-1}⟨{\eta }_2{\eta }_2^{″}(a,a,c_r)(z),\cos z⟩)\\ & -\frac{13}{48}{\kappa }^2{a}^{-1}(⟨{\eta }_2^{\prime }(a,a,c_r){(z)}^2,\cos z⟩-⟨{\eta }_2^{\prime }(a,a,c_r)(z),\sin 2z⟩),\end{array}$$

and 〈⋅,⋅〉 means the $L^2(\mathbb{T})$ inner product. We claim that π_r is well defined. As a matter of fact, a⁻¹η₂ is not singular for $a\in \mathbb{R}$ and |a| sufficiently small by (2.19). Clearly, π_r and, hence, π_± depend analytically on its arguments. As π_r(0,0)=∂π_r/∂c_r(0,0)=0 by (2.19), it follows from the implicit function theorem that a unique solution c_r=c₁(a) exists for π₊(a,a,c_r)=0 and, hence, the latter equation of (2.18) near c_r=0 for $a\in \mathbb{R}$ and |a| sufficiently small. Clearly, c₁ depends analytically on a.

To recapitulate,

$${\eta }_r={\eta }_2(a,a,c_1(a))\text{and}{c}_r=c_1(a)$$

uniquely solve (2.18) for $a\in \mathbb{R}$ and |a| sufficiently small, and by virtue of (2.14),

$$\eta (a)(z)={\eta }_0+a\cos z+{\eta }_2(a,a,c_1(a))(z)\text{and}c(a)=c_0+c_1(a)$$ 2.22

uniquely solve (2.6)–(2.7) and, hence, (2.2) for $a\in \mathbb{R}$ and |a| sufficiently small. Note that η is 2π periodic and even in z. Moreover, $\eta \in H^{\mathrm{\infty }}(\mathbb{T})$.

For $a,b\in \mathbb{R}$ and |a|,|b| sufficiently small, we write that

$$\eta (a;b,\kappa ,T)(z):={\eta }_0(b,\kappa ,T)+a\cos z+a^2{\eta }_2(z)+a^3{\eta }_3(z)+\cdots$$ 2.23

and

$$c(a;b,\kappa ,T):=c_0(b,\kappa ,T)+a{c}_1+a^2{c}_2+\cdots ,$$ 2.24

where η₂,η₃,… are 2π periodic, even and smooth functions of z, and $c_1,c_2,\dots \in \mathbb{R}$.

We claim that c₁=0. As a matter of fact, note that (2.2) and, hence, (2.6)–(2.7) remain invariant under z↦z+π by (2.5). Since $\mathrm{\partial }\eta /\mathrm{\partial }a(0)(z)=\cos z$, however, η(z)≠η(z+π) must hold. Thus, ∂c/∂a(0)=0. This proves the claim. If ${⟨{\eta }_{j-1},{\eta }_j⟩}_{L^2(\mathbb{T})}=0$ for any integer j≥1, in addition, then c_2j−1=0 for any integer j≥1. Hence, c is even in a.

Substituting (2.23) and (2.24) into (2.2), we may calculate the small amplitude expansion. The proof is very similar to that in [10], for instance. Hence, we omit the details.

Below we summarize the conclusion.

Lemma 2.1 (Existence of sufficiently small and periodic travelling waves) —

For any T≥0, κ>0 satisfying (2.13), $b\in \mathbb{R}$ and |b| sufficiently small, a one-parameter family of solutions of (2.2) exists, denoted by η(a;b,κ,T) and c(a;b,κ,T), for $a\in \mathbb{R}$ and |a| sufficiently small; $\eta \in H^{\mathrm{\infty }}(\mathbb{T})$ and it is even in z; η and c depend analytically on a, and b, κ. Moreover,

$$\eta (a;b,\kappa ,T)(z)=b(1-c_{\mathrm{ww}}(\kappa ;T))+a\cos z+a^2(h_0+h_2\cos 2z)+O(a{(a+b)}^2)$$ 2.25

and

$$c(a;b,\kappa ,T)=c_{\mathrm{ww}}(\kappa ;T)+b(\frac{3}{2}-\frac{5}{12}{\kappa }^2)(1-c_{\mathrm{ww}}(\kappa ;T))+a^2{c}_2+O(a{(a+b)}^2),$$ 2.26

as a,b→0, where

$$\begin{array}{r}{h}_0=(\frac{3}{8}-\frac{7}{96}{\kappa }^2)\frac{1}{c_{\mathrm{ww}}(\kappa ;T)-1},h_2=(\frac{3}{8}-\frac{11}{32}{\kappa }^2)\frac{1}{c_{\mathrm{ww}}(\kappa ;T)-c_{\mathrm{ww}}(2\kappa ;T)}\end{array}$$ 2.27

and

$$\begin{array}{r}{c}_2=(\frac{3}{2}-\frac{5}{12}{\kappa }^2)h_0+(\frac{3}{4}-\frac{1}{2}{\kappa }^2)h_2-\frac{3}{32}.\end{array}$$ 2.28

3. Modulational instability index

For T≥0, κ>0 satisfying (2.13), $a,b\in \mathbb{R}$ and |a|,|b| sufficiently small, let η=η(a;b,κ,T) and c=c(a;b,κ,T), denote a sufficiently small and 2π/κ periodic travelling wave of (2.1) and (1.6), respectively, whose existence follows from the previous section. We address its modulational stability and instability.

Linearizing (2.1) about η in the coordinate frame moving at the speed c, we arrive at

$${\zeta }_t+\kappa {\mathrm{\partial }}_z(c_{\mathrm{ww}}(\kappa |{\mathrm{\partial }}_z|;T)-c-3+3{(1+\eta )}^{1/2}+{\kappa }^2(\frac{5}{12}(\eta {\mathrm{\partial }}_z^2+{\eta }_{zz})+\frac{13}{24}{\eta }_z{\mathrm{\partial }}_z))\zeta =0,$$

where c_ww(κ|∂_z|;T) is in (1.6). Seeking a solution of the form ζ(z,t)=e^λκtζ(z), $\lambda \in \mathbb{C}$, we arrive at

$$\begin{array}{rl}\lambda \zeta & ={\mathrm{\partial }}_z(-c_{\mathrm{ww}}(\kappa |{\mathrm{\partial }}_z|;T)+c+3-3{(1+\eta )}^{1/2}-{\kappa }^2(\frac{5}{12}(\eta {\mathrm{\partial }}_z^2+{\eta }_{zz})+\frac{13}{24}{\eta }_z{\mathrm{\partial }}_z))\zeta \\ & =:\mathcal{L}(a;b,\kappa ,T)\zeta .\end{array}$$ 3.1

We say that η is spectrally unstable to square-integrable perturbation if the $L^2(\mathbb{R})$ spectrum of $\mathcal{L}$ intersects the open right half-plane of $\mathbb{C}$, and it is spectrally stable otherwise. Note that η is 2π periodic in z, but ζ needs not. Note that the spectrum of $\mathcal{L}$ is symmetric with respect to the reflections in the real and imaginary axes. Hence, η is spectrally unstable if and only if the spectrum of $\mathcal{L}$ is not contained in the imaginary axis.

It is well known (see [15], for instance, and references therein) that the $L^2(\mathbb{R})$ spectrum of $\mathcal{L}$ contains no eigenvalues. Rather, it consists of the essential spectrum. Moreover, a non-trivial solution of (3.1) does not belong to $L^p(\mathbb{R})$ for any $p\in [1,\mathrm{\infty })$. Rather, if $\zeta \in L^{\mathrm{\infty }}(\mathbb{R})$ solves (3.1), then, necessarily,

$$\zeta (z)={\mathrm{e}}^{\mathrm{i}\xi z}\varphi (z),\text{where}\hspace{0.17em}\varphi (z+2\pi )=\varphi (z),$$

for some ξ in the range $(-\frac{1}{2},\frac{1}{2}]$. It follows from Floquet theory (see [15], for instance, and references therein) that λ belongs to the $L^2(\mathbb{R})$ spectrum of $\mathcal{L}$ if and only if

$$\lambda \varphi ={\mathrm{e}}^{-\mathrm{i}\xi z}\mathcal{L}(a;b,\kappa ,T)\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}\xi z}\varphi =:\mathcal{L}(\xi )(a;b,\kappa ,T)\varphi$$ 3.2

for some $\xi \in (-\frac{1}{2},\frac{1}{2}]$ and $\varphi \in L^2(\mathbb{T})$. Thus,

$${\text{spec}}_{L^2(\mathbb{R})}(\mathcal{L}(a;b,\kappa ,T))=\bigcup _{\xi \in (-1/2,1/2]}\hspace{0.17em}{\text{spec}}_{L^2(\mathbb{T})}(\mathcal{L}(\xi )(a;b,\kappa ,T)).$$

Note that, for any ξ∈(−1/2,1/2], the $L^2(\mathbb{T})$ spectrum of $\mathcal{L}(\xi )$ comprises eigenvalues of finite multiplicities. Thus the essential spectrum of $\mathcal{L}$ may be characterized as a one parameter family of point spectra of $\mathcal{L}(\xi )$ for ξ∈(−1/2,1/2]. Note that

$${\text{spec}}_{L^2(\mathbb{T})}(\mathcal{L}(\xi ))=({\text{spec}}_{L^2(\mathbb{T})}(\mathcal{L}{(-\xi )))}^{\ast }.$$

Hence, it suffices to take $\xi \in [0,\frac{1}{2}]$.

Note that ξ=0 corresponds to the same period perturbations as η. Moreover, ξ>0 and small corresponds to long-wavelength perturbations, whose effects are to slowly vary the period and other wave characteristics. They furnish the spectral information of $\mathcal{L}$ in the vicinity of the origin in $\mathbb{C}$; see [15], for instance, for details. Therefore, we say that η is modulationally unstable if the $L^2(\mathbb{T})$ spectra of $\mathcal{L}(\xi )$ are not contained in the imaginary axis near the origin for ξ>0 and small, and it is modulationally stable otherwise.

For an arbitrary ξ, one must in general study (3.2) by means of numerical computation. But, for ξ>0 and small for λ in the vicinity of the origin in $\mathbb{C}$, we may take a spectral perturbation approach in [5,10,11], for instance, to address it analytically.

Throughout the section, T≥0 and κ>0 satisfying (2.13) are suppressed for simplicity of notation, unless specified otherwise. We assume b=0. For non-zero b, one may explore in like manner. But the calculation becomes lengthy and tedious. We use

$$\mathcal{L}(\xi ,a)=\mathcal{L}(\xi )(a;0,\kappa ,T).$$ 3.3

(a). Spectra of $\mathcal{L}(\xi ,0)$ and $\mathcal{L}(0,a)$

For a=0—namely, the rest state—a straightforward calculation reveals that

$$\mathcal{L}(\xi ,0)\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}nz}=\mathrm{i}\omega (n+\xi )\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}nz}\text{for }n\in \mathbb{Z}\text{ and }\xi \in [0,\frac{1}{2}],$$ 3.4

where

$$\omega (n+\xi )=(\xi +n)(c_{\mathrm{ww}}(\kappa ;T)-c_{\mathrm{ww}}(\kappa (n+\xi );T)).$$ 3.5

For ξ=0, ω(1)=ω(−1)=ω(0)=0 and ω(n)≠0 otherwise. Hence, zero is an $L^2(\mathbb{T})$ eigenvalue of $\mathcal{L}(0,0)$ with multiplicity three. Moreover,

$$\cos z,\sin z\text{and}1$$ 3.6

are the associated eigenfunctions, real-valued and orthogonal to each other. For ξ>0 sufficiently small, iω(±1+ξ) and iω(ξ) are the $L^2(\mathbb{T})$ eigenvalues of $\mathcal{L}(\xi ,0)$ in the vicinity of the origin in $\mathbb{C}$, and (3.6) are the associated eigenfunctions.

For $a\in \mathbb{R}$ and |a| sufficiently small for ξ=0, zero is an $L^2(\mathbb{T})$ eigenvalue of $\mathcal{L}(0,a)$ with algebraic multiplicity three and geometric multiplicity two, and

$$\begin{array}{rl}{\varphi }_1(z)& :=({\eta }_a-\frac{c_a}{c_b}{\eta }_b)(a;0,\kappa ,T)(z)=\cos z+a{p}_1+2a{h}_2\cos 2z+O(a^2),\\ {\varphi }_2(z)& :=-\frac{1}{a}{\eta }_z(a;0,\kappa ,T)(z)=\sin z+2a{h}_2\sin 2z+O(a^2)\\ \text{and}{\varphi }_3(z)& :=\frac{1}{1-c_{\mathrm{ww}}(\kappa ;T)}{\eta }_b(\kappa ,a,0;T)(z)=1+O(a^2),\end{array}\}$$ 3.7

are the associated eigenfunctions, where

$$p_1=2h_0-\frac{24c_2}{18-5{\kappa }^2}=\frac{1}{18-5{\kappa }^2}(\frac{9}{4}-\frac{3}{16}\frac{(3-2{\kappa }^2)(12-11{\kappa }^2)}{c_{\mathrm{ww}}(\kappa ;T)-c_{\mathrm{ww}}(2\kappa ;T)})$$ 3.8

and h₂ is defined in (2.27). The proof is nearly identical to that in [5], for instance. Hence, we omit the details.

(b). Spectral perturbation calculation

Recall that, for ξ>0 and sufficiently small for a=0, the $L^2(\mathbb{T})$ spectrum of $\mathcal{L}(\xi ,0)$ contains three purely imaginary eigenvalues iω(±1+ξ) and iω(ξ) in the vicinity of the origin in $\mathbb{C}$, and (3.6) spans the associated eigenspace. For ξ=0 for $a\in \mathbb{R}$ and |a| sufficiently small, the spectrum of $\mathcal{L}(0,a)$ contains three eigenvalues at the origin, and (3.7) spans the associated eigenspace.

For ξ>0, $a\in \mathbb{R}$ and ξ,|a| sufficiently small, it follows from perturbation theory (see [16], for instance, for details) that the $L^2(\mathbb{T})$ spectrum of $\mathcal{L}(\xi ,a)$ contains three eigenvalues in the vicinity of the origin in $\mathbb{C}$, and (3.7) spans the associated eigenspace. Let

$$\mathbf{\text{L}}(\xi ,a)={\left(\frac{⟨\mathcal{L}(\xi ,a){\varphi }_j,{\varphi }_k⟩}{⟨{\varphi }_j,{\varphi }_j⟩}\right)}_{j,k=1,2,3}\text{and}\mathbf{\text{I}}(a)={\left(\frac{⟨{\varphi }_j,{\varphi }_k⟩}{⟨{\varphi }_j,{\varphi }_j⟩}\right)}_{j,k=1,2,3},$$ 3.9

where ϕ₁,ϕ₂,ϕ₃ are in (3.7). Throughout the subsection, 〈⋅ ,⋅〉 means the $L^2(\mathbb{T})$ inner product. Note that L represents the action of $\mathcal{L}$ on the eigenspace, spanned by ϕ₁,ϕ₂,ϕ₃, and I is the projection of the identity onto the eigenspace. It follows from perturbation theory (see [16], for instance, for details) that for ξ>0, $a\in \mathbb{R}$ and ξ,|a| sufficiently small, the eigenvalues of $\mathcal{L}(\xi ,a)$ agree in location and multiplicity with the roots of $det(\mathbf{\text{L}}-\lambda \mathbf{\text{I}})$ up to terms of order a.

For $a\in \mathbb{R}$ and |a| sufficiently small, a Baker–Campbell–Hausdorff expansion reveals that

$$\mathcal{L}(\xi ,a)=\mathcal{L}(0,a)+\mathrm{i}\xi [\mathcal{L}(0,a),z]-\frac{1}{2}{\xi }^2[[\mathcal{L}(0,a),z],z]+O({\xi }^3)$$

as ξ→0, where [⋅ ,⋅] means the commutator. We merely pause to remark that $[\mathcal{L},z]$ and $[[\mathcal{L},z],z]$ are well defined in the periodic setting even though z is not. We use (3.1), (3.2) and (2.25), (2.26) to write

$$\begin{array}{rl}\mathcal{L}(\xi ,a)& =\mathcal{M}-a{\mathrm{\partial }}_z(\frac{3}{2}\cos z+{\kappa }^2(\frac{5}{12}\cos z({\mathrm{\partial }}_z^2-1)-\frac{13}{24}\sin z{\mathrm{\partial }}_z))\\ & -\mathrm{i}\xi a(\frac{3}{2}\cos z+{\kappa }^2(\frac{5}{12}(2{\mathrm{\partial }}_z\cos z{\mathrm{\partial }}_z+\cos z({\mathrm{\partial }}_z^2-1))-\frac{13}{24}(\sin z{\mathrm{\partial }}_z+{\mathrm{\partial }}_z\sin z)))\\ & +O({\xi }^3+{\xi }^{2}a+a^2)\end{array}$$ 3.10

as ξ,a→0, where

$$\mathcal{M}=\mathcal{L}(0,0)+\mathrm{i}\xi [\mathcal{L}(0,0),z]-\frac{1}{2}{\xi }^2[[\mathcal{L}(0,0),z],z]$$

agrees with $\mathcal{L}(\xi ,0)$ up to terms of order ξ² as ξ→0. We then resort to (3.4), (3.5), and we make an explicit calculation to find that

$$\begin{array}{rl}\mathcal{L}(\xi ,0)\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}nz}& =\mathrm{i}n(c_{\mathrm{ww}}(\kappa ;T)-c_{\mathrm{ww}}(n\kappa ;T))\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}nz}\\ & +\mathrm{i}\xi (c_{\mathrm{ww}}(\kappa ;T)-c_{\mathrm{ww}}(n\kappa ;T)-\kappa c_{\mathrm{ww}}^{\prime }(n\kappa ;T))\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}nz}\\ & -\frac{1}{2}{\xi }^2(2\kappa c_{\mathrm{ww}}^{\prime }(n\kappa ;T)+{\kappa }^2{c}_{\mathrm{ww}}^{″}(n\kappa ;T))\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}nz}+O({\xi }^3)\end{array}$$

as ξ→0. Therefore, for T≥0 but $T\ne \frac{1}{3}$, for κ>0 satisfying (2.13), $\mathcal{M}1=\mathrm{i}\xi (c_{\mathrm{ww}}(\kappa ;T)-1)$,

$$\begin{array}{rl}\mathcal{M}\left\{\begin{array}{c}\cos z\\ \sin z\end{array}\right\}& =-\mathrm{i}\xi \kappa c_{\mathrm{ww}}^{\prime }(\kappa ;T)\left\{\begin{array}{c}\cos z\\ \sin z\end{array}\right\}\pm \frac{1}{2}{\xi }^2(2\kappa c_{\mathrm{ww}}^{\prime }(\kappa ;T)+{\kappa }^2{c}_{\mathrm{ww}}^{″}(\kappa ;T))\left\{\begin{array}{c}\sin z\\ \cos z\end{array}\right\},\\ \mathcal{M}\left\{\begin{array}{c}\cos 2z\\ \sin 2z\end{array}\right\}& =\mp 2(c_{\mathrm{ww}}(\kappa ;T)-c_{\mathrm{ww}}(2\kappa ;T))\left\{\begin{array}{c}\sin 2z\\ \cos 2z\end{array}\right\}\\ & +\mathrm{i}\xi (c_{\mathrm{ww}}(\kappa ;T)-c_{\mathrm{ww}}(2\kappa ;T)-2\kappa c_{\mathrm{ww}}^{\prime }(2\kappa ;T))\left\{\begin{array}{c}\cos 2z\\ \sin 2z\end{array}\right\}\\ & \pm \frac{1}{2}{\xi }^2(2\kappa c_{\mathrm{ww}}^{\prime }(2\kappa ;T)+{\kappa }^2{c}_{\mathrm{ww}}^{″}(2\kappa ;T))\left\{\begin{array}{c}\sin 2z\\ \cos 2z\end{array}\right\}.\end{array}$$

For $T=\frac{1}{3}$, one must calculate higher-order terms in the expansion of $\mathcal{L}(\xi ,0)$. We do not pursue this here.

We use (3.10), (3.7) and the above formula for $\mathcal{M}$, and we make a lengthy but explicit calculation to find that

$$\begin{array}{rl}\mathcal{L}{\varphi }_1& =-\mathrm{i}\xi \kappa c_{\mathrm{ww}}^{\prime }(\kappa ;T)\cos z\\ & -\mathrm{i}\xi a(\frac{3}{4}-\frac{7}{48}{\kappa }^2-p_1(c_{\mathrm{ww}}(\kappa ;T)-1))\\ & +\mathrm{i}\xi a(-\frac{3}{4}+\frac{33}{16}{\kappa }^2+2h_2(c_{\mathrm{ww}}(\kappa ;T)-c_{\mathrm{ww}}(2\kappa ;T)-2\kappa c_{\mathrm{ww}}^{\prime }(2\kappa ;T)))\cos 2z\\ & +\frac{1}{2}{\xi }^2(2\kappa c_{\mathrm{ww}}^{\prime }(\kappa ;T)+{\kappa }^2{c}_{\mathrm{ww}}^{″}(\kappa ;T))\sin z+O({\xi }^3+{\xi }^{2}a+a^2),\\ \mathcal{L}{\varphi }_2& =-\mathrm{i}\xi \kappa c_{\mathrm{ww}}^{\prime }(\kappa ;T)\sin z\\ & +\mathrm{i}\xi a(-\frac{3}{4}+\frac{33}{16}{\kappa }^2+2h_2(c_{\mathrm{ww}}(\kappa ;T)-c_{\mathrm{ww}}(2\kappa ;T)-2\kappa c_{\mathrm{ww}}^{\prime }(2\kappa ;T)))\sin 2z\\ & -\frac{1}{2}{\xi }^2(2\kappa c_{\mathrm{ww}}^{\prime }(\kappa ;T)+{\kappa }^2{c}_{\mathrm{ww}}^{″}(\kappa ;T))\cos z+O({\xi }^3+{\xi }^{2}a+a^2),\\ \mathcal{L}{\varphi }_3& =a(3-\frac{5}{6}{\kappa }^2)\sin z+\mathrm{i}\xi (c_{\mathrm{ww}}(\kappa ;T)-1)-\mathrm{i}\xi a(\frac{3}{2}-\frac{23}{24}{\kappa }^2)\cos z+O({\xi }^3+{\xi }^{2}a+a^2)\end{array}$$

as ξ,a→0, where p₁ is in (3.8) and h₂ is in (2.27).

To proceed, we take the $L^2(\mathbb{T})$ inner product of the above and (3.7), and we make a lengthy but explicit calculation to find that

$$\begin{array}{rl}⟨\mathcal{L}{\varphi }_1,{\varphi }_1⟩& =⟨\mathcal{L}{\varphi }_2,{\varphi }_2⟩=-\frac{1}{2}\mathrm{i}\xi \kappa c_{\mathrm{ww}}^{\prime }(\kappa ;T)+O({\xi }^3+{\xi }^{2}a+a^2),\\ ⟨\mathcal{L}{\varphi }_1,{\varphi }_2⟩& =-⟨\mathcal{L}{\varphi }_2,{\varphi }_1⟩=\frac{1}{4}{\xi }^2(2\kappa c_{\mathrm{ww}}^{\prime }(\kappa ;T)+{\kappa }^2{c}_{\mathrm{ww}}^{″}(\kappa ;T))+O({\xi }^3+{\xi }^{2}a+a^2),\\ ⟨\mathcal{L}{\varphi }_1,{\varphi }_3⟩& =\mathrm{i}\xi a(-\frac{3}{4}+\frac{36}{48}{\kappa }^2+p_1(c_{\mathrm{ww}}(\kappa ;T)-1))+O({\xi }^3+{\xi }^{2}a+a^2),\\ ⟨\mathcal{L}{\varphi }_2,{\varphi }_3⟩& =0+O({\xi }^3+{\xi }^{2}a+a^2),\\ ⟨\mathcal{L}{\varphi }_3,{\varphi }_1⟩& =\mathrm{i}\xi a(-\frac{3}{4}+\frac{23}{48}{\kappa }^2+p_1(c_{\mathrm{ww}}(\kappa ;T)-1))+O({\xi }^3+{\xi }^{2}a+a^2),\\ ⟨\mathcal{L}{\varphi }_3,{\varphi }_2⟩& =a(\frac{3}{4}-\frac{5}{24}{\kappa }^2)+O({\xi }^3+{\xi }^{2}a+a^2),\\ ⟨\mathcal{L}{\varphi }_3,{\varphi }_3⟩& =\mathrm{i}\xi (c_{\mathrm{ww}}(\kappa ;T)-1)+O({\xi }^3+{\xi }^{2}a+a^2)\end{array}$$

as ξ,a→0, where p₁ is in (3.8). Moreover, we take the $L^2(\mathbb{T})$ inner products of (3.7), and we make an explicit calculation to find that

$$\begin{array}{rl}⟨{\varphi }_1,{\varphi }_1⟩& =⟨{\varphi }_2,{\varphi }_2⟩=\frac{1}{2}+O({\xi }^3+{\xi }^{2}a+a^2),\\ ⟨{\varphi }_1,{\varphi }_2⟩& ,⟨{\varphi }_2,{\varphi }_3⟩=0+O({\xi }^3+{\xi }^{2}a+a^2),\\ ⟨{\varphi }_1,{\varphi }_3⟩& =a{p}_1+O({\xi }^3+{\xi }^{2}a+a^2),\\ ⟨{\varphi }_3,{\varphi }_3⟩& =1+O({\xi }^3+{\xi }^{2}a+a^2)\end{array}$$

as ξ,a→0, where p₁ is in (3.8). Together, (3.9) becomes

$$\begin{array}{rl}\mathbf{\text{L}}(\xi ,a)& =a(\frac{3}{4}-\frac{5}{24}{\kappa }^2)\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 1& 0\end{array}\right)\\ & +\mathrm{i}\xi \left(\begin{array}{ccc}-\kappa c_{\mathrm{ww}}^{\prime }(\kappa ;T)& 0& 0\\ 0& -\kappa c_{\mathrm{ww}}^{\prime }(\kappa ;T)& 0\\ 0& 0& c_{\mathrm{ww}}(\kappa ;T)-1\end{array}\right)\\ & +\mathrm{i}\xi a\left(\begin{array}{ccc}0& 0& -\frac{3}{4}+\frac{7}{24}{\kappa }^2+2p_1(c_{\mathrm{ww}}(\kappa ;T)-1)\\ 0& 0& 0\\ -\frac{3}{4}+\frac{23}{48}{\kappa }^2+p_1(c_{\mathrm{ww}}(\kappa ;T)-1)& 0& 0\end{array}\right)\\ & +{\xi }^2(\kappa c_{\mathrm{ww}}^{\prime }(\kappa ;T)+\frac{1}{2}{\kappa }^2{c}_{\mathrm{ww}}^{″}(\kappa ;T))\left(\begin{array}{ccc}0& 1& 0\\ -1& 0& 0\\ 0& 0& 0\end{array}\right)+O({\xi }^3+{\xi }^{2}a+a^2)\end{array}$$ 3.11

and

$$\mathbf{\text{I}}(a)=\mathbf{\text{I}}+a{p}_1\left(\begin{array}{ccc}0& 0& 2\\ 0& 0& 0\\ 1& 0& 0\end{array}\right)+O(a^2)$$ 3.12

as ξ,a→0, where p₁ is in (3.8) and I is the 3×3 identity matrix.

(c). Modulational instability index

For any T≥0 but $T\ne \frac{1}{3}$, κ>0 satisfying (2.13), for ξ>0, $a\in \mathbb{R}$ and ξ,|a| sufficiently small, we turn the attention to the roots of

$$det(\mathbf{\text{L}}-\lambda \mathbf{\text{I}})(\xi ,a;\kappa ,T)=:(p_3{\lambda }^3+\mathrm{i}{p}_2{\lambda }^2+p_1\lambda +i{p}_0)(\xi ,a;\kappa ,T),$$

where L and I are in (3.11) and (3.12). Details are found in [5], for instance. Hence, we merely hit the main points.

Let

$$q(-\mathrm{i}\xi \lambda )(\xi ,a;\kappa ,T)=(\mathrm{i}{\xi }^3(q_3{\lambda }^3-q_2{\lambda }^2-q_1\lambda +q_0)(\xi ,a;\kappa ,T),$$

where p_j=ξ^3−jq_j, j=0,1,2,3. Note that q₀,q₁,…,q₃ are real-valued and depend analytically on ξ,a and κ for any ξ>0 and |a| sufficiently small for any κ>0 satisfying (2.13). Moreover, they are odd in ξ and even in a. For any T≥0 but $T\ne \frac{1}{3}$, κ>0 satisfying (2.13), $a\in \mathbb{R}$ and |a| sufficiently small, a periodic travelling wave η(a;0,κ,T) and c(a;0,κ,T) of (2.1) and (1.6) is modulationally unstable, provided that q possesses a pair of complex roots or, equivalently,

$${\mathrm{\Delta }}_0(\xi ,a;\kappa ,T):=(18q_3{q}_2{q}_1{q}_0+q_2^2{q}_1^2+4q_2^3{q}_0+4q_3{q}_1^3-27q_3^2{q}_0^2)(\xi ,a;\kappa ,T)<0$$

for ξ>0 and small, and it is modullationally stable if Δ₀>0. Note that Δ₀ is even in ξ and a. Hence, we write that

$${\mathrm{\Delta }}_0(\xi ,a;\kappa ,T)=:{\mathrm{\Delta }}_0(\xi ,0;\kappa ,T)+a^2\mathrm{\Delta }(\kappa ;T)+O(a^2({\xi }^2+a^2))$$

as a→0 for ξ>0 and small. We then use (3.11) and (3.12), and we make a Mathematica calculation to show that

$${\mathrm{\Delta }}_0(\xi ,0;\kappa ,T)={\kappa }^2{i}_1^2(\kappa ;T){(\xi i_2^2+\frac{1}{4}{\xi }^3{\kappa }^2{i}_1^2)}^2(\kappa ;T)>0$$

as ξ→0. Therefore, if Δ<0, then Δ₀<0 for ξ>0 and sufficiently small, depending on $a\in \mathbb{R}$ and |a| sufficiently small, implying modulational instability, whereas if Δ>0, then Δ₀>0 for ξ>0, $a\in \mathbb{R}$ and ξ,|a| sufficiently small, implying modulational stability. We use (3.11) and (3.12), and we make a Mathematica calculation to find Δ explicitly.

Below we summarize the conclusion.

Theorem 3.1 (Modulational instability index) —

For any T≥0 but $T\ne \frac{1}{3}$, for any κ>0 satisfying (2.13), a sufficiently small and 2π/κ periodic travelling wave of (2.1) and (1.6) is modulationally unstable, provided that

$$\mathrm{\Delta }(\kappa ;T):=\frac{i_1{i}_2}{i_3}{i}_4(\kappa ;T)<0,$$ 3.13

where

$$\begin{array}{rl}{i}_1(\kappa ;T)& =(\kappa c_{\mathrm{ww}}{(\kappa ;T))}^{″},\\ i_2(\kappa ;T)& =(\kappa c_{\mathrm{ww}}{(\kappa ;T))}^{\prime }-1,\\ i_3(\kappa ;T)& =c_{\mathrm{ww}}(\kappa ;T)-c_{\mathrm{ww}}(2\kappa ;T),\\ i_4(\kappa ;T)& =(3i_2-i_2{i}_3+6i_3-\frac{1}{12}{\kappa }^2(57i_2+34i_3)+\frac{1}{108}{\kappa }^4(198i_2+35i_3))(\kappa ,T),\end{array}\}$$ 3.14

and c_ww(κ;T) is in (1.6). It is modulationally stable if Δ (κ;T)>0.

Theorem 3.1 reveals four resonance mechanisms which contribute to the sign change in Δ and, hence, to the change in the modulational stability and instability in (2.1) and (1.6). Note that

$$c_{\mathrm{ww}}(\kappa ;T)=\text{the phase speed}\text{and}(\kappa c_{\mathrm{ww}}{(\kappa ;T))}^{\prime }=\text{the group speed}$$

in the linear theory. Specifically,

• (R1) i₁(κ;T)=0 at some κ; the group speed achieves an extremum at the wave number κ;

• (R2) i₂(κ;T)=0 at some κ; the group speed at the wave number κ coincides with the phase speed in the limit as κ→0, resulting in the ‘resonance of short and long waves;’

• (R3) i₃(κ;T)=0 at some κ; the phase speeds of the fundamental mode and the second harmonic coincide at the wave number κ, resulting in the ‘second harmonic resonance;’

• (R4) i₄(κ;T)=0 at some κ.

Resonances (R1), (R2) and (R3) are determined from the dispersion relation. For instance, i₁, i₂ and i₃ appear in an index formula for the Whitham equation; see [11] for details. Resonance (R4), on the other hand, results from the resonance of the dispersion and nonlinear effects, and it depends on the nonlinearity of the equation.

4. Results

For T=0, as (κc_ww(κ;0))′<1 for any κ>0 and it decreases monotonically over the interval $(0,\mathrm{\infty })$ by brutal force, i₁(κ;0)<0 and i₂(κ;0)<0 for any κ>0. As c_ww(κ;0)>0 for any κ>0 and it decreases monotonically over the interval $(0,\mathrm{\infty })$ (figure 1a), i₃(κ;0)>0 for any κ>0. Moreover, a straightforward calculation reveals that

$$\underset{\kappa \to 0+}{lim}\frac{i_4(\kappa )}{{\kappa }^2}=\frac{9}{2}\text{and}\underset{\kappa \to \mathrm{\infty }}{lim}{i}_4(\kappa )=-\mathrm{\infty }.$$

Hence, Δ(κ;0)>0 for κ>0 sufficiently small, implying the modulational stability, and it is negative for κ>0 sufficiently large, implying the modulational instability. The intermediate value theorem asserts a root of i₄. Moreover, a numerical evaluation of (3.14) reveals a unique root κ_c, say, of i₄ over the interval $(0,\mathrm{\infty })$ such that i₄(κ)>0 if 0<κ<κ_c and it is negative if ${\kappa }_c<\kappa <\mathrm{\infty }$. Upon close inspection (figure 2), κ_c=1.420… .

Figure 2.

The graph of i₄(κ) for κ∈(0,1.5). (Online version in colour.)

Therefore, a sufficiently small and 2π/κ periodic travelling wave of (2.1) and (1.2) is modulationally unstable if κ>κ_c, where κ_c=1.420… is a unique root of i₄ in (3.14) over the interval $(0,\mathrm{\infty })$. It is modulationally stable if 0<κ<κ_c. The result qualitatively agrees with the Benjamin–Feir instability of a Stokes wave (see [6–9], for instance) and that for the Whitham equation [5]. The critical wave number compares reasonably well with that in [7,8], for instance. The critical wave number for the Whitham equation is 1.146… [5].

For $T>\frac{1}{3}$, since c_ww(κ;T) and (κc_ww(κ;T))′ increase monotonically over the interval $(0,\mathrm{\infty })$ and since (κc_ww(κ;T))′ does not possess an extremum (figure 1b), i₁, i₂ and i₃ do not vanish over the interval $(0,\mathrm{\infty })$. Moreover, a numerical evaluation reveals that i₄ changes its sign once over the interval $(0,\mathrm{\infty })$. Together, a sufficiently small and periodic travelling wave of (2.1) and (1.6) is modulationally unstable, provided that the wave number is greater than a critical value, and modulationally stable otherwise, similarly to the gravity wave setting. A numerical evaluation reveals that the critical wave number κ_c(T), say, satisfies

$$\underset{T\to \mathrm{\infty }}{lim}{\kappa }_{\mathrm{c}}(T)\sqrt{T}\approx \mathrm{1.283.}$$

For $0<T<\frac{1}{3}$, on the other hand, (κc_ww(κ;T))′ achieves a unique minimum over the interval $(0,\mathrm{\infty })$. Moreover, i₂ and i₃ each takes one transverse root over the interval $(0,\mathrm{\infty })$ (figure 1c). Hence, i₁ through i₄ each contributes to the change in the modulational stability and instability.

Figure 3 illustrates in the κ and $\kappa \sqrt{T}$ plane the regions of modulational stability and instability for a sufficiently small and periodic travelling wave of (2.1) and (1.6). Along Curve 1, i₁=0 and the group speed achieves an extremum at the wave number κ. Curve 2 is associated with i₂=0, along which the group speed coincides with the phase speed in the limit as κ→0. In the deep water limit, as $\kappa \to \mathrm{\infty }$ while $\kappa \sqrt{T}$ is fixed, it is asymptotic to $\kappa =\frac{9}{4}{\kappa }^{2}T-\frac{3}{4}$. Curve 3 is associated with i₃=0, along which the phase speeds of the fundamental mode and the second harmonic coincide. In the deep water limit, it is asymptotic to $k^{2}T=\frac{1}{2}$. Moreover, along Curve 4, i₄ vanishes because of the resonance of the dispersion and nonlinear effects. The ‘lower’ branch of Curve 4 passes through κ=1.420…, the critical wave number for T=0. The ‘upper’ branch passes through $\kappa \sqrt{T}=1.283\dots$, the limit of strong surface tension.

The result qualitatively agrees with those in [12,13], for instance, from formal asymptotic expansions of the physical problem, and it improves upon that in [11] for the Whitham equation, for which ${\kappa }_{\mathrm{c}}(T)\sqrt{T}\to \mathrm{\infty }$ as $T\to \mathrm{\infty }$.

5. Discussion

(a). The Camassa–Holm equation

We may write (1.7), in the case of $a=\frac{1}{12}$, after normalization of parameters, as

$${\eta }_t+c_{\text{CH}}(|{\mathrm{\partial }}_x|){(2{(1+\eta )}^{3/2}-2\eta +\frac{7}{24}\eta {\eta }_{xx}+\frac{7}{48}{\eta }_x^2)}_x=0,$$ 5.1

where

$$\widehat{c_{\mathrm{CH}}(|{\mathrm{\partial }}_x|)f}(\kappa )=\frac{12-{\kappa }^2}{12+{\kappa }^2}\hspace{0.17em}\hat{f}(\kappa ).$$ 5.2

Note that c_CH(κ) approximates (1.2) for κ≪1.

For any κ>0, we may repeat the argument in §2 to determine sufficiently small and 2π/κ periodic travelling waves. Specifically, a two-parameter family of periodic travelling waves of (5.1) and (5.2) exists, denoted by η(a,b;κ)(z), where z=κ(x−c(a,b;κ)t), for $a,b\in \mathbb{R}$ and |a|,|b| sufficiently small; η is 2π periodic, even and smooth in z. Moreover,

$$\begin{array}{rl}\eta (a,b;\kappa )(z)& =b(1-c_{\mathrm{CH}}(\kappa ))+a\cos z+a^2(h_0+h_2\cos 2z)+O(a(a^2+b^2)),\\ c(a,b;\kappa )& =c_{\mathrm{CH}}(\kappa )+b(\frac{3}{2}-\frac{7}{24}{\kappa }^2)c_{\mathrm{CH}}(\kappa )(1-c_{\mathrm{CH}}(\kappa ))+a^2{c}_2+O(a(a^2+b^2))\end{array}$$

as a,b→0, where

$$\begin{array}{rl}{h}_0& =\frac{36-7{\kappa }^2}{96(c_{\mathrm{CH}}(\kappa )-1)},h_2=\frac{(12-7{\kappa }^2)c_{\mathrm{CH}}(2\kappa )}{32(c_{\mathrm{CH}}(\kappa )-c_{\mathrm{CH}}(2\kappa ))},\\ c_2& =c_{\mathrm{CH}}(\kappa )(\frac{36-7{\kappa }^2}{24}{h}_0+\frac{12-7{\kappa }^2}{16}{h}_2-\frac{3}{32}).\end{array}$$

We then proceed as in §3 to determine a modulational instability index

$${\mathrm{\Delta }}_{\mathrm{CH}}(\kappa ):=\frac{i_1(\kappa )i_2(\kappa )}{i_3(\kappa )}{i}_4(\kappa ),$$

where

$$\begin{array}{rl}{i}_1(\kappa )& =(\kappa c_{\mathrm{CH}}{(\kappa ))}^{″},\\ i_2(\kappa )& =(\kappa c_{\mathrm{CH}}{(\kappa ))}^{\prime }-1,\\ i_3(\kappa )& =c_{\mathrm{CH}}(\kappa )-c_{\mathrm{CH}}(2\kappa ),\\ i_4(\kappa )& =1296(c_{\mathrm{CH}}(2\kappa )i_2(\kappa )+2i_3(\kappa ))-432i_2(\kappa )i_3(\kappa )\\ & -1512{\kappa }^2{c}_{\mathrm{CH}}(2\kappa )i_2(\kappa )+49{\kappa }^4(9c_{\mathrm{CH}}(2\kappa )i_2(\kappa )-2i_3(\kappa )).\end{array}$$

We omit the details.

A straightforward calculation reveals that (i₂i₄/i₃)(κ)<0 for any κ>0 while i₁(κ) changes its sign from negative to positive across κ=6. Therefore, a sufficiently small and 2π/κ periodic travelling wave of (5.1) and (5.2) is modulationally unstable if κ>6. For other values of a in (1.7), the result is qualitatively the same. The Camassa–Holm equation thus seems to predict the Benjamin–Feir instability. But Resonance (R1) following theorem 3.1 results in the instability in (5.1) and (5.2), whereas it does not take place in (1.3), (2.1) and in the water wave problem.

(b). The velocity equation

The FDCH equation for the average horizontal velocity, after normalization of parameters,

$$u_t+c_{\mathrm{ww}}(|{\mathrm{\partial }}_x|)u_x+\frac{3}{2}u{u}_x=-(\frac{5}{12}u{u}_{xxx}+\frac{23}{24}{u}_x{u}_{xx}),$$ 5.3

where c_ww(|∂_x|) is in (1.6), differs from (2.1) by higher-power nonlinearities. We may repeat the arguments in §§2 and 3 to derive the modulational instability index, where (3.14) is replaced by

$$i_4(\kappa ;T)=(i_2+2i_3+\frac{1}{36}{\kappa }^2(57i_2+34i_3)+\frac{1}{324}{\kappa }^4(198i_2+35i_3))(\kappa ;T).$$ 5.4

We omit the details. The index formula for (5.3) and (1.6) agrees with that for the Whitham equation (see [5,11] for details) except the terms in (5.4) explicitly depending on κ² and κ⁴, which higher-derivative nonlinearities of (5.3) seem to contribute.

For T=0, it is straightforward to show that a sufficiently small and 2π/κ periodic travelling wave of (5.3) and (1.6) is modulationally unstable if κ>0.637… . Thus the FDCH equation for the average horizontal velocity predicts the Benjamin–Feir instability of a Stokes wave. For T>0, the modulational instability result for (5.3) and (1.6) qualitatively agrees with that in [11] for the Whitham equation (figure 4). In particular, ${\kappa }_{\mathrm{c}}(T)\sqrt{T}\to \mathrm{\infty }$ as $T\to \mathrm{\infty }$, where κ_c(T) is a critical wave number, depending on T, whereas the limit is finite in the FDCH equation for the fluid surface displacement and the water wave problem. Therefore, we learn that the higher-power nonlinearities of (2.1) capture the effects of surface tension on the modulational stability and instability, not the higher-derivative nonlinearities.

Figure 4.

Stability diagram for sufficiently small and periodic wave trains of (5.3) and (1.6). (Online version in colour.)

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Authors' contributions

The authors declare no specific contributions because the work was split evenly.

Competing interests

We declare we have no competing interests.

Funding

V.M.H. is supported by the National Science Foundation under the Faculty Early Career Development (CAREER) Award DMS-1352597, a Simons Fellowship in Mathematics and the University of Illinois at Urbana-Champaign under the Arnold O. Beckman Research Award RB16227.