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A Lagrangian approximation to the water-wave problem
PERGAMON
Applied Mathematics Letters
Applied Mathematics Letters 14 (2001) 789-795
www.elsevier.nl/locate/aml
A Lagrangian A p p r o x i m a t i o n to the Water-Wave P r o b l e m A. CONSTANTIN Department of Mathematics, Lund University P.O. Box 118, S-22100 Lurid, Sweden adrian, coast ant in.hath,
lu. se
(Received October 2000; accepted December 2000)
Communicated by B. Fuchssteiner Abstract--We d e s c r i b e t h e derivation by a vaxiational approach of the Camassa-Holm model for periodic shallow water waves. ~) 2001 Elsevier Science Ltd. All rights reserved.
K e y ' w o r d s - - W a t e r waves, Variational method. 1. I N T R O D U C T I O N Due to the degree of mathematical intractability of the generalwater-wave problem, m a n y approximate models for water waves have been suggested on the basis of various physical circumstances. These models capture one aspect or another of the general water-wave problem, the simplified problem allowing us to draw rigorous conclusions that explain rather satisfactorily a range of specific physical situations which m a y be observed experimentally. A celebrated model is the Korteweg-de Vries (KdV) equation ut + uux + ux~x = 0, written in normalized form, which models solitons and is a completely integrable infinite-dimensional Hamiltonian system (cf. [1]). Here u ( t , x ) is the horizontal velocity component of the water, or equivalently, the water's free surface over a flat bottom. The K d V equation does not model the occurrence of breaking for water waves (cf. [2]). The concept of wave breaking is intriguing because some water waves will certainly break, so t h a t eventual failure (solutionsexisting for only a short time) m a y be a quite legitimate feature of a water-wave model. Camassa and Holm [3] proposed the equation 1 ut - utxx + 2wu~ + 3uux = 2u~uxx + uux~x,
t > O,
x E ~,
(1.1)
written in normalized form, as a model for the unidirectional propagation of waves on water, the constant w ¢ 0 being related to the critical shallow water speed. Equation (1.1) models b o t h soliton interaction (see [3]) and wave breaking (see [3,5]) and is a completely integrable infinite-dimensional Hamiltonian system [6]2. There are numerous recent papers devoted to the s t u d y of (1.1). However, the approximation procedure by which the Camassa-Holm equation was 1The same equation was found using the method of recursion operators by Fuchssteiner and Fokas [4] as a bi-Hamiltonian equation with an infinite number of conserved functionals. 2In this context, let us mention that (1.1) is a counterexaxnple to a famous conjecture o n t h e complete integrability of nonlinear partial differential equations, the Painlev~ test (see [7]). 0893-9659/01/$ - see front matter © 2001 Elsevier Science Ltd. All rights reserved. PII: S0893-9659(01)00045-3
Typeset by A~cIS-TEX
790
A. CONSTANTIN
obtained originally [3] appears not to be fully satisfactory--for the various shortcomings and for a consistent derivation by the method of asymptotic expansions, we refer to [8]. The present paper is concerned with an alternative derivation of the Camassa-Holm equation for periodic waves, using a variational approach in the Lagrangian formalism. The main impetus for the results reported here comes from the observation that, as a m a t t e r of common experience, the waves observed on a beach or in a channel are approximately periodic. This motivates the need for a theory of shallow water waves t h a t are periodic. As some of these waves b r e a k - - a phenomenon t h a t linear wave theory cannot a c c o m m o d a t e - - t h e relevant approach has to be nonlinear. We submit t h a t the Camassa-Holm equation is of relevance in this theory. 2. G O V E R N I N G EQUATIONS FOR PLANE PROGRESSIVE WAVES As a starting point, we recall the general problem for the unidirectional propagation of plane water waves in order to bring out several points of interest. We consider the propagation of a periodic plane progressive wave of small amplitude in the x-direction over water of constant depth z = 0. The water is stationary in its undisturbed state 3, z = h0 (for a constant h0 > 0) being the undisturbed water surface, and the effects of surface tension are ignored. Let (u(t, x, z), v(t, x, z)) be the velocity of the w a t e r - - n o motion takes place in the y-direction, so t h a t there is only one horizontal component (in the direction of propagation) and the vertical component of the velocity field to be taken into account. The periodicity assumption means a periodic dependence in the x-variable (of, say, period one) of the velocity field. Let z = h0 + ~(t, x) be the water's free surface. Incompressibility (constant density p) is a good approximation for water (cf. [9]) so t h a t we have the equation of mass conservation in the form
ux + Vz = O.
(2.1)
Under the assumption t h a t water is inviscid, the equations of motion are Euler's equations (see [2])
Du - Dt
-
1 p
Px,
Dv Dt
-
1 p
Pz
-
g,
(2.2)
where P(t, x, z) denotes the pressure, g is the gravitational acceleration constant, and D is the material time derivative, ~ t = OfOt + u, °o~ x + v, °o-~z. The continuity equation (2.1) and Euler's equations (2.2) are the exact equations of motion for the velocity field. Let us now present the b o u n d a r y conditions which select the water-wave problem from all other possible solutions of equations (2.1),(2.2). The dynamic boundary condition P = P0,
on z = h0 + ~(t, x),
(2.3)
P0 being the constant atmospheric pressure, decouples the motion of the air from t h a t of the water. T h e first kinematic boundary condition is v = Ut + u~x,
on z = h0 + ~(t, x),
(2.4)
expressing the fact t h a t the same particles always form the free water surface, i.e., D (z -- h0 ~) = 0. Finally, there is no flow normal to the horizontal bed so that we obtain the second kinematic b o u n d a r y condition v = 0,
on z = 0.
3The wave being created by, say, the action of the wind blowing over the still water surface.
(2.5)
Water-Wave Problem
791
T h e general description of the p r o p a g a t i o n of a plane progressive wave is e n c o m p a s s e d by equations (2.1)-(2.5). T h e p r o b l e m is nondimensionalised using a typical wavelength L and a typical a m p l i t u d e of the wave a (cf. [2]). T h a t is, we define the set of nondimensional variables t~tL x ~ L x , z ~ hoz, - -gv~o ' u H u v / ~ o ,
I
] v ~ tvhovlgho L '7?~--~arl '
it being u n d e r s t o o d t h a t x is replaced by L x , so t h a t afterwards, the s y m b o l x stands for a nondimensional variable, etc.; this way we avoid new notations. We write the pressure in the new nondimensional variables as P = Po + pgho(1 - z) + pghop, with the nondimensional pressure variable p m e a s u r i n g the deviation from the hydrostatic pressure distribution: if the water would be s t a t i o n a r y (u - v = 0), the second equation in (2.2) coupled with (2.3) yields P = P0 + p g h o ( 1 - z). We obtain the following b o u n d a r y value p r o b l e m in nondimensional variables: Ut -I- UUx + VUz = --Px, 6 ~ (vt + uv~ + vvz) = - p z , Ux -4- Vz = 0 , p=e~
and
(2.6)
v = e(~t + u ~ ) , v = O,
on z = 1 + e V ,
on z = O,
where ¢ = a / h o is the a m p l i t u d e p a r a m e t e r and 6 = h o / L is the shallowness p a r a m e t e r . F r o m the fourth line in (2.6), it is a p p a r e n t t h a t b o t h v and p, if evaluated on z = 1 + c~], are essentially p r o p o r t i o n a l to e. This is consistent with the fact t h a t as e --* 0, we must have v --~ 0 and p --* 0 (with no disturbance, the free surface becomes a horizontal surface on which v = p = 0). W i t h t h e scaling [p H ep, (u, v) ~-~ ¢(u, v)], of the nondimensional variables (avoiding again the i n t r o d u c t i o n of a new notation), p r o b l e m (2.6) becomes ut + e(uux + VUz) = - P x ,
62{~ + e(uvx + V~z)} = -pz, u z + Vz = 0,
p=rl
and
V=rh+eUrlx,
v--O,
3. T H E
(2.7) onz=l+e~,
onz=0.
APPROXIMATION
PROCEDURE
T h e nondimensionalisation and scaling presented above will be useful in obtaining a scheme of a p p r o x i m a t i o n of the governing equations and b o u n d a r y conditions to obtain the single periodic C a m a s s a - H o l m equation ut -- u t z x + 3 u u z = 2 u z U z x + UUxxx,
t >0,
X E S.
(3.1)
Here S stands for the circle of length one (identified with ]R modulo 1) and the c o n s t a n t w was scaled out by defining u ( t , x) = g(t, x - w t ) + w with g(t, x) satisfying (1.1). For shallow water, we impose the condition 6 --* 0, so Pz = O(52) from the second e q u a t i o n in (2.6). To leading order (ignoring corrections of order 62), we make the simplifying a s s u m p t i o n t h a t p is independent of z. Then, from the first equation in (2.6), we infer t h a t the horizontal acceleration ~ t = 0 for all particles in a plane x = constant. Since the water was once at rest (in its u n d i s t u r b e d state), it follows t h a t u is the same for all particles in this plane, which remains a plane with its x - c o o r d i n a t e constant, so t h a t u is independent of z, i.e., u = u ( t , x ) . T h e third and last e q u a t i o n in (2.7) yields v = -zuz, (3.2)
792
A. CONSTANTIN
t h a t is, the vertical velocity of any particle is proportional to its height above the bottom. The r u d i m e n t a r y approximation on the free surface of (2.7) for small e, taking into account (3.2), is ut + r/~ = 0 and rh + uz = 0. The general solution of this simple system is rl = u = F(x - t), where F is any differentiable function, and thus, rl ~ u
(3.3)
to a first approximation. By consistently neglecting the c contribution, we will derive equation (3.1) governing unidirectional propagation. Our approach is based on the use of variational methods in the Lagrangian formalism. T h e motion of a mechanical system is described by a path of diffeomorphisms ~(t, .) of the ambient space. The knowledge of ~'(t, .) gives the configuration of the particles at time t. The Lie group 79 of smooth orientation-preserving diffeomorphisms of the circle ~ represents the configuration space for the spatially periodic motion of one-dimensional mechanical systems. T h e material velocity field is defined by (t, x) H ~/t(t, x), while the spatial velocity field is given by w(t,y) = ~/t(t,x), where y = 7(t,x), i.e., w(t,.) = ~/t o ~/-1. In terms of w, we have the Eulerian description (from the viewpoint of a fixed observer), while in terms of (% 7t), we have the Lagrangian description (the motion as seen from one of the particles). The velocity phase space is the tangent bundle T79 of 79. Let G be the tangent space at the identity Id. The Lagrangian is a scalar function £ : T79 --* R and the action along a path {7(t), 0 < t < T} in 79 is defined as a(~/) = f ? £(~/, ~/t) dr. The action principle states t h a t the equation of motion is the equatipn satisfied by a critical point of the action in the space of curves on 79, the paths {7(t), 0 < t < T} over which we are extremizing satisfying the fixed-end conditions 7(0) = "70 and 7(T) = 71- Note the following right-invariance property: if we replace the path t ~-* ~/(t) by t ~-~ 7 ( t ) o ¢ for a fixed time-independent ~ E 79, then the spatial velocity w = ~/t o 7 -1 is unchanged. This suggests that we consider a right-invariant Lagrangian/2 : T79 --~ IR by transporting a certain inner product K : G -~ lI~ to all tangent spaces of 79 by means of right translations. The most natural choice for K is the kinetic energy 4. For a progressive plane wave, no motion takes place in the y direction and, as a particle on the surface will always stay on the surface, we may regard the motion as that of a one-dimensional compressible m e m b r a n e - i n c o m p r e s s i b i l i t y in one dimension would mean linear motion and the incompressibility of the water is expressed in the relation (3.2) if we look at the whole water body. Taking into account (3.2), the kinetic energy on the surface (over one period) is K = l f s (u 2 + v 2)
= 19~S (u + (1 +
dx
lj~
+
to the order of our approximation. In accordance with the above considerations, we transform K to a right-invariant Lagrangian £; the corresponding action on a path {7(t), 0 < t < T} on D being .(.y)
=
1/Tfg{(~'toT-l)2-1-
[Oz(3'to')'-1)]2} dxdt.
Let us now find the condition for a p a t h 7 : [0, T] --* 79, parameterized by arc length, to be a critical point of the action in the space of paths with fixed endpoints, i.e., d +
I =0 = 0,
4With this choice, for the case of a perfect fluid moving in a bounded smooth domain M C R k, k = 2, 3, having the group of all volume-preserving diffeomorphisms of M as configuration space, the critical point condition for the action is Euler's equation (el. [10]).
Water-Wave
Problem
793
for every path ‘p : [0, T] --+ Coo(S) with endpoints at zero and such that y+ccp is a small variation of y on D. We have that
-$(y
+ cP)I,=s = 2;
dl
Yt + W)
ssaT s{(
0 (Y + ELp)-1}2 dzdtl,=o
T s0
+z2
2
1 (Ytz+~(Ptzbh+wY~
/Is
=~T~hv-l)
(lz + WI) O (7 + V-l {$
[h
+ vt)
0 (Y + VI-‘]
IcTo
I
dx dtl,,,
I dxdt
T +
4
ss0
(rt
0 r-l)
. L--/z 0 7-l
s
{
I Y~~+E(P~~)o(~+E(P)-~]I~=~
z
T
+
ss0
da:& 1
II >
1 9
8,
(rt Or-‘)
. (Ytz
0 7-l)
J!-
de
{
[
(rz + ~5)
0 (Y + ~)-l
t=O
dx dt.
Note that [( Yt + Wt) O (Y + Wl]
& ,
= (Pt 0-f-l
- (rtr 0-f-l)
lecO = ‘Pt O7-l ‘POP Yx 0 7-l
+ (79%OY -‘>
;
(7 + 4-11,=o (3.4)
= lpt 0 y-l
- wY-lPz(Yt
wl),
as differentiation with respect to E in the relation (y + E(P)o (y + ccp)-’ = Id leads to Yr O (Y + e&l
. ~(Y+~~)-l+r~~o(T+‘is)-l
~(1’+~~)-l+v^O(T+~r’)-l=O.
and therefore, $ (y + ~(p)-~l~,c = -(cp o r-‘)/(~~ we see that f
[( Yta: + vtz)
0 (Y + 4-l]
0 y-l).
IEZO= vtx 0 7-l
Along the lines used to prove (3.4),
+ (Ytzz 0 Y -‘> . f
(7 + E’p)-ll,=o
-1
(Ptx 0 7-l
=
- (Ytm 0 r-l)
Jz”oy,_,
= (rz 0 r-l)
& (cpt 0 r-l) [
-(‘POY-l)a~(YtOY-l)-(“itzOY-l)(YzzOY-l)
pay-1 (rz 0 7-lj3
(3.5)
1 .
Similarly, f
[( x-c + VZ) 0 (Y + W-l]
lecO = CPZO7-l
+ (YZZ OY-‘> . -$ (7 + 4-1J,=o = cpx 0 7-l
- (rm 0 r-‘)
v-Y-l YZ 07-l’
cpxoY-l-(YzzoY-l )bPv-')/(Yz OY-l) z c-h fwz) O(7 +v)-l E=o =II h~oy-1~2
so that
d
1
[
From (3.4)-(3.6),
we infer that T
$a
+
oT IS
(7 + 41e=0
=
o ss
s(Ytv-l)
[WY1
Ba,(~toY-1).[a,(~toy-1)-(~oy-1)~~(”ito~-1)-a,(~oy-‘)n,(-itor-1)]
-
(‘pv+)
&
(rt
or-‘)]
dxdt
kc&.
794
A. CONSTANTIN
Denoting Vt o ~,-1 __ u, the above expression becomes
U[C~t(~OV-1)--?~x(~OV-1)-~-?~Ox(~DO~y-1)] dxdt
~ . (v + ~)1~=o =
/
since
0t ( ~ o ~ -1) = ~ o~-1 + ~ o ~ - 1 . 0 t (~-1) = ~ o~-1 _ ~ / so t h a t
o ~ - 1 . ~t o~-1 ~'x ° V - 1 '
/
~t o ~-1 = 0~ (~ o ~-1) + uax (~ o ~-1). Indeed, differentiating/the relation ~/o V -1 = Id with respect to time, we get Vt o V-1 + 7x o 7 -1 • Ot(~ -1) = 0, which ~ v e s the desired expression for at(~/-1). Ifitegrating by parts with respect to t and x in the above formula for the derivative of the action functional, we obtain
.
(~ +/~)I~=0 =
-
(~ °
~-')
[u~ - u ~ =
+ 3uux
-
2u:u~
- uux~x]
dz dr.
The resulting Euler-Lagrange equation is (3.1), where u = Vt o V-1 and t ~-* v(t) E D is the curve (parameterized by arc length) yielding the critical point of the action functional in the space of paths. This completes the derivation of the periodic Camassa-Holm equation by an approximation procedure which allows its deduction from a variational principle. We would like to point out the fact t h a t '(3.1) is a reexpression of geodesic flow in the group of smooth diffeomorphisms of the circle--a fact already noted in [11], which can be seen as the relevant point in our derivation. Notice t h a i the length of a path does n o t depend on its parameterization and that for a p a t h parameterized by arc length, its length is proportional to a(V). J
/
Generated by a water-wave problem, the Camassa-Holm equation began to make its appearance in other branches of applied mathematics and physics also. Equation (3.1) was found by Dai [12] to be a model for/~onlinear waves in cylindrical hyperelastic rods with u(t, x) representing the radial stretch relgtive to a prestressed state. The viscous three-dimensional generalization of the Camassa-Holm equation can be used as the basis for a turbulence closure model [13] and was considered and studied in the theory of second-grade fluids [14]. REMARK.
/
REFERENCES 1. P. Drazin and R. Johnson, Solitons: An Introduction, Cambridge University Press, Cambridge, (1992). 2. R.S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge University Press, (1997). 3. R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Letters 71, 1661-1664 (1993). 4. B. Fuchssteiner and A. Fokas, Symplectic structures, their B~cklund transformation and hereditary symmetries, Physica D 4, 47-66 (1981). 5. A. Constantin and J: Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Mathematica 181, 229-243 (1998). 6. A. Constantin and H.P. McKean, A shallow water equation on the circle, Comm. Pure Appl. Math. 52, 949-982 (1999). 7. C. Gilson and A. Pickering, Factorization and Painlev6 analysis of a class of nonlinear third-order partial differential equations, J. Phys. A 28, 2871-2888 (1995). 8. R.S. Johnson, Camassa-Holm, Korteweg-deVries and related models for water waves, J. Fluid Mech. (submitted). 9. G. Crapper, Introduction to Water Waves, Ellis Horwood, Chichester, (1984). 10. V. Arnold, Sur la g6ometrie diff@rentielle des groupes de Lie de dimension infinie et ses applications l'hydrodynamique des fluides parfaits, Ann. Inst. Fourier (Grenoble) 16, 319-361, (1966).
Water-Wave Problem
795
11. G. Misiolek, A shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys. 24, 203-208 (1998). 12. H.-H. Dai, Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod, Acta Mechanica 127, 293-308 (1998). 13. S. Chen, C. Foias, E. Olson, E. Titi and S. Wynne, A connection between the Camassa-Holm equations and turbulent flows in channels and pipes, Phys. Fluids 11, 2343-2353 (1999). 14. D. Cioranescu and V. Girault, Solutions variationnelles et classiques d'une famille de fiuides de grade deux, C. R. Acad. Sci. Paris 322, 1163-1168 (1996).
Applied Mathematics Letters
Applied Mathematics Letters 14 (2001) 789-795
www.elsevier.nl/locate/aml
A Lagrangian A p p r o x i m a t i o n to the Water-Wave P r o b l e m A. CONSTANTIN Department of Mathematics, Lund University P.O. Box 118, S-22100 Lurid, Sweden adrian, coast ant in.hath,
lu. se
(Received October 2000; accepted December 2000)
Communicated by B. Fuchssteiner Abstract--We d e s c r i b e t h e derivation by a vaxiational approach of the Camassa-Holm model for periodic shallow water waves. ~) 2001 Elsevier Science Ltd. All rights reserved.
K e y ' w o r d s - - W a t e r waves, Variational method. 1. I N T R O D U C T I O N Due to the degree of mathematical intractability of the generalwater-wave problem, m a n y approximate models for water waves have been suggested on the basis of various physical circumstances. These models capture one aspect or another of the general water-wave problem, the simplified problem allowing us to draw rigorous conclusions that explain rather satisfactorily a range of specific physical situations which m a y be observed experimentally. A celebrated model is the Korteweg-de Vries (KdV) equation ut + uux + ux~x = 0, written in normalized form, which models solitons and is a completely integrable infinite-dimensional Hamiltonian system (cf. [1]). Here u ( t , x ) is the horizontal velocity component of the water, or equivalently, the water's free surface over a flat bottom. The K d V equation does not model the occurrence of breaking for water waves (cf. [2]). The concept of wave breaking is intriguing because some water waves will certainly break, so t h a t eventual failure (solutionsexisting for only a short time) m a y be a quite legitimate feature of a water-wave model. Camassa and Holm [3] proposed the equation 1 ut - utxx + 2wu~ + 3uux = 2u~uxx + uux~x,
t > O,
x E ~,
(1.1)
written in normalized form, as a model for the unidirectional propagation of waves on water, the constant w ¢ 0 being related to the critical shallow water speed. Equation (1.1) models b o t h soliton interaction (see [3]) and wave breaking (see [3,5]) and is a completely integrable infinite-dimensional Hamiltonian system [6]2. There are numerous recent papers devoted to the s t u d y of (1.1). However, the approximation procedure by which the Camassa-Holm equation was 1The same equation was found using the method of recursion operators by Fuchssteiner and Fokas [4] as a bi-Hamiltonian equation with an infinite number of conserved functionals. 2In this context, let us mention that (1.1) is a counterexaxnple to a famous conjecture o n t h e complete integrability of nonlinear partial differential equations, the Painlev~ test (see [7]). 0893-9659/01/$ - see front matter © 2001 Elsevier Science Ltd. All rights reserved. PII: S0893-9659(01)00045-3
Typeset by A~cIS-TEX
790
A. CONSTANTIN
obtained originally [3] appears not to be fully satisfactory--for the various shortcomings and for a consistent derivation by the method of asymptotic expansions, we refer to [8]. The present paper is concerned with an alternative derivation of the Camassa-Holm equation for periodic waves, using a variational approach in the Lagrangian formalism. The main impetus for the results reported here comes from the observation that, as a m a t t e r of common experience, the waves observed on a beach or in a channel are approximately periodic. This motivates the need for a theory of shallow water waves t h a t are periodic. As some of these waves b r e a k - - a phenomenon t h a t linear wave theory cannot a c c o m m o d a t e - - t h e relevant approach has to be nonlinear. We submit t h a t the Camassa-Holm equation is of relevance in this theory. 2. G O V E R N I N G EQUATIONS FOR PLANE PROGRESSIVE WAVES As a starting point, we recall the general problem for the unidirectional propagation of plane water waves in order to bring out several points of interest. We consider the propagation of a periodic plane progressive wave of small amplitude in the x-direction over water of constant depth z = 0. The water is stationary in its undisturbed state 3, z = h0 (for a constant h0 > 0) being the undisturbed water surface, and the effects of surface tension are ignored. Let (u(t, x, z), v(t, x, z)) be the velocity of the w a t e r - - n o motion takes place in the y-direction, so t h a t there is only one horizontal component (in the direction of propagation) and the vertical component of the velocity field to be taken into account. The periodicity assumption means a periodic dependence in the x-variable (of, say, period one) of the velocity field. Let z = h0 + ~(t, x) be the water's free surface. Incompressibility (constant density p) is a good approximation for water (cf. [9]) so t h a t we have the equation of mass conservation in the form
ux + Vz = O.
(2.1)
Under the assumption t h a t water is inviscid, the equations of motion are Euler's equations (see [2])
Du - Dt
-
1 p
Px,
Dv Dt
-
1 p
Pz
-
g,
(2.2)
where P(t, x, z) denotes the pressure, g is the gravitational acceleration constant, and D is the material time derivative, ~ t = OfOt + u, °o~ x + v, °o-~z. The continuity equation (2.1) and Euler's equations (2.2) are the exact equations of motion for the velocity field. Let us now present the b o u n d a r y conditions which select the water-wave problem from all other possible solutions of equations (2.1),(2.2). The dynamic boundary condition P = P0,
on z = h0 + ~(t, x),
(2.3)
P0 being the constant atmospheric pressure, decouples the motion of the air from t h a t of the water. T h e first kinematic boundary condition is v = Ut + u~x,
on z = h0 + ~(t, x),
(2.4)
expressing the fact t h a t the same particles always form the free water surface, i.e., D (z -- h0 ~) = 0. Finally, there is no flow normal to the horizontal bed so that we obtain the second kinematic b o u n d a r y condition v = 0,
on z = 0.
3The wave being created by, say, the action of the wind blowing over the still water surface.
(2.5)
Water-Wave Problem
791
T h e general description of the p r o p a g a t i o n of a plane progressive wave is e n c o m p a s s e d by equations (2.1)-(2.5). T h e p r o b l e m is nondimensionalised using a typical wavelength L and a typical a m p l i t u d e of the wave a (cf. [2]). T h a t is, we define the set of nondimensional variables t~tL x ~ L x , z ~ hoz, - -gv~o ' u H u v / ~ o ,
I
] v ~ tvhovlgho L '7?~--~arl '
it being u n d e r s t o o d t h a t x is replaced by L x , so t h a t afterwards, the s y m b o l x stands for a nondimensional variable, etc.; this way we avoid new notations. We write the pressure in the new nondimensional variables as P = Po + pgho(1 - z) + pghop, with the nondimensional pressure variable p m e a s u r i n g the deviation from the hydrostatic pressure distribution: if the water would be s t a t i o n a r y (u - v = 0), the second equation in (2.2) coupled with (2.3) yields P = P0 + p g h o ( 1 - z). We obtain the following b o u n d a r y value p r o b l e m in nondimensional variables: Ut -I- UUx + VUz = --Px, 6 ~ (vt + uv~ + vvz) = - p z , Ux -4- Vz = 0 , p=e~
and
(2.6)
v = e(~t + u ~ ) , v = O,
on z = 1 + e V ,
on z = O,
where ¢ = a / h o is the a m p l i t u d e p a r a m e t e r and 6 = h o / L is the shallowness p a r a m e t e r . F r o m the fourth line in (2.6), it is a p p a r e n t t h a t b o t h v and p, if evaluated on z = 1 + c~], are essentially p r o p o r t i o n a l to e. This is consistent with the fact t h a t as e --* 0, we must have v --~ 0 and p --* 0 (with no disturbance, the free surface becomes a horizontal surface on which v = p = 0). W i t h t h e scaling [p H ep, (u, v) ~-~ ¢(u, v)], of the nondimensional variables (avoiding again the i n t r o d u c t i o n of a new notation), p r o b l e m (2.6) becomes ut + e(uux + VUz) = - P x ,
62{~ + e(uvx + V~z)} = -pz, u z + Vz = 0,
p=rl
and
V=rh+eUrlx,
v--O,
3. T H E
(2.7) onz=l+e~,
onz=0.
APPROXIMATION
PROCEDURE
T h e nondimensionalisation and scaling presented above will be useful in obtaining a scheme of a p p r o x i m a t i o n of the governing equations and b o u n d a r y conditions to obtain the single periodic C a m a s s a - H o l m equation ut -- u t z x + 3 u u z = 2 u z U z x + UUxxx,
t >0,
X E S.
(3.1)
Here S stands for the circle of length one (identified with ]R modulo 1) and the c o n s t a n t w was scaled out by defining u ( t , x) = g(t, x - w t ) + w with g(t, x) satisfying (1.1). For shallow water, we impose the condition 6 --* 0, so Pz = O(52) from the second e q u a t i o n in (2.6). To leading order (ignoring corrections of order 62), we make the simplifying a s s u m p t i o n t h a t p is independent of z. Then, from the first equation in (2.6), we infer t h a t the horizontal acceleration ~ t = 0 for all particles in a plane x = constant. Since the water was once at rest (in its u n d i s t u r b e d state), it follows t h a t u is the same for all particles in this plane, which remains a plane with its x - c o o r d i n a t e constant, so t h a t u is independent of z, i.e., u = u ( t , x ) . T h e third and last e q u a t i o n in (2.7) yields v = -zuz, (3.2)
792
A. CONSTANTIN
t h a t is, the vertical velocity of any particle is proportional to its height above the bottom. The r u d i m e n t a r y approximation on the free surface of (2.7) for small e, taking into account (3.2), is ut + r/~ = 0 and rh + uz = 0. The general solution of this simple system is rl = u = F(x - t), where F is any differentiable function, and thus, rl ~ u
(3.3)
to a first approximation. By consistently neglecting the c contribution, we will derive equation (3.1) governing unidirectional propagation. Our approach is based on the use of variational methods in the Lagrangian formalism. T h e motion of a mechanical system is described by a path of diffeomorphisms ~(t, .) of the ambient space. The knowledge of ~'(t, .) gives the configuration of the particles at time t. The Lie group 79 of smooth orientation-preserving diffeomorphisms of the circle ~ represents the configuration space for the spatially periodic motion of one-dimensional mechanical systems. T h e material velocity field is defined by (t, x) H ~/t(t, x), while the spatial velocity field is given by w(t,y) = ~/t(t,x), where y = 7(t,x), i.e., w(t,.) = ~/t o ~/-1. In terms of w, we have the Eulerian description (from the viewpoint of a fixed observer), while in terms of (% 7t), we have the Lagrangian description (the motion as seen from one of the particles). The velocity phase space is the tangent bundle T79 of 79. Let G be the tangent space at the identity Id. The Lagrangian is a scalar function £ : T79 --* R and the action along a path {7(t), 0 < t < T} in 79 is defined as a(~/) = f ? £(~/, ~/t) dr. The action principle states t h a t the equation of motion is the equatipn satisfied by a critical point of the action in the space of curves on 79, the paths {7(t), 0 < t < T} over which we are extremizing satisfying the fixed-end conditions 7(0) = "70 and 7(T) = 71- Note the following right-invariance property: if we replace the path t ~-* ~/(t) by t ~-~ 7 ( t ) o ¢ for a fixed time-independent ~ E 79, then the spatial velocity w = ~/t o 7 -1 is unchanged. This suggests that we consider a right-invariant Lagrangian/2 : T79 --~ IR by transporting a certain inner product K : G -~ lI~ to all tangent spaces of 79 by means of right translations. The most natural choice for K is the kinetic energy 4. For a progressive plane wave, no motion takes place in the y direction and, as a particle on the surface will always stay on the surface, we may regard the motion as that of a one-dimensional compressible m e m b r a n e - i n c o m p r e s s i b i l i t y in one dimension would mean linear motion and the incompressibility of the water is expressed in the relation (3.2) if we look at the whole water body. Taking into account (3.2), the kinetic energy on the surface (over one period) is K = l f s (u 2 + v 2)
= 19~S (u + (1 +
dx
lj~
+
to the order of our approximation. In accordance with the above considerations, we transform K to a right-invariant Lagrangian £; the corresponding action on a path {7(t), 0 < t < T} on D being .(.y)
=
1/Tfg{(~'toT-l)2-1-
[Oz(3'to')'-1)]2} dxdt.
Let us now find the condition for a p a t h 7 : [0, T] --* 79, parameterized by arc length, to be a critical point of the action in the space of paths with fixed endpoints, i.e., d +
I =0 = 0,
4With this choice, for the case of a perfect fluid moving in a bounded smooth domain M C R k, k = 2, 3, having the group of all volume-preserving diffeomorphisms of M as configuration space, the critical point condition for the action is Euler's equation (el. [10]).
Water-Wave
Problem
793
for every path ‘p : [0, T] --+ Coo(S) with endpoints at zero and such that y+ccp is a small variation of y on D. We have that
-$(y
+ cP)I,=s = 2;
dl
Yt + W)
ssaT s{(
0 (Y + ELp)-1}2 dzdtl,=o
T s0
+z2
2
1 (Ytz+~(Ptzbh+wY~
/Is
=~T~hv-l)
(lz + WI) O (7 + V-l {$
[h
+ vt)
0 (Y + VI-‘]
IcTo
I
dx dtl,,,
I dxdt
T +
4
ss0
(rt
0 r-l)
. L--/z 0 7-l
s
{
I Y~~+E(P~~)o(~+E(P)-~]I~=~
z
T
+
ss0
da:& 1
II >
1 9
8,
(rt Or-‘)
. (Ytz
0 7-l)
J!-
de
{
[
(rz + ~5)
0 (Y + ~)-l
t=O
dx dt.
Note that [( Yt + Wt) O (Y + Wl]
& ,
= (Pt 0-f-l
- (rtr 0-f-l)
lecO = ‘Pt O7-l ‘POP Yx 0 7-l
+ (79%OY -‘>
;
(7 + 4-11,=o (3.4)
= lpt 0 y-l
- wY-lPz(Yt
wl),
as differentiation with respect to E in the relation (y + E(P)o (y + ccp)-’ = Id leads to Yr O (Y + e&l
. ~(Y+~~)-l+r~~o(T+‘is)-l
~(1’+~~)-l+v^O(T+~r’)-l=O.
and therefore, $ (y + ~(p)-~l~,c = -(cp o r-‘)/(~~ we see that f
[( Yta: + vtz)
0 (Y + 4-l]
0 y-l).
IEZO= vtx 0 7-l
Along the lines used to prove (3.4),
+ (Ytzz 0 Y -‘> . f
(7 + E’p)-ll,=o
-1
(Ptx 0 7-l
=
- (Ytm 0 r-l)
Jz”oy,_,
= (rz 0 r-l)
& (cpt 0 r-l) [
-(‘POY-l)a~(YtOY-l)-(“itzOY-l)(YzzOY-l)
pay-1 (rz 0 7-lj3
(3.5)
1 .
Similarly, f
[( x-c + VZ) 0 (Y + W-l]
lecO = CPZO7-l
+ (YZZ OY-‘> . -$ (7 + 4-1J,=o = cpx 0 7-l
- (rm 0 r-‘)
v-Y-l YZ 07-l’
cpxoY-l-(YzzoY-l )bPv-')/(Yz OY-l) z c-h fwz) O(7 +v)-l E=o =II h~oy-1~2
so that
d
1
[
From (3.4)-(3.6),
we infer that T
$a
+
oT IS
(7 + 41e=0
=
o ss
s(Ytv-l)
[WY1
Ba,(~toY-1).[a,(~toy-1)-(~oy-1)~~(”ito~-1)-a,(~oy-‘)n,(-itor-1)]
-
(‘pv+)
&
(rt
or-‘)]
dxdt
kc&.
794
A. CONSTANTIN
Denoting Vt o ~,-1 __ u, the above expression becomes
U[C~t(~OV-1)--?~x(~OV-1)-~-?~Ox(~DO~y-1)] dxdt
~ . (v + ~)1~=o =
/
since
0t ( ~ o ~ -1) = ~ o~-1 + ~ o ~ - 1 . 0 t (~-1) = ~ o~-1 _ ~ / so t h a t
o ~ - 1 . ~t o~-1 ~'x ° V - 1 '
/
~t o ~-1 = 0~ (~ o ~-1) + uax (~ o ~-1). Indeed, differentiating/the relation ~/o V -1 = Id with respect to time, we get Vt o V-1 + 7x o 7 -1 • Ot(~ -1) = 0, which ~ v e s the desired expression for at(~/-1). Ifitegrating by parts with respect to t and x in the above formula for the derivative of the action functional, we obtain
.
(~ +/~)I~=0 =
-
(~ °
~-')
[u~ - u ~ =
+ 3uux
-
2u:u~
- uux~x]
dz dr.
The resulting Euler-Lagrange equation is (3.1), where u = Vt o V-1 and t ~-* v(t) E D is the curve (parameterized by arc length) yielding the critical point of the action functional in the space of paths. This completes the derivation of the periodic Camassa-Holm equation by an approximation procedure which allows its deduction from a variational principle. We would like to point out the fact t h a t '(3.1) is a reexpression of geodesic flow in the group of smooth diffeomorphisms of the circle--a fact already noted in [11], which can be seen as the relevant point in our derivation. Notice t h a i the length of a path does n o t depend on its parameterization and that for a p a t h parameterized by arc length, its length is proportional to a(V). J
/
Generated by a water-wave problem, the Camassa-Holm equation began to make its appearance in other branches of applied mathematics and physics also. Equation (3.1) was found by Dai [12] to be a model for/~onlinear waves in cylindrical hyperelastic rods with u(t, x) representing the radial stretch relgtive to a prestressed state. The viscous three-dimensional generalization of the Camassa-Holm equation can be used as the basis for a turbulence closure model [13] and was considered and studied in the theory of second-grade fluids [14]. REMARK.
/
REFERENCES 1. P. Drazin and R. Johnson, Solitons: An Introduction, Cambridge University Press, Cambridge, (1992). 2. R.S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge University Press, (1997). 3. R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Letters 71, 1661-1664 (1993). 4. B. Fuchssteiner and A. Fokas, Symplectic structures, their B~cklund transformation and hereditary symmetries, Physica D 4, 47-66 (1981). 5. A. Constantin and J: Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Mathematica 181, 229-243 (1998). 6. A. Constantin and H.P. McKean, A shallow water equation on the circle, Comm. Pure Appl. Math. 52, 949-982 (1999). 7. C. Gilson and A. Pickering, Factorization and Painlev6 analysis of a class of nonlinear third-order partial differential equations, J. Phys. A 28, 2871-2888 (1995). 8. R.S. Johnson, Camassa-Holm, Korteweg-deVries and related models for water waves, J. Fluid Mech. (submitted). 9. G. Crapper, Introduction to Water Waves, Ellis Horwood, Chichester, (1984). 10. V. Arnold, Sur la g6ometrie diff@rentielle des groupes de Lie de dimension infinie et ses applications l'hydrodynamique des fluides parfaits, Ann. Inst. Fourier (Grenoble) 16, 319-361, (1966).
Water-Wave Problem
795
11. G. Misiolek, A shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys. 24, 203-208 (1998). 12. H.-H. Dai, Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod, Acta Mechanica 127, 293-308 (1998). 13. S. Chen, C. Foias, E. Olson, E. Titi and S. Wynne, A connection between the Camassa-Holm equations and turbulent flows in channels and pipes, Phys. Fluids 11, 2343-2353 (1999). 14. D. Cioranescu and V. Girault, Solutions variationnelles et classiques d'une famille de fiuides de grade deux, C. R. Acad. Sci. Paris 322, 1163-1168 (1996).