Trimodal Steady Water Waves

TRIMODAL STEADY WATER WAVES

MATS EHRNSTRÖM AND ERIK WAHLÉN

Abstract. We construct three-dimensional families of small-amplitudegravity-driven rotational steady water waves on finite depth. The so-lutions contain counter-currents and multiple crests in each minimalperiod. Each such wave generically is a combination of three differentFourier modes, giving rise to a rich and complex variety of wave pat-terns. The bifurcation argument is based on a blow-up technique, takingadvantage of three parameters associated with the vorticity distribution,the strength of the background stream, and the period of the wave.

1. Introduction

An explicit example of a rotational wave was constructed by Gerstneralready in 1802 [10]. In spite of this, irrotational flows continued to drawmuch attention, and the first existence proof for small-amplitude solutionswith a general distribution of vorticity, due to Dubreil-Jacotin, was publishedfirst in 1934 [5]. Only with the paper [3] by Constantin & Strauss in 2004a general result for large-amplitude waves appeared. Using bifurcation anddegree theory they constructed a global continuum of waves with an arbitraryvorticity distribution. Since then, much attention has been brought to therichness of phenomena appearing in rotational flows.

Whereas existence proofs for irrotational waves typically involve a pertur-bation argument starting from a still, or uniform, stream, a general vorticitydistribution allows for perturbing very involved background streams, includ-ing such with stagnation and so-called critical layers. Critical layers, whichare regions of the fluid consisting entirely of closed streamlines, can be ruledout in the case of irrotational currents. In contrast, there are classes of rota-tional currents allowing for arbitrarily many internal stagnation points andcritical layers, the exact number depending on the vorticity distribution andthe values of certain parameters.

The first rigorous existence proof for exact steady water waves with a crit-ical layer was given in [25], soon to be followed by the investigation [4] byConstantin and Varvaruca. Both investigations used bifurcation theory toestablish the existence of small-amplitude waves of constant vorticity. Thepaper [25] is a rigorous justification of linear waves with critical layers foundin [8] as a natural extension of the exact waves constructed in [3]; it maybe extended to more general vorticity distributions. The theory developed

2010 Mathematics Subject Classification. 76B15, 35Q35, 35B32, 76B47.Key words and phrases. Steady water waves; Multi-modal waves; Critical layers; Vor-

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2 MATS EHRNSTRÖM AND ERIK WAHLÉN

in [4] focuses on the case of constant vorticity and encompasses free sur-faces that are not graphs of functions, thereby allowing for the possibility ofoverhanging waves.

Since linear shear flows (constant vorticity) allow for at most one criti-cal layer, the authors of [6] and [7] considered flows in which the relativehorizontal velocity is oscillating. As it turns out, an affine vorticity distribu-tion is enough to induce several critical layers. The solution classes for suchvorticity distributions were investigated in [6]. The corresponding existencetheory was pursued in [7]. In that paper, a construction of bimodal waveswith critical layers and two crests in each minimal period is contained, aclass of waves previously known to exist only in the irrotational setting (see[2] and [12, 13, 24]). A later contribution to the theory of bimodal waves is[19].

The current paper answers the following natural question appearing in[7]. Does higher-dimensional bifurcation occur—i.e., can one find additionalparameters in order to obtain n-modal waves, where n � 3?

As it turns out, the wave-length parameter can be made to play exactlythis desired role (the other parameters appearing naturally in the problemare too closely related, cf. [7, Remark 4.10]), and we are able to answer thisquestion affirmatively. The situation is, however, much more complicatedthan in lower-dimensional bifurcation, and a precise investigation of the dif-ferent solution sets in necessary to understand the possible combinations ofdifferent wave modes that may appear (see Section 6). Whether one couldfind waves of even more complicated form—in the shape, for example, ofquadmodal surface profiles—is left as an open problem. The current settingseems not to allow for this, but it would most probably be possible to addsurface tension or stratification to achieve this goal, if desired.

Some comments on related work: A different continuation of the the-ory on waves with critical layers is the consideration of arbitrary vorticitydistributions while still allowing for internal stagnation. Such an investi-gation was initiated by Kozlov and Kuznetsov in [16], where they examineand classify the running streams (with flat, but free, surfaces) induced byLipschitz-continuous vorticity distributions. In [17] the same authors giveseveral a priori bounds for waves associated with such a general vorticitydistribution, and in [18] they describe the local bifurcation of waves withcritical layers and a general vorticity distribution. An additional recent con-tribution in this field is the work by Shatah, Walsh and Zeng [22], in whichexact travelling water waves with compactly supported vorticity distributionare constructed, a new addition in the theory of waves with critical layers.

The above works on exact waves with critical layers are all on gravity-driven waves in water of finite depth and constant density. A theory forrotational waves in the presence of stratification — i.e. for a fluid of non-constant density — was recently developed by Walsh [26]. In the investiga-tion [9] Escher, Matioc and Matioc combine ideas from [25] and [26] to provethe existence of stratified waves with a linear density distribution and criti-cal layers. This yields the existence of waves whose properties are not thatdistant from those of constant vorticity, but that may still include a second

TRIMODAL STEADY WATER WAVES 3

critical layer. Two further developments in this direction, establishing wavesof larger amplitude, are [20] and [11].

One of the interesting open questions in this field is the existence of waveswith an even more complicated surface profile than the ones constructed inthis or any of the above-mentioned papers. In the case of constant vorticity,numerical studies [15, 21, 23] indicate the existence of large-amplitude waveswith overhanging profiles and critical layers, as well as non-symmetric surfaceprofiles with several peaks. So far there are no rigorous results confirmingthese intriguing phenomena.

The outline of this work is as follows: In Section 2 we present the problemin a setting appropriate for our aims, and in Section 3 the correspondingfunctional-analytic framework. Section 4 contains the construction of theactual three-dimensional kernels, and Section 5 the Lyaounov–Schmidt re-duction and following proof of trimodal gravity water waves. Finally, inSection 6 we investigate the structure of solution sets appearing in the dif-ferent Fourier regimes (the waves obtained depend on which modes interact).A few illustrations and a concrete example showing the complexity of thewaves are given.

2. Mathematical formulation

We consider steady (travelling) waves in two dimensions. Let B = {y = 0}denote the flat bed and S := {y = 1 + ⌘(x)} the free surface. The fluiddomain is naturally defined by ⌦ := {(x, y) 2 R2

: 0 < y < 1 + ⌘(x)}, andthe steady water-wave problem is to find a stream function such that

{� , } = 0 in ⌦,

= m0

on B,

= m1

on S,

1

2

|r |2 + ⌘ = Q on S.

(2.1a)

Here {f, g} := fx

gy

� fy

gx

denotes the Poisson bracket, � := @2x

+ @2y

is theLaplace operator, and m

0

, m1

and Q are arbitrary constants. The formula-tion (2.1) is equivalent to the Euler equations for a gravity-driven, inviscid,incompressible fluid of constant density and finite depth [6], and it allows forboth rotation, i.e. � 6= 0, and stagnation, i.e. r = 0. We consider thecase

� = ↵ in ⌦, (2.1b)when the vorticity distribution is linear in ; affine distributions may beincorporated by translation of .

Waves exist for any wavelength (see [7]), so we choose to normalise theperiod to 2⇡ by a transformation x 7! x. In a similar fashion we map thevertical variable onto one of unit range, so that

(x, y) 7! (q, s) :=

✓

x,y

1 + ⌘(x)

◆

describes the transformation of ⌦ onto the strip ˆ

⌦ := {(q, s) 2 R2

: s 2(0, 1)}. Let ˆ (q, s) := (x, y). Since all x-derivates in (2.1) appear in pairs,


it is natural to introduce a wavelength parameter ⇠ := 2. In the newcoordinates the problem (2.1) takes the form

⇠

ˆ q

� s⌘q

ˆ s

1 + ⌘

!

q

� ⇠s⌘q

1 + ⌘

ˆ q

� s⌘q

ˆ s

1 + ⌘

!

s

+

ˆ ss

(1 + ⌘)2= ↵ ˆ in ˆ

⌦,

ˆ = m0

on s = 0,

ˆ = m1

on s = 1,

⇠

2

ˆ q

� s⌘q

ˆ s

1 + ⌘

!

2

+

ˆ 2

s

2(1 + ⌘)2+ ⌘ = Q on s = 1.

(2.2)We pose the problem for ⌘ 2 C2+�

even(S,R) and 2 C2+�

per,even(⌦,R), where thesubscripts per and even denotes 2⇡-periodicity and evenness in the horizontalvariable, and we have identified 2⇡-periodic functions with functions definedon the unit circle S. We also require that min ⌘ > �1. The parameter ↵influences the nature of in a substantial way (see [6]), and in order toobtain the desired triply-periodic waves we shall assume that ↵ < 0.

Laminar flows. Laminar flows are simultaneous solutions of (2.1) and (2.2)for which ⌘ = 0 and is independent of q. They are given by the formula

0

(·;µ,�,↵) := µ cos(✓0

(·� 1) + �), (2.3)where µ,� 2 R are arbitrary constants and

✓k

:= |↵+ ⇠k2|1/2, k 2 Z.The values of Q = Q(µ,↵,�), m

0

= m0

(µ,↵,�) and m1

= m1

(µ,�) aredetermined from (2.1), i.e.

Q(µ,�,↵) :=µ2✓2

0

sin

2

(�)

2

(2.4)

andm

1

(µ,�) := µ cos(�), m0

(µ,↵,�) := µ cos(�� ✓0

)

Since the laminar solutions 0

are the same in the variables (q, s) and (x, y),they are also independent of the wavelength parameter ⇠.

3. Functional-analytic framework

In this section we describe the framework developed in [25] and [7], whichshall be used for the three-dimensional bifurcation. Note that the theoryin [7] is written in the variables (x, s), whereas here we use the wavelengthparameter ⇠.

The map F . We shall linearise the problem (2.2) around a laminar flow 0

,whence we introduce a disturbance ˆ� through ˆ =

0

+

ˆ�, and the functionspace

X := X1

⇥X2

:= C2+�

even(S)⇥n

ˆ� 2 C2+�

per, even(ˆ

⌦) :

ˆ�|s=1

=

ˆ�|s=0

= 0

o

.

Furthermore, we define the target space

Y := Y1

⇥ Y2

:= C1+�

even(S)⇥ C�

per, even(ˆ

⌦),


and the setsO := {(⌘, ˆ�) 2 X : min ⌘ > �1}

andU := {(µ,↵,�, ⇠) 2 R4

: µ 6= 0,↵ < 0, sin(�) 6= 0, ⇠ > 0},which conveniently captures all necessary assumptions on the parameters(cf. [7]). Then O ⇢ X is an open neighbourhood of the origin in X, and theembedding X ,! Y is compact. Elements of Y will be written w := (⌘, ˆ�)and elements of U will be written ⇤ := (µ,↵,�, ⇠), and ⇤0

:= (µ,↵,�) toindicate independence of ⇠. Define the operator F : O ⇥ U ! Y by

F(w,⇤) := (F1

(w,⇤),F2

(w,⇤))

where

F1

(w,⇤) :=1

2

2

4⇠

ˆ�q

� s⌘q

(( 0

)

s

(s;⇤0) +

ˆ�s

)

1 + ⌘

!

2

+

(( 0

)

s

(s;⇤0) +

ˆ�s

)

2

(1 + ⌘)2

3

5

s=1

+ ⌘ �Q(⇤),

and

F2

(w,⇤) := ⇠

ˆ�q

� s⌘q

(( 0

)

s

(s;⇤0) +

ˆ�s

)

1 + ⌘

!

q

� ⇠s⌘q

1 + ⌘

ˆ�q

� s⌘q

(( 0

)

s

(s;⇤0) +

ˆ�s

)

1 + ⌘

!

s

+

( 0

)

ss

(s;⇤0) +

ˆ�ss

(1 + ⌘)2� ↵

⇣

0

(s;⇤0) +

ˆ�⌘

.

The problem F((⌘, ˆ�),⇤) = 0, (⌘, ˆ�) 2 O, is then equivalent to the water-wave problem (2.1), the map (⌘, ˆ�) 7! (⌘, ˆ ) is continuously differentiable,and F is real analytic O ⇥ U ! Y [7, Lemma 3.1].

Linearization. The Fréchet derivative of F at w = 0 is given by the pair

D

w

F1

(0,⇤0)w =

h

( 0

)

s

ˆ�s

� ( 0

)

2

s

⌘ + ⌘i

�

�

�

s=1

, (3.1)

D

w

F2

(0,⇤)w = ⇠ ˆ�qq

+

ˆ�ss

� ↵ˆ�� ⇠s( 0

)

s

⌘qq

� 2( 0

)

ss

⌘. (3.2)

Let ˜X2

:=

n

� 2 C2+�

per, even(ˆ⌦) : �|s=0

= 0

o

and ˜X :=

n

(⌘, ˆ�) 2 X1

⇥ ˜X2

o

.

Lemma 3.1 ([7]). The bounded, linear operator T (⇤

0) :

˜X2

! X given by

T (⇤

0)� :=

✓

� �|s=1

( 0

)

s

(1)

,�� s( 0

)

s

�|s=1

( 0

)

s

(1)

◆

is an isomorphism. Define L(⇤) := D

w

F(0,⇤)T (⇤

0) :

˜X2

! Y . Then

L(⇤)� =

✓

( 0

)

s

�s

�✓

( 0

)

ss

+

1

( 0

)

s

◆

�

�

�

�

�

�

s=1

, (⇠@2q

+ @2s

� ↵)�

◆

.

(3.3)


When the dependence on the parameters is unimportant, we shall forconvenience refer to D

w

F(0,⇤), L(⇤) and T (⇤) simply as D

w

F(0), L andT . Via T , elements � 2 ˜X

2

can be “lifted” to elements (⌘, ˆ�) 2 ˜X using thecorrespondence induced by the first component of T �.

Lemma 3.2 ([7]). The mapping ⌘(·) defined by

˜X2

3 �⌘(·)7! ⌘

�

= � �|s=1

( 0

)

s

(1)

2 X1

,

is linear and bounded, whence � 7! (⌘�

,�) is linear and bounded

˜X2

! ˜X.

Any property of the operator D

w

F(0) can be conveniently studied usingthe operator L. In particular, since ranD

w

F(0) = ranL and kerD

w

F(0) =

T kerL, the following lemma shows that D

w

F(0) : X ! Y is Fredholm ofindex 0.

Lemma 3.3 ([7]). The operator L :

˜X2

! Y is Fredholm of index 0. Its

kernel, kerL, is spanned by a finite number of functions of the form

�k

(q, s) =

(

cos(kq) sin⇤(✓k

s)/✓k

, ✓k

6= 0,

cos(kq)s, ✓k

= 0,k 2 Z, (3.4)

where we have used the notation

sin

⇤(✓

k

s) :=

(

sin(✓k

s), ⇠k2 + ↵ < 0,

sinh(✓k

s), ⇠k2 + ↵ > 0.

Define Z := {(⌘�

,�) : � 2 kerL} ⇢ ˜X ⇢ Y . Then the range of L, ranL, is

the orthogonal complement of Z in Y with respect to the inner product

hw1

, w2

iY

:=

ZZ

ˆ

⌦

ˆ�1

ˆ�2

dq ds+

Z

⇡

�⇡

⌘1

⌘2

dq, w1

, w2

2 Y.

The projection ⇧

Z

onto Z along ranL is given by

⇧

Z

w =

X hw, wk

iY

kwk

k2Y

wk

,

where the sum ranges over all wk

= (⌘�k,�

k

) 2 Z, with �k

of the form (3.4).

The abbreviation sin

⇤ in Lemma 3.3 will be used analogously for othertrigonometric and hyperbolic functions.

4. Existence of three-dimensional kernels

To ease notation, when ✓k

= 0 we interpret sin⇤(✓k

s)/✓k

as s and ✓k

cot

⇤(✓

k

)

as 1.

Lemma 4.1 (Bifurcation condition). Let ⇤ = (µ,↵,�, ⇠) 2 U . For k 2 Z we

have that cos(kq) sin⇤(✓k

s)/✓k

2 kerL if and only if

✓k

cot

⇤(✓

k

) =

1

µ2✓20

sin

2

(�)+ ✓

0

cot(�). (4.1)


To find nontrivial even functions, we assume that k > 0. As shown in [7]the problem of finding several solutions k for some parameters ⇤ in (4.1) isa question only of the left-hand side of the same equation. This amountsto finding integer solutions of one or more transcendental equations. Dueto the global non-monotonicity of the function ✓ 7! ✓ cot(✓) this is quite anintricate question (see [7, Lemma 4.3]). However, if one allows the wave-length parameter ⇠ to vary, a different approach is possible. Using this, weshall prove that to any two-dimensional kernel in the ‘trigonometric’ regimeone may adjoin a third, ‘hyperbolic’, dimension. Such a two-dimensionalkernel can be constructed either as in [7, Lemma 4.3] or as in Lemma 4.5below.

Now, let t = |↵| and ⇠ > 0, so that ✓k

:= |⇠k2 � t|1/2. Let furthermorek2

> k1

be positive integers such that

t� ⇠k21

> t� ⇠k22

> ⇡2 and t� ⇠k2j

6= n2⇡2,

for all n 2 Z and j = 1, 2. We study the (implicit) equation

f(t, ⇠) = 0,

withf(t, ⇠) := ✓

k1 cot(✓k1)� ✓k2 cot(✓k2).

When f(t, ⇠) = 0, we set

a(t, ⇠) := ✓k1 cot(✓k1) = ✓

k2 cot(✓k2).

Proposition 4.2. Suppose that f(t0

, ⇠0

) = 0 and a(t0

, ⇠0

) > 1. There is

a neighbourhood of (t0

, ⇠0

) in which f(t, ⇠) = 0 with a(t, ⇠) > 1 if and only

if t = t(⇠), where ⇠ 7! t(⇠) is an analytic function defined near ⇠0

, which

satisfies

dt

d⇠=

✓2k1✓2k2

+ t(a2 � a)

⇠(a2 � a)>

t

⇠.

Proof. We have

ft

(t, ⇠) =1

2

✓

cot(✓k1)

✓k1

� cot

2

(✓k1)

◆

� 1

2

✓

cot(✓k2)

✓k2

� cot

2

(✓k2)

◆

=

a� a2

2✓2k1✓2k2

�

✓2k2

� ✓2k1

�

=

⇠

2✓2k1✓2k2

(k22

� k21

)(a2 � a),

and

f⇠

(t, ⇠) =�k2

1

2

✓

cot(✓k1)

✓k1

� cot

2

(✓k1)� 1

◆

+

k22

2

✓

cot(✓k2)

✓k2

� cot

2

(✓k2)� 1

◆

= �1

2

(k22

� k21

) + (a� a2)k21

✓2k2

� k22

✓2k1

✓2k1✓2k2

!

= �1

2

(k22

� k21

)

1 +

t

✓2k1✓2k2

(a2 � a)

!

.

The proposition now follows from the implicit function theorem and implicitdifferentiation. ⇤


Proposition 4.3. The function ⇠ 7! t(⇠) from Proposition 4.2 induces an

analytic diffeomorphism ⇠ 7! a(⇠) from a bounded interval (⇠min

, ⇠max

) ⇢ R+

onto (1,1) with lim

⇠&⇠min a(⇠) = 1.

Proof. Since

@⇠

✓2k1

= @⇠

(t(⇠)� ⇠k21

) = t0⇠

� k21

>t

⇠� k2

1

>⇡2

⇠, (4.2)

it follows that

@⇠

a = @✓k1

(✓k1 cot(✓k1)) @⇠✓k1 =

@⇠

✓k1

✓k1

(a� a2 � ✓2k1)

< � ⇡2

2⇠✓2k1

(a2 � a+ ✓2k1) < �⇡

2

2⇠.

(4.3)

This proves that ⇠ 7! a(⇠) defines a local analytic diffeomorphism. The re-lations (4.2) and (4.3) hold when ⇠ > 0, t > 0, a 2 (1,1) and the entities✓k1 and ✓

k2 satisfy the given assumptions. Consider then a maximally con-tinued parametrization ⇠ 7! (t(⇠), a(⇠)). As long as a 2 (1,1) along such aparametrization the assumptions on ✓

k1 , ✓k2 and t cannot be violated. Hence,we only need to determine the set of ⇠ for which ⇠ > 0 and a(⇠) 2 (1,1).This can be deduced from (4.2): the differential inequality @

⇠

✓2k1

> ⇡2/⇠implies that

lim

⇠%1✓2k1

= 1, lim

⇠&0

✓2k1

= �1,

the first of which violates a > 1, and the second ✓2k1

� 0. Hence there exists⇠min

> 0 and ⇠max

< 1 such that Proposition 4.3 holds. ⇤

Lemma 4.4 (Three-dimensional). For any positive integers k2

> k1

and

positive real numbers ⇠0

, t0

with t0

� ⇠0

k21

> t0

� ⇠0

k22

> ⇡2,

f(t0

, ⇠0

) = 0 and a(t0

, ⇠0

) > 1,

there exist an integer k3

> k2

and positive real numbers ⇠, t such that ⇠k23

�t >0 and

✓k1 cot(✓k1) = ✓

k2 cot(✓k2) = ✓k3 coth(✓k3).

The integer k3

may be replaced by any larger integer.

Proof. Choose k3

> k2

such that ⇠0

k23

� t0

> 0 andq

⇠0

k23

� t0

coth

✓

q

⇠0

k23

� t0

◆

> a(t0

, ⇠0

).

This is possible in view of that lim✓3!1 ✓

3

coth(✓3

) = 1. Let ⇠ 7! (t(⇠), a(⇠))be the smooth parametrization for which f(t(⇠), ⇠) = 0 and a(⇠) % 1 for⇠ & ⇠

min

. We consider ⇠min

< ⇠ < ⇠0

. Since, for such ⇠,

⇠k23

� t(⇠) < ⇠k23

< ⇠0

k23

,

the function

(⇠min

, ⇠0

) 3 ⇠ 7!q

⇠k23

� t(⇠) coth

✓

q

⇠k23

� t(⇠)

◆


is analytic and bounded from above for ⇠k23

� t(⇠) > 0. In view of thata(⇠) % 1 as ⇠ & ⇠

min

, it follows that there exists ⇠ 2 (⇠min

, ⇠0

) withq

⇠k23

� t(⇠) coth

✓

q

⇠k23

� t(⇠)

◆

= a(⇠). ⇤

The construction behind Lemma 4.4 does not rule out that the kernelhas more than three dimensions. The following result shows that one canconstruct kernels with exactly three dimensions.

Lemma 4.5. For any wavenumbers k1

, k2

, k3

2 Z>0

with k3

> k2

> k1

and

k23

� k22

k23

� k21

>9

16

,

there exist (µ,↵,�, ⇠) 2 U such that (4.1) holds for k = k1

, k2

, k3

and for no

other integer k � k1

.

Proof. Let a 2 (1,1). Since ✓ 7! ✓ cot ✓ spans (0,1) on the intervals(0, ⇡

2

)+n⇡, n 2 Z>0

, there are ✓1

= ✓1

(a) 2 (2⇡, 5⇡2

) and ✓2

= ✓2

(a) 2 (⇡, 3⇡2

)

witha = ✓

1

cot ✓1

= ✓2

cot ✓2

.

We are looking for t = t(a) > 0 and ⇠ = ⇠(a) > 0 such that

1 �k21

1 �k22

�

t⇠

�

=

✓21

✓22

�

.

The unique solution of this linear system is

t :=k22

✓21

� k21

✓22

k22

� k21

, ⇠ :=✓21

� ✓22

k22

� k21

, (4.4)

which are both easily seen to be positive. By choosing µ and � appropriatelythis yields two integer solutions k = k

1

, k2

of (4.1). Since ✓ cot ✓ < 1 for✓ 2 (0,⇡), and due to the local monotonicity of ✓ 7! ✓ cot ✓ on the intervals(0,⇡) + n⇡, n 2 N, there can be no other solutions k � k

1

with ↵+ ⇠k2 < 0.To find a solution k = k

3

with ⇠k23

� t > 0 we consider

⇠(a)k23

� t(a) =(k2

3

� k22

)✓21

(a) + (k21

� k23

)✓22

(a)

k22

� k21

.

This expression is positive and uniformly bounded away from zero with re-spect to a whenever k

1

, k2

, k3

satisfy the assumptions of the lemma. Sincethe mapping a 7! (✓

1

(a), ✓2

(a)) is bounded, uniformly for all a 2 (1,1), wethus get that

a 7! ✓3

(a) := (⇠(a)k23

� t(a))1/2

maps (1,1) into a compact interval in (0,1). Hence, as a spans (1,1)

there will be a value of a for which

a = ✓k3(a) coth(✓3(a)). ⇤


5. The Lyapunov–Schmidt reduction and existence of trimodalsteady water waves

Let ⇤⇤ denote a quadruple (µ⇤,↵⇤,�⇤, ⇠⇤) such that (4.1) holds and sup-pose that

kerL(⇤⇤) = span{�⇤

1

, . . . ,�⇤n

},with �⇤

j

= cos(kj

q) sin⇤(✓kjs)/✓kj and 0 < k

1

< · · · < kn

. Let w⇤j

= T (⇤

⇤)�⇤

j

.From Lemma 3.3 it follows that Y = Z � ranL(⇤⇤

). As in that lemma, welet ⇧

Z

be the corresponding projection onto Z parallel to ranL(⇤⇤). This

decomposition induces similar decompositions ˜X = Z� (ranL(⇤⇤)\ ˜X) and

X = kerFw

(0,⇤⇤)�X

0

, where X0

= R(ranL(⇤⇤) \ ˜X) in which

R(⌘,�) =

✓

⌘,�� s( 0

)

s

�|s=1

( 0

)

s

(1)

◆

.

Applying the Lyapunov–Schmidt reduction [14, Thm I.2.3] we obtain thefollowing lemma.

Lemma 5.1. [14] There exist open neighborhoods N of 0 in kerFw

(0,⇤⇤),

M of 0 in X0

and U⇤of ⇤

⇤in R4

, and a function 2 C1(N ⇥ U⇤,M),

such that

F(w,⇤) = 0 for w 2 N +M, ⇤ 2 U⇤,

if and only if w = w⇤+ (w⇤,⇤) and w⇤

= t1

w⇤1

+ · · ·+ tn

w⇤n

2 N solves the

finite-dimensional problem

�(t,⇤) = 0 for t 2 V, ⇤ 2 U⇤, (5.1)in which

�(t,⇤) := ⇧Z

F(w⇤+ (w⇤,⇤),⇤),

and V := {t 2 Rn

: t1

w⇤1

+· · ·+tn

w⇤n

2 N}. The function has the properties

(0,⇤) = 0 and Dw

(0,⇤) = 0.

Bifurcation from a three-dimensional kernel.

Theorem 5.2 (Three-dimensional bifurcation). Suppose that

dimkerD

w

F(0,⇤⇤) = 3,

and that

a := ✓k1 cot

⇤(✓

k1) = ✓k2 cot

⇤(✓

k2) = ✓k3 coth(✓k3) > 1. (5.2)

Assume also that the integers k1

< k2

< k3

are positive, and let t :=

(t1

, t2

, t3

). Then there exists for every � 2 (0, 1) a smooth family of small-

amplitude nontrivial solutions

S := {(w(t), µ(t),↵(t), ⇠(t)) : 0 < |t| < ", |t1

t2

t3

| > �|t|3} (5.3)of F(w, µ,↵,�⇤, ⇠) = 0 in O⇥R3

, passing through (w(0), µ(0),↵(0), ⇠(0)) =(0, µ⇤,↵⇤, ⇠⇤) with

w(t) = t1

w⇤1

+ t2

w⇤2

+ t3

w⇤3

+O(|t|2)Proof of Theorem 5.2. The first part of the proof is analogous to that of[7, Theorem 4.8] (see also [1]). The second part involves calculating thedeterminant of 3 ⇥ 3-matrix, the entries of which depend transcendentallyon the parameters of the problem. This can be done via thorough, butrudimentary, investigation.


Part I. Reduction to a 3⇥3 function-valued matrix. Define w⇤j

:=

(⌘�

⇤j,�⇤

j

) 2 ˜X, j = 1, 2, 3, and recall that Z = span{w⇤1

, w⇤2

, w⇤3

}. If ⇧j

� =

�

j

w⇤j

, where ⇧j

denotes the projection onto span{w⇤j

}, equation (5.1) takesthe form

�

j

(t,⇤) = 0, j = 1, 2, 3. (5.4)This is a system of three equations with seven unknowns, and we note thatit has the trivial solution (0,⇤) for all ⇤ 2 U⇤.

We introduce polar coordinates by writing t = rˆt with |ˆt| = 1. Then�

j

(rˆt,⇤) = r j

(rˆt,⇤),

where

j

(t,⇤) :=

Z

1

0

ˆt ·rt

�

j

(zrˆt,⇤) dz, j = 1, 2, 3,

since �j

(0,⇤) = 0. (5.4) is equivalent tor

j

(t,⇤) = 0, j = 1, 2, 3. (5.5)Since @

tj�(0,⇤⇤) = ⇧

Z

D

w

F(0,⇤⇤)w⇤

j

= 0, we have that

j

(0,⇤⇤) =

ˆt ·rt

�

j

(0,⇤⇤) = 0, j = 1, 2, 3. (5.6)

We therefore apply the implicit function theorem to at the point (0,⇤⇤),

by proving that the matrix

M :=

2

4

@µ

1

(0,⇤⇤) @

µ

2

(0,⇤⇤) @

µ

3

(0,⇤⇤)

@↵

1

(0,⇤⇤) @

↵

2

(0,⇤⇤) @

↵

3

(0,⇤⇤)

@⇠

1

(0,⇤⇤) @

⇠

2

(0,⇤⇤) @

⇠

3

(0,⇤⇤)

3

5 (5.7)

is invertible. We have that

⇧

Z

D

2

wµ

F(0,⇤⇤)w⇤

j

=

hD2

wµ

F(0,⇤⇤)w⇤

j

, w⇤j

iY

kw⇤j

k2Y

w⇤j

,

since D2

wµ

F(0,⇤⇤)w⇤

j

is orthogonal to w⇤i

for i 6= j. In view of that @µ

j

(0,⇤⇤) =

P

3

i=1

ˆti

@ti@µ�j

(0,⇤⇤) and @

ti@µ�(0,⇤⇤) = ⇧

Z

D

2

wµ

F(0,⇤⇤)w⇤

i

we thus havethat

@µ

j

(0,⇤⇤) =

3

X

i=1

ˆti

@ti@µ�j

(0,⇤⇤) =

ˆtj

hD2

wµ

F(0,⇤⇤)w⇤

j

, w⇤j

iY

kw⇤j

k2Y

. (5.8)

Recall that D

w

F(0,⇤) = L(⇤)T �1

(⇤

0). Thus,

D

2

wµ

F(0,⇤⇤)w⇤

j

= D

µ

L(⇤⇤)�⇤ + L(⇤⇤

)D

µ

T �1

(⇤

⇤)w⇤

j

.

Since L = D

w

F(0) � T the second term on the right-hand side belongs toranD

w

F(0,⇤⇤), and we find that hD2

wµ

F(0,⇤⇤)w⇤

j

, w⇤j

iY

= hDµ

L(⇤⇤)�⇤

j

, w⇤j

iY

.Thus

@µ

j

(0,⇤⇤) =

hDµ

L(⇤⇤)�⇤

j

, w⇤j

iY

kw⇤j

k2Y

, j = 1, 2, 3.

Similar arguments hold for @µ

j

(0,⇤⇤) and @

↵

j

(0,⇤⇤), and we find that

detM = Cˆt1

ˆt2

ˆt3

det

2

4

hDµ

L(⇤⇤)�⇤

1

, w⇤1

iY

hDµ

L(⇤⇤)�⇤

2

, w⇤2

iY

hDµ

L(⇤⇤)�⇤

3

, w⇤3

iY

hD↵

L(⇤⇤)�⇤

1

, w⇤1

iY

hD↵

L(⇤⇤)�⇤

2

, w⇤2

iY

hD↵

L(⇤⇤)�⇤

3

, w⇤3

iY

hD⇠

L(⇤⇤)�⇤

1

, w⇤1

iY

hD⇠

L(⇤⇤)�⇤

2

, w⇤2

iY

hD⇠

L(⇤⇤)�⇤

3

, w⇤3

iY

3

5 ,

(5.9)


where C = kw⇤1

k�2

Y

kw⇤2

k�2

Y

kw⇤3

k�2

Y

6= 0.

Part II. Determining det(M). Let j 2 {1, 2, 3}, and

±⇤:= sgn(⇠k2

j

+ ↵).

One has (see [7])

⌦

D

µ

L(⇤⇤)�⇤

j

, w⇤j

↵

Y

= A

✓

sin

⇤(✓

kj )

✓kj

◆

2

,

and⌦

D

↵

L(⇤⇤)�⇤

j

, w⇤j

↵

Y

= B

✓

sin

⇤(✓

kj )

✓kj

◆

2

+ f(kj

),

where A and B are constants, A is non-zero, and

f(kj

) := ±⇤ ⇡

2

✓kj � cos

⇤(✓

kj ) sin⇤(✓

kj )

✓3kj

, (5.10)

which is naturally extended to a continuous function of ✓kj . It follows

from (3.3) that D

⇠

L(⇤)� = (0, @2q

�), whence

D

⇠

L(⇤⇤)�⇤

j

= (0,�k2j

�⇤j

).

This is k2j

times the second component of D↵

L(⇤⇤)�⇤

j

, and straightforwardintegration shows that

⌦

D

⇠

L(⇤⇤)�⇤

j

, w⇤j

↵

Y

= k2j

f(kj

).

We are thus left with calculating the determinant

det

0

B

B

B

@

A(sin

⇤(✓k1

))

2

✓

2k1

A(sin

⇤(✓k2

))

2

✓

2k2

A(sin

⇤(✓k3

))

2

✓

2k3

B(sin

⇤(✓k1

))

2

✓

2k1

+ f(k1

)

B(sin

⇤(✓k2

))

2

✓

2k2

+ f(k2

)

B(sin

⇤(✓k3

))

2

✓

2k3

+ f(k3

)

k21

f(k1

) k22

f(k2

) k23

f(k3

)

1

C

C

C

A

= A

f(k1

)f(k2

)(k22

� k21

)

(sin

⇤(✓

k3))2

✓2k3

+ f(k1

)f(k3

)(k21

� k23

)

(sin

⇤(✓

k2))2

✓2k2

+ f(k2

)f(k3

)(k23

� k22

)

(sin

⇤(✓

k1))2

✓2k1

!

= A

3

Y

j=1

f(kj

)

(k22

� k21

)

(sin

⇤(✓

k3))2

✓2k3f(k

3

)

+ (k21

� k23

)

(sin

⇤(✓

k2))2

✓2k2f(k

2

)

+ (k23

� k22

)

(sin

⇤(✓

k1))2

✓2k1f(k

1

)

!

.

(5.11)The function ✓ 7! f(✓) defined by (5.10) is everywhere negative, whence weare left with investigating the expression

˜f(✓) :=⇡

2

(sin

⇤(✓))2

✓2f(✓)


= ±⇤ (sin⇤(✓))2

✓2

✓

✓3

✓ � cos

⇤(✓) sin⇤(✓)

◆

= ±⇤ a(sin⇤(✓))2

a� a cos

⇤(✓) sin

⇤(✓)

✓

=

a(sin⇤(✓))2

±⇤(a� 1)� (sin

⇤(✓))2

.

Note that the denominator is strictly negative: for ⇠k2j

+ ↵ < 0 since a > 1,and for ⇠k2

j

+ ↵ > 0 since ✓ coth ✓ � 1 � sinh

2

(✓) < 0 for all ✓ 6= 0. Letx := (sin

⇤(✓))2. For a > 1 the function x 7! ax

a�1�x

is strictly increasing withlimit �a as x ! 1, so we immediately get that

˜f(✓k3) < �1. (5.12)

When ⇠k2j

+↵ < 0 the function [0, 1] 3 x 7! ax

1�a�x

is strictly decreasing withimage [�1, 0]. Since

✓k1 cot(✓k1) = ✓

k2 cot(✓k2) < ✓k1 cot(✓k2),

with the left-hand side positive, we may square both sides to obtain that

sin

2

(✓k1) > sin

2

(✓k2).

Taking (5.12) into consideration, we thus conclude that˜f(✓

k3) <˜f(✓

k1) <˜f(✓

k2) < 0.

Returning to (5.11), this shows that the determinant of M is non-zero:

(k22

� k21

)

˜f(✓k3) + (k2

1

� k23

)

˜f(✓k2) + (k2

3

� k22

)

˜f(✓k1)

= (k22

� k21

)(

˜f(✓k3)� ˜f(✓

k2)) + (k23

� k22

)(

˜f(✓k1)� ˜f(✓

k2)) < 0.

The condition |t1

t2

t3

| > �|t|3 implies that |ˆt1

ˆt2

ˆt3

| > �, so that the determi-nant is uniformly bounded from below. This guarantees that the interval0 < |t| < " can be chosen uniformly. ⇤

6. The structure of the solution set

Theorem 5.2 does not give the full local solution set near the bifurca-tion point. In particular, since the solutions are bounded away from thecoordinate planes in (t

1

, t2

, t3

)-space, any solutions obtained by restrictingthe period and using bifurcation with a one- or two-dimensional kernel areexcluded. In this section we present a method which gives a more com-plete picture of the solution set by taking into account the number theoreticproperties of the integers k

1

, k2

, k3

. While the method is not guaranteed tofind the full local solution set, the solutions obtained by lower-dimensionalbifurcation are included.

For the purpose of the following discussion it is conveniant to ignore theorder of the numbers k

1

, k2

, k3

. We therefore relabel them as m1

,m2

,m3

,with no particular order, but assuming that gcd(m

1

,m2

,m3

) = 1. If m =

gcd(m1

,m2

,m3

) > 1, then by working in the subspace X(m) of 2⇡/m-periodic functions we can reduce the problem to the previous case. Wetherefore have the following four different cases.


gcd(m1

,m2

) > 1, gcd(m1

,m3

) > 1, gcd(m2

,m3

) > 1. (i)gcd(m

1

,m2

) = 1, gcd(m1

,m3

) > 1, gcd(m2

,m3

) > 1. (ii)gcd(m

1

,m2

) = 1, gcd(m1

,m3

) = 1, gcd(m2

,m3

) > 1. (iii)gcd(m

1

,m2

) = 1, gcd(m1

,m3

) = 1, gcd(m2

,m3

) = 1. (iv)

Each of the above cases have subcases determined by whether the differentm

j

divide each other or not. In what follows, all cases will be presented in theform of the reduced equations they give rise to, as well as the correspondingsolution set in (t

1

, t2

, t3

)-space (cf. Theorem 5.2 and Equation (5.3)). Werecall that this is a subset of the open ball

B := {(w(t), µ(t),↵(t), ⇠(t)) : 0 < |t| < "} (6.1)

of possible solutions. The method is illustrated with more details for thefirst cases, whereas the last and analogue cases are presented in a shortermanner.

6.1. Case (i): an open ball of solutions. In this case no mi

can divideanother m

j

. Indeed, assume e.g. that m1

| m2

. Then gcd(m1

,m3

) | m2

aswell, so that gcd(m

1

,m2

,m3

) = gcd(m1

,m3

) > 1, yielding a contradiction.A numerical example is given by (6, 10, 15).

The relations between the mj

imply that

�

1

(0, t2

, t3

) = 0,

�

2

(t1

, 0, t3

) = 0,

�

3

(t1

, t2

, 0) = 0,

(6.2)

where ⇤ has been supressed for convenience. Then (6.2) is equivalent to that

�

j

(t,⇤) =

Z

1

0

d

dz�

j

(ztj

; ti

|i 6=j

,⇤) dz = tj

j

(t,⇤) = 0, (6.3)

with

j

(t,⇤) :=

Z

1

0

D

tj �j

(ztj

; ti

|i 6=j

,⇤) dz,

both for j = 1, 2, 3. At the point (0,⇤⇤), the relation

j

(0,⇤⇤) = D

tj �j

(0,⇤⇤)

enables us to apply the implicit function theorem to (5.7) without any t-dependent coefficients appearing before the matrix in (5.9) (cf. (5.6)); theresult is a full three-dimensional ball of solutions,

S(i)

= B,as given in (6.1).

6.2. Case (ii). Using the same argument as above, we notice that m3

-m

1

,m2

. Also, m1

cannot divide m2

, or contrariwise, since this would givem

1

= 1 and therefore would contradict gcd(m1

,m3

) > 1. Up to relabeling,we therefore obtain three alternatives:


Figure 1. Left: The cases (ii) a, (ii) b and (ii) c, in orderfrom left to right. The illustrations show the qualitative in-tersection of the solution set S

(ii)↵, ↵ = a, b, c, with the ball

B of radius " > 0 in (t1

, t2

, t3

)-space.

(a) m1

| m3

, m2

| m3

. An example of this is (2, 3, 6). One obtains

�

1

(0, t2

, t3

) = 0,

�

2

(t1

, 0, t3

) = 0,

�

3

(0, 0, 0) = 0.

Here (6.3) holds for j = 1, 2, whereas for �3

we use spherical coor-dinates t = rˆt, writing

�

3

(t,⇤) = r 3

(t,⇤) with

3

(0,⇤⇤) =

ˆt ·rt

�

3

(0,⇤⇤).

An iteration of the proof on page 11, see especially (5.8), then yieldsthat the system

t1

1

(t,⇤) = 0,

t2

2

(t,⇤) = 0,

r 3

(t,⇤) = 0,

can be solved using the implicit function theorem whenever |ˆt3

| � �,i.e., whenever |t

3

| � �|t|. Hence, for every � ⌧ 1, the restriction

S(ii)a

: |t3

| � �|t|of the ball B describes a smooth family of small-amplitude nontriv-ial solutions. The solutions that may be found by one- and two-dimensional bifurcation by setting the different t

j

’s to zero, namely

t1

= 0, 0 < (t22

+ t23

)

1/2 < ", |t3

| � �|t2

|t2

= 0, 0 < (t21

+ t23

)

1/2 < ", |t3

| � �|t1

|,are included in this solution set; and for t

3

= 0 with t21

+ t22

6= 0 nosolutions are found.

(b) m1

| m3

, m2

- m3

. An example of this is (2, 9, 12). One finds

�

1

(0, t2

, t3

) = 0,

�

2

(t1

, 0, t3

) = 0,

�

3

(0, t2

, 0) = 0.

Again (6.3) holds for j = 1, 2, whereas for �3

we use cylindrical

coordinates t = (r1,3

ˆt1

, t2

, r1,3

ˆt3

), with ri,j

:= (t2i

+ t2j

)

1/2. This


yields

�

3

(t,⇤) =

Z

1

0

d

dz�

3

(zr1,3

(

ˆt1

, ˆt3

); t2

,⇤) dz

= r1,3

Z

1

0

(

ˆt1

, ˆt3

) ·r(t1,t3)

�

3

(zr1,3

(

ˆt1

, ˆt3

); t2

,⇤) dz

= r1,3

3

(t,⇤),

with

3

(0,⇤⇤) = (

ˆt1

, ˆt3

) ·r(t1,t3)

�

3

(0,⇤⇤),

and the system of equations now becomes

t1

1

(t,⇤) = 0,

t2

2

(t,⇤) = 0,

r1,3

3

(t,⇤) = 0.

This can be solved whenever |ˆt3

| � �, i.e., for |t3

| � �(t21

+ t23

)

1/2.Hence, for any � ⌧ 1, we obtain the family of solutions given by therestriction

S(ii)b

: |t3

| � �|t1

|to B. As in the case (ii)

a

, any ’lower-dimensional’ solutions,

t1

= 0, 0 < (t22

+ t23

)

1/2 < ",

t2

= 0, 0 < (t21

+ t23

)

1/2 < ", |t3

| � �|t1

|,t1

= t3

= 0, 0 < |t2

| < ";

are included in the above solution set; and for t3

= 0 with t1

6= 0

the implicit function theorem is inconclusive.

(c) m1

- m3

, m2

- m3

. An example is (4, 9, 30).

�

1

(0, t2

, t3

) = 0,

�

2

(t1

, 0, t3

) = 0,

�

3

(t1

, 0, 0) = �3

(0, t2

, 0) = 0.

The difference with respect to case (iii)b

is that we may express �3

using different cylindrical coordinates as either t = (r1,3

ˆt1

, t2

, r1,3

ˆt3

)

or t = (t1

, r2,3

ˆt2

, r2,3

ˆt3

), where ri,j

= (t2i

+ t2j

)

1/2. Thus, the originalsystem reduces to

tj

j

(t,⇤) = 0,

rj,3

3

(t,⇤) = 0,

for j = 1, 2. This can be solved whenever |t3

| � �(t2j

+ t23

)

1/2 foreither j = 1 or j = 2, and we obtain

S(ii)c

: |t3

| � �min(|t1

|, |t2

|).The solutions obtained from lower-dimensional bifurcation,

ti

= 0, 0 < (t2j

+ t23

)

1/2 < ", |t3

| � �|tj

|,ti

= t3

= 0, 0 < |tj

| < ",


Figure 2. Left: The cases (iii) a, (iii) b and (iii) c, in order from left toright. The illustrations show the qualitative intersection of the solution setS(iii)↵ , ↵ = a, b, c, with the ball B of radius " > 0 in (t1, t2, t3)-space.

for i, j = 1, 2, i 6= j, are all included in the larger solution set; andfor t

3

= 0 with t1

t2

6= 0 no solutions are found.

6.3. Case (iii). The only possibilities for some mi

to divide another mj

arethat m

1

= 1, in which case it divides both m2

and m3

, or that m2

dividesm

3

, or contrariwise. Without loss of generality, assume that m3

- m2

. Wethen obtain the following alternatives:

(a) m1

= 1, m2

| m3

. Then m1

| m2

| m3

. An example is (1, 2, 4).

�

1

(0, t2

, t3

) = 0,

�

2

(0, 0, t3

) = 0,

�

3

(0, 0, 0) = 0.

In this case we combine the techniques from the cases (i), (ii)a

and (ii)b

, by expanding �1

in t1

, �2

using cylindral coordinates(r

1,2

ˆt1

, r1,2

ˆt2

, t3

), and �3

using spherical coordinates. The resultingdeterminant will be non-zero whenever |t

2

| � �|t1

| and |t3

| � �|t|,which may be reduced to

S(iii)a

: |t3

| � �|t2

| � �2|t1

|,for some � ⌧ 1. The system of equations

t1

1

(t,⇤) = 0,

r1,2

2

(t,⇤) = 0,

r 3

(t,⇤) = 0,

has no further non-trivial solutions.

(b) m1

= 1, m2

- m3

. An example is (1, 4, 6). Using techniques as inthe previous examples, one obtains

�

1

(0, t2

, t3

) = 0,

�

2

(0, 0, t3

) = 0,

�

3

(0, t2

, 0) = 0,

withS(iii)b

: min(|t2

|, |t3

|) � �|t1

|.


Figure 3. Left: The cases (iv) a and (iv) b, in order from leftto right. The illustrations show the qualitative intersectionof the solution set S

(ii)↵, ↵ = a, b, c, with the ball B of radius

" > 0 in (t1

, t2

, t3

)-space. Note the difference between tocases, which is visible only in one axial direction.

(c) m1

> 1, m2

| m3

. An example is (2, 3, 9). We have

�

1

(0, t2

, t3

) = 0,

�

2

(t1

, 0, 0) = �2

(0, 0, t3

) = 0,

�

3

(t1

, 0, 0) = 0.

Here, the union of {|t3

| � �|t2

| � �2|t1

|} and {|t3

| � �|t2

| � �2|t3

|}gives

S(iii)c

: |t3

| � �|t2

| � �2min(|t1

|, |t3

|).

(d) m1

> 1, m2

- m3

. An example is (2, 15, 21). For

�

1

(0, t2

, t3

) = 0,

�

2

(t1

, 0, 0) = �2

(0, 0, t3

) = 0,

�

3

(t1

, 0, 0) = �3

(0, t2

, 0) = 0.

the solution set is the union

{|t3

| � �|t2

| � �2|t1

|}[ {|t

2

| � �|t3

| � �2|t1

|}[ {|t

3

| � �|t2

| � �2|t3

|}[ {min(t

3

, t2

) � �|t1

|},and we obtain

S(iii)d

: min(|t2

|, |t3

|) � �min(|t1

|,max(|t2

|, |t3

|)).

6.4. Case (iv). The only possiblities for some mi

to divide another mj

isthat m

i

= 1. Without loss of generality, assume that m1

= 1. There are twocases:

(a) m1

= 1. Then m1

| m2

,m3

. An example is (1, 2, 3). We have

�

1

(0, t2

, 0) = �1

(0, 0, t3

) = 0,

�

2

(0, 0, t3

) = 0,

�

3

(0, t2

, 0) = 0,


andS(iv)a

: min(|t3

|, |t2

|) � �|t1

| � �2min(|t3

|, |t2

|).(b) m

1

6= 1. Then m1

- m2

, m1

- m3

, m2

- m3

. An example is (2, 3, 5).The system

�

1

(0, t2

, 0) = �1

(0, 0, t3

) = 0,

�

2

(t1

, 0, 0) = �2

(0, 0, t3

) = 0,

�

3

(t1

, 0, 0) = �3

(0, t2

, 0) = 0.

allows all eight possible combinations of |ti

| � �|tj

|, with i = 1, 2, 3and j 6= i. Now, without loss of generality, say that |t

1

| � |t2

| �|t3

|. Then also |t1

| � �|t2

| and |t2

| � �|t3

|, so the only remainingcondition is that

|t3

| � �min(|t1

|, |t2

|).Since this is to hold for arbitrary indices, one may let |t

j

| := min

i

|ti

|to obtain the uniform condition

S(iv)b

: |tj

| � �min

i 6=j

|ti

|.Note that the set S

(iv)bincludes elements from the t

j

-axes, for whichti

= 0 for i 6= j, although not the complete coordinate planesthrough the origin (for which only one t

j

= 0).

Example 6.1. For an illustration of our results, consider the wavenumbersk1

= 6, k2

= 10 and k3

= 15, as in case (i) in Section 6.1. According toLemma 4.5, the bifurcation condition (4.1) holds for these wavenumbers,and for no other wave numbers larger than 6. In fact, one can check thatequality is obtained for ⇠ ⇡ 0.571 and ↵ ⇡ �69.9, with a ⇡ 7.65, and thatk = 1, 2, 3, 4, 5 do not satisfy the same bifurcation condition.

The resulting kernel is exactly three-dimensional and, via Theorem 5.2 andthe analysis pursued in Section 6.1, it gives rise to a full three-dimensionalball of solutions, attained in the horisontal direction as nonlinear perturba-tions of the linear hull of cos(6q), cos(10q) and cos(15q).

Figure 4 shows the envelope of three such waves; one where all coefficientsare positive and equal, one where the coefficient in front of the highest modeis zero (which gives a bimodal wave), and one where the coefficient in frontof the highest mode is negative (but of the same size as those for the modesk1

= 6 and k2

= 10). Note that although all of the waves constructed inthis paper are symmetric around the axis q = 0, they are not necessarily soaround their highest crest or lowest trough.

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!"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]_abcdefghijklmnopqrstuvwxyz{|}~

Figure 4. Three examples of waves obtained as nonlinear per-

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Department of Mathematical Sciences, Norwegian University of Scienceand Technology, 7491 Trondheim, Norway

E-mail address: [email protected]

Centre for Mathematical Sciences, Lund University, PO Box 118, 221 00Lund, Sweden

E-mail address: [email protected]

Trimodal Steady Water Waves

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