Archive for Rational Mechanics and Analysis manuscript No will be inserted by the editor

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Archive for Rational Mechanics and Analysis manuscript No. (will be inserted by the editor) Gerard Iooss & Pavel Plotnikov Asymmetrical three-dimensional travelling gravity waves Abstract We consider periodic travelling gravity waves at the surface of an infinitely deep perfect fluid. The pattern is non symmetric with respect to the propagation direction of the waves and we consider a general non resonant situation. Defining a couple of amplitudes ?1, ?2 along the basis of wave vec- tors which satisfy the dispersion relation, first we give the formal asymptotic expansion of bifurcating solutions in powers of ?1, ?2. Then, introducing an additional equation for the unknown diffeomorphism of the torus, associated with an irrational rotation number, which allows to transform the differential at the successive points of the Newton iteration method, into a differential equation with two constant main coefficients, we are able to use a descent method leading to an invertible differential. Then by using an adapted Nash Moser theorem, we prove the existence of solutions with the above asymptotic expansion, for values of the couple (?21, ?22) in a subset of the first quadrant of the plane, with asymptotic full measure at the origin. Key words: nonlinear water waves; small divisors; bifurcation theory; pseudodifferential operators; travelling gravity waves; asymmetric 3D waves; rotation number AMS: 76B15; 47J15; 35S15; 76B07 Gerard Iooss IUF, Universite de Nice, Labo J.

• nj ?


• gravity waves


• propagation direction


• velocity field


• waves


• all function orthogonal


• waves can


• diamond waves


• dimensional travelling

Sujets

• Gravity wave
• Flow velocity

Informations

• Publié par: pefav
• Nombre de lectures: 7
• Langue: English

Extrait

Archive for Rational Mechanics and Analysis manuscript No.
(will be inserted by the editor)
G´erard Iooss & Pavel Plotnikov
Asymmetrical three-dimensional
travelling gravity waves
Abstract We consider periodic travelling gravity waves at the surface of an
infinitelydeepperfectfluid.Thepatternisnonsymmetricwithrespecttothe
propagation direction of the waves and we consider a general non resonant
situation. Defining a couple of amplitudesε ,ε along the basis of wave vec-1 2
tors which satisfy the dispersion relation, first we give the formal asymptotic
expansion of bifurcating solutions in powers of ε ,ε . Then, introducing an1 2
additional equation for the unknown diffeomorphism of the torus, associated
with anirrationalrotationnumber,whichallowstotransformthe differential
at the successive points of the Newton iteration method, into a differential
equation with two constant main coefficients, we are able to use a descent
method leading to an invertible differential. Then by using an adapted Nash
Mosertheorem,weprovetheexistenceofsolutionswiththeaboveasymptotic
2 2expansion, for values of the couple (ε ,ε ) in a subset of the first quadrant1 2
of the plane, with asymptotic full measure at the origin.
Key words: nonlinear water waves; small divisors; bifurcation theory;
pseudodifferential operators; travelling gravity waves; asymmetric 3D waves;
rotation number
AMS: 76B15; 47J15; 35S15; 76B07
G´erard Iooss
IUF, Universit´e de Nice, Labo J.A.Dieudonn´e, 06108 Nice Cedex 02, France
E-mail: [e-mail]
Pavel Plotnikov
Lavrentyev Institute of Hydrodynamics, Lavryentyev pr. 15, Novosibirsk 630090,
Russia
E-mail: [e-mail]2 G´erard Iooss & Pavel Plotnikov
1 Introduction
1.1 Presentation and history of the problem
We consider small-amplitude three-dimensional doubly periodic travelling
gravity waves on the free surface of a perfect fluid. These wavesare steady in
a frame moving with the velocity of the wave (−c in the absolute reference
frame). The fluid layer is supposed to be infinitely deep, and the flow is
irrotational only subjected to gravity. The bifurcation parameters are the
horizontal phase velocity, and the direction of propagation of the travelling
waves, the infinite depth case being not essentially different from the finite
depth case.
In 1847 Stokes [27] gave a nonlinear theory of two-dimensional travel-
ling gravity waves, computing the flow up to the cubic order of the am-
plitude of the waves, and the first mathematical proofs for such periodic
two-dimensional waves are due to Nekrasov [22] and Levi-Civita [19]. Math-
ematical progress on the study of three-dimensional doubly periodic water
wavescamemuchlater.Inparticular,firstformalexpansionsinpowersofthe
amplitude of three-dimensional travelling waves can be found in papers [7]
and[26].Onecan find manyreferencesandresultsof researchonthis subject
in the review paper of Dias and Kharif [6]. Reeder and Shinbrot (1981)[24]
proved the existence of gravity-capillary waves with symmetric diamond pat-
′terns,resultingfrom(horizontal)wavevectorsbelongingtoalatticeΓ (dual
to the spatial latticeΓ of the doubly periodic pattern) spanned by two wave
vectors K and K with the same length, the velocity of the wave being in1 2
the direction of the bissectrix of these two wave vectors. This was completed
by Craigand Nicholls(2000)[3]who used the hamiltonian formulationintro-
duced by Zakharov [28], with a variational method. These waves appear in
litterature as ”short crested waves” (see Roberts and Schwartz [25], Bridges,
Dias, Menasce [1] for an extensive discussion on various situations and nu-
merical computations), and the fact that the surface tension is supposed not
tobetoosmallisessentialforbeingabletouseLyapunov-Schmidttechnique,
and the authors mention a small divisor problem if there is no surface ten-
sion. Asymmetrical ”simple” doubly periodic waves in the non resonantcases
were alsoconsidered by Craigand Nicholls (2002)[2] who gavethe principal
part of the formal Taylorseries, taking into account of the two-dimensionsof
the vector parameterc. They emphasize the fact that this expansion is only
formal in the absence of surface tension.
Another type of mathematical results are obtained in using ”spatial dy-
namics”, in which one of the horizontal coordinates (the distinguished di-
rection) plays the role of a time variable, as was initiated by Kirchga¨ssner
[17] and extensively applied to two-dimensional water wave problems (see a
review in [5]). The advantage of this method is that one does not choose the
behaviorofthesolutionsinthe directionofthedistinguished coordinate,and
solutions periodic in this coordinate are a particular case, as well as quasi-
periodic or localized solutions (solitary waves). In this framework one may a
priori assume periodicity in a direction transverse to the distinguished direc-
tion, and a periodic solution in the distinguished direction is automaticallyAsymmetrical three-dimensional travelling gravity waves 3
doubly periodic and non necessarily symmetric with respect to the propa-
gation direction. The first mathematical results obtained by this method,
containing 3-dimensional doubly periodic travelling waves, start with Hara-
gus, Kirchga¨ssner, Groves and Mielke (2001) [11], [9], [13], generalized by
Groves and Haragus (2003) [10]. They use a hamiltonian formulation and
centermanifold reduction. This is essentiallybasedon the fact that the spec-
trum of the linearized operator is discrete and has only a finite number of
eigenvalues on the imaginary axis. These eigenvalues are related with the
dispersion relation mentioned above. Here, one component (or multiples of
such a component) of the wave vectorK is imposed in a direction transverse
to the distinguished one, and there is no restriction for the component ofK
in the distinguished direction, which, in solving the dispersion relation, gives
the eigenvalues of the linearized operator on the imaginary axis. However, it
appears that the number of imaginary eigenvalues becomes infinite when the
surface tension cancels, which prevents the use of center manifold reduction
in the limiting case we are considering in the present paper.
One essential difficulty here, with respect to the existing literature, ex-
cept ourpreviouswork[16],isthatwe assume the absence of surface tension.
Indeed the surface tension plays a major role in all existing proofs for three-
dimensional travelling gravity-capillary waves, and when the surface tension
is very small, which is the case in many usual situations, this implies a re-
duced domain of validity of results strongly dependent on the existence of
a non small surface tension. In our previous work [16] on three-dimensional
travelling gravity waves, there is no surface tension, and we restricted the
study to the existence of diamond waves: the periodic lattice is a diamond
lattice, and there are equal amplitudes at the leading order for the two basic
wave vectors K ,K symmetric with respect to the propagation direction1 2
of the waves. We proved the existence of bifurcating diamond gravity waves,
symmetric with respect to the propagation direction of the waves. In this
case,because of the absenceof surfacetension, a smalldivisorproblem arises
and since the use of Nash-Moser theorem is necessary, one of the essential
technical ingredients for the preparation of the differential near the origin
which needs to be inverted, is that the integral curves of the horizontal pro-
jection of the velocity field V of particles, may be transformed by a suitable
diffeomorphism of the torus, into straight lines, parallel to the direction of
propagation.This diffeomorphism was given by the solution of a simple ordi-
nary differential equation, thanks to the requiredsymmetries of the solution.
In the present work, we consider asymmetrical travelling gravity waves,
which implies that the basic wave vectors K and K have not the same1 2
length, and given the two amplitudes on the basic modes, the direction of
propagation u of the waves is part of the unknown. We need to find a dif-
feomorphism for ”preparing” the differential we have to invert, and this is
now a serious new difficulty here, leading to the necessity to compute the
rotation number ρ of the velocity field V, (which was ρ = 1 in the symmet-
ric case) for which a diophantine condition is now necessary. This leads to
our choice to consider as unknowns in our problem: the main unknowns (the
velocity potential and the shape of the free surface) together with the un-
known diffeomorphism and the associated rotation number of V. This allows4 G´erard Iooss & Pavel Plotnikov
us to use only once the Nash-Moser implicit function theorem. Finally we
are able to prove the existence of asymmetrical gravity waves (see Theorem
2) for nearly all choices of angles θ ,θ made by the basis of non symmetric1 2
wave vectors K ,K with the direction u of the critical velocity, and for1 2 0