数理解析研究所講究録
第 740 巻 1991 年 208-242

208

Normal Forms of the Bifurcation Equations in the Problem
of Capillary-Gravity Waves*
HISASHI OKAMOTO1 AND MAYUMI SHOJI2
Abstract. We analyze two dimensional capillary-gravity waves on surface of irrotational flow. Two sets of algebraic equations are proposed as normal forms. We showthat these equations provide bifurcation diagrams, which are qualitatively the sameas those obtained numerically. Thus we establish a rigorous basis for the numericalresults in [1,2,17,18]. We also show that study of not only physically acceptable
solutions but also self-intersecting unphysical solutions is necessary for complete
understanding of the phenomena.

\S 1. Introduction.

The purpose of the present paper is to give a mathematical account of thebifurcation diagrams arising in the theory of progressive water waves of two dimensional irrotational flow. In this paper we consider nonlinear capillary-gravity
waves, which was first analyzed by Wilton [21]. By capillary-gravity waves wemean that we take into account of both surface tension and gravity. There isa well-established (both physical and mathematical) theory on this problem[9,13-16,19-21]). Recent researches, however, showed new phenomena in thecapillary-gravity waves.
The problem is a bifurcation problem and it is of a primary importance todetermine bifurcation diagrams, which are sometimes called solution diagrams.
Among many references, Chen and Saffman $[1,2]$ , Schwartz and Vanden-Broeck[17] and the second author of the present paper [18] computed capillary-gravity
waves numerically. In particular, [18] presented various bifurcation diagrams forvarious parameters. A mathematical analysis of the bifurcation was given by
Pierson and Fife [15], Reeder and Shinbrot [16] and Toland and Jones $[9,20]$ .
These authors gave a rigorous mathematical background to the physical theory by
Wilton [21] and others. Later [13] showed that some of the results in [9,15,16,20]
can be derived more simply by making use of the results in Fujii et al. $[4,5]$ .
However, the numerical computation in [18] shows that there are considerablycomplicated structures in the bifurcation diagrams, which can not be found in theprevious papers.
In this paper we give a normal form of the bifurcation equation, which turns
out to be a set of polynomials in two independent variables with a bifurcation
parameter. Our normal form is more degenerate than the bifurcation equations in$[9,20]$ . By this introduction of degeneracy we can explain a semi-global structure of
progressive waves in [18], which can not be found by the previous works. Actually,
our task is to examine the agreement of the numerically computed diagrams in$($’This work was partially supported by Ohbayashi Corporation
1Research Institute for Mathemathecal $Scienc6s$ , Kyoto University, Kyoto, 606 Japan
2Dept. of Mathematics, Fac. of Science, University of Tokyo, Bunkyo-ku, Tokyo 113, Japan

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[18] and those given by abstract bifurcation equations in the present paper. Our
normal forms of the bifurcation equations are derived by utilizing the theorems in
Golubitsky et al. [6,7,8]
We emphasize a rather unexpected fact that we should take account of solutions
in which the wave proMes have self-intersections. (We call such a solution a overlap
solution or an unphysical solution. ) In fact, in some bifurcation diagrams, someclasses of non-overlap (thus physical) solutions are disconnected from the otherclasses in the phase space, if we take a look at only physical solutions. If, however,
we consider both physical and unphysical solutions, all classes of the solutions areconnected. The authors think this is an important step towards a new developmentin the theory of capillary-gravity waves. For example, the disconnected part of thesolutions will not be traced if we neglect unphysical solutions.
This paper is composed of five sections. In \S 2 we introduce an equation originally
due to Levi-Civita [10], which is a fundamental equation for the progressive water
waves. At the end of this section, a rough description of our result is presented. In\S 3 we consider bifurcation equations and its simplification by means of its $O(2)-$equivariance. The bifurcation equations are classified by a pair of distinct positive
integers, which we call a mode. We consider only the cases of mode $(1,2)$ and $(1,3)$
We discuss the validity of our normal forms in \S 4 in the case of $(1,2)$ . The caseof $(1,3)$ is discussed in \S 5.

\S 2. The fundamental equation.
We consider a two dimensional irrotational flow of inviscid incompressible fluidwith a free surface. What we consider is a problem of progressive waves, by which
we mean a fluid motion with free surface whose configuration is constant if viewed
in a coordinate system moving with the same speed as that of wave. Accordingly,
in this moving frame, the fluid particles move in a opposite direction while thewave profile remains at rest. In this moving frame, we take $(x, y)$ coordinate
system with horizontally to the right and vertically upwards. We let denotethe propagation speed and let $y=h(x)$ represent the free surface. Our further
assumption is that the wave profile is periodic in with a period, say, and thatthe wave profile is symmetric with respect to the y-axis. We assume that the flow
is infinitely deep. By this and the periodicity assumption, we have only to consider
the fluid in $-L/2<x<L/2,$ $-\infty<y<h(x)$ . Then the problem is to find awave profile function $y=h(x)$ and a complex potential $f\equiv U+iV$ , where isa velocity potential and $V$ a stream function, such that $U+iV$ is a holomorphic
function of $z\equiv x+iy$ and satisfies
$x$

$c$

$y$

$L$

$x$

$U$

$U( \pm\frac{L}{2},$$y)= \pm\frac{cL}{2}$,on$- \infty<y\leq h(\pm\frac{L}{2})$$\frac{\partial U}{\partial n}=0$

on $y=h(x)$$\frac{df}{dz}arrow c$as$yarrow-\infty$2and, respectively,$y=0$ ,

210

(2.1)

$\frac{1}{2}|\frac{df}{dz}|^{2}+gy+\frac{T}{d}K=constant$on $y=h(x)$ ,,where,
denotes the outward normal derivative, is the gravity acceleration,
the surface tension coefficient, the mass density, and $K$ is the curvature of theboundary $y=h(x)$ . $K$ is represented as follows:$\partial/\partial n$

$g$

$T$

$d$

$K=-( \frac{h’}{\sqrt{1+h^{\prime 2}}})’$,

where the prime means the difFerentiation with respect to .
This is a free boundary problem, in which a boundary portion $y=h(x)$ should
be sought. There is a mathematical device found by Stokes [19], which makes
it considerably easy to handle the free boundary problem above. The idea is toregard as an independent variable rather than the dependent one. The new
formulation is described as follows:
$x$

$f$

Find a(2.2)$2\pi$-periodic function$\theta=\theta(\sigma)$such that$e^{2H\theta} \frac{dH\theta}{d\sigma}-pe^{-H\theta}\sin(\theta)+q\frac{d}{d\sigma}(e^{H\theta}\frac{d\theta}{d\sigma})=0$$(0\leq\sigma<2\pi)$,where $H$ is a linear operator defined through the Fourier series as follows:(2.3)$H( \sum_{n=1}^{\infty}(a_{n}\sin n\sigma+b_{n}\cos n\sigma))=\sum_{n=1}^{\infty}(-a_{n}\cos n\sigma+b_{n}\sin n\sigma)$.

Note that the equation (2.2) depends on two parameters and . These parameters
are defined as
$p$

$q$

$q= \frac{2\pi T}{dc^{2}L}$$p= \frac{gL}{2\pi c^{2}}$(see, [13]). The physical meaning restricts those parameters to be nonnegative:$(p, q)\in[0, \infty)\cross[0, \infty)$.

This formulation (2.2,3), which is equivalent to the original one, is derived in [13]and used for proving the existenxe of non-zero solution . Since we define andin Appendix by$u– iv=ce^{-i\theta+\tau}$ ,$\theta$$\theta$$\tau$

where $(u, v)$ is the velocity vector, the function represents the angle made by thetangent vector to the free boundary and the x-axis.
$\theta$

Although the derivation of (2.2) is given in [13], we give an outline of thederivation in the Appendix for the sake of the reader’s convenience.
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In order to apply a mathematical theory of bifurcation such as Golubitsky and
Schaeffer [6,7,8] or others, we use a mathematical setting with function spaces.
byNamely we define, for a nonnegative integer , Banach spaces$k$$X_{k}$$X_{k}= \{f=\sum_{n=1}^{\infty}(a_{n}\sin n\sigma+b_{n}\cos n\sigma)|$$\sum_{n=1}^{\infty}n^{2k}(|a_{n}|^{2}+|b_{n}|^{2})<\infty\}$or we can write $X_{k}=H^{k}(S^{1})/R$ . Sobolev’s imbedding theorem yields that$C^{k-1}(S^{1})$ for $k=1,2,$. We then define a nonlinear operator $F:X_{2}arrow X_{0}$ by$X_{k}\subset$$\cdots$.

$F(p, q; \theta)=e^{2H\theta}\frac{dH\theta}{d\sigma}-pe^{-H\theta}\sin(\theta)+q\frac{d}{d\sigma}(e^{H\theta}\frac{d\theta}{d\sigma})$

is a solution
Thus, our task is the determination of the zeros of $F$ . Clearly,
to $F=0$ for all the values of the parameters and . Our aim is, therefore, tocorresponds to $(u, v)=(c, 0)$ andfind non-zero solutions. We remark thatthe corresponding free boundary is flat. Using this $F,$ $[13]$ gave a simple proof of.
existence of non-zero solutions which bifurcate fromisreproduce
paper
present
In theour aim to
the numerical bifurcation diagramsin [18] in an abstract way. Therefore we reformulate our equation so that we
can compare the results here and those in [18]. We divide $F$ by and define
parameters and by $\mu=1/p$ and $\kappa=q/p$ . We then redefine $F$ as follows.
$\theta\equiv 0$

$p$

$q$

$\theta\equiv 0$$\theta\equiv 0$$p$$\kappa$

(2.4)

$\mu$

$F( \mu, \kappa;\theta)=\mu e^{2H\theta}\frac{dH\theta}{d\sigma}-e^{-H\theta}\sin(\theta)+\kappa\frac{d}{d\sigma}(e^{H\theta}\frac{d\theta}{d\sigma})$.

The following (A-C) are easy to prove:(A). Altho ugh $F$ is physically meaningful only for $0\leq\kappa<\infty,$ $0<\mu<\infty$ th
into
and satisfies
mapping $F$ is a well-defined smooth mapping from$e$$R^{2}\cross X_{2}$$F(\mu, \kappa;0)\equiv 0$$X_{0}$.

(B). Its Fr\’echet derivati ve at $\theta=0$ is given by$F_{\theta}( \mu, \kappa;0)w=\mu\frac{dHw}{d\sigma}-w+\kappa\frac{d^{2}w}{d\sigma^{2}}$$(w\in X_{2})$.

This formula yields(2.5)$F_{\theta}(\mu, \kappa;0)(\sin n\sigma)=(n\mu-1-n^{2}\kappa)\sin n\sigma$$(n\in \mathbb{N})$.

and a similax formula with $sin$ replaced by $cos$ .
(C). Therefore,(2.6)$F_{\theta}(\mu, \kappa;0)$

has a nontrivial null space if and only if$n\mu=1+n^{2}\kappa$

4$(\mu, \kappa)$

satisfies

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for some positive integer .
$n$Since the proof of these facts are easy we do not give them here (see [13],where the same result is proved in the context of the original $F$ using $(p, q))$ . We,
however, note that $H^{1}(S^{1})$ and $H^{2}(S^{1})$ are Banach algebras, which leads to (A).
which satisfies
By arguing just in the same way as in [13], we see that
(2.6) is a bifurcation point. Namely there is a nontrivial solution to $F=0$ in any. It is, however, very important to noticeinneighborhood ofthat some of the bifurcation points differ from others by the dimension of thenull space. In fact, somesatisfies (2.6) for two distinct $n’ s$ . For a
is a 4-dimensional
satisfying (2.6) for and $m(n\neq m)$ , the kernel of,,and. We call such a point a
space spanned bysimplebifurcation point. Thus a double
double bifurcation point and the other abifurcation point is characterized by$(\mu_{0}, \kappa_{0}; 0)$$(\mu_{0}, \kappa_{0},0)$$R^{2}\cross X_{2}$$(\mu, \kappa)$$(\mu, \kappa)$$F_{\theta}(\mu, \kappa;0)$$n$$\sin m\sigma$$\cos m\sigma$$m\mu=1+m^{2}\kappa$$\sin n\sigma$,$\cos n\sigma$$n\mu=1+n^{2}\kappa$,

where $m$ and are distinct positive integers. We call the pair of positive integers$m$ anda mode. We define
$n$

$n$

(2.7)

$\mu_{0}(m, n)=\frac{m+n}{nm}$(2.8)$\kappa_{0}(m, n)=\frac{1}{mn}$,.

Consequently the bifurcation points of mode $(m, n)$ is characterized as$(\mu_{0}(m, n),$ $\kappa_{0}(m, n);0)$ . In a neighborhood of the simple bifurcation point, we canonly prove the existence of waves whose profile has troughs and crests withequal height and depth (Fig. 1). On the other hand, any neighborhood of the
double bifurcation point has solutions in which the wave profiles are of mixed
nature (Fig. 2). This fact is essentially known early in this century (Wilton [21]. However, the global structure of the solution set has not been well understood
until the studies by [1,2,17,18].
$(\mu, \kappa;\theta)=$

$n$

$n$

$)$

Fig. 2

Fig. 1

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213

The result in this paper is summarized as follows:
The paper [18] computed zeros of $F$ numerically and found a number of new
bifurcation diagrams. In this paper, we consider the Lyapunov-Schmidt reduction
for $F=0$ and obtain a bifurcation equation. We then simplify the equation bychange of variables to obtain a normal form which turns out to be polynomials
see $(4\cdot 12,1S)$ and $(5.1S, 14))$ . We draw the figures of zeros of the normal form,changing two p\‘arameters called unfolding parameters. We then verify that thesefigures agree with the numeri $cal$ diagrams in [18].

$($\S 3. O(2)-equivariance.

In this section, we prove that $F$ satisfies a certain property called $O(2)-$to be of a special simple form
equivariance and that this property forcesas(3.5-8) below). We first define an action of the orthogonal group $O(2)$ on$\alpha\in[0,2\pi$) andfollows: let us recall that $O(2)$ is generated by rotations of anglea reflection. Accordingly,
$f$

$($

$X_{0}$$\gamma_{\alpha}\theta(\sigma)=\theta(\sigma-\alpha)$$(0\leq\alpha<2\pi)$$\gamma-\theta(\sigma)=-\theta(-\sigma)$

defines an action of $O(2)$ on
senting the rotation of angle, whererepresents the element of $O(2)$ repreand -the reflection. Then we have$X_{0}$$\alpha$$\gamma_{\alpha}$$\gamma$

PROPOSITION 3.1. The $m$ apping $F:X_{2}arrow X_{0}$ is $O(2)- eq$ uivariant, by which wemean$( \gamma\in O(2) )$ .
$F(\mu, \kappa;\gamma\theta)=\gamma F(\mu, \kappa;\theta)$

The proof is given in [13], though it is straightforward.
Proposition 3.1 enables us to simplify the bifurcation equation. We make a
be a double bifurcation point ofbifurcation equation as follows: Let
onto themode $(m, n)$ $(0<m<n)$ . Let denote the -projection from. Then,andfour dimensional subspace spanned by
the equation$(\mu_{0}, \kappa_{0}; 0)$$P$$L^{2}$$L^{2}(S^{1})$$\sin m\sigma,$ $\cos m\sigma,$ $\sin n\sigma$(3.1)$(I-P)F(\mu_{0}+\mu_{1},$ $\kappa_{0}+\kappa_{1}$$\cos n\sigma$; $x\sin m\sigma+y\cos m\sigma$$+z\sin n\sigma+w\cos n\sigma+\phi(\mu_{1}, \kappa_{1}; x, y, z, w))=0$

uniquely defines a $(I-P)X_{2}$ -valued mappingby$(0,0;0,0,0,0)$ . We define$\phi$.

in some open set containing

$G$

(3.2)

$G(\mu_{1}, \kappa_{1}; x, y, z, w)=PF(*)$

where the arguments of $F,$
bifurcation equation.

$*$,is the same as in (3.1). This mapping
6

$G$

is a

214

Remark. From now on, we write as $f$ : $Rarrow R$ , even when the defining domainof $f$ is some small open set of R. For instance, we consider mapping germs at theorigin, although we write as if it were defined in the whole space.
By Proposition 3.1 and the fact that the bifurcation equation inherits the groupequivariance from the basic differential equation ([7]), we see that too has an
O(2)-equivariance. To represent this more conveniently, we identify $(x, y, z, w)\in$with
in the way that $\xi=x+iy$ , $\zeta=z+iw$ . Therefore, we can. Similarly, we can
as a mapping on (some open subset of)regardbe the componentwise expression
regard that takes its value in . Letof in . We now have$G$

$R^{4}$$(\xi, \zeta)\in C^{2}$$R^{2}\cross C^{2}$$G$$C^{2}$$G$PROPOSITION 3.2. The $m$ apping
the following (3.3,4) hold $true$ .
(3.3)$(G_{1}, G_{2})$$C^{2}$$G$$G$

above is $O(2)$ -equivariant in the sense that

$G(\mu_{1}, \kappa_{1} ; e^{im\alpha}\xi, e^{in\alpha}\zeta)=$$(e^{im\alpha}G_{1}(\mu_{1}, \kappa_{1}; \xi, \zeta), e^{in\alpha}G_{2}(\mu_{1}, \kappa_{1} ; \xi, \zeta))$

(3.4)

,

$(\alpha\in[0,2\pi))$$G(\mu_{1}, \kappa_{1} ; \overline{\xi},\overline{\zeta})=(\overline{G_{1}(\mu_{1},\kappa_{1};\xi,\zeta)},\overline{G_{2}(\mu_{1},\kappa_{1};\xi,\zeta)})$For the Proof, see [13]. Proposition 3.2 forces the mapping
form. Let us prepare some symbols.

Definition. Let
O(2)-invariant if

$k$

$G$

.

to be of a special

be a positive integer. We call a function $f$ :$f(a;e^{im\alpha}\xi, e^{in\alpha}\zeta)\equiv f(a;\xi, \zeta)$$R^{k}\cross C^{2}arrow \mathbb{R}$$(\alpha\in[0,2\pi))$and$f(a;\overline{\xi},\overline{\zeta})\equiv f(a;\xi, \zeta)$.
are satisfied. Here $a\in R^{k},$The set of all germs (at the origin) of O(2)-invariant-functions is a commutative ring with a unit. Let denote this ring. The set of all the mapping$\xi,$$\zeta\in C$$c\infty$$\mathcal{E}$:satisfying (3.3,4) constitutes an -module. Let $E$ denote this-module. In order to give a simple expression to and $E$ , we need to introduce
two positive integers
and $m’$ . We define them to be coprime positive integers
satisfying $n/m=n’/m’$ . We now have$G$$\mathbb{R}^{2}\cross C^{2}arrow C^{2}$$\mathcal{E}$$\mathcal{E}$$\mathcal{E}$$n’$

PROPOSITION 3.3. Any elemen$tf\in \mathcal{E}$is of the following form

$f(a;\xi, \zeta)=g(a;u, v, s)$

where

$g$

is a

$c\infty$

fun ction of $k+3$ variables and $u,$
$u=|\xi|^{2}$,$v=|\zeta|^{2}$,$s=Re[\overline{\xi}^{n’}\zeta^{m’}]$

PROPOSITION 3.4. The module $E$ is generated overments:$X_{1}=(\xi, 0)$,$X_{2}=(0,\overline{\xi}^{n’-1}\zeta^{m’})$,are defined by$v,$ $s$$\mathcal{E}$$X_{3}=(0, \zeta)$,.

by the following four ele$X_{4}=(0, \xi^{n’}\overline{\zeta}^{m’-1})$

The proofs of these Propositions 3.3,4 can be found in [11]( see also [4,8]).

7

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215

COROLLARY 3.1. The mapping
following form

$G$

$G_{1}=f_{1}\xi+f_{2}\overline{\xi}\zeta$$f_{j}$

are of the following form.

$(j=1,2,3,4)$$f_{j}=f_{j}(\mu_{1}, \kappa_{1}; |\xi|^{2}, |\zeta|^{2}, Re[\overline{\xi}^{2}\zeta])$

COROLLARY 3.2. The $m$ apping
following form$G$at the bifurcation point of mode $(1, 3)$ is of the(3.7)$G_{1}=g_{1}\xi+g_{2}\overline{\xi}^{2}\zeta$(3.8)$G_{2}=g_{3}\zeta+g_{4}\xi^{3}$

where

$e$

,
$G_{2}=f_{3}\zeta+f_{4}\xi^{2}$ ,

(3.5)
(3.6)

where

at the bifurcation point of mode $(1, 2)$ is of th$g_{j}$,,

are of the following form
$g_{j}=g_{j}(\mu_{1}, \kappa_{1} ; |\xi|^{2}, |\zeta|^{2}, Re[\overline{\xi}^{3}\zeta])$.

$(j=1,2,3,4)$\S 4. Bifurcation equation of mode $(1,2)$ .

In this section we give a normal form of the bifurcation equation of mode $(1,2)$ .
We show that a theorem in [11] is applicable to the problem now in question.
:
is now written as (3.5,6). We
The bifurcation equation[6,7,8]germs
theoryin
in
which
use themapping
containing one parameter are
considered. Our mapping, however, has two parameters. In [18], is used asa bifurcation parameter and a splitting parameter. We thereby freeze . We
. We then regard as a mapping germ of
introduce A and consider. We can now write as$G$$\mathbb{R}^{2}\cross C^{2}arrow C^{2}$$\mu$$\kappa$$\kappa_{1}$$G$$G(\lambda, 0;\xi, \zeta)$$(\lambda, \xi, \zeta)$,$G_{2}=f_{3}(\lambda;u, v, s)\zeta+f_{4}(\lambda;u, v, s)\xi^{2}$ .

(4.1)$G_{1}=f_{1}(\lambda;u, v, s)\xi+f_{2}(\lambda;u, v, s)\overline{\xi}\zeta$(4.2)

If we have shown that this mapping is finitely determined and if we have computed
can be realized
universal unfoldings, then the equation for small but nonzero
by one of the unfolded mappings ([6,7,8]). Thus we are lead to the analysis of(4.1,2).
is a bifurcation equation, all the derivatives of first order vanishes atSince
the origin. Accordingly $f_{j}(0;0,0,0)=0$ $(j=1,3)$ . In order to go further, weneed to compute $f_{2}(0;0,0,0)$ and $f_{4}(0;0,0,0)$ . This is done in [13]. The result is,in the present notation,$\kappa_{1}$$G$$f_{2}(0;0,0,0)=-3/4$ ,$f_{4}(0;0,0,0)=-3$ .

8

216

See also Proposition 4.1 below.
In this situation we can use Theorem 3.2 and 3.3 of [11]. According to thistheorem, the qualitative behaviour of the set of the solutions is described by

,

(4.3)

$(\delta\lambda+\hat{b}v)\zeta+\xi^{2}=0$$(\epsilon\lambda+\alpha+bv)\xi+\overline{\xi}\zeta=0$,

whererare real constants.
We now turn to the original problem of the progressive water waves. As in[18], we assume that the wave profile is symmetric with respect to the y-axis.
This assumption is equivalent to the oddness of in ([13]). The oddness ofis written by. Consequently we. This is interpreted as$G(\lambda,\xi,\zeta)=0$. We therefore put $\xi=x$ , $\zeta=z$ in
consider only real solutions to(4.3). Thus we have to solve$\epsilon,$$\alpha,$$\delta,$$b,\hat{b}$$\theta$$\sigma$$(\overline{\xi},\overline{\zeta})=(\xi, \zeta)$$\gamma-\theta=\theta$$\theta$(4.4)$(\epsilon\lambda+\alpha+bz^{2})x+xz=0$

,

(4.5)

$(\delta\lambda+\hat{b}z^{2})z+x^{2}$

$=0$

.

We now see the numerical bifurcation diagrams. Fig. 3-7 are borrowed from[i8]. Each of Fig. 3-7 has different and is drawn in the frame of $(\mu, A_{1}, A_{2})$ . In
each figure, the picture below shows the diagram viewed in the direction of -axis.
decreases as the figure number increases. The variables and in this paper
correspond toandin [18], respectively. In these diagrams, there are twokinds of curves: solid curves and broken curves. The both represent zeros of $F=0$ .
Th difference is that the solutions on the broken curves have self-intersections in
the wave profile like:$\kappa$$A_{1}$$x$$\kappa$$A_{1}$$z$$A_{2}$$e$On the other hand, those on the solid curves do not have any self-intersection.
So, they look like Fig. 2. At the boundary of solid curve and broken curve, the9217wave has contact points, which $re$ sembles Crapper’s pure capillary wave ([3]).
Fig. 2 corresponds to the point in Fig.4. References [1,2,17] reached this point.
Before [18], however, both physicists and mathematicians neglected the solutions
to $F=0$ which have self-intersections simply because they represent unphysicalstates. However, [18] revealed that there are some limit points (turning points)
in the physically meaningful regions of the diagrams (see Fig. 5) while they do
not appear for relatively large (see Fig. 3,4). It indicates that total analysisof both physical and unphysical solutions is necessary for the explanation of theexistence or nonexistence of limit points. The necessity becomes clearer when we
consider waves of $mode(1,3)$ in the next section.
By a simple computation, we find that (4.4,5) yields a bifurcation diagram listed
in [4]. Fig. 8 is borrowed from [4]. In this figure, and correspond to andin our notation, respectively. The bifurcation parameter is taken vertically.
We notice that these diagrams are only a subset of those in [18]. Therefore, thecan reproduce solutions only a small portion of the numericalnormal form
solu tions in [18].
In order to capture wider class of solutions, we now use an idea in [4- 7]: Suppose
we are given a problem to find zeros of $F(a_{1}, \cdots a_{k};u)=0$ with parameters. If
there exists a set of solutions which is described by a normal form of codimension, then the problem requires more than parameters.
Following this principle, we assume as follows:$a$$\kappa$

$\xi$

$x$

$\eta$$z$$(4\cdot 4,5)$

$k$

$k$

$\geq k$and , there is a parameter, say, , and a smooth mapping
In addition to. In other words, our problem issuch thatembedded in an extended system.
$\kappa$

$\nu$

$\mu$

$\hat{F}(\kappa, \mu, 0;u)\equiv F(\kappa, \mu;u)$$\hat{F}(\kappa, \mu, \nu;u)$

We do not know what is the additional parameter. Rather than identifying theparameter, we continue our abstract analysis under this assumption. We can thenregard $f_{2}(0;0,0,0)$ and $f_{4}(0;0,0,0)$ depend on . We finally assume that:$f_{2}(0;0,0,0)$ vanishes at someand that $f_{4}(0;0,0,0)$ does not at $v=v_{0}$ .
by in some neighborhood
Under this assumption, we are allowed to divideof $(\kappa_{1}, \mu_{1}, \nu ; u)=(0,0, \nu_{0} ; 0)$ ([11]). Then it holds that$\nu$$\nu=\nu_{0}$$G_{2}$(4.6)$G_{1}=(\epsilon\lambda+au+bv+cs+f_{5})\xi+(\eta\lambda+d_{1}u+d_{2}v+d_{3}s+f_{6})\overline{\xi}\zeta$

(4.7)

where
order

$f_{4}$,$G_{2}=(\delta\lambda+\hat{a}u+\hat{b}v+\hat{c}s+f_{7})\zeta+\xi^{2}$$(j=1,2,3),a,$are real constants and. We are now in a position to quote some results in [11]:$\delta,$$\epsilon,$$\geq 2$$\eta,$$d_{j}$$b,c,\hat{a},\hat{b},\hat{c}$$f_{j}$

THEOREM 4.2. Under some nondegeneracy conditions on
mapping in (4.6,7) is $O(2)$ -equivalent to$\delta,$$\epsilon,$

$G$

(4.8)

$G_{1}=(\epsilon\lambda+a^{/}u+b’v+c’s)\xi+d’v\overline{\xi}\zeta$(4.9)$G_{2}=(\delta\lambda+\hat{a}^{/}u+\hat{b}’v)\zeta+\xi^{2}$

10

,

,

,

$a,$

are terms of$b,\hat{a},\hat{b}$and$d_{2}$

, the

218
and ‘ are real constants. A universal $O(2)$ -unfolding of thewhere $a’,$ $c’,$mapping (4.8,9) is given by$b’,$(4.10)$d’,\hat{a}’$$F_{1}=[\epsilon\lambda+\alpha+(a’+\tau_{1})u+(b’+\tau_{2})v+(c’+\tau_{3})s]\xi+[\beta+(d’+\tau_{4})v]\overline{\xi}\zeta$(4.11)

where$\hat{b}$$F_{2}=[\delta\lambda+\hat{a}’u+\hat{b}’v]\zeta+\xi^{2}$$\alpha,$$\beta,$$\tau_{1},$$\cdots\tau_{4}$,,are unfolding parameters.

This theorem enables us to neglect the higher order terms and some of thefirst order terms in $(u, v, s)$ without losing any information on the bifurcation
diagrams. As for the precise statement of the nondegeneracy assumption, see[11]. This assumption includes, etc. We assume that these holds.
Since the assumption is generic in the sense that it is expressed as “.. $.”\neq 0$ , ourassumption is a reasonable on .
We now consider real solutions, whence we have$\epsilon\delta\neq 0,\hat{b}\neq 0$$e$(4.12)$(\epsilon\lambda+\alpha+ax^{2}+bz^{2}+cx^{2}z)x+(\beta+dz^{2})xz=0$,(4.13)$(\delta\lambda+\hat{a}x^{2}+\hat{b}z^{2})z+x^{2}$

$=0$

.

In order for (4.12,13) to be a normal form of the bifurcation equations, we must
reproduce Fig. 3-7 by choosing parameters appropriately. We now show this ispossible. Fig. 9-13 are the set of zeros of (4.12, 13) for suitable parameters. These
are drawn by a personal computer and a lazer beam printer. We see that Fig 9-13
completely reproduce Fig. 3-7.

Remark. By the same idea, Fujii et al. [5] finds bifurcation diagrams similar to
Fig. 9-13 in the problem of reaction diffusion equations. They do not, however,obtain the normal form (4.12,13).
We finally remark that these figures do not automatically prove that (4.12,13)
has zeros like Fig. 3-7. In fact, we are dealing with a mapping germ defined by(4.12, 13). Therefore we must prove that figures like Fig. 3-7 can be drawn in any
small neighborhood of the origin by choosing appropriately small and . Thisis, however, checked by the method described in [12].
Although we believe that there is a parameter which makes our assumption
valid, we do not know the identity of . This is a $very\cdot week$ point of our theory.
We first thought that the aspect ratio serves as the parameter. The aspect ratio
is a parameter which represent the ratio of the wave length and the depth of theflow. This means that we consider a $ge$ neralized problem in which $H$ in (2.3) isreplaced by$\alpha$$\beta$

$\nu$

$\nu$

$H( \sum_{n=1}^{\infty}(a_{n}\sin n\sigma+b_{n}\cos n\sigma))=\sum_{n=1}^{\infty}\frac{1+r^{n}}{1-r^{n}}(-a_{n}\cos n\sigma+b_{n}\sin n\sigma)$

11

219

(see [13]). Here $r\in[0,1$ ) is a new parameter. As $rarrow 0$ , the depth becomesdeeper. When $r=0$ this generalized problem reduces to the original problemof flows of infinite depth. Although only $0\leq r<1$ is physically allowed, theformulation itself has a well-defined meaning for all $r\in(-1,1)$ . Therefore we
expect $ed$ the existence of some at which $f_{2}(0;0,0,0)$ vanish. The calculation,
however, showed:$r$

PROPOSITION 4.1. It holds that$f_{2}(0;0,0,0)=- \frac{3(1+r)}{4(1-r)}$,$f_{4}(0;0,0,0)=- \frac{3(1+r)}{1-r}$.

PROOF: We note that (2.6) is replaced by$n \frac{1+r^{n}}{1-r^{n}}\mu=1+n^{2}\kappa$.

Similarly, (2.7,8) are replaced by(4.14)$\kappa_{0}=\frac{(1-r)^{2}}{2+8r+2r^{2}}$$\mu_{0}=\frac{3(1-r^{2})}{2+8r+2r^{2}}$,since $m=1$ and $n=2$ . We note that$f_{2}(0;0,0,0)= \frac{\partial^{2}G_{1}}{\partial x\partial z}(0;0,0)$.

It holds that,

(4.15)$\frac{\partial G}{\partial x}(\lambda;\xi, \zeta)=PF_{\theta}(\mu_{0}+\lambda, \kappa_{0}, r;\#)(\sin\sigma+\phi_{x}.)$where $F$ now depends upon as well as and and denotes $x\sin\sigma+y\cos\sigma+$$z\sin 2\sigma+w\cos 2\sigma+\phi$ . Differentiating (4.15) in, we obtain
$r$

$\mu$

$\#$

$\kappa$$z$($\frac{\partial^{2}G}{\partial x\partial z}(0;0,0)=PF_{\theta}(\mu_{0}, \kappa_{0}, r;0)(\phi_{xz}^{0})+PF_{\theta\theta}$$\mu_{0},$$\kappa_{0},$$r$; O)(sin$\sigma,$$\sin 2\sigma$).

commutes with $P$ by (2.5),
Since is $(I-P)X_{2}$ -valued and since
the first term of the right hand side vanishes. Hence $f_{2}(0;0,0)$ is the coefficient of; O)(sin). We now compute the second order Fr\’echet(ofderivative of $F$ . We have
$F_{\theta}(\kappa_{0}, \mu_{0}, r;0)$$\phi$$\sin\sigma$$F_{\theta\theta}$$\mu_{0},$$\kappa_{0},$$r$$\sigma,$$\sin 2\sigma$$F_{\theta\theta}( \mu_{0}, \kappa_{0}, r;0)(f, g)=2\mu_{0}\frac{d}{d\sigma}(HfHg)+fHg+gHf+\kappa_{0}\frac{d}{d\sigma}(Hf\frac{dg}{d\sigma}+Hg\frac{df}{d\sigma})$for all $f,$ $g\in X_{2}$ . This $formu$] a yields$F_{\theta\theta}$($\mu_{0},$$\kappa_{0},$$r$; O)(sin$\sigma,$$\sin 2\sigma$)$=2 \mu_{0}\frac{1+r}{1-r}\frac{1+r^{2}}{1-r^{2}}\frac{d}{d\sigma}(\cos\sigma\cos 2\sigma)$$-( \frac{1+r^{2}}{1-r^{2}}\cos 2\sigma\sin\sigma+\frac{1+r}{1-r}\cos\sigma\sin 2\sigma))$$+ \kappa_{0}\frac{d}{d\sigma}(-\frac{1+r}{1-r}\cos\sigma\cdot 2\cos 2\sigma-\frac{1+r^{2}}{1-r^{2}}cos2\sigma\cos\sigma)$

12

.

220
Consequently we obtain$PF_{\theta\theta}$($\mu_{0},$$\kappa_{0},$$r$; O)(sin$\sigma,$$\sin 2\sigma$)$= \mu_{0}\frac{1+r}{1-r}\frac{1+r^{2}}{1-r^{2}}(-\sin\sigma)$$+( \frac{1}{2}\frac{1+r^{2}}{1-r^{2}}-\frac{1}{2}\frac{1+r}{1-r})\sin\sigma+\kappa_{0}(\frac{1+r}{1-r}+\frac{1}{2}\frac{1+r^{2}}{1-r^{2}})\sin\sigma$.

By this equality and (4.14) we have$f_{2}(0;0,0,0)=- \frac{3(1+r)}{4(1-r)}$.

In a similar way we obtain$f_{4}(0;0,0,0)=PF_{\theta\theta}$($\mu_{0},$$\kappa_{0},$$r$; O)(sin$\sigma,$$\sin\sigma$) $=- \frac{3(1+r)}{1-r}$ .

This completes the proof. 1
We are puzzled by this computation: we must look for another parameter.
Although there is a possibility that the parameter may be a one repres nting
vorticity, a one representing the viscosity or a density ratio in the two phaseproblem, we do not have any definite idea for the moment.
$e$\S 5. Bifurcation equation of mode $(1,3)$ .
In this section we consider (3.7,8) and introduce a certain degeneracy whichmakes (3.7,8) yield bifurcation diagrams in [18].
First of $aU$ , we assume that(H1)

$g_{2}(0;0,0,0)\neq 0$,$g_{4}(0;0,0,0)\neq 0$.

We divide (3.7) by , and (3.8) by. This operation preserves $O(2)-$equivariance (see [11]) and give rise to the following O(2)-equiva1ent mapping:$-g_{4}$$g_{2}$(5.1)$G_{1}=(\epsilon\lambda+au+bv+cs+g_{5})\xi+\overline{\xi}^{2}\zeta$(5.2)$G_{2}=(\delta\lambda+\hat{a}u+\hat{b}v+\hat{c}s+g_{6})\zeta-\xi^{3}$

,

,

where
are real constants andandare of order. Before doing
further simplification of the mapping we prepare some symbols for the notationalbe the set of allconvenience. Let
satisfying $g(0;0,0,0)=0$ . The symbol
implies the ideal of generated by. Using this notation,
we can say that. Let
be the set of all the-germs of function
$0\in
R$
at
. We designate by
the set of all
satisfying $\phi(0)=0$ . Notethat the elements of
is a germ of a function of only. The ideal in generatedbyis denoted by {.
$\delta,$$\epsilon,$$a,$$\cdots\hat{c}$$g_{5}$$\mathcal{M}$$g\in \mathcal{E}$$\mathcal{M}^{2}$$\{fg;f\in \mathcal{M},g\in \mathcal{M}\}$$\mathcal{E}$$g_{5},$$g_{6}\in \mathcal{M}^{2}$$c\infty$$\mathcal{E}_{\lambda}$$\phi(\lambda)$$\phi\in \mathcal{E}_{\lambda}$$\mathcal{M}_{\lambda}$$\lambda$$\mathcal{E}_{\lambda}$$f_{1},$$\cdots f_{r}$$\geq 2$$g_{6}$$f_{1},$$\cdots f_{r}\rangle$

13$\mathcal{E}$22\’iLet us writ as$e$,, where. We now perform the following change of variables:$g_{5}=\lambda^{2}\eta_{1}(\lambda)+g_{7}$$g_{7},g_{8}\in \mathcal{M}^{2}\cap\{u, v, s\}$$g_{6}=\lambda^{2}\eta_{2}(\lambda)+g_{8}$$(\xi, \zeta)arrow((1+\phi(\lambda))\xi,$$\eta_{1},$$\eta_{2}\in \mathcal{E}_{\lambda}$

and,$\zeta)$where
is to be determined later. This change preserves the $O(2)-$equivariance (see [11]). Applying this change and dividing (5.1) by $(1+\phi)^{2}$, (5.2) by $(1+\phi)^{3}$ , we obtain$\phi\in \mathcal{M}_{\lambda}$$G_{1}’= \frac{1}{1+\phi}(\epsilon\lambda+\lambda^{2}\eta_{1}+au+bv+cs+g_{9})\xi+\overline{\xi}^{2}\zeta$,$G_{2}’= \frac{1}{(1+\phi)^{3}}(\delta\lambda+\lambda^{2}\eta_{2}+\hat{a}u+\hat{b}v+\hat{c}s+g_{10})\zeta-\xi^{3}$where$g_{9},$$g_{10}\in \mathcal{M}^{2}\cap\{u, v, s\}$. We choose$\phi$,

such that$\frac{\lambda+\epsilon^{-1}\lambda^{2}\eta_{1}(\lambda)}{1+\phi(\lambda)}=\frac{\lambda+\delta^{-1}\lambda^{2}\eta_{2}(\lambda)}{(1+\phi(\lambda))^{3}}$Namely we define$\phi$

by.

$\phi(\lambda)=\sqrt{\frac{1+\delta^{-1}\lambda\eta_{2}}{1+\epsilon^{-1}\lambda\eta 1}}-1$

We then write
asassume that fas the following form:$(\lambda+\epsilon^{-1}\lambda^{2}\eta_{1})/(1+\phi)$$\lambda$. After these computations, we may$G$$G_{1}=(\epsilon\lambda+au+bv+cs+g_{12})\xi+\overline{\xi}^{2}\zeta$$G_{2}=(\delta\lambda+\hat{a}u+\hat{b}v+\hat{c}s+g_{12})\zeta-\xi^{3}$

,

,

where
.
We next consider the following change of variables, which also preserves $O(2)-$equivariance:$g_{11},g_{12}\in \mathcal{M}^{2}\cap\{u,$ $v,$

$s\rangle$$(\xi, \zeta)arrow(\xi+\alpha\overline{\xi}^{2}\zeta,$

where$\alpha$and$\beta$$\zeta+\beta\xi^{3})$are real constants. By this change,$G$,is transformed to(5.3)$G_{1}’=(\epsilon\lambda+au+bv+(c+2a\alpha+2b\beta)s+g_{13})\xi+(1+\mu_{1})\overline{\xi}^{2}\zeta$(5.4)$G_{2}’=(\delta\lambda+\hat{a}u+\hat{b}v+(\hat{c}+2\hat{a}\alpha+2\hat{b}\beta)s+g_{15})\zeta-(1+\mu_{2})\xi^{3}$

where
(H2)

,

$g_{13},$ $g_{15}\in \mathcal{M}^{2}\cap\{u, v, s\}$and$\mu_{1},$$\mu_{2}\in\langle u,$$a\hat{b}-\hat{a}b\neq 0$

14,$v,$ $s$

}. If

,

222
then we can make the coefFicient of vanish in (5.3,4). Accordingly we obtainafter dividing (5.3) by $1+\mu_{1}$ and (5.4) by $1+\mu_{2}$ ,
$s$

(5.5)

$G_{1}’’=(\epsilon\lambda+au+bv+g_{15})\xi+\overline{\xi}^{2}\zeta$(5.6)$G_{2}’’=(\delta\lambda+\hat{a}u+\hat{b}v+g_{16})\zeta-\xi^{3}$

wherechange of variables:$g_{16}\in \mathcal{M}^{2}\cap\{u,$$g_{15},$$v,$$p_{1}$$\ell_{2}\zeta)$,

and $1+\beta_{1}u+\beta_{2}v+\beta_{3}s$ , respectively,. Applying this transformation and dividing (5.5) by$1+\alpha_{1}u+\alpha_{2}v+\alpha_{3}s$$p_{2}$$\alpha_{j}$$l_{1}^{2}l_{2},$,. Our next simplification is obtained by the following$s\rangle$$(\xi, \zeta)arrow(l_{1}\xi,$

where and denotewith real constants
and$(5.6)$ by, we have,$\beta_{j}$$l_{1}^{3}$$H_{1}= \frac{1}{l_{1}^{2}\ell_{2}}(\epsilon\lambda+au+bv+g_{17})\xi+\overline{\xi}^{2}\zeta$$H_{2}= \frac{p_{2}}{\ell_{1}^{3}}(\delta\lambda+\hat{a}u+\hat{b}v+g_{18})\zeta-\xi^{3}$

,

,

where
}.
We wish to eliminate some of the second order terms. Let$g_{17},$$g_{18}\in \mathcal{M}^{2}\cap\langle u,$$v,$ $s$$g_{17}=A_{1}\lambda u+A_{2}\lambda v+A_{3}\lambda s+g_{19}$where$\lambda u,$$\lambda v,$$g_{19}$$\lambda s$,$g_{18}=B_{1}\lambda u+B_{2}\lambda v+B_{3}\lambda s+g_{20}$andbelong toare, respectively,inand$\langle u, v, s\rangle^{2}+\mathcal{M}^{3}\cap\langle u,$$g_{20}$$v,$ $s$,}. Then the coefficients of$G_{2}$$G_{1}$$A_{1}-\epsilon(\alpha_{1}+\beta_{1})$,$A_{2}-\epsilon(\alpha_{2}+\beta_{2})$,$A_{3}-\epsilon(\alpha_{3}+\beta_{3})$and$B_{1}-\delta(3\alpha_{1}-\beta_{1})$Choosing
possible if$\alpha_{j}$

and

(H3)

$\beta_{j}$,$B_{2}-\delta(3\alpha_{2}-\beta_{2})$,$B_{3}-\delta(3\alpha_{3}-\beta_{3})$.

appropriately, we can make these coefficients vanish. This is$\epsilon\neq 0$,

Thus we have proved that the original
lowing new :$\delta\neq 0$$G$.

in (5.1,2) is O(2)-equiva1ent to the fol-

$G$

(5.7)

$G_{1}=(\epsilon\lambda+au+bv+c_{1}u^{2}+c_{2}uv+c_{3}v^{2}+g_{21})\xi+\overline{\xi}^{2}\zeta$(5.8)$G_{2}=(\delta\lambda+\hat{a}u+\hat{b}v+c_{4}u^{2}+c_{5}uv+c_{6}v^{2}+g_{22})\zeta-\xi^{3}$

15

,

,

$+\{us, vs, s^{2}\}$ . We wish to prove that (5.7,8) is $O(2)-$where. If this is possible, we have a normal form
equivalent to itself within some sense and our task is reduced to consider$g_{21},$$g_{22}\in \mathcal{M}^{3}\cap\{u,$$v,$$s\rangle$$g_{21}\equiv 0,$ $g_{22}\equiv 0$(5.9)$G_{1}=(\epsilon\lambda+au+bv+c_{1}u^{2}+c_{2}uv+c_{3}v^{2})\xi+\overline{\xi}^{2}\zeta$(5.10)$G_{2}=(\delta\lambda+\hat{a}u+\hat{b}v+c_{4}u^{2}+c_{5}uv+c_{6}v^{2})\zeta-\xi^{3}$,,

However, the results in [11] is limited to the case of mode $(1,2)$ and no ready-madeand. Accordingly
result seems to be available concerning the elimination of$O(2)$-equivalent to (5,9,10). We only show that
we give up proving that (5.7,8) is(5.9,10) explains the figures in [18] we 11.
. Setting $y=w=0$ , we have
We now assume that$g_{21}$$g_{22}$$\epsilon=\delta=1$(5.11)$G_{1}=(\lambda+ax^{2}+bz^{2}+c_{1}x^{4}+c_{2}x^{2}z^{2}+c_{3}z^{4})x+x^{2}z$(5.12)$G_{2}=(\lambda+\hat{a}x^{2}+\hat{b}z^{2}+c_{4}x^{4}+c_{5}x^{2}z^{2}+c_{6}z^{4})z-x^{3}$,.

Numerical Results for mode $(1,3)$ . Let us see the numerical bifurcation diagrams. Fig.14-19 are cited from [18]. Notation is the same as in the case of mode$(1,2)$ . As the figure number increases, the value ofincreases. Let us explain Fig.
17, for instance. The diagrams are viewed in three directions. The figure in theright above is a view in the direction of the -axis. The figure in the left below isa view in the $A_{3}direction$ . Wave profiles are drawn in the right below. The index, etc. correspond to the points marked in the figure in the left above. We canobserve a pair of closed loops, which are symmetric with respect to the -axis.
As the parameter increases, they intersect, are tangent to or separate from theprimary pitchfork branch. Fig. 19 shows that the loops shrink to points. Here
we observe a very important fact about the connectedness of the branch. Some ofthe physical solutions on the loops are connected to the mode 3 branch throughunphysical solutions (see Fig. 18). Thus the physical solutions on the loops are
disconnected from other physical solutions if we neglect the unphysical solutions.
In computing the branch, [18] used Keller’s path-tracing method as is common in
the bifurcation problem. Therefore, disconnected part may well be missed, when
you look for only physical solutions. If, however, we consider all the solutions,both physical and unphysical, then all the solutions ar traced.
We introduce two unfolding parameters and in (5.11, 12):$\kappa$

$\mu$

$a,$

$b$

$\lambda$$\kappa$$e$$\alpha$$\beta$(5.13)$G_{1}=(\lambda+\alpha+ax^{2}+bz^{2}+c_{1}x^{4}+c_{2}x^{2}z^{2}+c_{3}z^{4})x+x^{2}z$(5.14)$G_{2}=(\lambda+\hat{a}x^{2}+(b+\beta)z^{2}+c_{4}x^{4}+c_{5}x^{2}z^{2}+c_{6}z^{4})z-x^{3}$,.

Here we assumed that is close to and introduced a small . In other words, weand we unfolded the bifurcation equation
as sum $ed$ that there is a at which
there.
The equation (5.13) implies $x=0$ or$\hat{b}$

$b$

$\nu$

(5.15)$\beta$$b=\hat{b}$$\lambda=-\alpha-ax^{2}-bz^{2}-c_{1}x^{4}-c_{2}x^{2}z^{2}-c_{3}z^{4}-xz$

16

.

224
plane: $x=0$ ,The former relation and (5.14) give a pitchfork in$-(b+\beta)z^{3}-c_{6}z^{5}$ . We assume that $b>0$ and $c_{6}>0$ . Substituting (5.15) into
(5.14), we have$\lambda z=$$(\lambda, z)$(5.16))$-\alpha z+f\hat{a}-a$ $x^{2}z+\beta z^{3}-x^{3}-xz^{2}+d_{1}x^{4}z+d_{2}x^{2}z^{3}+d_{3}z^{5}$. $=0$ .

where we have put $d_{1}=c_{4}-c_{1},$ $d_{2}=c_{5}-c_{2},$ $d_{3}=c_{6}-c_{3}$ . This equation does notcontain .
The roots of (5.13, 14) constitute of either the pitchfork ( $x=0$ &$c_{6}z^{4}=0)$ or the curves given by (5.15, 16).
Let us check whether the normal form above can reproduce these figures. Choosing small parameters and appropriately, we can draw various diagrams. Among
those, Fig.20-23 suit our purpose. In these figures, we fixed the following parameters:$\lambda$$\lambda z+\hat{b}z^{3}+$$\beta$$\alpha$$a=5.25$ ,$\hat{a}=1.0$,$b=0.2$ ,$c_{1}=0$$c_{2}=0$$c_{3}=0.101$,$c_{4}=0$$c_{5}=0$$c_{6}=0.001$$\hat{b}=1.0$

The values of unfolding parameter in Fig.20 23 are, 1.0, 1.6, 1.8, 1.85, respectively. These abstract diagrams explains Fig.16-19 completely. Fig.14,15 are more
complex and we can not reproduce them by the normal form.
-$\alpha$Remark. As in \S 4, it is necessary to prove that Fig. 20-23 can be drawn in any
neighborhood of the degenerate bifurcation point. So far, we can not prove this.

Concluding Remark. We consider two pairs of algebraic equations (4.12,13) and(5.13,14). These “ normal forms “ reproduce quite a large class of the numericalsolutions in [18]. The success of our quations suggests that another appropriate
parameter exists and that the degeneracy which we assumed exists. Unfortunately
we can not specify the parameter. We showed that the aspect ratio parameter is
not the one.
$e$

Appendix.
In this Appendix, we give an outline of the derivation of (2.2). As Stokes did, we
regard as a function of rather than regarding as a function of . Following
Levi-Civita [10], we introduce independent variable
$f$

$z$

$f$

$z$

$\zeta=\exp(-\frac{2\pi if}{cL})$, weand dependent variable $\omega=i\log(c^{-1}df/dz)$ . By the relation
regard as a function of . Note that runs in $0<|z|<1$ and that is analytic
in . As
approaches zero. Thus the origin is a removable singularity andis an analytic function in the whole disk $|\zeta|<1$ . Let and denote, respectively,
the real and the imaginary part of . Let
be the polar coordinates for ,i.e.,. We now rewrite (2.1) by means ofand: On the free$\zetarightarrow frightarrow zrightarrow\omega$$\zeta$$\omega$$\zeta$$\zetaarrow 0,$$\zeta$$\omega$$\omega$$\theta$$\omega$$\tau$$(\rho, \sigma)$$\omega$$\zeta=\rho e^{\sigma}$$\zeta$$\theta(p, \sigma)$

17$\tau(\rho, \sigma)$225boundary, we have $V=0$ . Therefore it holds that
other hand, by$\frac{df}{dz}=ce^{-i\theta+\tau}$$dz/df= \frac{\partial x}{\partial U}+i_{\partial U^{L}}^{\partial}\lrcorner$. On the,we have$\sigma=\frac{-2\pi U}{cL}$$\frac{\partial x}{\partial U}=\frac{e^{-\tau}}{c}\cos\theta$,$\frac{\partial y}{\partial U}=\frac{e^{-\tau}}{c}\sin\theta$,$\frac{dh}{dx}(x)=\tan\theta$

and$| \frac{df}{dz}|^{2}=c^{2}e^{2\tau}$,$\frac{\partial}{\partial x}=\frac{\partial\sigma}{\partial x}\frac{\partial}{\partial\sigma}=-\frac{2\pi e^{\tau}}{L\cos\theta}\frac{\partial}{\partial\sigma}$

By these formula we easily get to(A.1)$e^{2\tau} \frac{\partial\tau}{\partial\sigma}-pe^{-\tau}\sin\theta+q\frac{\partial}{\partial\sigma}(e^{\tau}\frac{\partial\theta}{\partial\sigma})=0$

on$\rho=1$

,

.
where
We thus obtain the following reformulation:$p= \overline{2}\pi^{\frac{L}{c^{2}},q}E=\frac{2\pi T}{c^{2}Ld}$

Find a function
which is continuous on$\{|\zeta|<1\}$ and satisfy (A.1) and $\omega(0)=0$ .
$\omega=\omega(\zeta)$$\{|\zeta|\leq 1\}$, is analytic in

Levi-Civita considered the problem in this formulation when $q=0$ .
Since an analytic function is completely determined by its boundary value, afurther reduction of the equation (A.1) is possible. In fact we can write (A.1) only$(0\leq\sigma<2\pi)$ . This is done by utilizing (2.3). We have $\tau(1, \sigma)=H(\theta^{*})$ ,by. The equation (A.1) is now written as (2.2).
where$\theta(1, \sigma)$$\theta^{*}(\sigma)=\theta(1, \sigma)$

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Keywords. capillary-gravity wave, bifurcation from double eigenvalue with O(2)-symmetry

19

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23

232

H. Fujii et al.lA picture of the global bifurcation diagram$(I2-eI$$p_{11}>0,$ $q_{20}<0$$\sigma_{1}$$\sigma_{1}$$\sigma_{2}>0$$\sigma_{2}=0$$\sigma_{2}<0$$\eta$$\eta$

Fig. 8

233

Fig. 9Fig. 10

234

Fig. 11

Fig. 12

235

Fig. 13

236

Fig. 14$\ovalbox{\tt\small REJECT}^{\backslash }$Fig. 15

Fig. 16$\ovalbox{\tt\small REJECT}^{\ovalbox{\tt\small REJECT}^{\mathfrak{g}}}Y$3/

$(^{-}\lfloor$-#$c$Fig. 17

$9Z$

4

$*1$

Fig. 1833

241

Fig. 19$>\neq$Fig. 20

Fig. 21

Fig. 23

Fig. 22$3f^{-}$