Stability properties of steady water waves with vorticity

Stability Properties of Steady Water Waves with Vorticity

ADRIAN CONSTANTINTrinity College Dublin

AND

WALTER A. STRAUSSBrown University

Abstract

We present two stability analyses for exact periodic traveling water waves with

vorticity. The first approach leads in particular to linear stability properties of

water waves for which the vorticity decreases with depth. The second approach

leads to a formal stability property for long water waves that have small vorticity

and amplitude although we do not use a small-amplitude or long-wave approxi-

mation. c© 2006 Wiley Periodicals, Inc.

1 Introduction

Water waves are generally expected to be unstable. However, in several approx-

imate models, such as Korteweg–de Vries, they enjoy remarkable stability proper-

ties. Thus the full water-wave equations ought to possess some vestige of these

stability properties. This is what we investigate in this article. We do not make

any small-amplitude or shallow-water (long-wave) approximation but consider the

full two-dimensional periodic water wave problem, which consists of the Euler

equations under the influence of gravity with the classical dynamic and kinematic

boundary conditions on the free surface.

We present two quite different approaches to stability. In the first one, we begin

with a temporally invariant functional H, which is the energy suitably modified

by a vorticity term and subject to mass and momentum constraints. Following

[6], its first variation gives the full water-wave equations including the boundary

conditions, provided the vorticity is a strictly decreasing function of the depth.

Thus irrotational flow is excluded from this analysis. We easily calculate its second

variation around a steady flow under general perturbations of the same period. It is

positive, and the water wave is linearly stable, if either (i) the surface is unperturbed

or (ii) the velocity on the surface is perturbed only normally or (iii) the tangential

velocity is a sufficiently small perturbation.

Our second approach to stability has a very different point of departure. We

take another functional, which is essentially the dual of H but is not an invariant. It

is then transformed following [8] by essentially exchanging the roles of the vertical

Communications on Pure and Applied Mathematics, Vol. LX, 0911–0950 (2007)c© 2006 Wiley Periodicals, Inc.

912 A. CONSTANTIN AND W. A. STRAUSS

variable y and the stream function ψ . The first variation of the functional J then

gives the exact equations for steady water waves in the transformed variables. It

is these equations that were used in [7, 8] to construct global continua of steady

water waves. There is a curve of trivial solutions with flat surface profiles and

also a pitchfork bifurcating curve C of nontrivial solutions. The stability of the

trivial solutions switches exactly at the bifurcation point. We prove an “exchange of

stability” theorem, namely, that the second variation of J , the linearized operator,

is positive along C provided that both the depth and the vorticity are sufficiently

small. This result is proven using the classical Crandall-Rabinowitz theorem [9]

together with a tedious explicit computation of derivatives at the bifurcation point.

It appears to be a new result even in the irrotational case.

There is an immense literature that deals with small-amplitude, shallow-water

models of water waves, such as the Korteweg–deVries equation. A modern discus-

sion of such approximations can be found in [13]. Some of these model equations

are integrable Hamiltonian systems whose periodic steady solutions are remark-

ably stable (see, e.g., [14, 17]). In this paper we do not discuss such approximate

models. However, the models suggest that some stability properties might carry

over to the full water-wave equations.

Several authors have discussed the stability of water waves. All of them con-

sider only the irrotational case. The most significant analysis is that of Benjamin

and Feir [2], who assumed a small-amplitude approximation in the irrotational case

and showed that there always is a “sideband” instability, meaning that the pertur-

bation has a different period than the steady wave. Their work was made mathe-

matically rigorous by Bridges and Mielke [3]. On the other hand, in the present

paper we only consider perturbations of the same period as the steady wave.

Zakharov [19] and MacKay and Saffman [16] discussed linear stability for the

Hamiltonian system that arises with the use of the velocity potential in the irrota-

tional case. Such a formulation is not available in the presence of vorticity. Further-

more, their analysis made heavy use of a small-amplitude approximation. Mielke

[18] and Buffoni [4] used the same Hamiltonian structure to rigorously prove the

nonlinear stability of exact irrotational solitary (not periodic) water waves under

the assumption that there is nontrivial surface tension (capillary waves). It is well-

known that surface tension has a stabilizing influence. Measurements of physical

water waves show that surface tension is usually quite small or negligible. In the

present paper we consider waves with vanishing surface tension.

After fixing our notation in Section 2, we compute in Section 3 the second

variation of the temporal invariant H, which leads to a brief discussion of linear

stability. In the much simpler case of a fixed boundary, our calculation reduces

to that of Arnold [1]. In Section 4 we consider the transformed problem and the

functional J . We discuss its relationship to bifurcation theory and compute the

second variation of J near the bifurcation point for small vorticity. This leads to

our conclusions concerning formal stability.

STABILITY PROPERTIES OF STEADY WATER WAVES 913

2 Preliminaries

The water waves that one typically sees propagating on the surface of a body

of water are, as a matter of common experience, locally approximately periodic

and two-dimensional. That is, the motion is identical in any direction parallel to

the crest line. To describe such waves it suffices to consider a cross section of

the flow that is perpendicular to the crest line. Choose coordinates (x, y) with the

y-axis pointing vertically upwards, the x-axis horizontal, and the origin at mean

water level. Let (u(t, x, y), v(t, x, y)) be the velocity field of the flow, y = −d

the flat bed, and y = η(t, x) the free surface (Figure 2.1). Homogeneity (constant

density), a physically reasonable assumption for gravity waves [15], implies the

equation of mass conservation

(2.1) ux + vy = 0.

Also appropriate for gravity waves is the assumption of inviscid flow [12], so that

the motion of the water is given by Euler’s equation

(2.2)

{ut + uux + vuy = −Px

vt + uvx + vvy = −Py − g,

where P(t, x, y) denotes the pressure and g the gravitational constant. The free

surface decouples the motion of the water from that of the air so that the dynamic

boundary condition

(2.3) P = P0 on y = η(t, x)

holds in the absence of surface tension, where P0 is the atmospheric pressure.

Moreover, since the same particles always form the free surface, we have the kine-

matic boundary condition

(2.4) v = ηt + uηx on y = η(t, x).

On the flat bed we have the kinematic boundary condition

(2.5) v = 0 on y = −d,

expressing the fact that the bed is impenetrable.

We assume the flow is periodic in the sense that the velocity field (u, v), the

pressure P , and the free surface η all have period (wavelength) L in the x-variable.

Let

�(t) = {(x, y) ∈ R2 : 0 < x < L , −d < y < η(t, x)}

denote the fluid domain within a wavelength at time t ≥ 0. We assume that u, v,

P , and η are all C2 in the variables (t, x, y) as (x, y) ∈ �(t) where t varies in

some interval [0, T ) with T > 0. The incompressibility condition (2.1) enables us

to introduce a stream function ψ(t, x, y) defined up to a constant by

(2.6) ψx = −v, ψy = u.

Thus �ψ = −ω where ω = vx − uy is the vorticity.


y

L

y= η (t, x)

x

− d

0

FIGURE 2.1. The profile of a steady periodic wave.

Among the periodic waves there are the steady periodic waves that travel at a

given wave speed c > 0; that is, they depend only on x − ct and y. On S the

function ψ − cy is constant. We can choose this constant arbitrarily; we choose

ψ − cy = 0 on S. For such waves Euler’s equation (2.2) implies that ψ − cy and

ω are functionally dependent since their gradients are parallel. For simplicity, we

assume that this functional dependence is global; that is, there exists a C1 function

γ , called the vorticity function, such that

(2.7) −�ψ = ω = γ (ψ − cy)

throughout the fluid. The dependence is global if, for instance, the steady wave sat-

isfies the condition u < c throughout the fluid domain [8]. In this paper we assume

this condition, which is physically justified by measurements for water waves that

are not near breaking [15]. Then the governing equations can be reformulated in

the moving frame as

(2.8)

�ψ = −γ (ψ − cy) on − d < y < η(x)

|∇(ψ − cy)|2 + 2 g(y + d) = Q at y = η(x)

ψ − cy = 0 at y = η(x)

ψ − cy = p0 at y = −d,

where p0 is a constant. This constant p0 = ∫ η(x)

−d(u − c) dy, which is independent

of x , is called the relative mass flux. The nonlinear boundary condition at y =η(x) is an expression of Bernoulli’s law. The constant Q can be expressed as

Q = 2(E + gd), where E is the hydraulic head of the flow. For more details we

refer to [8].


Among the conservation laws for the governing equations (2.1)–(2.5) are the

total energy

(2.9) E =∫∫�(t)

[u2 + v2

2+ g(y + d)

]dy dx,

the mass

(2.10) m =∫∫�(t)

dy dx,

the horizontal component of the momentum

(2.11) M =∫∫�(t)

u dy dx,

and

(2.12) F =∫∫�(t)

F(ω)dy dx

for any C1-function F (see [6]).

3 Linear Stability

Consider periodic steady two-dimensional waves as described above. Let F :R → R be any C2-function for which F ′′ < 0. That is, F is strictly concave. We

define the C1-function γ by

(3.1) γ = (F ′)−1.

Now the stream function ψ and the free surface profile η completely determine the

steady flow. Since ψy < c throughout the fluid domain, the strict concavity of F is

equivalent to the condition γ ′ < 0 or, in view of (2.7), to ωy > 0. The functionals

E , F , m, and M may be written entirely in terms of ψ and η. They are defined on

the space of functions

(ψ, η) ∈ C2per(R × [−d,∞)) × C1

per(R)

that are smooth with period L in the variable x . Note that ψ(x, y) is defined for

−d ≤ y < ∞, part of which lies above the fluid domain. Normally ψ and η are

coupled via the boundary conditions. However, if the existence of a critical point

is not an issue, then we can extend a smooth stream function across a smooth free

boundary, and we are allowed to consider independent variations with respect to

these two functions. Let � = {(x, y) | −d < y < η(x), 0 < x < L} denote the

fluid domain.


3.1 Variations of the Constrained Energy

In [6] we proved that any critical point of E − F , subject to the constraints of

fixed mass m and horizontal momentum M, is a steady water wave with vorticity

function γ . When one identifies the Lagrange multipliers, the functional of which

we take variations is

(3.2) H(ψ, η) =∫∫�

{ |∇(ψ − cy)|22

+ g(y + d) − Q

2− F(−�ψ)

}dy dx .

Now we calculate the second variation. Beginning with a critical point (ψ, η), we

denote a pair of variations of ψ by ψ1 and ψ2, and similarly for variations of η.

Furthermore, we denote ω = −�ψ , ω1 = −�ψ1, and ω2 = −�ψ2.

THEOREM 3.1 The second variation of H is

δ2H =∫S

(∂ψ

∂y− c

){∂ψ2

∂nη1 + ∂ψ1

∂nη2

}dl

+∫∫�

{∇ψ2 · ∇ψ1 − F ′′(ω)ω1ω2

}dy dx

+∫S

{g + 1

2

∂

∂y[|∇(ψ − cy)|2]

}η1η2 dx .

PROOF: The first variation was calculated in [6] as δH = δI + δII + δIII where

δI =∫∫�

{(ψ − cy)ω1 − F ′(ω)ω1}dy dx,

δII =∫S

{ |∇(ψ − cy)|22

+ g(y + d) − Q

2− F(ω)

}η1 dx

and

δIII =∫S

(ψ − cy)∂ψ1

∂ndl −

∫B

(ψ + cd)∂ψ1

∂ydx .

Starting from these formulas for the first variation, we calculate another varia-

tion of each term. First,

δ2I =∫∫�

{ψ2ω1 − F ′′(ω)ω2ω1}dy dx +∫S

{(ψ − cy)ω1 − F ′(ω)ω1}η2 dx .

A common factor in the last integrand is ψ − cy = F ′(ω) = 0 on S. So the whole

surface integral vanishes. Writing ω1 = −�ψ1, we integrate the first term by parts


and thereby obtain

δ2I =∫∫�

{∇ψ2 · ∇ψ1 − F ′′(ω)ω2ω1}dy dx −∫S

ψ2

∂ψ1

∂ndl +

∫B

ψ2

∂ψ1

∂ydx .

In the calculations below, it is convenient to notice that

δ

∫S

f dx =∫S

(δ f + ∂ f

∂yδη

)dx .

The second part of our second variation is

δ2II =∫S

{∇(ψ − cy) · ∇ψ2 − F ′(ω)ω2}η1 dx

+∫S

{∇(ψ − cy) · ∇(ψy − c) + g − F ′(ω)ωy}η2η1 dx .

But F ′(ω) = ψ − cy = 0 on S and

∇(ψ − cy) · ∇ψ2 dx = (ψy − c)(−ηxψ2x + ψ2y)dx = (ψy − c)∂ψ2

∂ndl

so that

δ2II =∫S

(ψy − c)∂ψ2

∂nη1 dl +

∫S

{[1

2|∇(ψ − cy)|2

]y

+ g

}η2η1 dx .

The third part reduces to

δ2III =∫S

{ψ2 + (ψy − c)η2}∂ψ1

∂ndl −

∫B

ψ2

∂ψ1

∂ydx

because all the other terms are integrals over S with the factor ψ − cy, which

vanishes on S. Combining all the terms, we obtain

δ2H =∫∫�

{∇ψ2 · ∇ψ1 − F ′′(ω)ω2ω1}dy dx

+∫S

{[1

2|∇(ψ − cy)|2

]y

+ g

}η2η1 dx

+∫S

(ψy − c)

{∂ψ2

∂nη1 + ∂ψ1

∂nη2

}dl

because of the cancellation of the terms ± ∫Sψ2(∂ψ1/∂n)dl from δ2I and δ2III, as

well as the integrals on the bottom. �


Our goal is to find conditions under which this symmetric quadratic form is

nonnegative. For this purpose we take ψ1 = ψ2 and η1 = η2. Then

δ2H =∫∫�

{|∇ψ2|2 − F ′′(ω)|ω2|2}dy dx

+∫S

{[1

2|∇(ψ − cy)|2

]y

+ g

}|η2|2 dx + 2

∫S

(ψy − c)∂ψ2

∂nη2 dl.

(3.3)

By the concavity of F , which follows from the assumption that γ ′ < 0, the first

integral is nonnegative.

3.2 Stability Conditions

By linear stability of the traveling wave (ψ, η) we mean that the quadratic form

δ2H is nonnegative. As above, we assume that u = ψy < c in �. The following

lemma states that the second integral in (3.3) is positive. Recall [8] that if

(p) =∫ p

0

γ (−s)ds for p ∈ [p0, 0],

then Bernoulli’s law ensures that the quantity

1

2|∇(ψ − cy)|2 + gy + P − (cy − ψ)

is constant throughout the fluid domain �.

LEMMA 3.2 On S we have

{P − (cy − ψ)}y < 0

if also γ (p0) ≥ 0.

PROOF: Using the basic equation −�ψ = ω = γ (ψ − cy) satisfied by ψ , we

calculate {1

2|∇(ψ − cy)|2

}y

+ g

= g + ψxψxy + (ψy − c)ψyy

= g + ψxψxy − (ψy − c)ψxx − (ψy − c)γ (ψ − cy).

(3.4)

However, the vertical component of the Euler equation is

−Py = g + vt + uvx + vvy = g + ψxψxy − (ψy − c)ψxx .

So {1

2|∇(ψ − cy)|2

}y

+ g = −Py − (ψy − c)γ (ψ − cy)

= −{P − (cy − ψ)}y .

(3.5)


On the other hand, the Euler equation also implies

�P = −2v2y − 2uyvx = −2ψ2

xy − ψ2xx − ψ2

yy + (�ψ)2

so that

�{P − (cy − ψ)} = [γ (ψ − cy)]2 − ψ2xx − ψ2

yy − 2ψ2xy

+ γ ′(ψ − cy)|∇(ψ − cy)|2 + γ (ψ − cy)�ψ

= −ψ2xx − ψ2

yy − 2ψ2xy + γ ′(ψ − cy) |∇(ψ − cy)|2 ≤ 0

because γ is decreasing. Thus P − (cy − ψ) is superharmonic and its minimum

can only be attained on the top S or bottom B (unless it is a constant); cf. [10].

On the bottom we have ψx = v = 0 and ψ − cy = p0, so that by (3.4)–(3.5)

−{P − (cy − ψ)}y = g + ψxψxy − (ψy − c)[ψxx + γ (ψ − cy)]= g − (u − c)γ (p0).

Since u < c by assumption, this expression is strictly positive if γ (p0) ≥ 0. Thus

the minimum of P − (cy − ψ) must be attained on S.

However, this function is a constant on S so that it is minimized at every point

of S. Therefore

0 > {P − (cy − ψ)}y

∣∣S

= −{

1

2|∇(ψ − cy)|2

}y

∣∣∣∣S

− g

by the Hopf form of the maximum principle [10] and by (3.5). �

Another way to state Lemma 3.2 is that the upward normal derivative on S of

P − (cy − ψ) is negative.

We remark that Lemma 3.2 is also true if −γ (p0)(c − u(q, p0)) < g instead of

γ (p0) ≥ 0. This is evident by inspection of the above proof.

The above considerations lead to the following stability result under smooth

perturbations supported in a neighborhood of the free surface S = {y = η(x)} and

periodic in x :

THEOREM 3.3 Assume that ωy > 0 and γ (−p0) ≥ 0. Then

(i) Any traveling wave is linearly stable if the surface is unperturbed.

(ii) Any traveling wave is linearly stable if the velocity on the surface is per-

turbed only normally.

(iii) Any traveling wave is linearly stable provided the velocity on the surface is

perturbed only tangentially and |u −c| is uniformly sufficiently small on S.

(iv) Any traveling wave is linearly stable under perturbations (ψ2, η2) satisfy-

ing the inequality

(3.6)

∫S

∣∣∣∣∂ψ2

∂n

∣∣∣∣2

dl ≤ µ2

∫S

|ψ2|2 dl

provided µ = supS {|u − c|} is sufficiently small.


PROOF: As already pointed out, the hypothesis ωy > 0 is equivalent to F ′′ < 0

and to γ ′ < 0. By Lemma 3.2 the first and second integrals in (3.3) are nonnegative.

For the surface to be unperturbed, it means that η2 = 0, whence only the first

integral in (3.3) remains and therefore δ2H ≥ 0. This proves (i).

For the velocity on the surface to be perturbed only normally, it means that

the tangential component of the velocity perturbation vanishes. But this means

that ∂ψ2/∂n = 0 on S. Therefore the third term in (3.3) vanishes. Once again,

δ2H ≥ 0. This proves (ii).

The velocity on the surface is perturbed only tangentially if and only if the

normal component of the velocity perturbation vanishes, that is, ∂ψ2/∂τ = 0 on

S. This is equivalent to the requirement that ψ2 be constant on S. To prove (iii) we

will estimate the third term in (3.3) by the others. Define

a = supS

{|ψy − c|}, b = infS

12(|∇(ψ − cy)|2)y + g√

1 + η2x

, θ = supp∈[p0,0]

|γ ′(p)|.

Then

δ2H ≥∫∫�

|∇ψ2|2 dy dx + b

∫S

η22 dl + 1

θ

∫∫�

|�ψ2|2 dy dx

− 2a

∫S

∣∣∣∣∂ψ2

∂n

∣∣∣∣ |η2|dl

≥∫∫�

|∇ψ2|2 dy dx + 1

θ

∫∫�

|�ψ2|2 dy dx − a2

b

∫S

∣∣∣∣∂ψ2

∂n

∣∣∣∣2

dl ≥ 0,

if a is sufficiently small, by the standard trace inequality applied to ∇ψ2.

To prove (iv), with the same notation, observe that

δ2H ≥∫∫�

|∇ψ2|2 dy dx + b

∫S

η22 dl − 2a

∫S

∣∣∣∣∂ψ2

∂n

∣∣∣∣ |η2|dl

≥∫∫�

|∇ψ2|2 dy dx − a2

b

∫S

∣∣∣∣∂ψ2

∂n

∣∣∣∣2

dl

≥∫∫�

|∇ψ2|2 dy dx − a2µ2

b

∫S

|ψ2|2 dl ≥ 0

by the standard trace inequality (noting that ψ2 = 0 on B), provided aµ is small

enough. �

Example 1. The smallness conditions in Theorem 3.3(iii) and (iv) apply to certain

trivial solutions. Indeed, for trivial flows with a flat surface S = {y = 0} and


velocity components independent of x and with u < c, γ (0) = 0, and γ ′ < 0, we

have

(c − ψy)∣∣

S=

√λ > 0,

1

2[|∇(ψ − cy)|2]y

∣∣S+ g = ψyy(ψy − c)

∣∣S+ g = −γ (0)

√λ + g = g,

for some constant λ > 0. Thus a =√

λ and b = g. Notice that if fx = 0 on S,

then ∫ L

0

( f 2x + f 2

y )dx

∣∣∣∣y=0

= 2

∫ L

0

∫ 0

−d

( fx fxy + fy fyy)dy dx

= 2

∫∫�

fy{ fyy − fxx}dy dx

≤∫∫�

[g

λf 2

y + λ

g( fyy − fxx)

2

]dy dx

=∫∫�

[g

λf 2

y + λ

g(� f )2 − 4λ

gfxx fyy

]dy dx

=∫∫

�

[g

λf 2

y + λ

g(� f )2 − 4λ

gf 2xy

]dy dx

≤∫∫�

[g

λ|∇ f |2 + λ

g(� f )2

]dy dx

provided fx = fy = 0 on the flat bed y = −d. Thus∫∫�

f 2y dx ≤

∫∫�

[g

λ|∇ f |2 + λ

g(� f )2

]dy dx

for all smooth functions f , periodic in x , with fx = 0 on y = 0 and f ≡ 0 in a

neighborhood of the flat bed B = {y = −d}. Thus, setting f = ψ2,

δ2H ≥∫∫�

|∇ψ2|2 dy dx + 1

θ

∫∫�

|�ψ2|2 dy dx

+ g

∫S

η22 dx − 2

√λ

∫S

∣∣∣∣∂ψ2

∂y

∣∣∣∣ |η2|dx

≥∫∫�

|∇ψ2|2dy dx + 1

θ

∫∫�

|�ψ2|2dy dx − λ

g

∫S

∣∣∣∣∂ψ2

∂y

∣∣∣∣2

dx

≥(

1

θ− λ2

g2

)∫∫�

|�ψ2|2 dy dx


so that the condition

(3.7) supp∈[p0,0]

|γ ′(p)| = θ ≤ g2

λ2

ensures linear stability provided the velocity on S is perturbed only tangentially.

As for (iv), let us now assume that the perturbation ψ2 satisfies the inequality

(3.6). Then

2√

λ

∫S

∣∣∣∣∂ψ2

∂n

∣∣∣∣ |η2|dx ≤ g

∫S

η22 dx + λ

g

∫S

∣∣∣∣∂ψ2

∂n

∣∣∣∣2

dx

≤ g

∫S

η22 dx + λµ2

g

∫S

ψ22 dx .

But

ψ2(x, 0) =∫ 0

−d

∂ψ2

∂ydy, 0 ≤ x ≤ L ,

so ∫S

ψ22 dx =

∫ L

0

(∫ 0

−d

∂ψ2

∂ydy

)2

dx ≤ d

∫ L

0

∫ 0

−d

(∂ψ2

∂y

)2

dy dx

≤ d

∫∫�

|∇ψ2|2 dy dx .

Combining the previous two inequalities we get

δ2H ≥∫∫�

|∇ψ2|2 dy dx + g

∫S

η22 dx − 2

√λ

∫S

∂ψ2

∂yη2 dx

≥∫∫�

|∇ψ2|2 dy dx − λµ2

g

∫S

ψ22 dx

≥(

1 − dλµ2

g

) ∫∫�

|�ψ2|2 dy dx .

Since a =√

λ, we deduce that the condition

(3.8) aµ ≤√

g

d

ensures linear stability under perturbations satisfying (3.6).

Example 2. Theorem 3.3 applies to certain nontrivial flows as well. Indeed, it was

proven in [8] that there is a smooth curve of nontrivial wave solutions traveling at

speed c that bifurcate from the curve of trivial flows at a specific point λ∗ > 0. If

the inequalities in (3.7) and (3.8) were strict at λ = λ∗, then by a small alteration

of the previous inequalities we would infer the partial linear stability of nontrivial


waves of small-amplitude as stated in Theorem 3.3 (i)–(iv). Using this approach,

let us take as an example the vorticity function γ (ψ) = −ε ψ . Because of (3.7),

Theorem 3.3(iii) applies if we prove λ∗2 < g2/ε. We will show that this is the case

if

(3.9) 0 < ε ≤(

16g2

p40

)1/3

.

Indeed, from [8] we know that λ∗ < λ0, where λ0 > 0 is the unique solution of the

equation ∫ 0

p0

dp

(λ0 + 2(p))3/2= 1

g.

Thus it suffices to prove λ20 ≤ g2/ε. Now 2(p) = εp2, so the identity defining λ0

becomes1

λ0

−p0√λ0 + εp2

0

= 1

g

or g2 p20 = λ2

0(λ0 + εp20). But then g2 p2

0 ≥ 2 ε1/2|p0| λ5/2

0 so that

(λ∗)2 < λ20 ≤

(g2|p0|2√

ε

)4/5

≤ g2

ε

by (3.9).

4 Formal Stability of Some Bifurcating Solutions

In this section we will prove a formal stability property of small-amplitude,

symmetric, irrotational, steady periodic waves (periodic Stokes waves). A pertur-

bative argument will extend this stability result to flows with small vorticity. The

meaning of this formal stability property is that the linearized operator in stream

coordinates around the steady wave is nonnegative.

4.1 Another Functional and the Transformed Problem

We introduce another functional

L(ψ) =∫∫�

{1

2|∇(ψ − cy)|2 − g(y + d) + 1

2Q + (cy − ψ)

}dy dx,

where γ is any C1-function and ′(p) = γ (−p), (0) = 0. Since ψ will be

constant on the free surface S and on the flat bottom y = −d , it is natural to

introduce new independent variables

q = x, p = cy − ψ(x, y),


x πy=−d−π

y

y= (x)η

p

q πp=p0

−π

p=0

q=x

p= − ψ

FIGURE 4.1. A coordinate transformation.

that have the effect of fixing the free surface (see Figure 4.1). The dependent

variables in (2.8) are replaced by the single function h(q, p) = y + d , periodic in

the q-variable, so that

hq = ψx

c − ψy

, hp = 1

c − ψy

.

The function h is the height above the flat bottom. (We can recover the free surface

as η(x) = h(q, 0) and the velocity as u = ψy, v = −ψx .) In these coordinates the

functional L takes the form

J (h) =∫∫

R

{1 + h2

q

2h2p

− gh + Q

2+ (p)

}hp dq dp

where R = (0, L) × (p0, 0) is a rectangle in R2. The domain of J is the set of

all h ∈ C2(R̄) that are L-periodic and even in q with h = 0 for p = p0 and with

hp �= 0 in R̄. Throughout this section the waves are assumed to be symmetric

with respect to the crest line. It was proven in [5] that if the vorticity function is

decreasing, any wave with a profile that is monotone between crests and troughs

has to be symmetric. For any γ each solution constructed in [8] has the same

property. In this section the vorticity function is any smooth function, later to be

assumed sufficiently small.

A straightforward calculation of the first variation of J gives

〈δJ (h), k〉 = −∫∫

R

G1(h)h−3p k dq dp − 1

2

∫T

G2(Q, h)h−2p k dq,

where

G1(h) = hqqh2p − 2hqhphpq + (1 + h2

q)hpp + γ (−p) h3p


and

G2(Q, h) =1 + h2

q

h2p

+ 2gh − Q.

Thus any critical point of J satisfies the equations G1(h) = 0 in R, G2(Q, h) = 0

on T . As we proved in [6], these equations are precisely the water-wave equations

transformed to the new coordinates.

The second variation, calculated at any critical point h with nonvanishing hp, is

〈δ2J (h)l, k〉 = −∫∫

R

G1h(h)l · kh−3p dq dp − 1

2

∫T

G2h(Q, h)l · kh−2p dq

where the subscript h denotes the Fréchet derivative. Thus δ2J (h) represents the

linearized operator (G1h,G2h) around the solution h. Clearly δ2J (h) provides a

symmetric bilinear form on the space X of all functions k ∈ H 1(R̄) that are L-

periodic and even in q with∫

Tk dq = 0 and k = 0 at p = p0. The integral

condition means we retain the same average free surface level. We define h to be

formally stable if 〈δ2J (h)k, k〉 ≥ 0 for all k ∈ X .

4.2 Trivial Solutions

The trivial solutions are those with a flat surface. Given the constants L and p0,

such solutions form a one-parameter family given by h(q, p) = H(p) with

(4.1) H(p) = H(p, λ) =∫ p

p0

ds√λ + 2(s)

, p0 ≤ p ≤ 0,

where the parameter λ > −2min equals the square of the fluid velocity at the

surface in the moving frame. Furthermore, the parameter Q is a function of λ

given by

Q = λ + 2g

∫ 0

p0

dp√λ + 2(p)

.

In [8] we proved that there exists a bifurcation at a certain value λ∗ provided that

p0 satisfies the inequality

(4.2)

∫ 0

p0

[4π2(p − p0)

2

L2(2(p) − 2min)

1/2 + (2(p) − 2min)3/2

]dp < gp2

0.

PROPOSITION 4.1 The trivial solutions H( · , λ) are formally stable if and only if

λ ≥ λ∗.

PROOF: Calculating along the trivial curve h = H(p) by expanding

k(q, p) =∞∑

n=0

kn(p) cos

(2πnq

L

),


we find that

〈δ2J (H)k, k〉 = L

2

∞∑n=1

{ ∫ 0

p0

( [k ′n(p)]2

H 3p

+(

2πn

L

)2k2

n(p)

Hp

)dp − gk2

n(0)

}

+ L

{ ∫ 0

p0

[k ′0(p)]2

H 3p

dp − gk20(0)

}

since ∫ L

0

cos2

(2πnq

L

)dq =

{L/2 for n ≥ 1

L for n = 0.

Thus δ2J (H) is positive definite if and only if

gk2n(0) ≤

∫ 0

p0

{ [k ′n(p)]2

H 3p

+(

2πn

L

)2k2

n(p)

Hp

}dp for all n ≥ 0.

We know from [8] that

µ(λ) = infk

−gk2(0) + ∫ 0

p0H−3

p (k ′)2 dp∫ 0

p0H−1

p k2 dp

≥ −(

2π

L

)2

if and only if λ ≥ λ∗.

We deduce that

(4.3) 〈δ2J (H)k, k〉 ≥ 0 if and only if λ ≥ λ∗

since the condition∫

Tk dq = 0 is equivalent to k0(0) ≡ 0. �

Remark 4.2. If we permit∫

Tk dq �= 0, then the trivial solutions are formally stable

if and only if λ ≥ λ0, where λ0 > 0 is the unique solution of∫ 0

p0

(λ0 + 2(p))−3/2 dp = 1

g.

Indeed, in addition to λ ≥ λ∗, we need that

gk2(0) ≤∫ 0

p0

[k ′(p)]2

H 3p

dp for all k ∈ H 1(p0, 0) satisfying k(p0) = 0.

According to lemma 3.5 in [8], the above inequality holds if and only if λ ≥ λ0.

4.3 Nontrivial Solutions

Now we consider the local bifurcating curve of nontrivial solutions.

THEOREM 4.3 The nontrivial solutions that are sufficiently near the bifurcation

point are formally stable provided both the vorticity and the depth are sufficiently

small.


(λ , 0)

( λ (s) ,w(s))

( λ*, 0 )

FIGURE 4.2. The bifurcating solution curve.

To prove Theorem 4.3, we must study how the spectrum of the linearized op-

erator varies along the bifurcating curve near the bifurcation point (Figure 4.2).

Our basic tool is the exchange of stability theorem of Crandall and Rabinowitz [9],

which we now state. We denote by N (�) and R(�) the null space and range,

respectively, of any operator �.

DEFINITION 4.4 Let �, K : X → Y be two bounded linear operators from a

real Banach space X to another one Y . A complex number β is called a K -simple

eigenvalue of � if

dimN (� − βK ) = 1 = codimR(� − βK )

and

Kϕ∗ /∈ R(� − βK ) for 0 �= ϕ∗ ∈ N (� − βK).

THEOREM 4.5 (Crandall-Rabinowitz) Let X and Y be real Banach spaces and

let K : X → Y be a bounded linear operator. Let F : R × X → Y be C2 near

(λ∗, 0) ∈ R×X with F(λ, 0) = 0 for |λ∗−λ| small. Let � = −Fw(λ∗, 0). If β = 0

is a K -simple eigenvalue of � and a −Fλw(λ∗, 0)-simple eigenvalue of �, then

there exists locally a curve (λ(s), w(s)) ∈ R × X such that (λ(0), w(0)) = (λ∗, 0)

and F((λ(s), w(s)) = 0. Moreover, if F(λ,w) = 0 with w �= 0 and (λ,w)

near (λ∗, 0), then (λ,w) = (λ(s), w(s)) for some s �= 0. Furthermore, there are

eigenvalues β(s), βtriv(λ) ∈ R with eigenvectors ϕ(s), ϕtriv(λ) ∈ X such that

−Fw(λ(s), w(s))ϕ(s) = β(s)Kϕ(s),

−Fw(λ, 0)ϕtriv(λ) = βtriv(λ)Kϕtriv(λ),

with

β(0) = βtriv(λ∗) = 0 and ϕ(0) = ϕtriv(λ

∗) = ϕ∗.


Each curve is C1 (respectively, C2) if F is C2 (respectively, C3), with

dβtriv

dλ

∣∣∣∣λ=λ∗

�= 0 and lims→0, β(s) �=0

sλ′(s)β(s)

= − 1

β ′triv(λ

∗).

PROOF OF THEOREM 4.3: We shall apply Theorem 4.5 with the spaces

X = {w ∈ C3+α(R̄) | w = 0 on B, w even and L-periodic in q,∫ 2π

0w(q, 0)dq = 0

},

Y = C1+αper (R̄) × C2+α

per (T ),

where T = {p = 0} is the top and B = {p = p0} is the bottom of the rectangle

R = (0, 2π) × (p0, 0). Furthermore, we define

F(λ,w) = G(Q(λ), H(λ) + w).

Thus the operator F : (−2min,∞)× X → Y is given explicitly by F = (F1,F2)

with

F1(λ,w) = (1 + w2q)(Hpp + wpp) − 2wq(Hp + wp)wpq

+ (Hp + wp)2wqq + γ (−p)(Hp + wp)

3(4.4)

and

(4.5) F2(λ,w) = 1 + w2q + (2g(H + w) − Q)(Hp + wp)

2.

Now we define

Kw = (w, 0) ∀w ∈ X and � = −Fw(λ∗, 0).

Notice that Fw = (F1w,F2w) with

F1w(λ, 0) = ∂2p + H 2

p ∂2q + 3γ (−p)H 2

p ∂p in R

and

F2w(λ, 0) = 2(λ−1g − λ1/2∂p)∣∣T,

whereas

(4.6) Fλw(λ, 0) = (−a−4∂2q − 3γ a−4∂p , (−2a−4g − a−1∂p)

∣∣T

)where

a(p) = {λ + 2(p)}1/2.

We proved in [8] the existence of some bifurcation value λ∗ > −2min such that

0 is a −Fλw(λ∗, 0)-simple eigenvalue of �. Moreover, we claim that 0 is a K -

simple eigenvalue of �. This means that Kφ∗ = (φ∗, 0) �∈ R(−Fw(λ∗, 0)). In

view of the characterization of R(�) given in [8, lemma 3.8], this is equivalent to

the statement that ∫∫R

[ϕ∗]2a3 dq dp > 0 for ϕ∗ �≡ 0,

which is obvious.


Defining the spectrum as

�(λ,w) = {β ∈ C | −Fw(λ,w) − βK is not invertible from X to Y },it is known [8, lemma 4.4] that �(λ,w) consists entirely of eigenvalues of finite

multiplicity with no finite accumulation point and there is a neighborhood N of

[0,∞) in C such that �(λ,w) ∩ N is a finite set. Along the trivial curve (w ≡ 0),

we have h(q, p) = H(p) and

−Fw(λ, 0) = −(a−3[∂p(a

3∂p) + ∂q(a∂q)], 2λ−1[g − λ3/2∂p])

is a self-adjoint operator in L2(R) with the weight function a−3. Therefore �(λ, 0)

is real, with the eigenvalues having ∞ as a limit point, and with its smallest eigen-

value simple. Hence for small w �= 0 the spectrum �(λ,w), which generally

consists of complex numbers, has a unique simple eigenvalue β(λ,w) with small-

est real part. Thus h is formally stable if and only if β(λ, h − H(λ)) ≥ 0. We know

by Proposition 4.1 that β(λ∗, 0) = 0 and β(λ, 0) > 0 if and only if λ > λ∗.

We now claim that along the curve (λ(s), w(s)) of nontrivial solutions near the

bifurcation point (λ∗, 0), with λ∗ = λ(0), linear stability (respectively, instability)

holds if λ′′(0) < 0 (respectively, λ′′(0) > 0). Since ∂β(λ∗, 0)/∂λ �= 0 according to

Theorem 4.5, we deduce that

(4.7)∂β(λ∗, 0)

∂λ> 0.

Below (see Lemma 4.6) we will prove that λ′(0) = 0 for all vorticity functions

γ . Assuming this, we write λ′(s) = s λ′′(0) + O(s2). In combination with Theo-

rem 4.5, this yields

(4.8) lims→0

β(λ(s), w(s))

s2= −∂β(λ∗, 0)

∂λ· λ′′(0).

Since β(λ∗, 0) = 0, from (4.7)–(4.8) we infer that β(λ(s), w(s)) > 0 for |s| small

if λ′′(0) < 0. But β(λ∗, 0) is the smallest eigenvalue of −Fw(λ∗, 0) by Proposi-

tion 4.1, so that β(λ(s), w(s)) is the smallest eigenvalue of −Fw(λ(s), w(s)) and

our claim follows. Theorem 4.3 thus follows once we establish the validity of

Lemma 4.6 below. �

LEMMA 4.6 For all vorticity functions γ we have λ′(0) = 0. If γ ≡ 0, given

L > 0 and p0 < 0, for d > 0 sufficiently small we have that λ′′(0) < 0, whereas

d ′′(0) > 0 if d > 0 is sufficiently large. This pattern persists if ω is sufficiently

close to 0 throughout the domain.

4.4 Proof of Lemma 4.6

To simplify the intricate calculations we will assume that L = 2π throughout

this proof. After showing that λ′(0) = 0 for all vorticity functions γ , we will pro-

ceed with the calculations of λ′′(0) in the particular case of the irrotational setting.


The proof is complete once the result concerning the sign of λ′′(0) is established

in the irrotational case. Indeed, λ′′(0) is a continuous function of γ so that its sign

persists for uniformly small γ .

In fact, we will prove that, due to a symmetry of the problem, the bifurcation

must be of the pitchfork type. This observation was pointed out to us by M. Golu-

bitsky. It implies automatically that λ′(0) = 0 but not the more difficult assertion

concerning the sign of λ′′(0), for which we must go through all the details anyway.

See Remark 4.8 at the end of this section.

Explicitly writing (4.4)–(4.5), we get

(1 + w2q)(Hpp + wpp) − 2wq(Hp + wp)wpq

+ (Hp + wp)2wqq + γ (−p)(Hp + wp)

3 = 0 in R,(4.9)

1 + w2q + (2g(H + w) − Q)(Hp + wp)

2 = 0 on T .(4.10)

Step 1. Taking partial derivatives with respect to s and denoting ′ = ∂s , we

obtain

2wqw′q(Hpp + wpp) + (1 + w2

q)(H ′pp + w′

pp)

− 2w′q(Hp + wp)wpq − 2wq(H ′

p + w′p)wpq

− 2wq(Hp + wp)w′pq + 2(Hp + wp)(H ′

p + w′p)wqq

+ (Hp + wp)2w′

qq + 3(Hp + wp)2γ (−p)(H ′

p + w′p) = 0 in R,

(4.11)

2wqw′q + (2g(H ′ + w′) − Q′)(Hp + wp)

2

+ 2(2g(H + w) − Q)(Hp + wp)(H ′p + w′

p) = 0 on T .(4.12)

Evaluating (4.11)–(4.12) at s = 0 yields

H ′pp + w′

pp + H 2p w′

qq + 3 γ (−p) H 2p (H ′

p + w′p) = 0 in R,(4.13)

(2g(H ′ + w′) − Q′)H 2p + 2(2gH − Q)Hp(H ′

p + w′p) = 0 on T,(4.14)

since w(0) ≡ 0. The previous two relations can be simplified considerably. Indeed,

since

Hpp = −γ (−p) H 3p in R,

we have

H ′pp = −3γ (−p)H 2

p H ′p in R,

so that (4.13) becomes

(4.15) w′pp + H 2

p w′qq + 3γ (−p)H 2

p w′p = 0 in R (at s = 0).

On the other hand, the Bernoulli condition

(2gH − Q)H 2p + 1 = 0 on T


yields

(2gH ′ − Q′)H 2p + 2Hp H ′

p(2gH − Q) = 0 on T,

so (4.14) becomes

(4.16) gH 2p w′ + (2gH − Q)Hpw

′p = 0 on T .

In combination with Bernoulli’s condition and the fact that Hp = λ−1/2 on T , the

previous relation leads to

(4.17) gw′ = λ3/2w′p on T (at s = 0).

We can summarize (4.15)–(4.17) as

(4.18) Fw(λ(0), w′(0)) = (0, 0).

This last relation is not accidental as the local bifurcation approach pursued in [8]

ensures that lims→0 w′(s, q, p) = M(p) cos q with

(4.19) M(p0) = 0, M(p) > 0, for p ∈ (p0, 0],satisfying

(4.20)

{Mpp − H 2

p M = 0 in (p0, 0)

M(p0) = 0, gM(0) = λ3/2 Mp(0).

Step 2. Differentiating (4.11)–(4.12) with respect to s, we get

2(w′q)

2(Hpp + wpp) + 2wqw′′q(Hpp + wpp) + 4wqw

′q(H ′

pp + w′pp)

+ (1 + w2q)(H ′′

pp + w′′pp) − 2w′′

q(Hp + wp)wpq − 4w′q(H ′

p + w′p)wpq

− 4w′q(Hp + wp)w

′pq − 2wq(H ′′

p + w′′p)wpq − 4wq(H ′

p + w′p)w

′pq

− 2wq(H ′p + w′

p)w′pq − 2wq(Hp + wp)w

′′pq + 2(H ′

p + w′p)

2wqq

+ 2(Hp + wp)(H ′′p + w′′

p)wqq + 4(Hp + wp)(H ′p + w′

p)w′qq

+ (Hp + wp)2w′′

qq + 6(Hp + wp)(H ′p + w′

p)2γ (−p)

+ 3γ (−p)(Hp + wp)2(H ′′

p + w′′p) = 0 in R,

(4.21)

whereas on the top

2wqw′′q + 2(w′

q)2 + (2g(H ′′ + w′′) − Q′′)(Hp + wp)

2

+ 4(2g(H ′ + w′) − Q′)(Hp + wp)(H ′p + w′

p)

+ 2(2g(H + w) − Q)(H ′p + w′

p)2

+ 2(2g(H + w) − Q)(Hp + wp)(H ′′p + w′′

p) = 0 on T .

(4.22)

Evaluation of the previous two equations at s = 0 leads to the equations for w′′(0),

2(w′q)

2 Hpp + (H ′′pp + w′′

pp) − 4w′q Hpw

′pq

+ 4Hp(H ′p + w′

p)w′qq + H 2

p w′′qq + 6Hp(H ′

p + w′p)

2γ (−p)

+ 3γ (−p)H 2p (H ′′

p + w′′p) = 0 in R,

(4.23)


2(w′q)

2 + H 2p (2g(H ′′ + w′′) − Q′′)

+ 4(2g(H ′ + w′) − Q′)Hp(H ′p + w′

p) + 2(2gH − Q)(H ′p + w′

p)2

+ 2(2gH − Q)Hp (H ′′p + w′′

p) = 0 on T .

(4.24)

Differentiating the equation Hpp = −γ (−p) H 3p twice with respect to the pa-

rameter s, we get

(4.25)

{H ′

pp = −3γ (−p)H ′p H 2

p

H ′′pp = −3 γ (−p) H 2

p H ′′p − 6γ (−p) Hp(H ′

p)2 in R,

and (4.23) becomes

2(w′q)

2 Hpp + w′′pp − 4w′

q Hpw′pq + 4Hp(H ′

p + w′p)w

′qq

+ H 2p w′′

qq + 12Hp H ′pw

′pγ (−p) + 6γ (−p) Hp(w

′p)

2

+ 3γ (−p)H 2p w′′

p = 0 in R (at s = 0).

(4.26)

On the other hand, differentiating the Bernoulli condition (2gH − Q)H 2p + 1 = 0

on T twice yields

(2gH ′′ − Q′′)H 2p + 4(2gH ′ − Q′)Hp H ′

p + 2(H ′p)

2(2gH − Q)

+ 2Hp H ′′P(2gH − Q) = 0 on T,

and (4.24) becomes

2(w′q)

2 + 2gH 2p w′′ + 8gw′ Hpw

′p + 8gw′ Hp H ′

p

+ 4(2gH ′ − Q′)Hpw′p + 2(2gH − Q) (w′

p)2

+ 2(2gH − Q)Hpw′′p = 0 on T (at s = 0).

(4.27)

Multiplying (4.26) by H−3p w′ in order to bring the expression into a self-adjoint

form and integrating over R, we obtain

2

∫∫R

(w′q)

2 Hpp H−3p w′ dq dp +

∫∫R

w′′ppw

′ H−3p dq dp(4.28)

− 4

∫∫R

H−2p w′

qw′pqw

′ dq dp + 4

∫∫R

H−2p H ′

pw′qqw

′ dq dp

+ 4

∫∫R

H−2p w′

pw′qqw

′ dq dp +∫∫

R

H−1p w′′

qqw′ dq dp +


+ 12

∫∫R

γ (−p) H−2p H ′

pw′pw

′ dq dp

+ 6

∫∫R

γ (−p)H−2p (w′

p)2w′ dq dp

+ 3

∫∫R

γ (−p) H−1p w′′

pw′ dq dp = 0 at s = 0.

Multiplying relation (4.15) by H−3p w′′, we get

(H−3p w′

p)pw′′ + (H−1

p w′q)q = 0 in R (at s = 0),

which, due to its self-adjoint form, yields by integration∫∫R

(H−3

p w′′ppw

′ + 3H−1p γ (−p)w′′

pw′ + H−1

p w′′qqw

′)dq dp

+∫T

(H−3

p w′pw

′′ − H−3p w′w′′

p

)dq = 0 at s = 0.

In view of the previous equation, (4.28) becomes

2

∫∫R

(w′q)

2 Hpp H−3p w′ dq dp − 4

∫∫R

H−2p w′

qw′pqw

′ dq dp

+ 4

∫∫R

H−2p H ′

pw′qqw

′ dq dp + 4

∫∫R

H−2p w′

pw′qqw

′ dq dp

+ 12

∫∫R

γ (−p)H−2p H ′

pw′pw

′ dq dp

+ 6

∫∫R

γ (−p)H−2p (w′

p)2w′ dq dp

−∫T

H−3p (w′

pw′′ − w′w′′

p)dq = 0 at s = 0.

(4.29)

Since

(4.30) Hp = (λ + 2(p))−1/2

implies

(4.31) Hpp = −γ (−p)H 3p , H ′

p = −λ′

2H 3

p ,


with w′ = M(p) cos(q), we find that (4.29) becomes

2πλ′∫ 0

p0

Hp M2 dp − 6πλ′∫ 0

p0

γ (−p)Hp M Mp dp

−∫T

H−3p (w′

pw′′ − w′w′′

p)dq = 0 at s = 0(4.32)

in view of (4.30) and

(4.33)

{∫ 2π

0sin2 q cos q dq = ∫ 2π

0sin q cos2 q dq = ∫ 2π

0cos3 q dq = 0∫ 2π

0cos2 q dq = π.

On the other hand, multiplying (4.20) by H 2p M , an integration on [p0, 0] yields

3

∫ 0

p0

γ (−p) Hp M Mp dp

=∫ 0

p0

Hp M2 dp −∫ 0

p0

H−1p M Mpp dp

=∫ 0

p0

Hp M2 dp − H−1p (0)M(0)Mp(0) +

∫ 0

p0

γ (−p)Hp M Mp dp

if we integrate by parts and take into account (4.30). Using this in (4.32), we obtain∫T

H−3p (w′

pw′′ − w′w′′

p)dq

= πλ′(

3H−1p (0)M(0)Mp(0) −

∫ 0

p0

Hp M2 dp

)at s = 0.

(4.34)

Differentiating the Bernoulli condition

(4.35) 2gH − Q + 1

H 2p

= 0 on T

with respect to s and taking into account (4.30), we get

(4.36) 2gH ′ − Q′ =2H ′

p

H 3p

= −λ′.

With w′(0) = M(p) cos(q) in (4.28), we therefore obtain

2M2 sin2(q) + 2gH 2p w′′ + 8gHp M Mp cos2(q)

− 4gλ′(0)H 4p M cos(q) − 4λ′(0)Hp Mp cos(q) − 2H−2

p M2p cos2(q)

− 2H−1p w′′

p = 0 on T (at s = 0).


Multiplying the previous relation by 12w′(0) H−2

p (0) and integrating over a period,

we get

g

∫T

w′w′′ dq − 2πgλ′ H 2p M2 − 2πλ′ H−1

p M Mp −∫T

H−3p w′′

pw′ dq = 0

at s = p = 0. In combination with (4.20), this yields

∫T

H−3p (w′

pw′′ − w′′

pw′) dq

= 2gπλ′M2 H 2p + 2πλ′ H−1

p M Mp at s = p = 0.

(4.37)

From (4.34) and the previous relation we infer that

λ′(0)

(∫ 0

p0

Hp M2 dp + 2gM2(0)H 2p (0) − H−1

p (0)M(0)Mp(0)

)= 0.

But H−1p (0)M(0)Mp(0) = gM2(0)H 2

p (0) by (4.20) so that

λ′(0)

(∫ 0

p0

Hp M2 dp + gM2(0)H 2p (0)

)= 0.

This yields

(4.38) λ′(0) = 0.

Step 3(a). Now differentiating (4.21) with respect to s, we get

6(w′q)

2(H ′pp + w′

pp) + 6w′qw

′′q(Hpp + wpp) + 6wqw

′q(H ′′

pp + w′′pp)

+ 6wqw′′q(H ′

pp + w′pp) + 2wqw

′′′q (Hpp + wpp) + (1 + w2

q)(H ′′′pp + w′′′

pp)

− 2w′′′q wpq(Hp + wp) − 4w′′

qw′pq(Hp + wp) − 6w′′

qwpq(H ′p + w′

p)

− 12w′qw

′pq(H ′

p + w′p) − 6w′

qwpq(H ′′p + w′′

p) − 6w′qw

′′pq(Hp + wp)

− 6wqw′pq(H ′′

p + w′′p) − 2wqwpq(H ′′′

p + w′′′p ) − 6wqw

′′pq(H ′

p + w′p)

− 2wqw′′′pq(Hp + wp) + 6(H ′

p + w′p)(H ′′

p + w′′p)wqq + 6(H ′

p + w′p)

2w′qq

+ 2(Hp + wp)(H ′′′p + w′′′

p )wqq + 6(Hp + wp)(H ′′p + w′′

p)w′qq

+ 2(Hp + wp)(H ′p + w′

p)w′′qq + (Hp + wp)

2w′′′qq + 6γ (−p)(H ′

p + w′p)

3

+ 18γ (−p)(Hp + wp)(H ′p + w′

p)(H ′′p + w′′

p)

+ 3γ (−p)(Hp + wp)2(H ′′′

p + w′′′p ) = 0 in R.


Since w ≡ 0 at s = 0, we infer that

6(w′q)

2(H ′pp + w′

pp) + 6w′qw

′′q Hpp + H ′′′

pp + w′′′pp − 6w′′

qw′pq Hp

− 12w′qw

′pq(H ′

p + w′p) − 6w′

qw′′pq Hp + 6w′

qq(H ′p + w′

p)2

+ 6Hp(H ′′p + w′′

p)w′qq + 6Hp(H ′

p + w′p)w

′′qq + H 2

p w′′′qq

+ 6γ (−p)(H ′p + w′

p)3 + 18γ (−p)Hp(H ′

p + w′p)(H ′′

p + w′′p)

+ 3γ (−p)H 2p (H ′′′

p + w′′′p ) = 0 in R at s = 0.

We also have the partial differential equation for H in R, which after three differ-

entiations with respect to s yields at s = 0 precisely the previous relation without

terms involving w′, w′′, or w′′′. This observation leads to the following simplifica-

tion of the previous equation:

6(w′q)

2(H ′pp + w′

pp) + 6w′qw

′′q Hpp + w′′′

pp − 6w′′qw

′pq Hp

− 12w′qw

′pq(H ′

p + w′p) − 6w′

qw′′pq Hp + 6w′

qq(H ′p + w′

p)2

+ 6Hp(H ′′p + w′′

p)w′qq + 6Hp(H ′

p + w′p)w

′′qq + H 2

p w′′′qq

+ 18γ (−p)(H ′p)

2w′p + 18γ (−p)H ′

p(w′p)

2 + 18γ (−p)Hp H ′pw

′′p

+ 6γ (−p)(w′p)

3 + 18γ (−p)Hp H ′′p w′

p + 18γ (−p)Hpw′pw

′′p

+ 3γ (−p)H 2p w′′′

p = 0 in R at s = 0.

This is the equation satisfied by w′′. Multiplying it by H−3p w′ and taking into

account the fact that

H ′p = H ′

pp = 0 at s = 0,

in view of (4.31), (4.38), and (4.25), we obtain by integrating over R that

6

∫∫R

H−3p Hppw

′qw

′′qw

′ dq dp + 6

∫∫R

H−3p (w′

q)2w′

ppw′ dq dp

+∫∫

R

w′′′pp H−3

p w′ dq dp − 6

∫∫R

H−2p w′′

qw′pqw

′ dq dp

− 12

∫∫R

H−3p w′

qw′pqw

′pw

′ dq dp − 6

∫∫R

H−2p w′

qw′′pqw

′ dq dp

+ 6

∫∫R

H−3p w′

qq(w′p)

2w′ dq dp + 6

∫∫R

H−2p H ′′

p w′qqw

′ dq dp

+ 6

∫∫R

H−2p w′

qqw′′pw

′ dq dp + 6

∫∫R

H−2p w′′

qqw′pw

′ dq dp

+∫∫

R

H−1p w′′′

qqw′ dq dp + 6

∫∫R

γ (−p)H−3p (w′

p)3w′ dq dp +


+ 18

∫∫R

γ (−p)H−2p H ′′

p w′pw

′ dq dp

+ 18

∫∫R

γ (−p)H−2p w′

pw′′pw

′ dq dp

+ 3

∫∫R

γ (−p)H−1p w′′′

p w′ dq dp = 0 at s = 0.

On the other hand, writing (4.15) in the self-adjoint form

(H−3p w′

p)p + (H−1p w′

q)q = 0

and multiplying by w′′′, after integration over R, we get∫∫R

(H−3p w′′′

ppw′ + 3γ (−p)H−1

p w′′′p w′ + H−1

p w′′′qqw

′)dq dp

−∫T

(H−3p w′

pw′′′ − H−3

p w′w′′′p )dq = 0 at s = 0.

Combining the previous two relations, we deduce that

(4.39)

6

∫∫R

H−3p Hppw

′qw

′′qw

′ dq dp + 6

∫∫R

H−3p (w′

q)2w′

ppw′ dq dp

− 6

∫∫R

H−2p w′′

qw′pqw

′ dq dp − 12

∫∫R

H−3p w′

qw′pqw

′pw

′ dq dp

− 6

∫∫R

H−2p w′

qw′′pqw

′ dq dp + 6

∫∫R

H−3p w′

qq(w′p)

2w′ dq dp

+ 6

∫∫R

H−2p H ′′

p w′qqw

′ dq dp + 6

∫∫R

H−2p w′

qqw′′pw

′ dq dp

+ 6

∫∫R

H−2p w′′

qqw′pw

′ dq dp + 6

∫∫R

γ (−p)H−3p (w′

p)3w′ dq dp

+ 18

∫∫R

γ (−p)H−2p H ′′

p w′pw

′ dq dp

+ 18

∫∫R

γ (−p)H−2p w′

pw′′pw

′ dq dp

−∫T

(H−3

p w′pw

′′′ − H−3p w′w′′′

p

)dq = 0 at s = 0.


Step 3(b). On the other hand, differentiating (4.22) with respect to s, we get

on the top T

2w′qw

′′q + 2wqw

′′′q + 4w′

qw′′q + (2g(H ′′′ + w′′′) − Q′′′)(Hp + wp)

2

+ 6(2g(H ′′ + w′′) − Q′′)(Hp + wp)(H ′p + w′

p)

+ 6(2g(H ′ + w′) − Q′)(Hp + wp)(H ′′p + w′′

p)

+ 6(2g(H ′ + w′) − Q′)(H ′p + w′

p)2

+ 6(2g(H + w) − Q)(H ′p + w′

p)(H ′′p + w′′

p)

+ 2(2g(H + w) − Q)(Hp + wp)(H ′′′p + w′′′

p ) = 0.

In the previous relation, all terms involving no w′, w′′, or w′′′ can be eliminated

since we can differentiate three times with respect to s along the trivial bifurcation

branch of flat surfaces. Evaluating the outcome at s = 0 and taking into account

the fact that w(0) ≡ 0 in R, as well as the fact that

2gH ′ − Q′ = H ′p = 0 at s = 0

in view of (4.36), we obtain

6w′qw

′′q + 2gw′′′ H 2

p + 6(2gH ′′ − Q′′)Hpw′p

+ 12gHpw′′w′

p + 12gw′(w′p)

2 + 12gw′ Hp H ′′p + 12gw′ Hpw

′′p

+ 6(2gH − Q)H ′′p w′

p + 6(2gH − Q)w′pw

′′p

+ 2(2gH − Q)Hpw′′′p = 0 on T (at s = 0).

Notice that (4.16) multiplied by H ′′p H−1

p (respectively, by w′′p H−1

p ) yields

(2gH − Q)w′p H ′′

p + gw′ Hp H ′′p = (2gH − Q)w′

pw′′p + gw′ Hpw

′′p = 0 on T

at s = 0, which, used in the preceding relation, leads to

3w′qw

′′q + gw′′′ H 2

p + 3(2gH ′′ − Q′′)Hpw′p + 6gHpw

′′w′p + 6gw′(w′

p)2

+ 3gw′ Hp H ′′p + 3gw′ Hpw

′′p + (2gH − Q)Hpw

′′′p = 0 on T (at s = 0),

after division by a factor of 2. Taking into account (4.35) and the relations

(4.40) H ′′p = −λ′′

2H 3

p , 2gH ′′ − Q′′ = −λ′′ on T at s = 0,

which are direct consequences of (4.36)–(4.38), we deduce that

3w′qw

′′q + gw′′′ H 2

p − 3λ′′ Hpw′p + 6gHpw

′′w′p + 6gw′(w′

p)2

− 3g

2λ′′w′ H 4

p + 3gw′ Hpw′′p − H−1

p w′′′p = 0 on T at s = 0.


Multiply this by H−2p w′ to get

gw′′′w′ − H−3p w′′′

p w′

= − 3H−2p w′w′

qw′′q + 3λ′′ H−1

p w′w′p − 6gH−1

p w′w′′w′p

− 6gH−2p (w′)2(w′

p)2 + 3g

2λ′′(w′)2 H 2

p − 3gH−1p (w′)2w′′

p on T

at s = 0. Since

gw′′′w′ = H−3p w′

pw′′′ on T (at s = 0),

in view of (4.17), integration of the above yields∫T

H−3p (w′

pw′′′ − w′′′

p w′) dq = −3

∫T

H−2p w′w′

qw′′q dq

+ 3λ′′∫T

H−1p w′w′

p dq − 6g

∫T

H−1p w′w′′w′

p dq

− 6g

∫T

H−2p (w′)2(w′

p)2 dq + 3g

2λ′′

∫T

(w′)2 H 2p dq

− 3g

∫T

H−1p (w′)2w′′

p dq at s = 0.

(4.41)

Combining (4.39) with (4.41) and also using (4.31) and (4.40), we deduce that

− 6

∫∫R

γ (−p)w′qw

′′qw

′ dq dp + 6

∫∫R

H−3p (w′

q)2w′

ppw′ dq dp(4.42)

− 6

∫∫R

H−2p w′′

qw′pqw

′ dq dp − 12

∫∫R

H−3p w′

qw′pqw

′pw

′ dq dp

− 6

∫∫R

H−2p w′

qw′′pqw

′ dq dp + 6

∫∫R

H−3p w′

qq(w′p)

2w′ dq dp

− 3λ′′∫∫

R

Hpw′qqw

′ dq dp + 6

∫∫R

H−2p w′

qqw′′pw

′ dq dp

+ 6

∫∫R

H−2p w′′

qqw′pw

′ dq dp + 6

∫∫R

γ (−p)H−3p (w′

p)3w′ dq dp

+ 18

∫∫R

γ (−p)H−2p H ′′

p w′pw

′ dq dp

+ 18

∫∫R

γ (−p)H−2p w′

pw′′pw

′ dq dp +


+ 3

∫T

H−2p w′w′

qw′′q dq − 3λ′′

∫T

H−1p w′w′

p dq

+ 6g

∫T

H−1p w′w′′w′

p dq + 6g

∫T

H−2p (w′)2(w′

p)2 dq

− 3g

2λ′′

∫T

(w′)2 H 2p dq + 3g

∫T

H−1p (w′)2w′′

p dq = 0 at s = 0.

Step 4. At this point we leave the general setting and concentrate on the

irrotational case, in which case we can write w′′ explicitly. For γ ≡ 0 we have

Hp = λ−1/2 throughout R, and (4.42) becomes

6λ3/2

∫∫R

(w′q)

2w′ppw

′ dq dp − 6λ

∫∫R

w′′qw

′pqw

′ dq dp

− 12λ3/2

∫∫R

w′qw

′pqw

′pw

′ dq dp − 6λ

∫∫R

w′qw

′′pqw

′ dq dp

+ 6λ3/2

∫∫R

w′qq(w

′p)

2w′ dq dp − 3λ−1/2λ′′∫∫

R

w′qqw

′ dq dp

+ 6λ

∫∫R

w′qqw

′′pw

′ dq dp + 6λ

∫∫R

w′′qqw

′pw

′ dq dp

+ 3λ

∫T

w′w′qw

′′q dq − 3λ1/2λ′′

∫T

w′w′p dq

+ 6gλ1/2

∫T

w′w′′w′p dq + 6gλ

∫T

(w′)2(w′p)

2 dq

− 3g

2λλ′′

∫T

(w′)2 dq + 3gλ1/2

∫T

(w′)2w′′p dq = 0 at s = 0.

(4.43)

Since w(0) = M(p) cos(q) and Hp = λ−1/2 in R, we deduce that at s = 0,

(4.44)

w′′pp + λ−1w′′

qq − 4λ−1/2 M Mp = 0 in R

w′′ = 0 on p = p0

gw′′ − λ3/2w′′p = −λM2 sin2 q − 3λ1/2gM Mp cos2 q on p = 0.

Indeed, the partial differential equation in (4.44) follows at once. The first bound-

ary condition is a consequence of differentiating twice with respect to s the equa-

tion w = 0, valid on p = p0. Since (4.35)–(4.36) transform (4.27) into

2M2 sinq +2gλ−1w′′ + 8gλ−1/2 M Mp cos2 q − 2λ−1 M2p cosq −2λ1/2w′′

p = 0


on T at s = 0, while (4.20) ensures that gM = λ3/2 Mp on T , the previous equation

multiplied by λ/2 is precisely the second boundary condition of (4.44). Because

each w(s) is an even function of q (cf. [8]), we have a Fourier expansion

(4.45) w′′(p) =∑n≥0

wn(p) cos(nq), p ∈ [p0, 0], q ∈ [0, 2π].

Since sin2 q = (1 − cos(2q))/2 and cos2 q = (1 + cos(2q))/2, we can rewrite the

second boundary condition in (4.44) as

gw′′ − λ3/2w′′p = −λ

2M2 − 3

2λ1/2gM Mp +

(λ

2M2 − 3

2λ1/2gM Mp

)cos(2q).

By (4.45) we thus infer that

(4.46)

∂2pw0 = 4λ−1/2 M Mp in (p0, 0)

w0 = 0 on p = p0

gw0 − λ3/2∂pw0 = −λ2

M2 − 32λ1/2gM Mp on p = 0,

whereas

(4.47)

∂2pwn − λ−1n2wn = 0 in (p0, 0)

wn = 0 on p = p0

gwn − λ3/2∂pwn = 0 on p = 0

for n ≥ 1, n �= 2,

and

(4.48)

∂2pw2 − 4λ−1w2 = 0 in (p0, 0)

w2 = 0 on p = p0

gw2 − λ3/2∂pw2 = λ2

M2 − 32λ1/2gM Mp on p = 0.

By (4.47) we infer at once that

(4.49) wn ≡ 0 for n ≥ 1, n �= 2.

Recall [8] that

(4.50) M(p) = sinh

(p − p0√

λ

), p ∈ [p0, 0],

where λ = λ(0) > 0 is the unique solution of the equation

(4.51) λ = g tanh

( |p0|√λ

).

Using (4.50), we compute

4λ−1/2 M Mp = 2λ−1 sinh

(2(p − p0)√

λ

).


Therefore, the general solution w0 of the inhomogeneous differential equation in

(4.46) with w0(p0) = 0 is

(4.52) w0(p) = 1

2sinh

(2(p − p0)√

λ

)+ A(p − p0)

with A ∈ R. We determine A from the Robin boundary condition in (4.46) as

(4.53) A = λ

2

1 + 2 cosh2(

p−p0√λ

)gp0 + λ3/2

< 0.

The sign of A is due to the fact that λ = λ(0) < (g|p0|)3/2; cf. [8]. On the other

hand, the general solution w2 of the differential equation in (4.48) with w2(p0) = 0

is

(4.54) w2(p) = B sinh

(2(p − p0)√

λ

).

We determine B from the Robin boundary condition in (4.48) by taking into ac-

count (4.50)–(4.51) as

(4.55) B =2 cosh

(p−p0√

λ

)+ 1

4 sinh2(

p−p0√λ

) > 0.

From (4.45), (4.49), (4.52), and (4.54), we deduce that

(4.56) w′′ = 1

2sinh

(2(p − p0)√

λ

)+ A(p − p0) + B sinh

(2(p − p0)√

λ

)cos(2q)

with A and B given by (4.53) and (4.55), respectively.

Step 5. Knowing the explicit formula for w′′, since w′ = M(p) cos(q) with

M specified at (4.50), we can analyze (4.43) carefully to determine the sign of

λ′′(0). The coefficient of λ′′ in (4.43) is

−3λ−1/2

∫∫R

w′qqw

′ dq dp − 3λ1/2

∫T

w′pw

′ dq − 3g

2λ−1

∫T

(w′)2 dq.

Taking into account (4.33) and (4.50), this expression is precisely 3π times

λ−1/2

∫ 0

p0

sinh2

(p − p0√

λ

)dp − sinh

(−p0√λ

)cosh

(−p0√λ

)− g

2λsinh2

(−p0√λ

)

= p0

2√

λ− 1

2sinh

(−p0√λ

)cosh

(−p0√λ

)− g

2λsinh2

(−p0√λ

).

Recalling (4.51), the coefficient of λ′′ in (4.43) is found to be

(4.57) 3π

{p0

2√

λ− sinh

(−p0√λ

)cosh

(−p0√λ

)}< 0

as p0 < 0.


The terms in (4.43) not involving w′′ or λ′′ are

6λ3/2

∫∫R

M3 Mpp cos2(q) sin2(q)dq dp

− 12λ3/2

∫∫R

M2 M2p cos2(q) sin2(q)dq dp

− 6λ3/2

∫∫R

M2 M2p cos4(q)dq dp + 6gλ

∫T

M2 M2p cos4(q)dq.

Since

(4.58)

∫ 2π

0

sin2(q) cos2(q)dq = π

4,

∫ 2π

0

cos4(q)dq = 3π

4,

the above expression becomes

3π

4

{2λ1/2

∫ 0

p0

sinh4

(p − p0√

λ

)dp − 10λ1/2

∫ 0

p0

sinh2

(2(p − p0)√

λ

)dp

+ 3g sinh2

(−2p0√λ

)}.

Computing

∫ 0

p0

sinh2

(p − p0√

λ

)dp = p0

2+ 1

2λ1/2 sinh

(−p0√λ

)cosh

(−p0√λ

),

∫ 0

p0

sinh2

(2(p − p0)√

λ

)dp = p0

2+ λ1/2 sinh

(−p0√λ

)cosh3

(−p0√λ

)

− 1

2λ1/2 sinh

(−p0√λ

)cosh

(−p0√λ

),

∫ 0

p0

sinh4

(p − p0√

λ

)dp = −3p0

8+ 1

4λ

12 sinh

(−p0√λ

)cosh3

(−p0√λ

)

− 5

8λ1/2 sinh

(−p0√λ

)cosh

(−p0√λ

),

(4.59)

we infer that the contribution to (4.43) of the terms not involving w′′ or λ′′ is

(4.60)3π

2

{−λ1/2 p0 + 6g sinh2

(−p0√λ

)cosh2

(−p0√λ

)}

if we also take into account (4.51).


On the other hand, the double integral terms in (4.43) involving w′′ are

6λ

∫∫R

w′′q M Mp sin(q) cos(q)dq dp + 6λ

∫∫R

w′′pq M2 sin(q) cos(q)dq dp

− 6λ

∫∫R

w′′p M2 cos2(q)dq dp + 6λ

∫∫R

w′′qq M Mp cos2(q)dq dp.

Using (4.56) we transform this into

6λ

∫∫R

{−8λ−1/2 B

[sinh2

(p − p0√

λ

)+ cosh2

(p − p0√

λ

)]M2 sin2(q) cos2(q)

− λ−1/2

[sinh2

(p − p0√

λ

)+ cosh2

(p − p0√

λ

)]M2 cos2(q) − AM2 cos2(q)

− 2λ−1/2 B

[sinh2

(p − p0√

λ

)+ cosh2

(p − p0√

λ

)]M2 cos2(q)[cos2(q) − sin2(q)]

− 8λ−1/2 B sinh

(p − p0√

λ

)cosh

(p − p0√

λ

)M Mp cos4(q)

}dq dp.

Taking into account (4.33), (4.50), and (4.58), the above expression becomes

− 6πλ

∫ 0

p0

{A sinh2

(p − p0√

λ

)+ λ−1/2(1 + 9B) sinh2

(p − p0√

λ

)cosh2

(p − p0√

λ

)

+ λ−1/2(1 + 3B) sinh4

(p − p0√

λ

)}dq dp.

We compute this using (4.59) as

3πλ

2

{(λ−1/2 − 2A) p0

+ (12B + 3 − 2λ1/2 A) sinh

(−p0√λ

)cosh

(−p0√λ

)

− 2(6B + 1) sinh

(−p0√λ

)cosh3

(−p0√λ

)}.

(4.61)

Finally, the terms in (4.43) unaccounted for are boundary terms involving w′′,namely,

− 3λ

∫T

w′′q M2 sin(q) cos(q)dq + 6gλ1/2

∫T

w′′M Mp cos2(q)dq

+ 3gλ1/2

∫T

w′′p M2 cos2(q)dq.


Using (4.50)–(4.51), (4.33), and (4.58), the above expression becomes

π

{6λB sinh3

(−p0√λ

)cosh

(−p0√λ

)

+ 9g sinh2

(−p0√λ

)cosh2

(−p0√λ

)

+ 9gB sinh2

(−p0√λ

)cosh2

(−p0√λ

)

− 6g Ap0 sinh

(−p0√λ

)cosh

(−p0√λ

)

+ 3g Aλ1/2 sinh2

(−p0√λ

)+ 3g sinh4

(−p0√λ

)+ 3gB sinh4

(−p0√λ

)}.

(4.62)

Re-expressing (4.43) by taking into account the forms (4.57), (4.60), (4.61), and

(4.62) of its component parts, we obtain

3πλ′′(0)

{1

2p0λ

−1/2 − sinh

(−p0√λ

)cosh

(−p0√λ

)}

− 3πp0λA

(1 + 2 cosh2

(−p0√λ

))+ 9πgB sinh2

(−p0√λ

)

+ 9πg

2sinh2

(−p0√λ

)+ 9πg sinh2

(−p0√λ

)cosh2

(−p0√λ

)= 0

in view of (4.51). By means of (4.55) and dividing by 3π , we transform the previ-

ous relation into

λ′′(0)

{1

2p0λ

−1/2 − sinh

(−p0√λ

)cosh

(−p0√λ

)}

+ g

4

{1 + 2 cosh2

(−p0√λ

)}{3 + 6 sinh2

(−p0√λ

)− 4p0λA

g

}= 0.

(4.63)

On the other hand, (4.53) yields

(4.64) 2p0λA = λ2 p0

gp0 + λ3/2

(1 + 2 cosh2

(−p0√λ

)).

It is convenient to observe that [8]

(4.65) d = −p0√λ

.

Using (4.51) and (4.65), we compute

λ2 p0

gp0 + λ3/2=

λλ3/2 −p0√λ

−gp0 − λ3/2=

λ3/2gd sinh(d)

cosh(d)

g−p0√

λλ1/2 − λ3/2

=λ3/2gd sinh(d)

cosh(d)

cosh(d)

sinh(d)λ3/2d − λ3/2

=gd sinh(d)

cosh(d)

d cosh(d)

sinh(d)− 1

= gd sinh2(d)

d cosh2(d) − sinh(d) cosh(d).


The previous equality in combination with (4.63) and (4.64) leads to the equation

λ′′(0)

{1

2p0λ

−1/2 − sinh(d) cosh(d)

}+ g

4(1 + 2 cosh2(d))

·{

3 + 6 sinh2(d) − 2d sinh2(d)(1 + 2 cosh2(d))

d cosh2(d) − sinh(d) cosh(d)

}= 0

(4.66)

if we recall (4.65). Since

2d sinh2(d)(1 + 2 cosh2(d))

d cosh2(d) − sinh(d) cosh(d)= 2d sinh2(d)(1 + 2 cosh2(d))

(d cosh(d) − sinh(d)) cosh(d)

= 2d(d + d3

6+ · · · )2

(1 + 2(1 + d2

2+ · · · )2

)(d(1 + d2

2+ · · · ) − (d + d3

6+ · · · ))(1 + d2

2+ · · · )

≈ 6d3

(d + d3

2) − (d + d3

6)

= 18 for d ↓ 0,

whereas

2d sinh2(d)(1 + 2 cosh2(d))

d cosh2(d) − sinh(d) cosh(d)= 2 sinh2(d)

d(2 + 1

cosh2(d)

)d − sinh(d)

cosh(d)

≈ 4 sinh2(d) for d → ∞,

we deduce from the fact that the coefficient of λ′′(0) in (4.66) is always negative,

in accordance with (4.57), that

(4.67) λ′′(0) < 0 for d small enough,

whereas

(4.68) λ′′(0) > 0 for d large enough.

In other words, small-amplitude irrotational waves are stable in the long-wave

regime and unstable in the short-wave regime.

4.5 Further Remarks

Remark 4.7 (Large-Amplitude Solutions). As one moves along the curve of non-

trivial solutions away from the bifurcation point (λ∗, 0), the spectrum changes con-

tinuously. The solutions remain linearly stable/unstable until the lowest eigenvalue

β(s) becomes 0. Say β(s0) = 0 with s0 > 0. At that point 0 ∈ �(λ(s0), w(s0)).

That is, Fw(λ(s0), w(s0)) has a null space. At that point various scenarios might

occur. For instance, a secondary bifurcation or a multiple bifurcation might occur,

we could encounter a turning point, or the continuum might cease to be a curve and

become a considerably more complicated pattern (Figure 4.3). Beyond the point

(λ(s0), w(s0)), the question of stability can be answered only with an independent

analysis.


λλ∗

trivial curve

turning point

simple bifurcation

multiple

bifurcation

FIGURE 4.3. Continuation along the bifurcation curve (for simplicity,

the symmetry properties of the continuum are ignored).

Remark 4.8 (Symmetry). By recognizing the presence of symmetries in our bifur-

cation problem, some of the calculations needed in the proof of Lemma 4.6 are

considerably simplified, as is generally the case when a group of symmetries is

acting [11]. Let us exemplify this by presenting a more general and elegant proof

of the relation λ′(0) = 0. The fact that h(q, p) is even and 2π-periodic in the

q-variable suggests the symmetry G : h(q, p) �→ h(q + π, p) with G2 being

the identity operator. A direct computation confirms that if F(λ,w) = 0, then

F(λ, Gw) = 0 and that the operator Fww(λ, 0) commutes with G. Moreover,

w(−s) = Gw(s) and Gw′(s) = −w′(−s) for |s| small. This is consistent with a

pitchfork bifurcation (Figure 4.4).

Starting from F(λ(s), w(s)) = 0 and differentiating gives

(4.69) λ′(s)Fλ + Fww′(s) = 0.

Evaluation of (4.69) at s = 0 yields the eigenvalue equation Fw(λ∗, 0) w′(0) = 0

since Fλ(λ∗, 0) = 0. Differentiating (4.69) with respect to s, we obtain

λ′′Fλ + [λ′]2Fλλ + 2λ′Fλww′ + Fww[w′]2 + Fww′′ = 0,

which, evaluated at s = 0, yields

(4.70) 2λ′(0)Fλw(λ∗, 0)w′(0) + Fww(λ∗, 0)[w′(0)]2 + Fw(λ∗, 0)w′′(0) = 0


w

λ

w(s)

w(−s) = Gw(s)

FIGURE 4.4. A pitchfork bifurcation.

since Fλ(λ∗, 0) = Fλλ(λ

∗, 0) = 0. Applying the symmetry G to the equation

(4.70), we notice that w′ �→ −w′ while w′′ �→ w′′ so that (4.70) implies

(4.71) −2λ′(0)Fλw(λ∗, 0)w′(0) + Fww(λ∗, 0)[w′(0)]2 + Fw(λ∗, 0)w′′(0) = 0.

By (4.70)–(4.71) we get λ′(0)Fλw(λ∗, 0)w′(0) = 0. Since Fλw(λ∗, 0)w′(0) �= 0,

zero being a Fλw(λ∗, 0)-simple eigenvalue as mentioned above, we deduce that

λ′(0) = 0. This is the result of step 2 above.

Concerning the calculations involved subsequently in the proof of Lemma 4.6

towards the determination of the sign of λ′′(0), while the presence of the symmetry

is definitely useful, it appears that most of the intricate calculations performed

above are still necessary. Nevertheless, some further information can be obtained.

Indeed, differentiating (4.69) a second time with respect to s, we obtain

λ′′′Fλ + 3λ′λ′′Fλλ + (λ′)3Fλλλ + 3λ′′Fλww′ + 3(λ′)2Fλλww′

+ 2λ′Fλww(w′)2 + 2λ′Fλww′′ + λ′Fwwλ(w′)2 + Fwww(w′)3

+ 2Fwww′w′′ + λ′Fwλw′′ + Fwww′w′′ + Fww′′′ = 0.

We put s = 0 and use λ′(0) = 0 and Fλ(λ∗, 0) = Fλλ(λ

∗, 0) = Fλλλ(λ∗, 0) = 0 to

get

(4.72) 3λ′′Fλww′ + 3Fwww′w′′ + Fwww(w′)3 + Fww′′′ = 0 at s = 0.

Taking the weighted inner product 〈〈 · , · 〉〉 of (4.71) with w′ and with weight a3 in

R and weight 12a2 on T , we obtain

(4.73) κ1λ′′(0) + κ2 = 0.


The coefficient of λ′′(0) is

κ1 = 3〈〈Fλww′, w′〉〉

= −3

{ ∫∫R

a−1w′qqw

′ dq dp + 3

∫∫R

a−1γw′pw

′ dq dp

+ g

∫T

a−2(w′)2 dq + 1

2

∫T

aw′pw

′ dq

}

if we take into account the explicit formula for Fλw given by (4.6). This expression

is precisely 3� if we use the notation from [8, pp. 501–502]. There we proved that

� < 0. Thus κ1 < 0. Now the last term of κ2 is 〈〈Fww′′′, w′〉〉 = 〈〈w′′′,Fww′〉〉 = 0

as in lemma 3.8 of [8]. Therefore

κ2 = 3〈〈Fwww′w′′, w′〉〉 + 〈〈Fww(w′)3, w′〉〉 at s = 0.

Thus to infer the statement of Theorem 4.3 one only needs to know the sign of κ2.

It appears, however, that at this point the explicit calculations made earlier in the

case of irrotational flow cannot be avoided.

Acknowledgment. We thank M. Golubitsky for explaining to us the signifi-

cance of the symmetry. We are also indebted to J. Shatah for stimulating conver-

sations. This research was supported in part by Science Foundation Ireland Grant

04/BR/M0042 and by National Science Foundation Grant DMS-0405066.

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ADRIAN CONSTANTIN WALTER A. STRAUSS

Department of Mathematics Brown University

Trinity College Dublin Department of Mathematics

Dublin 2 Box 1917

IRELAND Providence, RI 02912

E-mail: adrian@ E-mail: wstrauss@

maths.tcd.ie math.brown.edu

Received July 2005.

Stability properties of steady water waves with vorticity

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