Modulational instability in the FDCH equation

# Modulational instability in the full-dispersion Camassa-Holm equation

Vera Mikyoung Hur Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801 USA  and  Ashish K. Pandey
July 12, 2019
###### Abstract.

We determine the stability and instability of a sufficiently small and periodic traveling 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 it improves upon that for the Whitham equation, correctly predicting the limit of strong surface tension. We discuss the modulational stability and instability in the Camassa-Holm equation and related models.

## 1. Introduction

In the 1960s, Whitham (see [Whi74], for instance) proposed

 (1.1) ηt+cww(√β|∂x|)ηx+(3√(1+αη)−3)ηx=0,

to argue for wave breaking in shallow water. That is, the solution remains bounded but its slope becomes unbounded in finite time. Here is proportional to elapsed time, and is the spatial variable in the primary direction of wave propagation; is the fluid surface displacement from the undisturbed depth,

 α=a typical amplitudethe undisturbed fluid depthandβ=(the undisturbed fluid depth)2(a typical wavelength)2.

Moreover, is a Fourier multiplier operator, defined as

 (1.2) ˆcww(|∂x|)f(κ)=√tanhκκˆf(κ).

Note that means the phase speed in the linear theory of water waves. For small amplitude waves satisfying , we may expand the nonlinearity of (1.1) up to terms of order to arrive at

 (1.3) ηt+cww(√β|∂x|)ηx+32αηηx=0.

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

 cww(√βk)=1−16βk2+O(β2).

Therefore, for small amplitude and long waves satisfying and , we arrive at the famous Korteweg-de Vries equation

 (1.4) ηt+ηx+16βηxxx+32αηηx=0.

As a matter of fact, for well-prepared 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 over the relevant time scale; see [Lan13], for instance, for details. But (1.3) and (1.2) offer improvements over (1.4) for short waves. Whitham conjectured wave breaking for (1.3) and (1.2). One of the authors [Hur15] recently proved this. In stark contrast, no solutions of (1.4) break.

Moreover, Johnson and one of the authors [HJ15a] showed that a sufficiently small and periodic traveling wave of the Whitham equation be spectrally unstable to long wavelength perturbations, provided that . In other words, (1.3) (or (1.1)) and (1.2) predict the Benjamin-Feir instability of a Stokes wave; see [BF67, BH67, Whi67] and [BM95], for instance. In contrast, periodic traveling waves of the Korteweg-de Vries equation are all modulationally stable. By the way, under the assumption that is small, we may modify (1.4) to arrive at the Benjamin-Bona-Mahony equation

 (1.5) ηt+ηx−16βηxxt+32αηηx=0.

It agrees with (1.4) for long waves but is preferable for short waves. Note that the phase speed for (1.5) is bounded for all frequencies. The authors [HP15] showed that a sufficiently small and periodic traveling wave of (1.5) be modulationally unstable if . Hence the Benjamin-Bona-Mahony equation seems to predict the Benjamin-Feir instability of a Stokes wave. But the instability mechanism is different from that in the Whitham equation or the water wave problem; see [HP15] for details.

Furthermore, in the presence of the effects of surface tension, Johnson and one of the authors [HJ15b] determined the modulational stability and instability of a sufficiently small and periodic traveling wave of (1.3) and

 (1.6) ˆcww(|∂x|;T)f(κ)=√(1+Tκ2)tanhκκˆf(κ),

where is the coefficient of surface tension. The result agrees by and large with those in [Kaw75, DR77], for instance, from formal asymptotic expansions of the physical problem. But it fails to predict the limit of “strong surface tension.” Perhaps, this is not surprising because (1.3) neglects higher order nonlinearities of the water wave problem. It is interesting to find an equation, which predicts the modulational stability and instability of a gravity capillary wave. This is the subject of investigation here.

By the way, the authors [HP16] recently extended the Whitham equation to include bidirectional propagation, and they showed that the “full-dispersion shallow water equations” correctly predict capillary effects on the Benjamin-Feir instability. But the modulation calculation is very lengthy and tedious. Here we seek higher order nonlinearities suitable for unidirectional propagation. Such a model is likely to be Hamiltonian and is potentially useful for other purposes.

As a matter of fact, for medium amplitude and long waves satisfying and , the Camassa-Holm equations for the fluid surface displacement

 (1.7) ηt+ηx+β(aηxxx+bηxxt)+32αηηx−38α2η2ηx+316α3η3ηx=−αβ(cηηxxx+dηxηxx)

and for the average horizontal velocity

 (1.8) ut+ux+β(auxxx+buxxt)+32αuux=−αβ(cuuxxx+duxuxx),

where

 0⩽a⩽16,b=a−16,c=32a+16,andd=92a+524,

extend the Korteweg-de Vries equation to include higher order nonlinearities, and they approximate the physical problem; see [Lan13], for instance, for details. In the case of , (1.7) reads

 ηt+ηx+112β(ηxxx−ηxxt)+32αηηx−38α2η2ηx+316α3η3ηx=−724αβ(ηηxxx+2ηxηxx),

which is particularly interesting because it predicts wave breaking; see [Lan13] and references therein. Note that

 3αη1+√1+αη=32αη−38α2η2+316α3η3+O(α4).

Lannes [Lan13] combined the dispersion relation of water waves and a Camassa-Holm equation, to propose the full-dispersion Camassa-Holm (FDCH) equation for the fluid surface displacement

 (1.9) ηt+cww(√β|∂x|)ηx+3αη1+√1+αηηx=−αβ(512ηηxxx+2324ηxηxx),

where is in (1.2), or (1.6) in the presence of the effects of surface tension. For relatively long waves satisfying , (1.9) and (1.2) agree with (1.7), where , up to terms of order . But, including all the dispersion of water waves, (1.9) and (1.2) may offer an improvement over (1.7) for short waves. For small amplitude waves satisfying , (1.9) agrees with (1.3) up to terms of order . But, including higher order nonlinearities, (1.9) may offer an improvement 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 introduce

 (1.10) ut+cww(√β|∂x|)ux+32αuux=−αβ(512uuxxx+2324uxuxx).

We follow along the same line as the arguments in [HJ15a, HJ15b, HP15] (see also [BHJ16]) and investigate the modulational stability and instability in the FDCH equation. A main difference lies in that the nonlinearities of (1.9) involve higher order derivatives and, hence, a periodic traveling wave 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 periodic traveling wave of (1.9) and (1.2) is spectrally unstable to long wavelength perturbations, provided that

 κ>1.420…,

and stable to square integrable perturbations otherwise. The result qualitatively agrees with the Benjamin-Feir instability of a Stokes wave (see [BF67, BH67, Whi67], for instance) and that for the Whitham equation (see [HJ15a]). The critical wave number compares reasonably well with in the Benjmain-Feir instability. Including the effects of surface tension, in the and plane, we determine the regions of modulational stability and instability for a sufficiently small and periodic traveling wave of (1.9) and (1.6); see Figure 4 for details. The result qualitatively agrees with those in [Kaw75, DR77], for instance, from formal asymptotic expansions of the physical problem, and it improves upon that in [HJ15b] for the Whitham equation. In particular, the limit of as , where is a critical wave number, whereas the limit is unbounded for the Whitham equation (see [HJ15b]).

Moreover, we show that a sufficiently small and periodic traveling wave of (1.7) is modulationally unstable if . To the best of the authors’ knowledge, this is new. The Camassa-Holm equation seems to predict the Benjamin-Feir instability of a Stokes wave. But the instability mechanism is different from that in (1.9) and (1.2), or the water wave problem. One may use the Evans function and other ODE methods to determine the modulational stability and instability for all amplitudes. This is an interesting direction of future research. The result herein indicates that the stability and instability depend on the carrier wave.

In the absence of the effects of surface tension, we show that a sufficiently small and periodic traveling wave of (1.10) and (1.2) is modulationally unstable if 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 (see [HJ15b]). In particular, it fails to predict the limit of strong surface tension. Therefore, we learn that the higher power nonlinearities of (1.9) improve the result, not the higher derivative nonlinearities.

It is interesting to explain breaking, peaking, and other phenomena of water waves in (1.9) and (1.2) (or (1.6)).

#### Notation

The notation to be used is mostly standard, but worth briefly reviewing. Let denote the unit circle in . We identify functions over with periodic functions over via and, for simplicity of notation, we write rather than . For in the range , let consist of real or complex valued, Lebesgue measurable, and periodic functions over such that

and if . Let consist of functions whose derivatives are in . Let .

For , the Fourier series of is defined by

 ∑n∈Zˆf(n)einz,whereˆf(n)=12π∫π−πf(z)e−inz dz.

If then its Fourier series converges to pointwise almost everywhere. We define the inner product as

 (1.11) ⟨f1,f2⟩L2(T)=12π∫π−πf1(z)f∗2(z) dz=∑n∈Zˆf1(n)ˆf2∗(n).

## 2. Sufficiently small and periodic traveling waves

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

 (2.1)

where is in (1.6), and we calculate their small amplitude expansion.

### Properties of cww(⋅;T)

For any , is even and real analytic, and . Note that for any , and for , ; see [HJ15a, HJ15b], for instance, for details.

Note that decreases to zero monotonically away from the origin. For , increases monotonically and unboundedly away from the origin. For , on the other hand, , and as . Hence possesses a unique minimum over the interval ; see Figure 1.

By a traveling wave of (2.1) and (1.6), we mean a solution of the form for some , the wave speed, where satisfies by quadrature

 (cww(|∂x|;T)−c−3)η+2(1+η)3/2−2+512ηηxx+1348η2x=(1−c)2b

for some . We seek a periodic traveling wave of (2.1) and (1.6). That is, is a periodic function of for some , the wave number, and it satisfies

 (2.2) (cww(κ|∂z|;T)−c−3)η+2(1+η)3/2−2+512κ2ηηzz+1348κ2η2z=(1−c)2b.

Note that

 (2.3) cww(κ|∂z|;0):Hs(T)→Hs+1/2(T),andcww(κ|∂z|;T):Hs+1/2(T)→Hs(T)

for , for any and . Note that

 (2.4) cww(κ|∂z|;T)einz=cww(nκ;T)einzfor n∈Z.

Note that (2.2) remains invariant under

 (2.5) z↦z+z0andz↦−z

for any . Hence we may assume that is even. But (2.2) does not possess scaling invariance. Hence we may not a priori assume that . Rather, the (in)stability result herein depends on the carrier wave number. Moreover, (2.2) does not possess Galilean invariance. Hence we may not a priori assume that . Rather, we exploit the variation of (2.2) in the variable in the instability proof. To compare, the Whitham equation for periodic traveling waves possesses Galilean invariance; see [HJ15b], for instance.

We follow along the same line as the arguments in [HJ15a, HJ15b, HP15], for instance, to construct periodic traveling waves of (2.1) and (1.6). A main difference lies in the lack of a priori smoothness of solutions of (2.2). We examine the mapping properties of various operators to construct smooth solutions.

For any and an integer , let

 F:Hk+2(T)×R+×R×R+→Hk(T)

denote

 (2.6) F(η,c;b,κ,T)=(cww(κ|∂z|;T)−c−3)η+2(1+η)3/2−2+512κ2ηηzz+1348κ2η2z−(1−c)2b.

It is well defined by (2.3) and a Sobolev inequality. We seek a solution , , and of

 (2.7) F(η,c;b,κ,T)=0.

Since is arbitrary, . Note that is invariant under (2.5). Hence we may assume that is even.

For any , and , , , note that

 Fη(η,c;b,κ,T)ζ=(cww(κ|∂z|;T)−c−3+3(1+η)1/2+κ2(512(ηzz+η∂2z)+1324ηz∂z))ζ:Hk+2(T)→Hk(T)

is continuous by (2.3) and a Sobolev inequality. Here a subscript means Fréchet differentiation. Moreover, for any , and , , , note that is continuous. Since and

 Fκ(η;κ,c,b):=c′ww(κ|∂z|;T)η+56κηηzz+1324κη2z

are continuous likewise, depends continuously differentiably on its arguments. Furthermore, since the Fréchet derivatives of with respect to , and , of all orders are zero everywhere by brutal force, and since is a real analytic function, is a real analytic operator.

### Bifurcation condition

For any , , for any , and sufficiently small, note that

 (2.8) η0(c;b,κ,T)=b(1−c)+O(b2)

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 for some then, necessarily,

 L0:=Fη(η0,c0;b,κ,T):Hk+2(T)→Hk(T),

where

 (2.9) L0=cww(κ|∂z|;T)−c0−3+3(1+η0)1/2+512κ2η0∂2z,

is not an isomorphism. Here depends on . But we suppress it for simplicity of notation. A straightforward calculation reveals that , , if and only if

 (2.10) c0=cww(nκ;T)−3+3(1+η0)1/2−512κ2n2η0.

For and, hence, by (2.8), it simplifies to . Without loss of generality, we restrict the attention to . For sufficiently small, (2.10) and (2.8) become

 (2.11) c0(b,κ,T)= cww(κ;T)+b(32−512κ2)(1−cww(κ;T))+O(b2) and (2.12) η0(b,κ,T)= b(1−cww(κ;T))+O(b2).

For , since for everywhere in (see Figure 1a), it is straightforward to verify that for any , and sufficiently small, the kernel of is two dimensional and spanned by . Moreover, the co-kernel of is two dimensional. Therefore, is a Fredholm operator of index zero.

Similarly, for , since for everywhere in (see Figure 1b), for any , and sufficiently small, is a Fredholm operator of index zero, whose kernel is two dimensional and spanned by .

For , on the other hand, for any integer , it is possible to find some such that (see Figure 1c). If

 (2.13) cww(κ;T)≠cww(nκ;T)for any n=2,3,…

then is likewise a Fredholm operator of index zero, whose kernel is two dimensional and spanned by . But if for some integer , resulting in the resonance of the fundamental mode and the -th harmonic, then the kernel is four dimensional. One may follow along the same line as the argument in [Jon89], for instance, to construct a periodic traveling wave. But we do not pursue this here.

### Lyapunov-Schmidt procedure

For any , satisfying (2.13), and sufficiently small, we employ a Lyapunov-Schmidt procedure to construct non-constant solutions of (2.6)-(2.7) and, hence, (2.2) bifurcating from and , where and are in (2.12) and (2.11). Throughout the proof, , , and are fixed and suppressed for simplicity of notation.

Recall that , where is in (2.6), and , where is in (2.9). We write that

 (2.14) η(z)=η0+12(aeiz+a∗e−iz)+ηr(z)andc=c0+cr,

and we require that , be even and

 (2.15) ⟨ηr,e±iz⟩L2(T)=0,

and . Substituting (2.14) into (2.6)-(2.7), we use , , and we make an explicit calculation to arrive at

 L0ηr= (3(1+η0)1/2+cr)(12(aeiz+a∗e−iz)+ηr) −2(1+η0+12(aeiz+a∗e−iz)+ηr)3/2 −1348κ2(i2(aeiz−a∗e−iz)+η′r)2 −512κ2(12(aeiz+a∗e−iz)+ηr)(−12(aeiz+a∗e−iz)+η′′r) (2.16) =: g(ηr;a,a∗,cr).

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

 g:Hk+2(T)×C×C×R→Hk(T).

Recall that is a real analytic operator. Hence depends analytically on its arguments. Clearly, for all .

Let denote the spectral projection, defined as

 Πf(z)=ˆf(1)eiz+ˆf(−1)e−iz.

Since by (2.15), we may rewrite (2.16) as

 (2.17) L0ηr=(1−Π)g(ηr;a,a∗,cr)and0=Πg(ηr;a,a∗,cr).

Moreover, for any , and satisfying (2.13), note that is invertible on . Specifically,

 L0−1f(z)=∑n≠±1ˆf(n)cww(κn;T)−cww(κ;T)+512κ2η0(1−n2)einz.

Hence we may rewrite (2.17) as

 (2.18) ηr=L−10(I−Π)g(ηr;a,a∗,cr)and0=Πg(ηr;a,a∗,cr).

Note that is bounded. We claim that

 L−10:(1−Π)Hk(T)→Hk+2(T)

is bounded. As a matter of fact,

 ∣∣ ∣∣n2ˆf(n)cww(κn;T)−cww(κ;T)+512κ2η0(1−n2)∣∣ ∣∣⩽C|ˆf(n)|

for some constant for and sufficiently large. Therefore, for any and ,

 L−10(1−Π)g:Hk+2(T)→Hk+2(T)

is bounded. Note that it depends analytically on its argument. Since for any , it follows from the implicit function theorem that a unique solution

 η2=ηr(a,a∗,cr)

exists to the former equation of (2.18) near for and sufficiently small for any . Note that depends analytically on its arguments and it satisfies (2.15) for sufficiently small for any . The uniqueness implies

 (2.19) η2(0,0,cr)=0for any cr∈R.

Moreover, since (2.6)-(2.7) and, hence, (2.18) are invariant under (2.5) for any , it follows that

 (2.20) η2(a,a∗,cr)(z+z0)=η2(aeiz0,a∗e−iz0,cr)andη2(a,a∗,cr)(−z)=η2(a,a∗,cr)(z)

for any for any and sufficiently small, and .

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

 Πg(η2(a,a∗,cr);a,a∗,cr)=0

for and sufficiently small for . This is solvable, provided that

 (2.21) π±(a,a∗,cr):=⟨g(η2(a,a∗,cr);a,a∗,cr),aeiz±a∗e−iz⟩L2(T)=0.

We use (2.20), where , and (2.21) to show that

 π−(a∗,a,cr)=π−(a,a∗,cr)=−π−(a∗,a,cr).

Hence holds for any and sufficiently small for any . Moreover, we use (2.20), where , and (2.21) to show that

 π+(a,a∗,cr)=π+(|a|,|a|,cr).

Hence it suffices to solve for any and sufficiently small.

Substituting (2.16) into (2.21), where , we make an explicit calculation to arrive at

 π+(a,a,cr)=a2(πcr+πr(a,cr)),

where

 πr(a,cr)= −2a−1⟨(1+η0+acosz+η2(a,a,cr)(z))3/2,cosz⟩ −512κ2(⟨η′′2(a,a,cr)(z)−η2(a,a,cr)(z),cos2z⟩−a−1⟨η2η′′2(a,a,cr)(z),cosz⟩) −1348κ2a−1(⟨η′2(a,a,cr)(z)2,cosz⟩−⟨η′2(a,a,cr)(z),sin2z⟩),

and means the inner product. We merely pause to remark that is well defined. As a matter of fact, is not singular for and sufficiently small by (2.19). Clearly, and, hence, depend analytically on its arguments. Since by (2.19), it follows from the implicit function theorem that a unique solution

 cr=c1(a)

exists to and, hence, the latter equation of (2.18) near for and sufficiently small. Clearly, depends analytically on .

To recapitulate,

 ηr=η2(a,a,c1(a))andcr=c1(a)

uniquely solve (2.18) for and sufficiently small, and by virtue of (2.14),

 (2.22) η(a)(z)=η0+acosz+η2(a,a,c1(a))(z)andc(a)=c0+c1(a)

uniquely solve (2.6)-(2.7) and, hence, (2.2) for and sufficiently small. Note that is periodic and even in . Moreover, .

For and sufficiently small, we write that

 (2.23) η(a;b,κ,T)(z):= η0(b,κ,T)+acosz+a2η2(z)+a3η3(z)+⋯ and (2.24) c(a;b,κ,T):= c0(b,κ,T)+ac1+a2c2+⋯,

where are periodic, even, and smooth functions of , and .

We claim that . As a matter of fact, note that (2.2) and, hence, (2.6)-(2.7) remain invariant under by (2.5). Since , however, must hold. Thus . This proves the claim. If for any integer , in addition, then for any integer . Hence is even in .

Substituting (2.23) and (2.24) into (2.2), we may calculate the small amplitude expansion. The proof is very similar to that in [HP15], for instance. Hence we omit the details.

Below we summarize the conclusion.

###### Lemma 2.1 (Existence of sufficiently small and periodic traveling waves).

For any , satisfying (2.13), and sufficiently small, a one parameter family of solutions of (2.2) exists, denoted and , for and sufficiently small; and it is even in ; and depend analytically on , and , . Moreover,

 (2.25) η(a;b,κ,T)(z)= b(1−cww(κ;T))+acosz+a2(h0+h2cos2z)+O(a(a+b)2), (2.26) c(a;b,κ,T)= cww(κ;T)+b(32−512κ2)(1−cww(κ;T))+a2c2+O(a(a+b)2)

as , where

 (2.27) h0=(38−796κ2)1cww(κ;T)−1,h2=(38−1132κ2)1cww(κ;T)−cww(2κ;T), and (2.28)

## 3. Modulational instability index

For , satisfying (2.13), and sufficiently small, let and , denote a sufficiently small and periodic traveling wave of (2.1) and (1.6), whose existence follows from the previous section. We address its modulational stability and instability.

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

 ζt+κ∂z(cww(κ|∂z|;T)−c−3+3(1+η)1/2+κ2(512(η∂2z+ηzz)+1324ηz∂z))ζ=0,

where is in (1.6). Seeking a solution of the form , , we arrive at

 (3.1) λζ =∂z(−cww(κ|∂z|;T)+c+3−3(1+η)1/2−κ2(512(η∂2z+ηzz)+1324ηz∂z))ζ =:L(a;b,κ,T)ζ.

We say that is spectrally unstable to square integrable perturbation if the spectrum of intersects the open right-half plane of , and it is spectrally stable otherwise. Note that is periodic in , but needs not. Note that the spectrum of is symmetric with respect to the reflections in the real and imaginary axes. Hence is spectrally unstable if and only if the spectrum of is not contained in the imaginary axis.

It is well known (see [BHJ16], for instance, and references therein) that the spectrum of contains no eigenvalues. Rather, it consists of the essential spectrum. Moreover, a nontrivial solution of (3.1) does not belong to for any . Rather, if solves (3.1) then, necessarily,

 ζ(z)=eiξzϕ(z)