The Camassa-Holm equation as the long-wave limit of the improved Boussinesq equation and of a class of nonlocal wave equations

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by  H. A. Erbay

THE CAMASSA-HOLM EQUATION AS THE LONG-WAVE LIMIT OF
THE IMPROVED BOUSSINESQ EQUATION AND OF A CLASS OF
NONLOCAL WAVE EQUATIONS
H.A. Erbay and S. Erbay
6 ∗
1
Department of Natural and Mathematical Sciences, Faculty of Engineering,
0
2 Ozyegin University,
Cekmekoy 34794,Istanbul, Turkey
n
a A. Erkip
J

9 Faculty of Engineering and Natural Sciences,

Sabanci University,
]
P Tuzla 34956,Istanbul, Turkey
A
.
h
Abstract
t
a
m In the present study we prove rigorously that in the long-wave
[ limit, the unidirectional solutions of a class of nonlocal wave equa-

tions to which the improved Boussinesq equation belongs are well
1
v approximated by the solutions of the Camassa-Holm equation over
4
a long time scale. This general class of nonlocal wave equations model
5
bidirectional wave propagation in a nonlocally and nonlinearly elastic
1
2 mediumwhoseconstitutiveequationisgivenbyaconvolutionintegral.
0
TojustifytheCamassa-Holm approximationweshowthatapproxima-
.
1 tion errors remain small over a long time interval. To bemore precise,
0
we obtain error estimates in terms of two independent, small, positive
6
1 parameters ǫ and δ measuring the effect of nonlinearity and disper-
:
v sion, respectively. We further show that similar conclusions are also
i validforthelower orderapproximations: theBenjamin-Bona-Mahony
X
approximation and the Korteweg-de Vries approximation.
r
a
1 Introduction
In the present paper we rigorously prove that, in the long-wave limit and
on a relevant time interval, the right-going solutions of both the improved
Boussinesq (IB) equation
u −u −δ2u −ǫ(u2) = 0, (1)
tt xx xxtt xx
1
and, more generally, the nonlocal wave equation
u = β ∗(u+ǫu2) (2)
tt δ xx
are well approximated by the solutions of the Camassa-Holm (CH) equation
3 5 3
w +w +ǫww − δ2w − δ2w − ǫδ2(2w w +ww ) = 0. (3)
t x x xxx xxt x xx xxx
4 4 4
In the above equations, u = u(x,t) and w = w(x,t) are real-valued func-
tions, ǫ and δ are two small positive parameters measuring the effect of
nonlinearity and dispersion, respectively, the symbol ∗ denotes convolution
in the x-variable, β (x) = 1β(x) is the kernel function. It should be noted
δ δ δ
that (3) can be written in a more standard form by means of a coordinate
transformation. That is, in a moving frame defined by x¯ = 2 (x− 3t) and
√5 5
t¯= 2 t, (3) becomes
3√5
6 9
v + v +3ǫvv −δ2v − ǫδ2(2v v +vv ) = 0, (4)
t¯ x¯ x¯ t¯x¯x¯ x¯ x¯x¯ x¯x¯x¯
5 5
withv(x¯,t¯) = w(x,t). Also,bytheuseofthescalingtransformationU(X,τ) =
ǫu(x,t), x = δX, t = δτ, (1)and(3)canbewritteninamorestandardform
with no parameters, but the above forms of (1) and (3) are more suitable to
deal with small-but-finite amplitude long wave solutions.
In the literature, there have been a number of works concerning rigorous
justification of the model equations derived for the unidirectional propaga-
tion of long waves from nonlinear wave equations modeling various physical
systems. One of these model equations is the CH equation [4, 14, 15] de-
rived for the unidirectional propagation of long water waves in the context
of a shallow water approximation to the Euler equations of inviscid incom-
pressible fluid flow. The CH equation has attracted much attention from
researchers over the years. The two main properties of the CH equation are:
it is an infinite-dimensional completely integrable Hamiltonian system and
it captures wave-breaking of water waves (see [5, 6, 7, 17] for details). A
rigorous justification of the CH equation for shallow water waves was given
in [7].
In a recent study [11], the CH equation has been also derived as an ap-
propriate model for the unidirectional propagation of long elastic waves in
an infinite, nonlocally and nonlinearly elastic medium (see also [12]). The
2
constitutive behavior of the nonlocally and nonlinearly elastic medium is de-
scribed by a convolution integral (we refer the reader to [9, 10] for a detailed
description of the nonlocally and nonlinearly elastic medium) and in the case
of quadratic nonlinearity the one-dimensional equation of motion reduces to
the nonlocal equation given in (2). Moreover, the nonlocal equation (that is,
the equation of motion for the medium) reduces to the IB equation (1) for a
particular choice of the kernel function appearing in the integral-type consti-
tutive relation (see Section 5 for details). In order to derive formally the CH
equation from the IB equation, an asymptotic expansion valid as nonlinear-
ity and dispersion parameters, that is ǫ and δ, tend to zero independently is
used in [11]. It has been also pointed out that a similar formal derivation of
the CH equation is possible by starting from the nonlocal equation (2).
The question that naturally arises is under which conditions the unidi-
rectional solutions of the nonlocal equation are well approximated by the
solutions of the CH equation and this is the subject of the present study.
Given a solution of the CH equation we find the corresponding solution of
the nonlocal equation and show that the approximation error, i.e. the dif-
ference between the two solutions, remains small in suitable norms on a
relevant time interval. We conclude that the CH equation is an appropri-
ate model equation for the unidirectional propagation of nonlinear dispersive
elastic waves. The methodology used in this study adapts the techniques in
[3, 7, 13].
We note that, in the terminology of some authors, our results are in fact
consistency-existence-convergence results for the CH approximation of the
IB equation and, more generally, of the nonlocal equation. We refer to [3]
and the references therein for a detailed discussion of these concepts.
As it is pointed above, the general class of nonlocal wave equations con-
tains the IB equation as a member. Therefore, to simplify our presentation,
we start with the CH approximation of the IB equation and then extend the
analysis to the case of the general class of nonlocal wave equations. Though
our analysis is mainly concerned with the CH approximations of the IB equa-
tion and the nonlocal equation, our results apply as well to the Benjamin-
Bona-Mahony (BBM) approximation. We also show how to use our results
to justify the Korteweg-de Vries (KdV) approximation.
The structure of the paper is as follows. In Section 2 we observe that
the solutions of the CH equation are uniformly bounded in suitable norms
for all values of ǫ and δ. In Section 3 we estimate the residual term that
arises when we plug the solution of the CH equation into the IB equation. In
3
Section 4, using the energy estimate based on certain commutator estimates,
we complete the proof of the main theorem. In Section 5 we extend our
consideration from the IB equation to the nonlocal equation and we prove a
similar theorem for the nonlocal equation. Finally, in Section 6 we give error
estimates for the long-wave approximations based on the BBM equation [2]
and the KdV equation [16].
Throughout this paper, we use the standard notation for function spaces.
The Fourier transform of u, defined by u(ξ) = u(x)e iξxdx, is denoted by
R −
the symbol u. The symbol kukLp represents thRe Lp (1 ≤ p < ∞) norm of u
on R. The symbol hu,vi represents thebinner product of u and v in L2. The
notation Hsb= Hs(R) denotes the L2-based Sobolev space of order s on R,
with the norm kuk = (1+ξ2)s|u(ξ)|2dξ 1/2. The symbol R in will
Hs R R
be suppressed. C is a ge(cid:0)nReric positive constan(cid:1)t. Partial differentiatioRns are
b
denoted by D , D etc.
t x
2 Uniform Estimates for the Solutions of the
Camassa-Holm Equation
In this section, we observe that the solutions wǫ,δ of the CH equation are
uniformly bounded in suitable norms for all values of ǫ and δ. This is a
direct consequence of the estimates proved by Constantin and Lannes in [7]
for a more general class of equations, containing the CH equation as a special
case.
For convenience of the reader, we rephrase below Proposition 4 of [7].
To that end, we first recall some definitions from [7]: (i) For every s ≥ 0,
the symbol Xs+1(R) represents the space Hs+1(R) endowed with the norm
|f|2 = kfk2 +δ2kf k2 , and (ii ) the symbol P denotes the index set
Xs+1 Hs x Hs
P = {(ǫ,δ) : 0 < δ < δ , ǫ ≤ Mδ}
0
for some δ > 0 and M > 0. Then, Proposition 4 of [7] is as follows:
0
Proposition 1. Assume that κ < 0 and let δ > 0, M > 0, s > 3, and
5 0 2
w ∈ Hs+1(R). Then there exist T > 0 and a unique family of solutions
0
4
wǫ,δ to the Cauchy problem
(ǫ,δ)
(cid:8) (cid:9) ∈P
w +w +κ ǫww +κ ǫ2w2w +κ ǫ3w3w +δ2(κ w +κ w )
t x 1 x 2 x 3 x 4 xxx 5 xxt
−ǫδ2(κ ww +κ w w ) = 0, (5)
6 xxx 7 x xx
w(x,0) = w (x) (6)
0
(withconstants κ (i = 1,2,...,7))boundedin C [0, T],Xs+1(R) ∩C1 [0, T],Xs(R) .
i ǫ ǫ
(cid:0) (cid:1) (cid:0) (cid:1)
We refer the reader to [7] for the proof of this proposition. Furthermore,
T of the existence time T/ǫ is expressed in [7] as
1
T = T δ ,M,|w | , ,κ ,κ ,κ ,κ > 0.
(cid:18) 0 0 Xδs0+1 κ5 2 3 6 7(cid:19)
Obviously, the CH equation (3) is a special case of (5) where κ = 1,
1
κ = κ = 0, κ = −3, κ = −5 and 2κ = κ = −3. In subsequent
2 3 4 4 5 4 6 7 2
sections we will need to use uniform estimates for the terms wǫ,δ(t)
Hs+k
and wtǫ,δ(t) with some k ≥ 1. Proposition 1 provides(cid:13)(cid:13)us with(cid:13)(cid:13)such
(cid:13) (cid:13)Hs+k−1
estim(cid:13)ates, ne(cid:13)vertheless to avoid the extra δ2 term in the Xs+1-norm, we will
(cid:13) (cid:13)
use a weaker version based on the inclusion Xs+k+1 ⊂ Hs+k. Furthermore,
for simplicity, we take δ = M = 1. We thus reach the following corollary:
0
Corollary 1. Let w ∈ Hs+k+1(R), s > 1/2, k ≥ 1. Then, there exist T > 0,
0
C > 0 and a unique family of solutions
T T
wǫ,δ ∈ C [0, ],Hs+k(R) ∩C1 [0, ],Hs+k 1(R)
−
(cid:18) ǫ (cid:19) (cid:18) ǫ (cid:19)
to the CH equation (3) with initial value w(x,0) = w (x), satisfying
0
wǫ,δ(t) + wǫ,δ(t) ≤ C,
Hs+k (cid:13) t (cid:13)Hs+k−1
(cid:13) (cid:13)
(cid:13) (cid:13)
(cid:13) (cid:13)
for all 0 < δ ≤ 1, ǫ ≤ δ and t ∈ [0, (cid:13)T]. (cid:13)
ǫ
3 Estimates for the Residual Term Correspond-
ing to the Camassa-Holm Approximation
Let wǫ,δ be the family of solutions mentioned in Corollary 1 for the Cauchy
problem of the CH equation with initial value w ∈ Hs+k+1(R). In this
0
5
section we estimate the residual term that arises when we plug wǫ,δ into the
IB equation. Obviously, the residual term f for the IB equation is
f = w −w −δ2w −ǫ(w2) , (7)
tt xx xxtt xx
where and hereafter we drop the indices ǫ, δ in u and w for simplicity.
Using the CH equation we now show that the residual term f has a
potential function. We start by rewriting the CH equation in the form
3 5 3
w +w = −ǫww + δ2w + δ2w + ǫδ2(2w w +ww ). (8)
t x x xxx xxt x xx xxx
4 4 4
Using repeatedly (8) in (7) we get
3 5 3 1
f =(D −D ) −ǫww + δ2w + δ2w + ǫδ2D ( w2 +ww )
t x (cid:20) x 4 xxx 4 xxt 4 x 2 x xx (cid:21)
−δ2w −ǫ(w2)
xxtt xx
w3 3
=ǫ2D2( )− ǫ2δ2 D2(w2 +2ww )
x 3 8 x x xx
(cid:2) (cid:3)
1
+ δ4 (D2D −3D3)(3w +5w )
16 x t x xxx xxt
(cid:2) (cid:3)
3
+ ǫδ4 (D3D −3D4)(w2 +2ww )
32 x t x x xx
(cid:2) (cid:3)
1
+ ǫδ2D (−3wD2 +2w +w D )(w +w ) . (9)
4 x x xx x x t x
(cid:2) (cid:3)
After some straightforward calculations we write f = F with
x
w3 1
F =ǫ2( ) − ǫ2δ2 3(w2 +2ww ) −3w(w2) +2w (w2) +w (w2)
3 x 8 x xx x xxx xx x x xx
(cid:2) (cid:3)
1
+ δ4 (D D −3D2)(3w +5w )
16 x t x xxx xxt
(cid:2) (cid:3)
1
+ ǫδ4 3(D2D −3D3)(w2 +2ww )
32 x t x x xx
(cid:2)
+2(−3wD2 +2w +w D )(3w +5w )
x xx x x xxx xxt
1 (cid:3)
+ ǫ2δ4 (−9wD3 +6w D +3w D2)(w2 +2ww ) .
32 x xx x x x x xx
(cid:2) (cid:3)
Note that, except for the term D3D2w, F is a combination of terms of the
x t
form Djw with j ≤ 5 or DlD w with l ≤ 4. By taking k = 5 it immediately
x x t
6
follows from Corollary 1 that all of the terms in F, except D3D2w, are
x t
uniformly bounded in the Hs norm. To deal with the term D3D2w, we first
x t
rewrite the CH equation in the form
3 3
w = Q −w −ǫww + δ2w + ǫδ2(2w w +ww ) , (10)
t x x xxx x xx xxx
(cid:20) 4 4 (cid:21)
where the operator Q is
1
5 −
Q = 1− δ2D2 . (11)
(cid:18) 4 x(cid:19)
Then, applying the operator D3D to (10) and using (8) we get
x t
3 3
D3D w =D3D Q −w −ǫww + δ2w + ǫδ2(2w w +ww )
x t t x t (cid:20) x x 4 xxx 4 x xx xxx (cid:21)
=D [−Q(w +ǫ(ww ) )
t xxxx x xxx
3 3
+ δ2QD2w + ǫδ2QD2(2w w +ww ) .
4 x xxxx 4 x x xx xxx x(cid:21)
We note that the operator norms of Q and Qδ2D2 are bounded:
x
4
kQk ≤ 1 and δ2QD2 ≤ .
Hs x Hs 5
(cid:13) (cid:13)
The use of these bounds and uniform e(cid:13)stimate(cid:13)for D3D2w yield
x t
D3D2w ≤ C D4w ≤ Ckw k . (12)
x t Hs x t Hs t Hs+4
(cid:13) (cid:13) (cid:13) (cid:13)
As all the terms(cid:13)in F hav(cid:13)e coefficie(cid:13)nts ǫ2,(cid:13)ǫ2δ2, δ4, ǫδ4 or ǫ2δ4 (with 0 < ǫ ≤
δ ≤ 1) we obtain the following estimate for the potential function
kF(t)k ≤ C ǫ2 +δ4 (kwk +kw k ). (13)
Hs Hs+5 t Hs+4
(cid:0) (cid:1)
Using Corollary 1 with k = 5, we obtain:
Lemma 3.1. Let w ∈ Hs+6(R), s > 1/2. Then, there is some C > 0
0
so that the family of solutions wǫ,δ to the CH equation (3) with initial value
w(x,0) = w (x), satisfy
0
w −w −δ2w −ǫ(w2) = F
tt xx xxtt xx x
with
kF (t)k ≤ C ǫ2 +δ4 ,
Hs
for all 0 < ǫ ≤ δ ≤ 1 and t ∈ [0, T]. (cid:0) (cid:1)
ǫ
7
4 Justification of the Camassa-Holm Approx-
imation
InthissectionweproveTheorem4.2givenbelow. Wehavethewell-posedness
result for the IB equation (1) in a general setting [8, 10]:
Theorem 4.1. Let u ,u ∈ Hs(R), s > 1/2. Then for any pair of param-
0 1
eters ǫ and δ, there is some Tǫ,δ > 0 so that the Cauchy problem for the IB
equation (1) with initial values u(x,0) = u (x), u (x,0) = u (x) has a unique
0 t 1
solution u ∈ C2 [0,Tǫ,δ],Hs(R) .
(cid:0) (cid:1)
The existence time Tǫ,δ above may depend on ǫ and δ and it may be
chosen arbitrarily large as long as Tǫ,δ < Tǫ,δ where Tǫ,δ is the maximal
max max
time. Furthermore, it was shown in [10] that the existence time, if it is finite,
is determined by the L blow-up condition
∞
lim supku(t)k = ∞.
L∞
t Tmǫ,aδx
→
We now consider the solutions w of the CH equation with initial data
w(x,0) = w . Then we take w (x) and w (x,0) as the initial conditions for
0 0 t
the IB equation (1), that is,
u(x,0) = w (x), u (x,0) = w (x,0).
0 t t
Let u be the corresponding solutions of the Cauchy problem defined for the
IB equation (1) with these initial conditions. Since w ∈ Hs+6(R), clearly
0
u(x,0),u (x,0) ∈ Hs(R). Recalling from Corollary 1 that the guaranteed
t
existence time for w is T/ǫ, without loss of generality we will take Tǫ,δ ≤ T/ǫ.
In the course of our proof of Theorem 4.2, we will use certain commutator
estimates. We recall that the commutator is defined as [K,L] = KL−LK.
We refer the reader to [17] (see Proposition B.8) for the following result.
Proposition 2. Let q > 1/2, s ≥ 0 and let σ be a Fourier multiplier of
0
order s.
1. If 0 ≤ s ≤ q +1 and w ∈ Hq0+1 then, for all g ∈ Hs 1, one has
0 −
k[σ(Dx),w]gkL2 ≤ CkwxkHq0kgkHs−1,
8
2. If −q < r ≤ q +1−s and w ∈ Hq0+1 then, for all g ∈ Hr+s 1, one
0 0 −
has
k[σ(Dx),w]gkHr ≤ CkwxkHq0kgkHr+s−1.
Forthereader’sconvenience wenowrestatethetwoestimatesoftheabove
proposition as follows. Let Λs = (1−D2)s/2 and take w ∈ Hs+1, g ∈ Hs 1
x −
and h ∈ Hs. Then, for q = s, the first estimate above yields
0
h[Λs,w]g,Λshi ≤ CkwkHs+1kgkHs−1khkHs. (14)
Similarly, for q = s and −s < r ≤ 1, we obtain from the second estimate
0
that
hΛ[Λs,w]h,Λs 1gi ≤CkΛ[Λs,w]hk kΛs 1gk
− L2 − L2
≤Ck[Λs,w]hkH1kgkHs−1
≤CkwkHs+1khkHskgkHs−1. (15)
We are now ready to prove the main result for the CH approximation of the
IB equation (an extension of the following theorem to the nonlocal equation
will be given in Section 5 (see Theorem 5.2)):
Theorem 4.2. Let w ∈ Hs+6(R), s > 1/2 and suppose that wǫ,δ is the
0
solution of the CH equation (3) with initial value w(x,0) = w (x). Then,
0
there exist T > 0 and δ ≤ 1 such that the solution uǫ,δ of the Cauchy
1
problem for the IB equation
u −u −δ2u −ǫ(u2) = 0
tt xx xxtt xx
u(x,0) = w (x), u (x,0) = wǫ,δ(x,0),
0 t t
satisfies
kuǫ,δ(t)−wǫ,δ(t)k ≤ C ǫ2 +δ4 t
Hs
for all t ∈ 0, T and all 0 < ǫ ≤ δ ≤ δ . (cid:0) (cid:1)
ǫ 1
(cid:2) (cid:3)
Proof. We fix the parameters ǫ and δ such that 0 < ǫ ≤ δ ≤ 1. Let r = u−w.
We define
Tǫ,δ = sup t ≤ Tǫ,δ : kr(τ)k ≤ 1 for all τ ∈ [0,t] . (16)
0 Hs
(cid:8) (cid:9)
We note that either r Tǫ,δ = 1 or Tǫ,δ = Tǫ,δ. Moreover, in the latter
0 0
(cid:13) (cid:16) (cid:17)(cid:13)Hs
case we must have T(cid:13)ǫ,δ = Tǫ,δ(cid:13)= T/ǫ by the discussion above about for the
(cid:13)0 (cid:13)
9
maximal time Tǫ,δ . For the rest of the proof we will drop the superscripts
max
ǫ,δ to simplify the notation. Henceforth, we will take t ∈ [0,Tǫ,δ]. Obviously,
0
the function r = u−w satisfies the initial conditions r(x,0) = r (x,0) = 0.
t
Furthermore, it satisfies the evolution equation
1−δ2D2 r −r −ǫ r2 +2wr = −F ,
x tt xx xx x
(cid:0) (cid:1) (cid:0) (cid:1)
with the residual term F = w −w −δ2w −ǫ(w2) that was already
x tt xx xxtt xx
estimated in (13)). We define a function ρ so that r = ρ with ρ(x,0) =
x
ρ (x,0) = 0. This is possible since r satisfies the initial conditions r(x,0) =
t
r (x,0) = 0 (see [10] for details). In what follows we will use both ρ and r to
t
further simplify the calculation. The above equation then becomes
1−δ2D2 ρ −r −ǫ r2 +2wr = −F. (17)
x tt x x
(cid:0) (cid:1) (cid:0) (cid:1)
Motivated by the approach in [13], we define the ”energy” as
1
E2(t) = kρ (t)k2 +δ2kr (t)k2 +kr(t)k2 +ǫhΛs(w(t)r(t)),Λsr(t)i
s 2 t Hs t Hs Hs
(cid:0)ǫ (cid:1)
+ Λsr2(t),Λsr(t) . (18)
2
(cid:10) (cid:11)
Note that
|hΛs(wr),Λsri| ≤ Ckr(t)k2 , and Λsr2,Λsr ≤ kr(t)k3 ≤ kr(t)k2 ,
Hs Hs Hs
(cid:12)(cid:10) (cid:11)(cid:12)
where we have used (16). Thus, for suffi(cid:12) ciently sma(cid:12)ll values of ǫ, we have
1
E2(t) ≥ kρ k2 +δ2kr k2 +krk2 ,
s 4 t Hs t Hs Hs
(cid:0) (cid:1)
which shows that E2(t) is positive definite. The above result also shows that
s
an estimate obtained for E2 gives an estimate for kr(t)k2 . Differentiating
s Hs
E2(t) with respect to t and using (17) to eliminate the term ρ from the
s tt
resulting equation we get
d d ǫ
E2 = ǫhΛs(wr),Λsri+ Λsr2,Λsr −ǫ Λs(r2 +2wr),Λsr
dt s dt 2 t
(cid:16) (cid:17)
(cid:10) (cid:11) (cid:10) (cid:11)
−hΛsF,Λsρ i
t
=ǫ[hΛs(w r),Λsri−hΛs(wr),Λsr i+hΛsr,Λs(wr )i+hΛs(rr ),Λsri
t t t t
1
− Λsr2,Λsr −hΛsF,Λsρ i. (19)
t t
2 (cid:21)
(cid:10) (cid:11)
10