CONTRIBUTION OF HIGHER ORDER TERMS TO THE NONLINEAR SHALLOW WATER WAVES
H. DEMIRAY1§
Abstract. In this work, by utilizing the scaled multiple-space expansion method, we studied the propagation of weakly nonlinear waves in shallow water and obtained the gov-erning evolution equations of various order terms in the perturbation expansion. Seeking a progressive wave solution to these evolution equations we obtained the speed correction terms so as to remove some possible secularities. The result obtained here is exactly the same with that of obtained by the modified reductive perturbation method [12]. We also proposed a method for the evolution equation governing then thorder term in the perturbation expansion. By defining a single time parameter we showed the connection of the modified reductive perturbation method to the scaled multiple-space expansion method.
Keywords: Shallow water waves, scaled multiple-space expansion, solitary waves. AMS Subject Classification: 74J35
1. Introduction
The studies of nonlinear waves of various fields in physics and engineering, by use of the reductive perturbation method in the long-wave approximation, lead to the Korteweg-deVries equation as the evolution equation ( Davidson [1], Antar and Demiray [2]). The examination of the higher order terms in the perturbation expansion by use of the reduc-tive perturbation method gives some secularities( Ichikawa et al. [3], Aoyama and Ichikawa [4]). To remove such secularities Sugimoto and Kakutani [5] introduced additional slow variables both in space and time in reductive perturbation theory, Kodama and Taniuti [6] presented the renormalization procedure of the velocity of the KdV soliton. Nevertheless, the latter approach remains somewhat obscure, since there is no reasonable connection between the smallness parameters of the stretched variables and the one used in the per-turbation expansion of the field variables. Another attempt to remove such secularities was made by Kraenkel and Manna [7] for long water waves by use of multiple-time scale expansion method but they could not obtain explicitly the speed correction terms. In order to remove this uncertainty, Malfielt and Wieers [8] presented a dressed solitary wave approach, which is based on the assumption that the field variables admit localized travel-ling wave solution. Then, for the long-wave limit, they expanded the field quantities into a power series of the wave number, which is assumed to be the only smallness parameter, and obtained the explicit solution for various order terms in the perturbation expansion. However, this approach is successful when one studies the progressive wave solution to
1 Department of Mathematics, Faculty of Arts and Sciences, Isik University, Sile 34980 Istanbul,Turkey
e-mail: [email protected]
§ Manuscript received May 12, 2012.
TWMS Journal of Applied and Engineering Mathematics Vol.2 No.2 c⃝I¸sık University, Department of Mathematics 2012; all rights reserved.
the original nonlinear field equations and it does not give any idea about the form of the evolution equations governing various order terms in the perturbation expansion. The result obtained here is completely different from that of [4]. Another method, so called the ” modified reductive perturbation method” is presented by Demiray [9-11] in which a scaling parameter, that assumes a perturbation expansion, is presented to balance the higher order nonlinearities with higher order dispersive effects so as to remove some possi-ble secularities in the solution. The result obtained in this work is exactly the same with that of Malfliet and Wieers [8], but different from that of [4].
In this work, by utilizing the scaled multiple-space expansion method, we studied the propagation of weakly nonlinear waves in shallow water and obtained a set of Korteweg-de Vries equations as the evolution equations. By seeking a progressive wave solution to these evolution equations we obtained the speed correction terms so as to remove some possible secularities. The result obtained here is exactly the same with that of given by the modified reductive perturbation method [12].We also proposed a method for the evolution equation governing then thorder term in the perturbation expansion. By defining a single time parameter we showed the connection between the modified reductive perturbation method and the multiple-time scale expansion method.
2. MULTIPLE-TIME SCALE FORMALISM FOR WATER WAVES
We consider two dimensional incompressible inviscid fluid in a constant gravitational fieldgacting in the negativez∗ - direction. The space coordinates are denoted by (x∗, z∗) and the corresponding velocity components by (u∗, w∗). The equations of motion describ-ing such a fluid are:
∂u∗ ∂x∗ +
∂w∗
∂z∗ = 0, (incompressibility), (1) ∂u∗
∂t∗ +u
∗ ∂u∗
∂∂x∗ +w
∗∂u∗
∂z∗ + 1 ρ
∂P∗
∂x∗ = 0, (2)
∂w∗ ∂t∗ +u
∗ ∂w∗
∂∂x∗ +w
∗∂w∗
∂z∗ + 1 ρ
∂P∗
∂z∗ +g= 0, (3)
wheret∗ is the time parameter,ρis the mass density andP∗ is the fluid pressure function. Assuming that the flow is ir-rotational, the velocity components can be expressed in terms of scalar potentialϕ∗ as
u∗ = ∂ϕ
∗
∂x∗, w
∗= ∂ϕ∗
∂z∗. (4)
Then, the incompressibility condition becomes
∂2ϕ∗ ∂x∗2 +
∂2ϕ∗
∂z∗2 = 0, (5)
and the Euler equation reads
P∗−P0∗
ρ +
∂ϕ∗ ∂t∗ +
1 2[(
∂ϕ∗ ∂x∗)
2+ (∂ϕ∗ ∂z∗)
2] +gz∗ = 0, (6)
whereP0∗ is the atmospheric pressure.
We consider the case of fluid of height h0 , bounded above by a steady atmospheric pressure P0∗. Let the upper surface be described by z∗ = η∗(x∗, t∗). The kinematical boundary condition on this surface can be expressed as:
∂ϕ∗ ∂z∗ =
∂η∗ ∂t∗ +
∂ϕ∗ ∂x∗
∂η∗
∂x∗ = 0, on z
From the equation (6), the dynamical boundary condition on this surface reads ∂ϕ∗
∂t∗ + 1 2[(
∂ϕ∗ ∂x∗)
2+ (∂ϕ∗ ∂z∗)
2] +gη∗ = 0, on z∗ =η∗. (8) Finally, the lower boundary is supposed to be rigid horizontal plane. Therefore, at z∗ = −h0, the normal component of the velocity must vanish, i.e.,
∂ϕ∗
∂z∗ = 0, at z
∗=−h0. (9)
At this stage it is convenient to introduce the following non-dimensional quantities:
x∗=h0x, z∗ =h0z, t∗= h0 c0t, ϕ
∗=c0h0ϕ, ηˆ ∗ =h0η,ˆ
P∗ =ρc20p, P0∗=ρc02p0, c0= (gh0)1/2. (10) Introducing (10) into equations (5)-(9) the following non-dimensional equations are ob-tained:
∂2ϕˆ ∂x2 +
∂2ϕˆ
∂z2 = 0, (11)
∂ϕˆ ∂z =
∂ηˆ ∂t +
∂ϕˆ ∂x
∂ηˆ
∂x = 0, at z= ˆη, (12)
∂ϕˆ ∂t +
1 2[(
∂ϕˆ ∂x)
2+ (∂ϕˆ ∂z)
2] + ˆη= 0, atz= ˆη, (13) ∂ϕˆ
∂z = 0, at z=−1. (14)
Now, we shall consider the long-wave in shallow water approximation to the above equa-tions by applying the scaled multiple- space expansion formalism. For that purpose we shall propose the following slow variables:
ξ=ϵ1/2(t−x), τn=en+3/2x, (n= 0,1,2,3, ...) (15)
whereϵis a small parameter characterizing the smallness of certain physical entities. For our future purposes, we introduce the following new variables:
ˆ
ϕ=ϵ1/2ϕ, ηˆ=ϵη. (16)
Introducing (15) and (16) into (11)-(14) we have
∂2ϕ ∂z2 +ϵ
∂2ϕ ∂ξ2 −2ϵ
2 ∂2ϕ ∂ξ∂τ0
+ϵ3(∂ 2ϕ ∂τ02 −2
∂2ϕ ∂ξ∂τ1
) +O(ϵ4) = 0, (17) ∂ϕ
∂z =ϵ ∂η ∂ξ +ϵ
2(−∂ϕ ∂ξ +
∞ ∑
n=0
ϵn+1∂ϕ ∂τn
)(−∂η ∂ξ +
∞ ∑
n=0
ϵn+1 ∂η ∂τn
) = 0, at z=ϵη, (18)
∂ϕ ∂ξ +
ϵ 2(−
∂ϕ ∂ξ +
∞ ∑
n=0
ϵn+1∂ϕ ∂τn
)2+1 2(
∂ϕ ∂z)
2+η= 0, at z=ϵη, (19) ∂ϕ
∂z = 0, at z=−1. (20)
Now, we expand the functionsϕand η into a suitable power series in the parameterϵas:
ϕ=ϕ0+ϵϕ1+ϵ2ϕ2+...,
O(1) equations:
∂ϕ0
∂z2 = 0. (22)
and the boundary conditions
∂ϕ0
∂z = 0 at z=−1, ∂ϕ0
∂z |z=0 = 0, [ ∂ϕ0
∂ξ + 1 2(
∂ϕ0 ∂z )
2]|
z=0+η0= 0. (23) O(ϵ) equations:
∂2ϕ1 ∂z2 +
∂2ϕ0
∂ξ2 = 0, (24)
and the boundary conditions
∂ϕ1
∂z = 0, at z=−1, ∂ϕ1
∂z |z=0− ∂η0
∂ξ = 0,
[∂ϕ1 ∂ξ +
1 2(
∂ϕ0 ∂ξ )
2+∂ϕ0 ∂z
∂ϕ1
∂z ]|z=0+η1 = 0. (25) O(ϵ2)equations:
∂2ϕ2 ∂z2 +
∂2ϕ1 ∂ξ2 −2
∂2ϕ0
∂ξ∂τ0 = 0, (26)
and the boundary conditions
∂ϕ2
∂z = 0, at z=−1, [ ∂ϕ2
∂z +η0 ∂2ϕ1
∂z2 ]|z=0− ∂η1
∂ξ − ∂ϕ0
∂z |z=0 ∂η0
∂ξ = 0,
[∂ϕ2 ∂ξ +η0
∂2ϕ1 ∂z∂ξ +
1 2(
∂ϕ1 ∂z )
2+∂ϕ0 ∂z
∂ϕ2 ∂z +
∂ϕ0 ∂ξ (
∂ϕ1 ∂ξ −
∂ϕ0 ∂τ0
]|z=0+η2 = 0. (27) O(ϵ3)equations:
∂2ϕ3 ∂z2 +
∂2ϕ2 ∂ξ2 −2
∂2ϕ1 ∂ξτ0
+ ∂ 2ϕ
0 ∂τ02 −2
∂2ϕ0 ∂ξ∂τ1
= 0, (28)
and the boundary conditions:
∂ϕ3
∂z = 0, at z=−1,
[∂ϕ3 ∂z +η0
∂2ϕ2 ∂z2 +η1
∂2ϕ1
∂z2 ]|z=0− ∂η2
∂ξ − ∂ϕ0
∂ξ ( ∂η1
∂ξ − ∂η0 ∂τ0
)−∂η0 ∂ξ (
∂ϕ1 ∂ξ −
∂ϕ0 ∂τ0
)|z=0= 0,
[∂ϕ3 ∂ξ +η0
∂2ϕ2 ∂ξ∂z+η1
∂2ϕ1 ∂ξ∂z +
1 2η
2 0
∂3ϕ1
∂z2∂ξ]|z=0+ [ ∂ϕ1
∂z ( ∂ϕ2
∂z +η0 ∂2ϕ1
∂z2 ]|z=0
+1 2(
∂ϕ1 ∂ξ −
∂ϕ0 ∂τ0)
2|
z=0+ ∂ϕ0
∂ξ (η0 ∂2ϕ1 ∂ξ∂z+
∂ϕ2 ∂ξ −
∂ϕ1 ∂τ0 −
∂ϕ0
2.1. Solution of the field equations. From the solution of the sets (22) and (23) one obtains
ϕ0 =φ0(ξ, τn), η0 =− ∂φ0
∂ξ (n= 0,1,2,3, ...), (30) whereφ0(ξ, τn) is an unknown function of its argument whose evolution equation will be
obtained later.
Introducing (30) into (24) and (25), the solution of O(ϵ) equations gives the following result
ϕ1=−1 2
∂2φ 0 ∂ξ2 (z
2+ 2z) +φ1(ξ, τ
n), η1 =−
∂φ1 ∂ξ −
1 2(
∂φ0 ∂ξ )
2, (31)
whereφ1(ξ, τn) (n= 0,1,2,3, ..) is another unknown function whose governing evolution
equation will be obtained from the higher order perturbation expansion.
Introducing (30) and (31) into equations (26) and (27), the solution of O(ϵ2) equations may be obtained as
ϕ2= 1 24
∂4φ0 ∂ξ4 (z
4+ 4z3) + (∂2φ0 ∂ξ∂τ0 −
1 2
∂2φ1 ∂ξ2])z
2
+(−1 3
∂4φ0 ∂ξ4 −
∂2φ1 ∂ξ2 + 2
∂2φ0 ∂ξ∂τ0
)z+φ2(ξ, τn),
η2 =− ∂φ2
∂ξ − ∂φ0
∂ξ ∂3φ0
∂ξ3 − 1 2(
∂φ0 ∂ξ )
2−∂φ0 ∂ξ
∂φ1 ∂ξ +
φ0 ∂ξ
∂φ0 ∂τ0
, (32)
where φ2(ξ, τn) is another unknown function whose evolution equation will be obtained
from the higher order perturbation expansion. The use of the second boundary condition in (31) yields
∂2φ0 ∂ξ∂τ0
+3 2
∂φ0 ∂ξ
∂2φ0 ∂ξ2 −
1 6
∂4φ0
∂ξ4 = 0. (33)
Noting the relation∂φ0/∂ξ=−η0, the equation (33) reduces to the following Korteweg-de Vries (KdV) equation
∂η0 ∂τ0 −
3 2η0
∂η0 ∂ξ −
1 6
∂3η0
∂ξ3 = 0. (34)
The equation (34) gives the evolution ofη0 withτ0and ξ. The remaining spatial variables remain as some parameters. The evolution ofη0withτn(n= 1,2,3, ....) should be obtained
from the higher order perturbation expansion.
To obtain the solution for O(ϵ3) equations we introduce (30), (31) and (32) into (28) and (29), which results in
∂2ϕ3 ∂z2 +
1 24
∂6φ0 ∂ξ6 (z
4+ 4z3) + (2 ∂4φ0 ∂ξ3∂τ
0 − 1 2
∂4φ1 ∂ξ4 )z
2
+(−1 3
∂6φ0 ∂ξ6 −
∂4φ1 ∂ξ4 + 4
∂4φ0 ∂ξ3∂τ0)z+
∂2φ2 ∂ξ2 −2
∂2φ1 ∂ξ∂τ0 +
∂2φ0 ∂τ02 −2
∂2φ0
∂ξ∂τ1 = 0, (35) and the boundary conditions
∂φ3
∂z = 0 at z=−1, ∂ϕ3
∂z |z=0−4 ∂φ0
∂ξ ∂2φ0 ∂ξ∂τ0 −
2∂ 2φ
0 ∂ξ2
∂φ0 ∂τ0 +3 ∂
∂ξ( ∂φ0
∂ξ ∂φ1
∂ξ ) + 3 2(
∂φ0 ∂ξ )
2∂2φ0 ∂ξ2 +
∂φ0 ∂ξ
∂4φ 0 ∂ξ4 + 2
∂2φ 0 ∂ξ2
∂3φ 0 ∂ξ3 +
∂2φ 2
The solution of (35) after the use of the first boundary condition in (36) gives
ϕ3 =− 1 720
∂6φ0 ∂ξ6 (z
6+ 6z5)− 1 12(2
∂4φ0 ∂ξ3∂τ0 −
1 2
∂4φ1 ∂ξ4 )z
4
+( 1 18
∂6φ0 ∂ξ6 +
1 6
∂4φ1 ∂ξ4 −
2 3
∂4φ0 ∂ξ3∂τ 0
)z3−1 2(
∂2φ2 ∂ξ2 −2
∂2φ1 ∂ξ∂τ0 −
2∂ 2φ 0 ∂ξ∂τ1 +∂ 2φ 0 ∂τ02 )z
2
+(−2 15
∂6φ 0 ∂ξ6 −
1 3
∂4φ 1 ∂ξ4 +
4 3
∂4φ 0 ∂ξ3∂τ0 + 2
∂2φ 1 ∂ξ∂τ0 + 2
∂2φ 0 ∂ξ∂τ1 −
∂2φ 0 ∂τ02 −
∂2φ 2
∂ξ2 )z+φ3, (37) where φ3(ξ, τn) is another unknown function whose evolution equation will be obtained
from the higher order perturbation expansion. Employing the last boundary condition in (36) the following evolution equation is obtained
∂2φ1 ∂ξ∂τ0 + 3 2 ∂ ∂ξ( ∂φ0 ∂ξ ∂φ1 ∂ξ )− 1 6
∂4φ1
∂ξ4 =S(φ0), (38)
where the functionS(φ0) is defined by
S(φ0) = 1 15
∂6φ0 ∂ξ6 −
2 3
∂4φ0 ∂ξ3∂τ
0 − ∂2φ0
∂ξ2 ∂3φ0
∂ξ3 − 1 2
∂φ0 ∂ξ
∂4φ0 ∂ξ4 +2∂φ0
∂ξ ∂2φ0 ∂ξ∂τ0
+∂ 2φ0 ∂ξ2
∂φ0 ∂τ0 −
3 4(
∂φ0 ∂ξ )
2∂2φ0 ∂ξ2 +
1 2
∂2φ0 ∂τ2
0
− ∂2φ0 ∂ξ∂τ1
. (39)
Setting ∂φ0/∂ξ = −η0, ∂φ1/∂ξ = −η1−η20/2, the equations (38) and (39) takes the following form
∂η1 ∂τ0 −
3 2
∂
∂ξ(η0η1)− 1 6
∂3η1
∂ξ3 =T(η0), (40) whereT(η0) is defined by
T(η0) = 1 15
∂5η0 ∂ξ5 −
7 12
∂3η0 ∂ξ2∂τ
0 +4
3 ∂η0
∂ξ ∂2η0
∂ξ2 + 2 3η0
∂3η0 ∂ξ3 +
3 4η 2 0 ∂η0 ∂ξ − 9 4η0 ∂η0 τ0 −
∂η0 ∂τ1
. (41)
The equation (40) is the degenerate (linearized) KdV equation with the non-homogeneous termT(η0) . The left side of (40) gives the evolution ofη1with respect toτ0 andξ, whereas the right side is the evolution of η0 with respect to τ1, τ0 and ξ . These evolutions are related to each other through the equation (40). For this order, the slow variables τn (n= 2,3, ...) remain as some parameters.
2.2. Solitary waves. In this sub-section we shall study the localized travelling wave solution to the evolution equations (34) and (40). For that purpose we introduce
ηi=ηi(ζ, τn), ζ =α(τn)[ξ+g(τ0, τn)], (i= 0,1), (n= 1,2, ...), (42)
where the parameters α(τn) and g(τ0, τn) (n= 1,2,3, ...) are to be determined from the
solution. As far as the equation (34) is concerned, only the variables areξ and τ0 ; and τn (n= 1,2,3, ..) appears to be some parameters.
Introducing (42) for i= 0 into the evolution equation (34) we have
∂g ∂τ0η
′
0− 3 2η0η
′
0− α2
6 η
′′′
0 = 0, (43)
where the prime denotes the differentiation of the corresponding quantity with respect to ζ . Integrating (43) with respect to ζ and utilizing the localization condition, i.e.,η0 and its various order derivatives vanish asζ → ±∞, we obtain
∂c ∂τ0η0−
3 4η 2 0− α2 6 η ′′
The equation (44) admits the solitary wave solution of the form
η0 =asech2ζ, (45)
where a is the amplitude of the solitary wave, and, in general, it is a function of the parameters τn. Inserting (45) into (44) and setting the coefficients of various powers of
sechζ equal to zero we obtain
α= (3a 4 )
1/2, g= a
2τ0+θ(τn), (n= 1,2,3, ...). (46) To obtain the solution for the evolution equation (40) we introduce (42) for i= 1 into (40), which results in
αa 2 η ′ 1− 3α 2 (η0η1)
′−α3
6 η
′′′
1 =− 1 a
∂a ∂τ1η0−
1 α
∂α ∂τ1ζη
′
0−α ∂g ∂τ1η ′ 0+ α5 15η (5) 0 −7aα3
24 η ′′′ 0 + 4α3 3 η ′
0η0′′+ 2α3
3 η0η
′′′
0 + 3α
4 η 2 0η0′ −
9aα 8 η0η
′
0. (47)
Integrating (47) with respect toζ and utilizing the localization condition we have
αa 2 η1−
3α
2 (η0η1)− α3
6 η
′′
1 =− 1 α
∂α ∂τ1
ζη0−(1 a
∂a ∂τ1 −
1 α ∂α ∂τ1 ∫ η0dζ
−α∂g ∂τ1
η0+ α5 15η (4) 0 − 7aα3 24 η ′′ 0 + α3 3 (η ′
0)2+ 2α3
3 η0η
′′
0 + α 4(η0)
3−9aα 16 (η0)
2. (48) Since we are concerned with the localized waves, in order to remove the secularity in η1, the coefficient ofζη1 must vanish; which yields
∂α ∂τ1 =
∂a
∂τ1 = 0. (49)
This equation states that the coefficientsα and aare independent ofτ1. In a similar way one can prove thatα anda are independent of τn (n= 1,2,3, ...). The remaining part of
the equation (48) takes the following form
η′′1 + (12
aη0−4)η1 = 8 a
∂g ∂τ1η0−
3a 10η (4) 0 + 7a 4 η ′′
0 −2(η′0)2−4η0η′′0 − 2 aη 3 0 + 9 2η 2
0. (50) Noting the relations
η0′′= 4η0− 6 aη
2
0, (η′0)2 = 4η02− 4 aη
3 0, η
(4)
0 = 16η0− 120
a η 2 0+
120 a2 η
3
0, (51) the equation (50) becomes
η1′′+ (12
aη0−4)η1 = ( 8 a
∂g ∂τ1
+ 11a
5 )η0+ 6η 2 0 −
6 aη
3
0. (52)
The homogeneous differential equation obtained from (52) admitsη′0 as one of the funda-mental solutions. Therefore, the term proportional to η0 on the right hand side of (52) causes the secularity inη1. In order to remove this secularity the coefficient of η0 must vanish, which yields
8 a
∂g ∂τ1
+11a
5 = 0, or g= a 2τ0−
11a2
40 τ1+θ1(τn), (n= 2,3,4, ...). (53) This result is exactly the same with that of [12] in which the modified reductive pertur-bation method was employed. The remaining part of equation (52) becomes
η′′1+ (12
aη0−4)η1 = 6η 2 0 −
6 aη
3
The particular solution of equation (54) gives
η1 =− a 2η0+
3 4η
2
0. (55)
Since, in the present work, we are concerned with the contribution of zeroth order term to the higher order terms in the perturbation expansion, the non-homogeneous term Sn(η0, η1, ....ηn−1) will depend only on η0 , i. e., Sn(η0). Thus, in studying the
pro-gressive wave solution, the linearized KdV equation for then th order term will take the following form
ηn′′+ (12
a η0−4)ηn= ( 8 a
∂g ∂τn
+an)η0+
n+1
∑
k=1
dkη0k+1, (56) where the coefficient an, dk (k = 1,2, ..., n + 1) are to be calculated from the higher
order perturbation expansion. Again, the first term on the right hand side causes to the secularity inηn; thus, this coefficient must vanish
8 a
∂g ∂τn
+an= 0. (57)
From this equation one can determine the dependence ofc on τn .
Since the solution obtained here is exactly the same with that of [12], then, one may raise the question whether a connection exists between the modified reductive perturbation method and the scaled multiple-space expansion method. The answer to this question is ”yes” provided that the following substitution is made
τ =ϵ3/2cx=c0τ0+c1τ1+c2τ2+ 33τ3+c4τ4+...., (58) where c is the scale parameter defined in [12] and assumes a perturbation expansion of the form
c=c0+ϵc1+ϵ2c2+ϵ3c3+ϵ4c4+...., (59) and the coefficientsci are related to g through
ci =
∂g ∂τi
, (i= 0,1,2,3, ...). (60)
3. Conclusion
The study of the effects of the first order term to higher order terms in the reductive perturbation method leads to some secularities [3, 4]. To remove such secularities various methods, like the renormalization method by Kodama and Tanuiti [6] , multiple scale expansion method by Kraenkel and Manna [7] and the modified reductive perturbation method by Demiray [9]. The methods presented in [6] and [7] are quite complicated as compared to one given in [9].
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