Nonlinearly Resonant Surface Waves and Homoclinic Bifurcation

ADVANCES I N A P P L I E D MECHANICS, VOLUME

26

Nonlinearly Resonant Surface Waves and Homoclinic Bifurcation KLAUS KIRCHGASSNER~ Marh. Instifur A Uniuersifat Stuttgarf Sfuftgarf,Federal Republic of Germany

................

I. Introduction

135

11. The Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. Transformations and Symmetry . . . . . . . IV. The Method ...................................................

142

V. Reduction and Results.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

152

A. Capillary-Gravity Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Capillary-Gravity Waves under Periodic Forcing ......................... C. Capillary-Gravity Waves under Local Forcing ........................... D. Forced Gravity Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

152 159 163 168

VI. The Mathematics . . . . . . . . . . . . . . . . . . . . .

144 147

172

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

179

I. Introduction The behavior of steady nonlinear water waves on the surface of an inviscid heavy fluid layer has received much attention during the past century, both from the mathematical and from the physical side. Despite a persistent scientific effort involving some of the best names of the time, a number of fundamental questions have not been completely answered. For instance, do solitary waves exist in the presence of surface tension, or, how does a fluid react to a localized pressure distribution moving over its surface with constant speed? Similarly, what are the possible motions under a moving pressure distribution which is spatially periodic? These problems are related to determining the flow of an inviscid fluid through a channel with a bump in its bottom, or its periodic dislocation. How far upstream, one may ask, t Research partially supported by the Deutsche Forschungsgemeinschaft under Ki 131/3-1. 135 Copyright @> 1Y8X Academic Pres Inc All rights of reproduction In any form rererved ISBN 0.12 OO2OZh 2

136

Klaus Kirchgassner

can the bump be felt, and how does this distance depend on the size of the obstacle? In this survey, recent results answering the time-independent aspects are described for motions of moderate amplitude. We do this from a mathematical point of view by exhibiting the totality of solutions of the underlying equations of motion, the Euler equations. Only two-dimensional flows which are steady, i.e., time-independent in a moving frame, are considered. The motion is then described by a quasilinear system of elliptic equations in an infinite cylindrical domain and, if the unbounded variable is interpreted as a dynamical variable, then the bounded solutions, obeying the appropriate boundary conditions, are orbits in an infinite-dimensional phase space. This space consists of functions living on the cross section of the cylinder. Solitary waves appear as homoclinic orbits and cnoidal waves as closed orbits. Restricting the flow to moderate amplitudes, the orbits live on a lowdimensional surface-the center-manifold-and are described by low-order ordinary differential equations. These are the analogues of the Landau equations and they contain all moderate amplitude solutions. The limiting case of a layer of infinite depth is excluded as well as instationary motions of any kind, e.g., the difficult stability question. Thus, the scope of this paper is rather narrow, but we treat the problems with rigor. Our main intention is to present a new mathematical method by analyzing a few examples yet unsolved in the literature, and to show how this method could be used for a wider range of problems. In its long history, the analysis of nonlinear surface waves has been promoted by scientists of various backgrounds, and a vast literature is available for the unforced case, i.e., when the pressure at the surface is constant. The reader who is interested in tracing back the main ideas and results is referred to the classical monographs of Stoker (1957) and Whitham (1974), and to the mathematical book of Zeidler (1971a, 1977). We also mention the article by Yuen and Lake (1984) as one of the more recent excellent surveys on the whole spectrum of this field. The influence of an external pressure distribution has already been analyzed in a linearized model in Stoker (1957) and, for an analogous situation-periodically deformed bottom of the fluid layer-in Zeidler (1971a) for some nonlinear cases (cf. also the literature given there). Among the many contributions which have appeared in the meantime, the brilliant discussion and justification of Stoker’s conjecture on the shape of surface waves of extreme form has to be mentioned. Here, Amick, Fraenkel and Toland (1981, 1982) seriously attack global aspects of a

Nonlinearly Resonant Surface Waves

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qualitative theory, and Amick et al. (1984) complement these results by describing the extreme form of internal waves. As for the stability theorywhich we otherwise neglect here-we mention some important recent developments based on Hamiltonian formulations of the problem by Zakharov (1968) and Miles (1977), concerning the behavior of critical eigenvalues leaving the imaginary axis (cf. MacKay and Saffman, 1985). Attempts to describe the nonlinear interaction between external pressure waves and surface waves have been undertaken by Wu and Wu (1982), Akylas (1984), and by Grimshaw et al. (1985). As anticipated by the linear analysis, the resonant case, when the pressure speed coincides with the critical wave speed, becomes of particular importance and difficulty. All these authors try to calculate the far field behavior in space and time by treating model equations, e.g., forced KdV equations. Let us finally mention the mathematical results on cnoidal capillarygravity waves by Ter-Krikorov (1963), Beckert (1963), Zeidler (1971b), Beale (1979), and Jones and Toland (1986). Here we analyze this nonlinearly resonant interaction for the Euler equation in full generality with the sole assumption of moderate wave amplitudes (order of the mean depth of the fluid layer). We describe the space-like modulations for a periodic pressure wave as well as one having compact support, i.e., vanishing identically outside some bounded interval. The method with which we achieve this goal is based on ideas from the theory of dynamical systems as applied to elliptic equations. The idea to use “dynamical” arguments for solving nonlinear elliptic problems in a strip goes back in the linear case to Burak (1972) and was developed by Scheurle and the author (1981, 1982a, b). Later, Amick and the author (1987) showed how to incorporate free boundary problems, when proving that solitary surface waves exist in the presence of not-too-small surface tension, i.e., when the Bond number is greater than 1/3. The extension of the method to nonautonomous semilinear systems and finally to quasilinear systems is due to Mielke (1986a, 1 9 8 6 ~ )He . also proposed some of the transformations which we use throughout this paper. The method itself is a nonlinear separation of variables by which one can reduce the elliptic system to an ordinary differential system of minimal order, if bounded solutions of restricted amplitude are considered. There are recent global versions of this method for nonlinear parabolic systems known as “inertial manifolds” due to to Foias, Sell and TCmam (1986), but the necessary increase in the distance of consecutive eigenvalues of the linearized operator does not occur for the problems under consideration here.

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Klaus Kirchgassner

The plan in this article is to present this reduction method for nonlinear surface waves in two dimensions. They are characterized by three parameters: A = g h / c 2 ,the inverse square of the Froude number, b = T/pgh', the Bond number, and by E , the dimensionless amplitude of the external pressure. T denotes the coefficient of surface tension, h the mean depth of the layer, p the density and g the gravity. For technical reasons we treat irrotational motions only, although the general case could be included without any principal difficulty, thus incorporating work of Beyer (1971) and Zeidler (1973). Capillary-gravity waves are described for h > 0. We have to distinguish between b < $and b > i. In the latter case, our reduction leads, in lowest order in p = A - 1, to the second order equation

where - a , ( x ) , x E R is the form of the free surface and & p 0 ( x )the external pressure distribution. For the complete equation see (5.4). Similarly one obtains for b = 0, the case of pure gravity waves, in lowest order of p = 1 -A,

a:

= 3 pa,

+ ;a; + 3 &PO.

For the complete equations see (5.36). The case b < 4 leads to a fourth-order system, which we have given explicitly at the end of Section V.A. However, its analysis, even for the unforced case E = 0, is essentially open. In both of the above equations, a , is of order O ( I/*./). The lowest-order approximation-i.e., all terms of order Ipl-of the velocity field v is given by u , = a,, v2 = 0. To determine approximations of higher order, one has to incorporate two almost identical transformations, first proposed by Benjamin (1966) and Mielke (1987c), and described here in Chapter 111. Up to this point, po has only to be bounded and smooth. The special assumptions of the local (compact support) and also of the global (periodic) nature of p o are made only for the discussion of (1.1) and (1.2). We could incorporate, for example, quasi-periodic forcing and apply recent results by Scheurle (1986). The above equations contain all bounded solutions of moderate amplitude of the original problem. For E = 0 they are reversible, i.e., invariant under x -+ -x. The breaking of reversibility by p o generates a nonstandard bifurcation. For a thorough discussion of the role of symmetry in this realm, see Zufiria (1986). In the local case, which has been analyzed for a related problem by Mielke (1986b), the limiting equation (cf. 5.26)), from which all solutions

Nonlinear!,, Resonant Surface Waues

139

can be constructed by perturbation, leads to the intersection of two homoclinic orbits being shifted along the a:,-axis in the two-dimensional phase space, The complete set of solutions can be depicted from Figure 7 in Section V.C, for p = A - 1 > 0, 6 > A similar analysis could be d o n e for p < 0 a n d for 6 = 0. Observe that the solutions shown in Figure 7 have a j u m p in the first derivative being proportional to the integral of p o . We also give a perturbation argument for higher-order approximations in the class of continuous a n d piecewise twice continuously differentiable functions with a possible j u m p at x = 0. However, for p f 0, these are smooth classical solutions of the full equations (5.26) (for previous work, cf. Keady, 1971). In the global case, when p o is periodic, the appearance of horseshoes has to be expected whenever a homoclinic orbit exists. Then solitary waves are broken into a spacelike chaos. Similar phenomena can occur for heteroclinic solutions in the form of bores. They have been analyzed for a two-phase-flow in a channel by Mielke (1987~).For the homoclinic case, see Turner (1981). The existence of a transverse homoclinic point, which is the cause for the chaotic behavior here, is implied by the Melnikov condition, a certain scalar condition in E a n d the free phase p of t h e homoclinic orbit. To show its validity here, we have to use the period of p o as a n extra parameter. The final conclusions are contained in Propositions 5.3 and 5.4. We also mention the existence of subharmonic bifurcations near the homoclinic orbit. The analysis of the nonlinear resonant reaction of cnoidal waves t o p o has been completely suppressed. It leads to a bifurcation of tori in a set of positive measure in the phase space. The flow on these tori is quasiperiodic a n d requires for its existence the application of K A M theory (cf. Moser, 1973; Scheurle, 1987). The appearance of chaotic phenomena in nonlinear wave motion has been predicted by several authors, e.g., Pumir et al. (1983) a n d Abarbanel (1983), who treat model equations for different physical systems. Previous work of the author on this subject can be found in Kirchgassner (1984,1985). In order to make this survey accessible to a wider readership, 1 have minimized the technicalities wherever the procedure is formally explained. Sometimes I have included sketches of proofs which seemed necessary for a deeper understanding of the subsequent material; there the mathematics is a bit more demanding. The real mathematical justifications are summarized in Section VI without compromise. Much of the material I have presented here in comprised form is d u e to A. Mielke, to whom I a m very indebted. My deep gratitude extends to J. K. Hale, to whom I owe my knowledge in the field of dynamical systems, a n d

:.

Klaus Kirchgassner

140

to C. Amick and R. Turner for introducing me to the field of nonlinear surface waves. I express my sincere thanks also to T. Y. Wu for his persistent encouragement and kind patience. Major parts of this manuscript were written when I was visiting the Department of Mathematics of the University of Utah, where I profited from the inspiring atmosphere and many helpful discussions with Frank Hoppensteadt, Jorge Ize and Klaus Schmitt. My thanks extend to Ms. A. Hackbarth and to W. Pluschke for the careful preparation of the manuscript.

SELECTED SYMBOLS 1. Parameters

g = acceleration of gravity

h =mean height of fluid layer c = speed of wave T = coefficient of surface tension p = mass density

2. Coordinates and Transformations

(5,7 )Cartesian

coordinates in moving frame

v = ( u , , u2)‘=

7 = z( 5) free surface,

(::)

velocity vector &pO(5) external pressure

a,, a, a, denote partial derivatives 9:stream function, i.e., a,Vr

= -u2,

a T 9 = u, ,

(x, y ) transformed coordinates: x = 5,

D

=Rx

TI,,=^ = o

y = 9((, 7)

( 0 , l ) transformed flow domain

W = ( W ,, W2)’transformed velocity field: v = g ( W ) (1, W 2 ) ’ -( 1 , O ) ’

Nonlinearly Resonant Surface Waves

141

where

3 . Spaces C k ( R ,E ) : space of k-times continuously differentiable functions from R into the (real) normed vector-space E

C h ( R )= C k ( R ,R)

Ck(R, E ) : subspace of C k ( R , E ) consisting of those functions which, together with their derivatives up to order k, are bounded

Ck,,(R, E ) : subspace of Ck(R, E ) consisting of functions which, together with their derivatives, are uniformly continuous on R k2(0,1) = M O , 1) x LAO, 11,

Norm llWll0

W'(0, 1) = H ' ( 0 , 1 ) x H ' ( 0 , l),

X

= R x L2(0, l),

llwllx

= (w,

Norm llWlll scalarproduct (w,ti)

w)''~,

defined in (3.5') Z = D(A),

llwIIa = IIwIIx

+ IIAwIIx

4. Operators 71 =identity,

J=(;

A ( A ) : linearization in w = 0;

A = A(Ao), D ( A ) = R x W'(0,l) n M,

A.

A = (A,

= (1,O)

where M

-A)

={

E )

resp. (A, b, E )

resp. (1, b, 0)

W,(O)= 0, Wl(1) = a } for b = 0,

and M = { W,(O)= 0 , W,( 1) = p }

for b>O

A

So eigenprojection commuting with A corresponding to eigenvalues with Re (T = 0; So = X, a r q ,

sl=n-so,

X,=S,X,

z,=s,z

(T

Klaus Kirchgassner

142

Nonlinear operators:

11. The Problem

Here we describe the basic equations for the interaction of traveling nonlinear surface waves with in-phase external pressure waves. An inviscid fluid layer of mean depth h is considered under gravity g. On its free upper surface, where capillary forces may also act, it supports nonlinear surface waves of permanent form, traveling from right to left with constant speed c. In a moving frame this phenomenon is stationary, and so is the external pressure &p0.We write the equilibrium equations in nondimensional form for irrotational flow in a 6, 7-system, where 6 is the unbounded coordinate div v = curl v = 0, u* = 0,

$IvI’+ p

+ Az = const.

v,a*z - u* = 0

0<7
7=0

I

17 = 4 5 )

Nonlinearly Resonant Surface Waves

143

where v = ( v , , v J ’ ; z describes the free boundary, p the (constant) density, p the pressure on the free surface and A = g h / c 2= (Froude number)-’. We seek solutions which are bounded in the flow domain a={([, T)/[ER, 0 < 7 < z ( [ ) } . Due to the moving frame we have a normalized flux Q = 1 through any simple cross section of the layer. Solitary waves satisfy the asymptotic condition

For cnoidal waves, (2.2) is replaced by the requirement of periodicity. For forced motions, the asymptotics are more complicated, but will follow naturally from our analysis. We restrict ourselves to moderate-amplitude solutions, i.e., some suitable norm is a priori bounded from above. This upper bound could be estimated explicitly by quantitatively showing the validity of subsequently described, almost identical transformations and estimates on noncritical eigenvalues of the operator A(A, b ) . We shall treat a number of what we think are instructive problems which are distinguished by the form of p , the pressure on the free surface. The Bernoulli constant C in (2.1) is set to A + $ and the following cases are considered. 1 . Forced gravity waves: p = E P ~ , , where p , ( [ ) is given. We distinguish between local forcing, i.e., E # 0 and p o has compact support (vanishes outside a bounded interval), and global forcing, i.e., E # 0, and p o is a given periodic function. The period is considered as an extra parameter.

2. Forced capillary-gravity waves:

where po is given as in 1. The dimensionless quantity b is the Bond-number

As the reader can see throughout the analysis, the assumption of irrotational flow could be removed at the expense of additional technical work. Also we could include more general types of forcing, e.g., quasiperiodic p o .

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Klaus Kirchgassner

111. Transformations and Symmetry

Here we formulate (2.1) in the canonical domain D = R x (0, 1) using a semi-Langrangean transformation proposed by Benjamin ( 1966). Then we apply an almost identical transformation of the velocity components to bring the boundary conditions into a suitable form for our method. This transformation is due to Mielke (1987~). Define the stream-function ? by

a,v=v,,

ag?=--uz,

then, since Q = 1, we have ?(& (x, y) E D via

x = 5,

~ ( 5 )=)1.

?I,=,,=o; We introduce new variables

y = w 5 , 771,

and set

Then (3.1) is invertible for small 121, I V1l, I V,l. We obtain for V = ( V , , V,)’

axV = M(V)a,,V, VI+$I V12+ p

+ AZ = 0

(1+ V,)a,Z- v , = o

(x, y ) E D, (3.1)

y=l,

where

M ( V )= Divide the first equation in (3.1) by ( 1 + V,)’ and integrate in y to obtain

v 1+ , v, --ax -

[l:vl]. ~

d.(Z+[&])=O,

y=l,

(3.2)

=I,!, Vl(y) dy.

where [ V , ]

The fact that, if external forces are not present, (3.1) is reflectionally

Nonlinearly Resonant Surface Waves

145

not distinguished, can be expressed by

symmetric in x, i.e., +aand -a

and

~ ( k vJ,J) iv)= -EM(V) a,v.

(3.3)

We call the vectorfield M ( V ) a, V reversible. 2 can be expressed by V , via (3.2). We simplify the first boundary condition at y = 1. This can be achieved by defining 2w,=(l+V,)’+v:-l,

w2=v*(1+v,)-’,

or, inversely, by

being valid for V , > -1. System (3.1) reads now a,W=K(W)a,.W, W,+p+hZ=0,

inD,

W2=0

fory=O

J,Z- W2=0,

fory=l

(3.4)

where

In addition, (3.4) implies via (3.2) (3.4’) is reversible with respect to I?.It is this formulation Observe, that K ( W ) a,>w which we are going to discuss. The idea behind our method is simple: we treat (3.4) as a dynamical system in the unbounded variable x, although the initial value problem is not solvable in general. However, with the “boundary condition” that W is bounded, (3.4) is well posed and we can apply concepts and ideas from the theory of dynamical systems. To obtain the final formulations we have to distinguish the two cases b = 0 and b > 0. The trace of W, resp. W2 at y = 1 is introduced as an extra variable.

Klaus Kirchgassner

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Case b = 0, no surface tension: Define

),

-AJ aWz( , W1)

&Ah=(

w=(i),

J = ( *0

- 1o)

Then, if the first boundary condition for y = 1 is differentiated i n x, (3.4) can be written d , w = A ( h ) w + P ( t , x, w),

We seek solutions of (3.5) in X

6)= a;+

(w,

=R x

wE

D(A).

(3.5)

L2(0, 1) with the scalar product

i,: w,G,+ w,IF2)

dy.

(

(3.5‘)

Define D ( A )= R x W’(0, 1) n { W2(0)= 0, W , (1 ) = a } . Here W’(0, 1) = ( H ’ ( 0 , 1))2is the usual Sobolev-space; the traces of W, on y = 0 or 1 are defined. We call W E Cb(R, X ) n C”,R, D ( A ) ) a bounded solution of (3.5). Here, the suffix “b” indicates the boundedness of w, i.e., sup{llaxw(x)ll. X€R

+IIw(x)IIA}<~,

It is easily seen that (3.5) is reversible for

E

+ IIAwllx

IIwIIA

= IIwIIx

= 0,

with respect to (3.6)

0 0

-1

i.e.,

&AIR

= -R&A),

P(o, R W )

=

-RP(o, w).

Having determined solutions of (3.5), we find the free boundary via (3.4’).

Remark: Differentiating the first boundary condition at y = 1 introduces an artificial degeneracy into the problem. As we will see, an extra 0 eigenvalue of A ( A ) is generated. This is the price for obtaining a dynamical formulation for the boundary values of w. Case b > 0:

Define

147

Nonlinearly Resonant Surface Waves moreover, for X = (A, b, e ) ,

Then the basic equation follows from (3.4): d , w = i ( A , h)w+

k(X,X, w),

WE

D(A).

(3.7)

We work in X with scalar product as i n (3.5‘), but a replaced by

p.

D ( A )= R xH’(0, I ) n {W,(O) =0, WA1) = P I .

The underlying symmetry for

F

R=

=0

:I.

is determined by

[-:: 0 0

(3.7a)

-1

The notion of a bounded solution is understood as in the first case. It is readily seen that this notion implies the usual solvability of the original equations if the norms are sufficiently small. For later use we give F explicitly up to quadratic terms

1: -(-(3[

p=

wl - 3 W,d, wl+O ( (w(‘(d,w ( ) - w,a,w,+ w,a, wz+ O(l W121d,Wl) w,d,

In the following sections we will use the abbreviations

a

1

Wfl+[ Wil~+&P,+O(IWl’))

=

A( A,, , h,,),

F

=

F + A(A, b ) A, -

‘

(3.8)

(3.9)

where ( A ” , b,) is a “critical” point in the (A, 6)-parameter space, and similarly for A(h,) when b = 0 is considered.

IV. The Method

In a purely formal way, equations (3.5) and (3.7) describe the “evolution” of nonlinear waves in the spatial variable x. But rigorous conclusions can

148

Klaus Kirchgassner

be drawn from this concept: a solitary wave, e.g., is a curve in D ( A ) emanating and returning to the equilibrium w = O , i.e., it is a homoclinic orbit. Similarly, cnoidal waves correspond to closed orbits. Our aim is to show that bounded solutions of moderate amplitude are bound to a lowdimensional manifold in the phase space D ( A ) . Its dimension can be determined and coincides with the order of the reduced system of ordinary differential equations. It is this reduction which can be considered as a nonlinear separation of variables. It is reminiscent of the center-manifold approach in ordinary differential equations. In this realm the method is not standard; therefore we separate the formal aspects given here from their mathematical justification in Chapter VI. In view of the rich structure we treat the case b>O first. We have to investigate the spectrum of i ( A , b). It consists of eigenvalues only; we denote them by CT. A R = - R g implies that the spectrum is invariant under CT+ -u as well as under CT+ Cr (c.c.) in view of the reality of Thus, non-real and non-imaginary eigenvalues appear in quadruples. Simple eigenvalues can never leave the real and the imaginary axis. The eigenvalue problem itself reads

A.

-a,,w, = u w , , a,.w,= u w ~ , W*(O)= 0,

W,( 1) = p.

Observe that w = (p, W , , W2)’can be considered as a function of y only. (4.1) yields ( A - bu’) sin u = u cos CT. (4.2) It is a neat exercise to show the validity of the following picture. The curves C , , . . . , C, are the loci of multiple imaginary resp. real eigenvalues which are close to the imaginary axis. The analytic form of C , , C, near A = 1, b=fisgivenby4(A - l ) = 5 ( 3 6 - 1)’+0((3b-1)3).Therest ofthespectrum is bounded away from the imaginary axis. Bifurcation from the rest state w = 0 occurs when a point in the parameter-space traverses one of the curves C2,C 3 ,C,, not C1.Hence, understanding the full solution picture requires a complete analysis of possible solutions near the singular point A = 1, b = f . We will give the reduced equations for this case. There are numerical experiments of Hunter and Vanden-Broeck ( 1983) covering these parameter values, but without a definite conclusion whether solitary waves exist for

Nonlinearly Resonant Surface Waves

149

FIG. 1. Critical spectrum of A ( h , b ) . Simple eigenvalues are denoted by ”.”, multiple ones by “x”.

b < f , A < 1 or not. Existence of cnoidal waves has been shown by Beyer (1971), Zeidler (1973), and Beale (1979), even for nonpotential flow. Existence of solitary waves for b > A > 1 has been proved by Amick and the author (1987). Everything said so far holds only for F = 0. The case E # 0 is unsolved in major parts. The linear dispersion relation for cnoidal waves corresponds to the imaginary eigenvalues; e.g., in region I11 set u = iq, q E R; then K = q / h , o = q c / h yields the dispersion relation given in Whitham (1974, p. 446).

4,

How to reduce near C, ?

A

We choose ( A o , b o ) E C, for some j and set = A(Ao, bo). The steps which have to be performed are listed below. Their justification is given in Chapter VI. Determine the “critical” eigenvalues u of A with R e u = 0 and its generalized eigenfunctions ‘PJ. Their span is denoted by 2 0 . Moreover, calculate the adjoint A* of in X and the generalized eigenfunctions G k to the critical eigenvalues such that (cp,, + A ) = 8; holds. Define the projections A

Snw=C (w,

+‘h~t,

s, = n so, -

Klaus Kirchgassner

150

and w,=S,w,

X,=S,X,

A,

Al,=A,,

j=O,l.

Then the S, commute with i.e., S,a c AS,; X = X , + X I .We set 2,= XIn D ( A ) . The equations (3.5) and (3.7) assume the form 1

+ F,@,

W" + w 1 ),

(4.3a)

axw, = A , w , + F , ( X , . , w o + w , ) ,

(4.3b)

a h W"

= Aow,

. 1

A

where A = (A, b, E ) and F is as defined in (3.9), F, = S,F. Equation (4.3b) can be inverted in 2, since the spectrum of is bounded away from the imaginary axis. Therefore, it is rather obvious that w, should be functionally dependent on wo. However, one can prove more: if we restrict ourselves to solutions (wo, w , ) ( x ) which are, for all x E R, in some suitable neighborhood U of 0 in Z, x Z , , then w, is a pointwise function of w,,, i.e., there exists a smooth function h ( X , x, w,,) mapping A(] x R x Z,, into Z , such that

A,

w , ( x ) = h ( A , x, W,(X)),

x E R,

(4.4)

holds for all solutions with w ( x ) E U, x E R. A,, denotes a neighborhood of A0 . We call h the reduction function. It satisfies h ( A 0 , . ,0) = d,,h(A(,, . , 0) = 0.

(4.5a)

- ~ ~ l + I b - b ~ ~ l ) I I w ~ ) l l + I I w ~ ) l l ~ (Since + I ~ I ) . Z, is finite Therefore h = o ( ( ~ A dimensional, we do not need to distinguish between different norms in Zo.) Define h,(X,w,)

= h ( A , .,WO)IF=o,

hI@, * , w , ) = ( h - h , ) ( A , . , w o ) .

(4.5b)

Then h,, is independent of x. If we decompose R into its action in 2, and Z,, R = R,+ R , , we obtain h,(X, ROW") = R,ho(A, wo).

(4.52)

Moreover, if po is periodic in x with period d > 0, then the same holds for F, and F , and also for h :

h(A, x + d , wo) = h(A, x, w,).

(4.5d)

If p o has compact support and if A E A,,, wo E U,, then, for every y > 0, there exists a constant c ( y , A,, U,) such that Ilh(A,

X,

W O ) I I AC (~Y , Ao, Uo) e-'IXI.

(4.5e)

151

Nonlinearly Resonant Surface Waves

We have listed only those properties of h in (4.5) which are needed in the subsequent analysis. Observe that h can be computed from (4.3) by

d , , h ( ~ o w o + Fo)= A , h + F , ,

F, = F,(X, ,wo+ h ) , *

j =0,l

(4.6)

to any algebraic order. Although we do not need higher-order approximations, we give some examples at the end of Chapter VI nevertheless. With the aid of the reduction function one is able to reduce (4.3) to the system of ordinary differential equations dxWO

= A l W ( l + f O ( ~ ,. ,wo),

(4.7)

where f o ( k .,wo)=Fo(& .,wo+h(A,

*,Wo)).

We decompose f o into a reversible and a nonreversible part by defining foo=foIF=O, fa0

= f o o ( A , 6, wo),

All

=fo-foo,

f0l

= f o , ( A , 6, E , x, wo).

(4.8)

For the construction of the projection S,, one has to determine the adjoint A* of = bo). It is an elementary exercise to verify, with the aid of the scalarproduct (3.57,

a A(&,

i

1

-W,(l) A*w = -8, W,+ W,( 1 ) , d,Wl

{

(4.9)

D ( A * )= R x W'(0,l) n W,(O) = 0, W,( 1 ) = --/3 bl

l

.

The (generalized) eigenfunctions of the critical eigenvalues of A* can be chosen to be biorthogonal to the eigenfunctions of and S,, is easily constructed. Finally we list the corresponding facts for the case 6 = 0 . Here the eigenvalue relation A ( h ) w= (TW reads (w= ( a ,W)')

A,

-A W,( 1) = (T W ,( 1)

-a, a,

w, = u w, , w, = uw,,

W,(O)= 0,

(4.10)

W,(1) = a.

The eigenvalue relation is A sin u = u cos u.

(4.11 )

152

Klaus Kirchgassner

(T = 0 is a simple eigenvalue for A # 1, and a triple eigenvalue for A = 1. For 0 < A < 1, all eigenvalues are real and simple, and for A > 1 there is a pair of imaginary simple eigenvalues whose values correspond to the dispersion relation for cnoidal waves; all other eigenvalues are real and simple. Therefore, A = 1 is the critical parameter value where bifurcation from the trivial solution w = O occurs. The reduced equation is of third order. The formal procedure is in full analogy to the one described for b>0. We define A = A ( 1 ) and F = . F + A ( A ) - A and obtain reduction via h = h , + h , as in (4.4). The final system corresponds to (4.7); b, however, is missing. To obtain the projection commuting with we use again the adjoint operator to here given by

A

A,

D(A*)= R x W'(0,l) n {a + W,( 1) = 0, W,(O)= 0},

(4.12)

w = ( a ,W)'.

V. Reduction and Results A. CAPILLARY-GRAVITY WAVES

With the method described in the last section we are able to determine the reduced equations of minimal order. We do this here for b > 0, E Z 0, i.e., for the case of forced capillary-gravity waves. In the discussion of the reduced equation we restrict ourselves to E = 0, postponing E # 0 to later sections. The available parameters (A, b ) are taken near the bifurcation curves C, and C3 of Figure 1. C, represents the simplest case, A = 1, b > f. We show the reduced phase space to be two-dimensional. A unique solitary wave exists for A > 1 as a wave of depression. In phase space it appears as a homoclinic orbit which is the envelope of a one-parameter family of periodic orbits (the cnoidal waves around the conjugate flow). For A < 1, 0 is a center, and a family of cnoidal waves bifurcates from 0. There is again a homoclinic envelope with its tail in the conjugate flow. But this has no physical meaning.

Nonlinearly Resonant Surface Waves

153

The situation near C3 is more complicated and essentially unsolved. The reduced phase space is four-dimensional and the existence of a homoclinic orbit is equivalent to the intersection of two curves-the stable and unstable manifold of O-in a four-dimensional space. Except for reversibility we have not found any inherent symmetry to guarantee such an intersection. Therefore we anticipate existence of solitary waves for (A, b ) on a curve in the parameter space. For completeness and for future research the explicit form of the reduced equations near A = 1, h = +is included. Reduction at C,: Set A = 1 + p, Ipl small, b > f . Since b is considered to be fixed, we suppress, if possible, its explicit notation. u=O is a double eigenvalue of = 1) with the generalized eigenfunctions

A A(

AQn=o,

Qn=(:).

AQI=Qo.

All other eigenvalues have nonzero real part. Observe, that R Q , = Q , , R Q , = - Q ~ ,R from (3.7a). We identify Z o , the linear span of Q , , ' p l ,with [w2 via W o = U o Q o + U , q 1 + + ( U n , u , ) ' . Then

The adjoint eigenfunctions are determined as

A*+'

= 0,

A*+" = + I ,

(Q,, + l k )

= 8:.

Observe that &R,= -RoAo, .f&)R,= -R,j;,,)holds. Therefore, f & must be odd in a , , fAo even. Using the explicit form of in (3.8), (4.5), (4.8) we obtain ( a : = a,a,) ab=al(l+3a,+rn,(y,a))+r",(~,~,a)

(5.la)

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Klaus Kirchgassner

where rk,= O(plal'+k+la12tk), r,, = O ( ~ p + ~ l a l k) ,= 0 , 1. The remainder terms rkn are even in a , . For small IpI, /&I, lal, a , can be determined as a function of a,, a; and the parameters, using (5.la), a1 =

where

Po0

a x 1 -3%+Pno(P*., (a,),)+Po,(E, P, (ao),)),

(5.2)

is even in ah and Poo=O(Plaoli+lanl:),

~ol=O(~~+~laol~).

Here, we use the notation ( a o ) ,= (a,, ah), Iaol, = 1 4+ lahl, (5.3) and similarly for (a,)2, when we include a,". Inserting (5.2) into (5.lb) finally yields

so, being even in ah, soo= 0(plaol:+la,12), sol= O ( E+~la,lJ. ~ It is this equation which we are going to discuss. It contains all information about the bounded, small-amplitude solutions of the original problem. Observe that (5.3) is autonomous for E = 0; explicit x-dependence is introduced by Po and s o , . We discuss (5.4) for E = 0. Since the phase space has dimension 2, all bounded solutions are either equilibria, connections of those, or closed orbits. There are only two rest points if Ipl and la& are small, namely

For p > 0, a, is a saddle, a,, a center, and for p < O vice versa. There is at most one saddle-saddle connection (homoclinic orbit). If it exists, it contains the center and all closed orbits in its interior (Poincark-Bendixson). Verifying this picture will yield the uniqueness of the constructed solutions. Scaling as follows

transforms (5.4), for

E

= 0,

into

A;'=signlpl. A,,-+A:+ R , ( p , (A,)*). R , is even in A&and satisfies R,,=O(p),with respect to IA&.

(5.6)

Nonlinearly Resonant Surface Waves

155

Proposition 5.1. Equation (5.6) has, for every suficiently small positive value of p, a unique nonzero even solution decaying to 0 at infinity.

The proof is elementary and could be left to the reader. But the argument is of some importance for the subsequent analysis; therefore we include it here. That there is at most one such solution follows from the fact that the stable and unstable manifold of (0,O) are one-dimensional. The assertion is true for p = 0 (inspect phase portrait). Call this solution Qo and set Q = Q,,+ z ; then z ” - z = -3Qoz+r(p,

*,

(z)~),

(5.7)

where r is a smooth function of its arguments and an even function of 5. Denote by C : the space of k-times continuously differentiable functions in R which, together with their derivatives up to order k, are bounded, and denote by I ( Z \ ( ~ the corresponding sup-norm. Then r obeys the estimate 11 rllnSc , p + ~~z~~~ for sufficiently small 1 1 ~ 1 1 ~Moreover . r maps even functions z into even functions. The left side of (5.7) has a bounded inverse from C t into C’, given by the kernelfunction

~ ( 5t ),= -4 For f

E

CE we denote

( K f ) ( 5 )=

e-lt-rl ,

5‘

.x

(5, t ) E R 2 .

K ( 5 , t l f ( t ) dt.

Also, K preserves exponential decay up to exponent 1; i.e., continuous functions decaying like exp(-al[)) are mapped into C2-functions with the same decay, if 0 5 (Y < 1. Applying K to (5.7) yields Z=

Lz+Kr(p;,z),

L=-3KQo.

(5.8)

L : C:-+ C’, is a continuous, linear operator. Since Qo decays like exp(-151), L maps bounded sets in C: into sets of uniform exponential decay in C’,. An easy extension of Arzeli-Ascoli’s theorem yields the compactness of L

in CB. Therefore the spectrum of L consists of eigenvalues p. If we could exclude p = 1, (5.8) would be uniquely solvable for small 1p1 and 1 1 ~ 1 1 ~ However, . p = 1 is a simple eigenvalue with eigenfunction Q&.The simplicity follows by multiplying z “ = z - 3KQOz by Q&and integrating between -a and f. One obtains d ( z / Q h ) / d t = 0. We take Q o as an even function; thus Q&is odd.

156

Klaus Kirchgassner

Now we eliminate 1 as an eigenvalue of L by a trick. Since r maps even functions into even functions and so does L, we see, by restricting (5.8) to Ci,e= {z E C’,/z(x) = z(-x)}, that 11- Lis continuouslyinvertible.Thus (5.8) is uniquely solvable in Cg,eif lpl and llzllr are sufficiently small. To obtain exponential decay for z at infinity, observe that the above arguments work for functions z, for which exp(l&l)z”’(()E C!, j = 0, 1,2; the assertion is thus proved. The above argument shows that it is the reversibility which makes (5.6) so stable. Its breaking by external forces changes the solution picture dramatically, as we will see. The proof has an additional consequence. Consider the perturbed version of (5.6) A,“= A(i-$A;+ R d p ,

(&I?)+ &f(O

(5.9)

where f~ CE. Under which condition is (5.8) solvable near Q, the unique even solution of Proposition 5.1? Q decays like exp(-Itl). If we set A. = Q + z we obtain z”-z = B(P, 0 ) z + &cL, 6, ( Z M + & f ( 5 ) , (5.10) where

and therefore

Invert (5.10), using K , to obtain

z - KB(Pu,Q ) z = K&P,

. , ( Z M + &Kf

(5.1 1)

It is easy to see that (5.11) is solvable if and only if ( q ,, K&(P, . , ( Z ) J + EKf)

(5.12)

= 0,

where q, is the adjoint eigenfunction to Q’ of the eigenvalue p relative to the scalar product

=1

of KB,

(5.13) In effect, K B is compact in C i and the simplicity of p = l is robust to changes in p. Normalize q , such that ( q , , Q’)= 1, and define z, = z - ( q , , z)Q‘. Then we can solve (5.11) for small lpl, I E ~ in the subspace

Non 1in ea rlji R eson a n t Su rface W aves

157

Ci,Lof Ci, with ( q l , z) = 0, if the right side is projected into C‘i,Las well, yielding z I = Z , ( E , p, zo), zo = ( q ,, z ) , llZ1112 = O( F + lzol’). Set Qf = K q , and observe that Qf = Q’+ O ( p ) , Q’= Q:,+ O ( p ) ;then we obtain from (5.12) the solvability condition for (5.1 1 )

(QT,

i o ( P , ‘ , zoQ’+z’,)+&f)=O.

Specialize to z,=O and replace QT by

Qh:

F(Q:I,.~)+O(&p++*)=0.

(5.14)

We could have included the case where z,), the projection of z on the span of Q’, is nonzero. But this does not yield new results. It is (5.14) which will lead to Melnikov’s condition in the next section. We summarize: Proposition 5.2. For the solvability of (5.9) in C i , it is suficient to solve the scalar equation (5.14) for small E and p.

We return to the discussion of (5.4) for F = 0 resp. its scaled version (5.6). According to Proposition 5.1 we know the solutions for p > 0. For p < 0, set a,= a:, + bo, where a:, = 2 p / 3 + O ( p 2 )as given in (5.5), and proceed for b, as for a, before. Thus, one obtains a complete description of all possible solutions, as indicated in Figure 2. Explicit formulae can be given by tracing back the transformations in 111. The free surface z ( x ) = 1 + Z ( x ) is then given by Z ( x ) = -lplA,(

(””’3 b)- 1

I/*

x)

+ O(p2).

(5.15)

For the solitary wave we have

Similarly, approximations for the cnoidal waves can be determined for < 0. Solve

p

A,”= - A o - $ A ; , w=1+6wI+

A,,( [ )= B ( w ( ) ,

...,

B=6B,+6*B2+

where 6 is an amplitude parameter and B can be considered even. One obtains in a straightforward way 62

A , ( ( ) = 6 cos((i -~6z)t)-:s2+-cos((i 4

(5.17) - ~ 6 ~ ) 2 5 )0(tj3), +

which yields the form of the free surface via (5.15) for each fixed p < O .

Klaus Kirchgassner

158 ob

t

t aI

n.

A

Finally we derive the reduced equations near A = 1, b =;. Define = A ( l , f ) , A* as in (4.9) and proceed as before. Again, a = O is the only eigenvalue of with R e a = 0. It has multiplicity 4.The generalized eigenfunctions are

a

with ai’pi= RI, = Ro

!I.

+, = 0. We identify Zo with R4 and obtain for Ao=[

0

0

1

0),

0 0 0 1 0 0 0 0 The adjoint eigenfunctions are

-A

0 0

0 1 0 0 Ro=[i

0 0

-1

=

Lo,

Nonlinearly Resonant Sugace Waves

159

B. CAPILLARY-GRAVITY WAVESU N D E R PERIODICFORCING External periodic forces interacting with nonlinear waves may lead to chaotic phenomena. This is the theme of this section. From the point of view of dynamical systems, chaos appears here as a consequence of a transverse homoclinic point bifurcating from a homoclinic orbit. Thus, the interactions of periodic forces with solitary waves are expected to yield chaotic behavior. For certain second-order equations, these phenomena have been analyzed (cf. Holmes and Marsden, 1982; Chow et al., 1980; Guckenheimer and Holmes, 1983). However, the situation is not as easy as model equations may suggest. The “dirt” generated by the real equations and hidden in so” and sol in (5.4) is small in E , p, la&, but it is only algebraically small. It turns out that the condition for a transverse homoclinic point to exist is not robust to perturbations of this sort. To overcome this difficulty, we introduce the period d of the external pressure as an additional parameter.

160

Klaus Kirchgassner

where P o ( 0 = PO(X), Observe that Po and R , have the period d(31p1/(3b- 1))”2= d, (cf. Theorem 6.1). Consider the case p > 0; then A,= A&= 0 is a saddle point of (5.18) for r] = 0. Such a critical point is robust under small bounded perturbations, i.e., given p > 0, then for all sufficiently small and S > 0, there exists a unique solution A; of (5.18) satisfying l(A:)I2< 6. This solution is periodic with period d,. It is given by A:

= -r]KPo+

O( v2).

(5.19)

The proof is a simple application of the implicit function theorem. We seek further solutions of (5.18) near Q, the unique even homoclinic solution of (5.18) for 77 = 0 , which was constructed in the last section. Taking advantage of the translational invariance of (5.18) for r] = 0 , we introduce a free phase P and define Ao(5) = Q(t+ P )+z(5+ P).

Now we can follow the analysis ofthe last section ((5.10) to (5.14)), replacing ~f by -r]P,+ R , . Thus we obtain the existence of a solution close to Q for r ] # 0, if

( Q &P, o ( ’ - P ) ) + O ( p +

r]) =

k ( P , p ) =0.

(5.20)

Let us set

--s

and observe that Q&and Po are explicitly known. If ko has a simple zero for some P = Po, p = 0, i.e., (5.21) we can solve (5.20) for (p, p ) near (Po,0). (5.21) is known as the “Melnikov condition.”

Nonlinearly Resonant Surface Waves

161

Before we solve the Melnikov condition, let us discuss its consequences. Since )Ao- QI2(.$)is small and since Q decays to 0 at infinity, is small for large 151. Thus A. must lie in the intersection of the stable and unstable manifold of A,*. In effect, this intersection is transverse, i.e., with linear independent tangent spaces (cf. Chow et al., 1980). The following figure shows the well-known intersection properties of these manifolds for the PoincarC map T, which takes a point y € R 2 into A,(d,,y), where A o = ( A o ,A:)) solves (5.18) with initial condition y at t = O . The points of intersections Fk, k E Z,satisfy TI;, = Fk+land thus form an orbit of the diffeomorphism T The set M := { Ph/ k E Z}u { P z } is T-invariant and compact. Here P z = ( A * ( O ) ,A*’(O)),P, = ( A ( & ) , A’(&,)). In each point there exists a natural local coordinate system given by the tangent vectors to the invariant manifolds at 4. The action of T is strictly contracting in the stable and strictly expanding in the unstable direction. Thus M forms what is called a hyperbolic set. The dynamics of such sets are well known (cf. Newhouse, 1980, for details). We extract from the wealth of possible consequences just one significant result: Attach to M a T-invariant neighborhood U (M ) and assume that there is a 6-pseudoorbit { Qk/k E h} in U ( M ) , i.e., there exists a positive 6 such that

I

Qkt l -

T ( Qk

)I < 8,

k Z;

then, for each sufficiently small positive 6 there exists an r > 0 and an orbit { Ph/ k c Z} c U (M ) such that IPk-QhI
holds for all k E Z .

This is known as the shadowing lemma. Its proof shows that r = O(6). In view of Figure 3 we can generate 8-pseudoorbits (Qk)in M by jumping from the stable to the unstable manifold in a &neighborhood of P $ . Since

FIG. 3

162

Klaus Kirchgassner

-

A, = Q + O( 7 )and Q exp( -161) one needs, starting at some for 6 = 0, an interval of order ln(1/6) for entering a 6-neighborhood of P Z . Being on the stable manifold, the orbit ( P k ) shadowing ( Q k )must follow M . Thus we obtain Proposition 5.3. Assume that (5.21) holds. Take 6, O < 6 < 1. To each natural number N a n d to each sequence ( a ,,. . . , a N - , ) ofpositive real numbers satisfying a k + ,- a k s In 1 / 6 , k = 1, there exists a solution of (5.18) with at least N extrema. The position of N extrema can be chosen such that the distance between two consecutive ones is at least a k .

Therefore we have found solutions of the original problem (5.4) with arbitrarily many extrema. Their amplitude is of order O( [ A - 11). For large 1x1 they follow the d-periodic solution a t , which is of order I E ~ . An example is shown in Figure4. It is to be expected that these solutions are unstable and form transient states in an otherwise chaotic motion. But this is pure speculation. As is well known, such a homoclinic bifurcation is accompanied by subharmonic bifurcations of the periodic orbits inside the homoclinic orbit (cf. Chow et al., 1980). These arguments carry over immediately to Equation ( 5 . 1 8 ) ;we leave the details to the interested reader. The case p < 0 can be treated by translating A . into the conjugate flow, which is a saddle then. The analysis is similar to the one given above. Now we solve k , ( P ) = 0. Take P o ( [ )= A cos w6,

FIG. 4

w

2T =-.

4

163

Nonlinearly Resonant Surface Waves

Then, since Qo is even, k,(O) = 0 . To calculate kh(0) we determine the residues of

Integration over a large rectangle with base [ - R , R ] and height up to i2k7r, k E N, and taking the limit R + CO, k + a,yields kb(0)=w2Re

-

-

laX

e-xw 1 - e-'""'

f ( 5 ) d5=2.rrw3

Since w d,' p - l I 2 ,kb vanishes of exponential order for p = 0 . Therefore we use the period d of po as an additional parameter. Set d = d o p - l / ' ; then the Melnikov condition is satisfied for p < p", 1771 < 77" implying Proposition 5.3 for all sufficiently large periods d. Proposition 5.4. Assumep,(x) = b c o s ( ( 2 . r r l d ) x ) Then . thereexistpositive numbers po, vo such that, if d > ( p o ) - ' / ' ,O < A - 1 < p o , I E ~< ~ ~ k ( Pp) has a simple zero for p = 0 and thus, Proposition 5.3 holds.

The above analysis shows that the behavior of liquid layers may become chaotic under periodic external pressure waves. The same is true for more general pressure distributions such as quasiperiodic ones (cf. Scheurle, 1986). We have also seen how delicate the dependence on parameters may be, which should introduce some scepticism towards the results obtained using model equations. Of course we have discussed very special cases only. We have not included, for example, a serious analysis of small amplitude effects for A < 1, i.e., the effect of an external pressure on cnoidal waves.

c . CAPILLARY-GRAVITY WAVES U N D E R

LOCAL FORCING

The external pressure is assumed to vanish outside some bounded interval. Moreover, we suppose (5.22)

~

,

Klaus Kirchgassner

164

It is shown that the lowest order approximation of every solution of (5.4) has a jump in the first derivative which is of order O ( E ) .We describe the complete solution set. The importance of the limiting equation (5.26) was discovered by Mielke (1986b), when he studied the steady flow through a channel with an obstacle. His analysis and conclusions carry over to our problem, since the reduced equations are the same. The set of solutions can be found by the intersection of two shifted homoclinic orbits and their interior, as will be seen below. As was pointed out in Section IV, h , inherits the exponential decay from Fo,. Therefore, if I(a)l,s y, IA - 115 y, F i ) I y, y sufficiently small, and if A > 0 is arbitrarily chosen, then there exists a c( A, y ) such that

1&1<

I I ~ , ( A ,E , x,ao)II 5 c(A, 7) e-""lI&l,

&().

(5.23)

This again implies a similar inequality for the remainder term so, in (5.4) IsOI(&, p, x, (aoL)l< c l ( ~Y) , e - A " ' ( l w l + l ~ Ila(,lz).

(5.24)

We define rl = "I/--"*(b -4)-I'*(Po),

as a parameter replacing

E.

Pil(5) = p o ( x )

Scaling as in (5.6), (5.4) leads to (5.25)

where the remainder terms satisfy ( A ' = A( 6 -4)) IRob,

(A")2)15

r"lP1 IAolz,

l ~ , ( pT ,, t , ( A , , ) ~ ) ~rS, ~ p ~ e"- A2' l~~ l~l ~~'

Observe that Po/(Po) converges to 0 for every 5 2 0 , and its mean is (Po/(Po))= 1. Therefore it is natural to consider the following limiting equation A,"- sign( p )A,,+ :A: - T&,

= 0,

(5.26)

where So is the Dirac functional concentrated at 0. It is this equation which governs the solution behavior of our problem. Mielke (1986b) has shown that (5.26) yields the complete unfolding of the original equations in the (A, &)-parameterspace. This requires a discussion of penetration properties of stable and unstable manifolds. Here we settle for a less ambitious task and show that every solution of (5.26) with A,(-m)=O can be extended by perturbations of the order

Nonlinearl-v Resonant Surface Waves

165

O(lpl”2) to a solution of (5.25) and thus to the full equations. We restrict the analysis to p > 0 and leave the analogous calculations for p < 0 to the interested reader. We work in the space Y of continuous functions which are twice continuously differentiable in (-a, 01 and [0, a), bounded, together with their derivatives, and decay to 0 at -alike exp([/2).

where [W+

= [0, a), [W- = (-m,

01, and

Solving (5.26) means finding A o € Y with A&(+O) -A&(-O) = 77 which satisfies (5.26) with 77 = 0 for all 5 # 0. Using the kernel K in Section V.B, and K S , = -exp( -(51)/2 we can write (5.25) as

Observe the validity of the following estimates:

(5.28) for bounded 7, p, [ A & . R, and R ,map Y into the space of functions being continuous and bounded in (-00,Ol and [O,CO) with decay exp([/2) at 5 = -03, but may have a jump at 0. As is easily seen, K maps these functions back into Y . The composed map is Lipschitz continuous, but not differentiable, in view of the possible jump in the derivatives of A,. Thus we have

166

Klaus Kirchgassner

Now take any solution Bo€ Y of (5.26). Set A,= B o + Z and obtain 2

+3KBoZ = -+KZ2+ 7

1

) + K ( Ro+ R l ) (Bo+

-KP,+ ie-Ic-'

(,Po)

2).

In view of (5.28), (5.29), the right side defines, for small p and llZll y, a contraction in Y. It remains to be shown that 1 + 3 K B o has a bounded inverse in Y. In fact, observe that B ; = d r B o lies in the nullspace. We can argue, as in Section V.A, that every function in the nullspace of 1+ 3 K B o is proportional to Bt, for 5 < 0. If we have in addition B , ( + a ) = 0, then the argument works for E > O also, and thus -3KB,, has a one-dimensional eigenspace to the eigenvalue 1. As will be shown when discussing (5.26) in detail, the only other case to be considered is Bo periodic for [> 0. Observe that -3KBo is still compact in C : by a direct application of Ascoli-Arzela's theorem. If Z + 3 K B o Z = 0 , Z E Y, multiply by B &and integrate from 0 to [ and obtain

( B & Z ' -B ; Z ) ( ( ) - ( BhZ' - B;IZ)(+O)= C for 5 > 0. Set 2 = BA W and conclude W ' = C / B,!;. If C # 0 , Z would have a singularity of the order (5- to) In([- 5,))whenever tois a simple zero of Bt, contradicting Z E Ci([O,a)). But all Bo being periodic for [ > 0 have infinitely many 5 where this holds. Thus, (5.27) has, for every solution Bee Y of (5.26) and for every sufficiently small p > 0, a unique solution in Y close to B o , and similarly for p
FIG.5. Solutions of limiting equation ( 5 . 2 6 ) for

p>O, O <

7 <4/3&

167

Nonlinearly Resonant Surface Waves

FIG. 6.

Solutions of limiting equation (5.26) for p > 0, -4/3&<

7 <0

Proposition 5.5. Given any of’thesolutions of (5.26), shown in Figures 5 to 7, then there exist positive constants po and y such that, f o r every p with lpl< p o , there exists a unique solution A. of (5.27) satisfying IIAo- Boll < ylpl”’. Moreover, if Bo has a limit for [++a(homoclinic or heteroclinic solutions), then so has Ao. Both are even functions.

It is easy to classify all possible cases. For 7 = 2/3&, p > 0, there exists a heteroclinic solution, being equal to fi for [> 0. Comparing the above results with the Propositions 5.1 and 5.2, we see that the orbits I always correspond to the response of A,,=O to the perturbation measured in the parameter 7,whereas I1 reveals the response of the homoclinic orbit (Proposition 5.2). Of particular interest are the solutions 111 in Figure 5 which are even, homoclinic and “bifurcating from infinity.” The distance of their

I FIG. 7.

Solutions of limiting equation (5.26) for p
Klaus Kirchgassner

168

maxima to [ = O is given by

where ((s) is the smallest positive 0 of small values of 9 > 0 a n d

€'-t3= '7'.

Therefore

5-

7 for

T ( v ) = l l n s1+0(1). Hence we obtain the following "paradox": The smaller the amplitude of the external pressure, the further upstream the influence of the local pressure distribution can be felt.

D. FORCEDGRAVITYWAVES The analysis of forced pure gravity waves ( h = 0) proceeds along the previously described route, except for two new features: the effect of a n artificial eigenvalue 0 a n d the need to consider a n integral of the system (3.5). Both problems have the same source, namely the fact that we use the Bernoulli equation on the free boundary in differentiated form. Since the methods by which we overcome these problems are of general interest, we present them here. The basic equations were given in (3.5). First we have to determine the spectrum of A(A)

~;,").'O~=UW,

A ( A ) w = ( -A Wz(1 )

w h e r e & = W l ( l ) , W,(O)=O.Weobtainasin(4.11): a c o s a = A sin c r ; a ~ @ , resulting in the following spectral picture.

0

<

1

x = 1

Fic;. 8

x

:,

1

Nonlinearly Resonant Surface Waves

169

The symmetry of this picture is caused by the commutation property (3.6) of A(A). Nonreal eigenvalues exist only for A > 1 as a pair of simple eigenvalues f iq. We have q / A + 1 for A + +a. Via the identifications w = qc/h, k = q / h , one obtains from (4.11) the dispersion relations (13.25), p. 438 in Whitham (1974). The only multiple eigenvalue occurs for A = 1; it has multiplicity 3. Therefore we unfold the solution set near A = 1 . Set

F=F+X(A)-A,

A=&i),

A = I - ~ :

then we have

F ( E, A,

X, W) =

(

1

PWAX, 1) - & & P O ( X ) Wy3, W , - 3 Wla,W , , -

w,a,wl+ w,a,,w,

up to terms of order O(lWl:+plWl:). We define as in (4.7), (4.8): A,= foo+fol. The generalized eigenfunctions of A to u = O are (A'p,= ( P ~ - ~ , 'p-1

=O)

.;i;-!

Their span 2, is identified with R'. The operators restricted to Zo read

"=(; ; ;), 0

The adjoint A* and are ( A * + k= Jlk+'; IJJ'

with ( q j ,+ k )

= 6;.

1

0

a and R

(cf. (3.6))

A was given in (4.12). Its generalized eigenfunctions =0 ) ,

Define

170

Klaus Kirchgassner

Acccording to (4.3), (4.4) we have w = wo

+ ha( A, a) + h,( E , A, x, a)

(5.32)

for all sufficiently small llwllA. Moreover, ho(A, Roa) = R,ho(A,a),

R , = RIZ,

ha = O(lA - I I la1+ la12),

In view of the reversibility, f we obtain

and

9

h , = O(E).

f i 0are odd in a , , and fA0 is even. Thus

ah= a , ( l + ~ p + $ u o - ~ u z + r , ) o ( pa))+&pb+ , r o , ( p ,E, x,a),

(5.33a)

a : = a 2 + & a : - a O a 2 -a:+ r I 0 ( p a , ) + r I I ( p E, , x, a),

(5.33b)

a ~ = a , ( 3 p - ~ a 2 + 9 a o + r z o ( p , a ) ) + 3 ~ p ~ , + r ~, , ,( xp , a )

(5.33c)

where A

and

= 1 - p,

ph = d,po, and where lk0=O(Ipl(al+la12),

k=0,2,

r l 0 = 0(Ip1la1~+la1~),

are even in a , ,

r k l = O(Elal+

w),

k

= 0, 1,

(5.33)'

2.

Due to the fact that we have used one of the basic equations in a differentiated form, the above equations are not independent. This can be seen by the validity of (5.34) which follows from (3.2) and (3.5a). First we observe that

Moreover we have from (5.30) and (5.32) W, = ao-;y2a2+ h i , W, = - Y U ,

+ h',

where h = ( h a , h i , h2)'. An elementary calculation leads to 4 ( 2 + p ) ~ 2 +~

~~+~u:+;u:-~u,,u,+ h'(~ 1) u~ [+h l ]

+ &pa+ O( laI3+ p (a12+ ~p+ E la1) = C.

(5.35)

Nonlinear1.v Resonant Surface Waves

171

Now (5.33a, b, c) imply (5.35) for some C. On the other hand, near a =0, E = p = C = 0, we can express a2 as a unique function of a,, a , , p, F and C. Thus we can replace ( 5 . 3 3 ~by ) (5.35) and reduce (5.33) to a second-order equation. Another consideration concerns the term h ’ ( l ) - [ h ’ ] in (5.35). It seems that one should calculate h up to order e+plal+la12. However, since h maps into Z , , and 2, is the orthogonal complement of the @, we obtain, using (+*, h ) = 0 , h’(1) - [ A ’ ] = 0. Concerning the free constant C, we observe that C = O for E = 0 , if a tends to 0 for ,$+-a. This includes homoclinic solutions for E = 0 as well as all solutions for F f 0 if p o has compact support. In the global case, p ( , being periodic, C may be of order F . But this constant can be incorporated into p o . Therefore we set C = 0. Using (5.35) for C = 0 and h’ - [ h ’ ] = 0 to eliminate a, in (5.33), we obtain (5.36)

to

where is of the same order as r,,, and r,, is as in (5.33)’. The similarity between (5.36) and (5.1) is evident. In particular, the justification of the scaling as well as the truncation can be accomplished as in Section V.A, B, C. Therefore we restrict our analysis to the main steps and the results. Scaling as follows a d x ) = IpIAdl),

a , ( x )= 1pI”’AI(5),

5 = (pII”x,

leads to the limiting equation 3.5 A , “ = 3 sign(p)A(,+;A;+yP,, P

(5.37)

where Po(5) = p o ( x ) . For E = 0, we can draw the same conclusions as in proposition 5.1 ( p > 0) and (5.17) ( p (0). However, here p > O means A < 1 and we have A,
(5.38)

Thus, for p > 0 we have a unique even solitary wave of elevation. But (5.38) is valid also for the one-parameter family of cnoidal waves bifurcating from 0 for p < 0.

172

Klaus Kirchgassner

. the propositions 5.3 If F # 0 and po is periodic, we set 7 = 3 & / p 2Then and 5.4 hold. In the local case, when po has compact support and ( p o ) # 0, we define 7 = 3 ~ ( p ~ ) / l pand l ~ proceed '~ as in Section V.C. Then proposition 5.5 holds, when the signs of A, in Figures 5 to 7 are interchanged. The formula (5.38) for the free surface is valid.

VI. The Mathematics

In this final chapter we describe the mathematical basis for the reduction method, which we have formally applied in the previous sections. Moreover we show how higher order approximations of the reduction function h can be computed. The reduction method described in Section IV and used in Section V is reminiscent of the center manifold approach for ordinary differential equations, which was first proved by Pliss (1964) and Kelley (1967) for autonomous (x-independent) equations. The extension to nonautonomous equations in the extended phase space can be found in Aulbach (1982). Generalization to partial differential equations are well known for semiflows, i.e., the parabolic case (cf. Henry, 1981), and for hyperbolic equations (cf. Carr, 1981). For elliptic systems, as they are considered here, this method was first formulated by the author (1982) in a special situation. Fischer (1984) proved its validity for general semilinear autonomous systems. The first application to free boundary value problems was given by Amick and the author (1987), the version for semilinear, nonautonomous elliptic systems by Mielke (1986a), and for quasilinear systems by Mielke (1987a). We shall use Mielke's formulation and prove two properties HI, H2 of the basic equations (3.5) and (3.7). One can use them as axioms for the validity of the reduction method. In the following lemma we shall treat-pars pro toto-the system (3.5) in the real Hilbert space X = R x L2(0,1) x L 2 ( 0 ,l ) , with the norm 11. Moreover Z = D ( A ) =R x H ' ( 0 , 1) x H ' ( O , 1) n { W,(0)= 0, W , (1) = a } and 11. / I A = 11 + IlA. The norm of a scalar function W in L, resp. HI is denoted b y 1 WI, resp. I WI,, and 1 * I is the Euclidean distance in R". We set X = (A, F ) , Xo = ( 1 , O ) and = i ( 1 ) . The operator is linear, closed and densely defined in X and has a compact inverse. Thus, its spectrum consists of eigenvalues with finite multiplicities. These properties are more or less trivial.

tix.

]Ix

[Ix.

A

A

Nonlinearly Resonant Surface Waves Lemma 6.1. Consider the natural complex$cation Then there exist positive numbers qo, yo such that

II(A

-

zI)-lyk+k

of

173

a in k

Yo

=X

+ iX. (6.1)

5-

IZI

holdsfor all z = iq, 1912 qo, q E R. ProoJ

We have to solve -

W,( 1 ) -ZW,(l) -a,w,-zw,= a,w,-zw,=

= a,

(6.2a)

VI,

(6.2b)

v,.

(6.2~)

From (6.2b) and ( 6 . 2 ~ we ) obtain

I VII;+ I VI, i= I

J,

WII i+ IJ,. WzI ?I+ lZ12(IWII

+ 1 W21;)

+2q I m ( r n W 2 ( 1 ) ) . Multiplying (6.2a) by W,( 1) yields

2q Im( W,(l) W,(l)) = -21 W2(1)12-2 Re(GW,(l)). We estimate [ W,] via ( 6 . 2 ~ )

and

(6.3)

174

Klaus Kirchgassner

where

c,=1+-(2+9,) lz12

(1+-d,)

,

Therefore, if qo> 0 is chosen sufficiently large, and e l , E? sufficiently small, one obtains for all IzI 2 qo, Rez = 0, v = (a, V , , V,)',

l~,wll:,+la,w2l:,~c:llvll'x. Moreover we obtain from (6.2a)

I wI( 1)I 5

1 IZI

(la I + I w2(111).

If we estimate I W2(l)l by (6.4), the assertion follows for any IzI 2 qo. The estimate (6.1) can be extended to a cone IRe z I 5 61Im zI for O < 6 < Y O 1 , 141= IIm ZI 3 4 0 .

II(A- z U ) - ~ I I ~ + =~ ]]((a - ( z

-

i q ) ( A- iqU)-')-'(A- iqU)-']\z+x

A.

Therefore, the line z = iq contains at most finitely many eigenvalues of Denote by Sothe real eigenprojection: S,,A c ASoand S , = U - SO,?()X = X o , S , X = X , , A , = A l x , , j = O , l . Then we have X = X , , O X , , a = A , O A , . A

H1:

I

The space XO has jinite dimension. I f u E Z &-the spectrum of Ao-then Re (T = 0. There exists a p > 0 such that (T E 1 implies )Re (TI 2 /I. To each positive p' < p there exists a y , ( / I ' ) such that the inequality

A,

holds for all z E C with (RezI s p'

Nonlinearly Resonant Surface Waves

175

Now, we turn to the nonlinearity in (3.5) which we write F(X, ., W ) = ( i ( A ) - i ( l ) ) w +

where h = ( A , E ) , ho=(l,O),and

P=(

-&axPo ( K ( W ) -m,w

Observe, that W E D ( A ) implies W E H ' ( 0 , 1) x H ' ( 0 , 1 ) . Since H ' ( 0 , 1 ) is embedded in Co[O,11, g ( W ) and thus K ( W ) defined in 111 are Ck-mappings from ( H ' ( 0 ,1))2 into (Co[O,11)' for each EN. Therefore, if p 0 € Ci,,,(R), the space of k-times differentiable functions in R with bounded and uniformly continuous derivatives, then F E Ci,,(A x R x D ( A ) ,X ) where A is some neighborhood of A,,= (1,O). Moreover, there exists a bounded function y 2 ( r ) for r > 0 such that, if J J w J < J Ar, we have IIF(h, . , w ) I l x s

-11

Y2(r)(lEI+lA

IIWIIA+IlW112A).

(6.6)

We can decompose the system (3.5) as follows: W", W I ) ,

(6.7a)

= ~ i , w , + f 1 ( X ,. r W O , W I ) ,

(6.7b)

a x w o = Aow,,+f,(& axw1

where wJ E X, n D ( A ) ,f; = SJF,j

H2:

*

1

= 0, 1.

There exist neighborhoods U I , c X o , U ~ DC( A ) n X l , A o f A O ~ R 2 , and some k E N such that f = ( f o , f , )E Ck,z'(Ax R x U ~ UX; , X 0 x XI) holds. Furthermore f(A,,, . ,0) = 0, d,f(A,, , . ,0) = 0.

Since the projections are continuous in X and in D ( A ) , we conclude from the previous considerations that H2 is fulfilled for the nonlinearity F from (3.5).The same can be shown for (3.7). The value Xo is given by A = 1, F = 0 in the first, and by A = 1 , b > 0 fixed, E = 0 in the second case. Let us remark that (6.5) cannot be improved in the power of IzI as is well known for elliptic systems. Using results of Burak (1972) one could show, by verifying that the Agmon condition for the boundary values holds, that projections S:, S; exist corresponding to the positive and negative part of Z and that A: := AISFgenerate holomophic semigroups for x s 0 resp. x 2 0. Thus, we could invert (6.7b) for f l E CB(R, X ) .

A,

176

Klaus Kirchgassner

-A,

However, the invertibility of a, can be shown without the projections S:. The inequality (6.5) leads to a logarithmic singularity of the inverse which can be handled. This approach has the advantage that the reduction method can be extended to the case where F maps into some closed subspace of X . For the implications of this generalization see Fischer (1984) and Mielke (1987b). Both systems, (3.5) as well as (3.7), have a quasilinear structure, i.e., the highest derivative d,W has coefficients depending on w. The inversion of d,-A, in (6.7b) leads to a loss of regularity due to the singularity of exp( -All[() at [ = 0. For semilinear systems, when F maps D ( A Y ) y, < 1, into X , this loss can be compensated by the gain in regularity between D ( A Y )and D ( A ) .Thus the extension from finite to infinite dimensions is relatively straightforward in the semilinear case (Fischer, 1984; Mielke, 1986a). For quasilinear systems, Mielke (1987a) has shown that this difficulty can be overcome by a result of “maximal regularity” for the linear equations

which correspond to (6.7). One constructs a space Y over X , x XI such that (6.8) is uniquely solvable for (v,,, go, g , ) E X o x Y with a solution satisfying (d,wo, d,w,) E Y. Mielke (1987b) constructed Sobolev-spaces with exponential weight leading to maximal regularity for (6.8). Thus he obtained the following: Theorem 6.1. Let the assumptions H 1 and H 2 be valid. Then there are neighborhoods of zero Uoc U ; c XI,, U2c U ; c D ( A ) n X I , a neighborhood A o c A of ha and a function

~ = ~ ( A , x , ~ ~ ) ~ C ~ U,) ( A ~ X R X U ~ , with the following properties: (i) The set

MA ={(x,wo, h ( X , x , w o ) ~ R x X o x ( D ( A ) n X , ) l ( x , w o Uo) )~~x is a local integral manifold for (6.7) for A E Ao. ( i i ) Every solution of (6.7) with A € A o and ( w o , w ~ ) ( x ) EU o x U 2 ,xER, belongs to M A . (iii) We have h ( A o , x, 0) = dw,,h(Ao,x, 0) = 0 for all x E R. (iv) Zf fo and f l in (6.7) are periodic in x, then so is h with the same period.

Nonlinearly Resonant Surface Waves

177

( v ) I f there are linear isometries R , :X , + X n , i = 0 , 1, and a constant K E {-I, l} such that

J;(&

KX,

ROWO, Riwi) = KR,J;(A,X , wo, w i ) ,

A,R,= KR,,&,

i =o, 1,

then h ( A , K X , Rowo)= R ,h ( A , x, wo) holds. For the proof see Mielke (1987a). The formulae (4.4) to (4.7) follow by setting

h,(A, b, wo) = h(A", w,,),

where l o = (A, b, E = O ) ,

h , ( A , b, &, x, wo) = h ( A , x, wo) - h(A", wo). The effect of reversibility is described by R, = R I , , K = -1. In Section IV we have claimed exponential decay of h in x of any order, if p o has compact support. This follows from Corollary 6.2. Assume there exists a function f ( A , w ) E Ci,T'(Ax U ~ X U i , X ) and some 0, d > O such that

~ l f (X ,~W,) -

f ( ~ w)llX , 5 D e-d'x',

x

E

R,

f o r all w = w,+ w , , W,,E U:,, w, E US. Then U,,, U, and A,, in Theorem 6.1 can be chosen such that a function L(A, wo)E C : ( A o x U,,, U 2 )exists satisfying Ilh(A, x, w d - i ( A , wO)IIx,5 A d ) e-"lx',

X E

R,

f o r some y ( d ) and all w,,E U,,, A E A,, . The proof follows from Theorem 3.3 i n Mielke (1986a) and the proof of Theorem 6.1. If, as in the case considered in Section V.C, f is independent of x for all sufficiently large 1x1, the choice of d is arbitrary.

Computational Aspects We close with some remarks concerning the computation of the reduction function h. Although h is not unique, in general, it has a unique Taylor expansion about wo = 0 and A = A,,, i.e., the coefficients of the Taylor jet of order k are uniquely determined by the properties o f h ; different h differ only in terms which are exponentially small in wo and /L (cf. Sijbrand, 1981).

178

Klaus Kirchgassner

The computation of this Taylor jet is conceptually simple but actually quite tedious. In the cases we have been considering here, where 0 is the only critical eigenvalue having geometric multiplicity 1, the calculations lead to a sequence of recursively solvable linear equations. As an example we treat the case of gravity waves ( b = 0). We wish to determine the terms of order O( E + plal+ of h. Remember that we identified W,,E Z , with a € R3, A = (A, E ) , A = 1 - p.

h ( A , w 0 ) = ~ h 0 ( x ) + p h i ( a ) + h 2 ( a ) +... , where h2(a) = ht2’(a,a), and h‘*’(a, b) is a symmetric bilinear form over R3 with values in D ( A )n X I . According to Theorem 6.1, wI(x)= h(A, x, W~,(X)) holds for all small bounded solutions. Inserting into (6.7b) and collecting the terms of O( 8 ) of F yields

Using the explicit forms of F = ( F 0 , Fl, F2)’

‘p,

+’ in (5.30) resp. (5.31) we obtain for

and

- 4 6 - A[ F1I + 3b’ Fi I 10

2Y ~

~

0

+

~

-

~

+

~

~

2

~

~

~

l

l

(6.10) + 3 ~

F~ - ~ Y [ Y F , I This implies FO1 - (1 5,

2 i +n

’ 0)’-

32 ~ ’ - ,

(6.11)

and thus (hy, h:) = = 0 implies @(x, 1) = [@](x) and vice versa. Thus we have to solve From (6.9) and (6.11) we conclude that d,h:=d,hy

(a,@, d y @ ) = V@ for some scalar function @. Moreover (+’,h’) V2@=(-3+3 10

d,@(x, 0) = 0,

2Y

2

)axPo

@(x,1 ) = [@](x).

(6.12)

The solution of (6.12) is uniquely determined up to a constant for bounded @, and thus we obtain h0 = (a,@(. , l ) , a,@, a,@)‘.

To calculate h’(a) we observe that hl defines a linear functional from X , into D ( A )n X I . Since X , is spanned by cp,, ‘pl, ‘p2 it suffices to determine

~

2

~

i

1

179

Nonlinearly Resonant Surface Waves

hf

= h'(9,).

Define for A,

=(1

- p,

0)

Then one obtains

Moreover, we conclude from (5.33): a:)= a , , a : = a2 and a; = 0 up to terms of order O(&+pIal+lal2). Therefore d , h l = A l h l + F : leads to

Alh:)=O, h;=O,

A,h:+FIl=O,

h:=(O,O,-&y+;y')',

h l2 -- ( - L

A,h:=h:; 27-L20Y

175, 1400

'+!8Y

9

O)'.

Similarly one could calculate the quadratic approximations h2(a),for which it suffices to determine h?, = h"'(cp,, c p , ) . Since Acp, = q l - l ,Q-, = 0, this can be achieved recursively. We leave the lengthy calculations to the reader as an exercise. The results are

References Abarbanel, H. D. I. (1983). Universality and strange attractors in internal-wave dynamics. J. Fluid Mech. 135, 407-434. Akylas, T. R. (1984). On the excitation of long nonlinear water waves by a moving pressure distribution. Y. FIuid Mech. 141, 455-466. Amick, C. J., and Toland, J. F. (1981). On solitary water-waves of finite amplitude. Arch. Rat. Mech. Anal. 76, 9-95.

Amick, C . J., Fraenkel, L. E., and Toland, J. F. (1982). On the Stokes conjecture for the wave of extreme form. Acra M a r h . 148, 193-214. Amick, C . J., and Toland, J. F. (1984). The limiting form of internal waves. Proc. Roy. Soc. London, A 394, 329-344.

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Amick, C. J., and Kirchgassner, K. (1987). Solitary water-waves in the presence of surface tension. Manuscript. Aulbach, B. (1982). A reduction principle for nonautonomous differential equations. Arch. Math. 39, 217-232. Beale, J. T. (1979). The existence of cnoidal water waves with surface tension. J. Dig Eqn. 31, 230-263. Beckert, H. (1963). Existenzbeweis fur permanente Kapillarwellen einer schweren Flussigkeit. Arch. Rat. Mech. Anal. 13, 15-45. Benjamin, T. B. (1966). Intenal waves of finite amplitude and permanent form. J. Fluid Mech. 25, 241-270. Benjamin, T. B. (1967). Internal waves of permanent form in fluids of great depth. J. Fluid Mech. 29, 559-592. Beyer, K. (1971). Existenzbeweise fur nicht wirbelfreie Schwerewellen endlicher und unendlicher Tiefe. Dissertation B, Leipzig. Burak, T. (1972). On semigroups generated by restrictions of elliptic operators to invariant subspaces. Israel J. Math. 12, 79-93. Carr, J. (1981). “Applications of center manifold theory.” Appl. Math. Sci., Vol. 3 5 . Springer, New York/Berlin. Chow, S. N., Hale, J. K., and Mallet-Paret, J. (1980). An example of bifurcation to homoclinic orbits. J. Dig Eqn. 37, 351-373. Fischer, G. (1984). Zentrumsmannigfaltigkeiten bei elliptischen Differentialgleichungen. Math. Nachr. 115, 137-157. Foias C., Nicolaenko, B., Sell, G. R., and TCmam, R. (1986). Inertial manifolds for the Kuramoto-Sivashinsky equation and an estimate of their lowest dimension. To appear in J. Maths. Pures et Applique‘es. Grimshaw, R. H. J., and Smyth, N. (1985). Resonant flow of a stratified fluid over topography. Res. Rep. No. 14, Dep. Math., Univ. Melbourne. Guckenheimer, J., and Holmes, Ph. (1983). “Nonlinear oscillations, dynamical systems, and bifurcations of vector fields.” Appl. Math. Sci., Vol. 42, Springer, New York/ Berlin. Henry, D. (1981 ). “Geometric theory of semilinear parabolic equations.” Lecture Notes in Mathematics, Vol. 840. Springer, New York/Berlin. Holmes, P. J., and Marsden, J. E. (1982). Horeshoes in perturbations of Hamiltonians with two degrees of freedom. Comrn. Math. Phys. 82, 523-544. Hunter, J. K., and Vanden-Broeck, J. M. (1983). Solitary and periodic gravity-capillary waves of finite amplitude. J. Fluid Mech. 134, 205-219. Jones, M., and Toland, J. (1986). Symmetry and bifurcation of capillary-gravity waves. Arch. Rat. Mech. Anal. 96, 29-54. Keady, G. (1971). Upstream influence in a two-fluid system. J. Fluid Mech. 49, 373-384. Kelley, A. (1967). The stable, center stable, center, center unstable and unstable manifolds. J. Dig Eqn. 3, 546-570. Kirchgassner, K. (1982a). Wave solutions of reversible systems and applications. J. Dig Eqn. 45, 113-127. Kirchgassner, K. (1982b). Homoclinic bifurcation of perturbed reversible systems. In “Lecture Notes in Mathematics” (W. Knobloch and K. Schmitt, eds.), Vol. 1017, pp. 328-363. Springer, New York/Berlin. Kirchgassner, K. (1984). Solitary waves under external forcing. In “Lecture Notes in Physics” (P. G. Ciarlet and M. Roseau, eds.), Vol. 195, pp. 211-234. Springer, New York/Berlin. Kirchgassner, K. (1985). Nonlinear wave motion and homoclinic bifurcation. In “Theoretical and Applied Mechanics” (F. I . Niordson and N. Olhoff, eds.), pp. 219-231. Elsevier Science Publ. B. V., IUTAM.

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