#### Superposition solutions to the extended KdV equation for water surface waves

Superposition solutions to the extended KdV equation for water surface waves
Piotr Rozmej 0 1
Anna Karczewska 0 1
Eryk Infeld 0 1
0 E. Infeld National Centre for Nuclear Research , Hoz ̇a 69, 00-681 Warszawa , Poland
1 A. Karczewska Faculty of Mathematics , Computer Science and Econometrics , University of Zielona Góra , Szafrana 4a, 65-246 Zielona Góra , Poland
The KdV equation can be derived in the shallow water limit of the Euler equations. Over the last few decades, this equation has been extended to include higher-order effects. Although this equation has only one conservation law, exact periodic and solitonic solutions exist. Khare and Saxena (Phys Lett A 377:27612765, 2013; J Math Phys 55:032701, 2014; J Math Phys 56:032104, 2015) demonstrated the possibility of generating new exact solutions by combining known ones for several fundamental equations (e.g., Korteweg-de Vries, nonlinear Schrödinger). Here we find that this construction can be repeated for higher-order, nonintegrable extensions of these equations. Contrary to many statements in the literature, there seems to be no correlation between integrability and the number of nonlinear one variable wave solutions.
Shallow water waves; Extended KdV equation; Analytic solutions; Nonlinear equations
1 Introduction
A long time ago, Stokes opened the field of nonlinear
hydrodynamics by showing that waves described by
nonlinear models can be periodic [
1
]. Although several
related results followed, it took half a century before
the Korteweg–de Vries equation became widely known
[
2
]. A more accurate equation system, Boussinesq, was
formulated in 1871. It is also the theme of several recent
papers [
3,4
]. Another direction research has gone in is
including perpendicular dynamics in KdV, e.g., [5].
The KdV equation is one of the most successful
physical equations. It consists of the mathematically
simplest possible terms representing the interplay of
nonlinearity and dispersion. This simplicity may be
one of the reasons for success. Here we investigate this
equation, improved as derived from the Euler inviscid
and irrotational water equations.
Just as for conventional KdV, two small
parameters are assumed: wave amplitude/depth (a/H ) and
depth/wavelength squared (H/ l)2. These
dimensionless expansion constants are called α and β. We take the
expansion one-order higher. The new terms will then be
of second order. This procedure limits considerations
to waves for which the two parameters are comparable.
Unfortunately some authors tend to be careless about
this limitation.
The next approximation to Euler’s equations for long
waves over a shallow riverbed is (η is the elevation
above a flat surface divided by H )
3 1 3 2 2
ηt + ηx + 2 α ηηx + 6 β η3x − 8 α η ηx
+ αβ
23 5
24 ηx η2x + 12 ηη3x
+ 31690 β2η5x = 0.
In (1) subsequently we use low indexes for derivatives
n
ηnx ≡ ∂∂xηn . This second-order equation was called
by Marchant and Smyth [
6,7
] the extended KdV. It was
also derived in a different way in [8] and [
9,10
]. We
call it KdV2. It is not integrable. However, by keeping
the same terms but changing one numerical coefficient
(specifically, replacing 2243 by 56 ) we can obtain an
integrable equation [
11,12
].
Not only is KdV2 non-integrable, it only seems to
have one conservation law (volume or mass) [
13
].
However, a simple derivation of adiabatically conserved
quantities can be found in [
14
].
Recently, Khare and Saxena [
15–17
] demonstrated
that for several nonlinear equations which admit
solutions in terms of elliptic functions cn(x , m), dn(x , m)
there exist solutions in terms of superpositions cn(x , m)
± √m dn(x , m). They also showed that KdV which
admits solutions in terms of dn2(x , m) also admits
solutions in terms of superpositions dn2(x , m) ±
√m cn(x , m) dn(x , m). Since then we found analytic
solutions to KdV2 in terms of cn2(x , m) [
18,19
], the
results of Khare and Saxena [
15–17
] inspired us to look
for solutions to KdV2 in similar form.
2 Exact periodic solutions for KdV2
First, we repeat shortly the results obtained by Khare
and Saxena [
15
], but formulating them for KdV in a
fixed frame, that is, for the equation
3 1
ηt + ηx + 2 α ηηx + 6 β η3x = 0.
Assuming solution in the form
η(x , t ) = A dn2[B(x − vt ), m]
one finds
4 B2β
A = 3 α
2
and v = 1 + 3 β B2(2 − m)
(1)
(2)
(3)
Next, the authors [
15
] showed that superpositions
1
η±(x , t ) = 2 A dn2[B(x − vt ), m]
± √m cn[B(x −vt ), m] dn[B(x −vt ), m]
(4)
are solutions to (2) with the same relation
between A and B, but for a different velocity,
v± = 1 + 61 β B2(5 − m).
Now, we look for periodic nonlinear wave solutions
of KdV2 (1). Introduce y := x − vt . Then η(x , t ) =
η(y), ηt = − vηy and Eq. (1) takes the form of an
ODE
(1 − v)ηy + 23 α ηηy + 61 β η3y − 83 α2η2ηy
+ αβ
+ 31690 β2η5y = 0.
23 5
24 ηy η2y + 12 ηη3y
2.1 Single periodic function dn2
(5)
(6)
First, we recall some properties of the Jacobi elliptic
functions (arguments are omitted)
sn2 + cn2 = 1,
dn2 + m sn2 = 1.
Their derivatives are
d sn d cn
dy = cn dn, dy = − sn dn,
dddyn = − m sn cn. (7)
Assume a solution of (1) in the same form as KdV
solution (3). Insertion of (3) into (5) yields
A Bm
180
cn dn sn F0 + F2 cn2 + F4 cn4
= 0.
(8)
Equation (8) holds for arbitrary arguments when
F0, F2, F4 vanish simultaneously. The explicit form of
this set of equations is following
F0 = 135α2 A2(m − 1)2 + 30α A(m − 1)
× β B2(63m − 20) + 18
Equation (11) is equivalent to the [10, Eq. (26)]
obtained for solitonic solutions to KdV2. Denoting
z := BA2αβ one obtains from (11) two possible solutions
z1 =
43 − √2305
152
< 0 and z2 =
43 + √2305
152
The case z = z1 leads to B2 < 0 and has to be rejected
as in previous papers [
10,18
]. Then for z = z2 the
amplitude A is
A =
43 + √2305 B2β
3 α
Despite the same form of solutions to KdV and
KdV2, there is a fundamental difference. KdV only
(9)
(10)
2.2 Comparison to KdV solutions
Is a solution of KdV2 much different from the KdV
solution for the same m? In order to compare
solutions of both equations, remember that the set of three
Eqs. (9)–(11) fixes all A, B, v coefficients for KdV2 for
given m. In the case of KdV, the equation analogous
to (8) only imposes two conditions on three
parameters. Therefore, one parameter, say amplitude A, can
be chosen arbitrary. Then we compare coefficients of
solutions to KdV2 and KdV choosing the same value
of A, that is, AKdV2. Such comparison is displayed in
Fig. 1 for α = β = 110 .
It is clear that vKdV2 and vKdV are very similar.
We have the following relations: for KdV B2
A = 43 βα ,
whereas for KdV2 B2
A
= βα z2. Since z2 ≈ 0.6,
BKdV/BKdV = 43z2 ≈ 1.12. The same relations hold
between KdV2 and KdV coefficients for superposition
solutions shown in Fig. 3.
The above examples for the case α = β = 110 show
that for somewhat small values of α the coefficients of
KdV2 dn2 solutions are not much different from those
of KdV.
However, physically relevant exact solutions of dn2
to KdV2 can be found for much larger values of the
1.0
0.8
0.6
0.4
0.2
0.0
0.0
0.1
0.2
0.3
0.4
0.5
parameter α. In Fig. 2, the amplitude A(α, m) given
by (15) is shown as a contour plot for the region
α ∈ [0.01, 0.5], m ∈ [
0, 1
]. It is clear that reasonable
amplitudes occur in wide regions of α and m values.
The amplitudes become too big only for α → 0, but in
such cases KdV works very well.
2.3 Superposition “ dn2 + √m cn dn”
Now assume the periodic solution to be in the same
form as the corresponding superposition solution of
KdV [
15
] function
η+(y) = 21 A dn2(B y, m) + √m cn(B y, m)
dn(B y, m) , (17)
where A, B, v are yet unknown constants. (m is the
elliptic parameter.) We will need
ηy = − 21 AB √m √m cn + dn 2 sn,
η2y = 21 AB2 √m
√m cn + dn 2
×
η3y = 21 AB3 √m √m cn + dn 2
η5y = − 21 AB5 √m √m cn + dn 2
× sn m cn2 + 6√m cn dn + dn2 − 4m sn2 , (20)
(21)
(22)
(23)
(24)
(25)
(26)
(27)
(29)
Equations (34) and (35) are equivalent and give the
same condition as (11). Solving (34) with respect to
B2, we obtain the same relations as in [10, Eq. (28)]
and
z2 =
It is clear that z1 < 0 and z2 > 0. B has to be
realvalued. This is possible for the case z = z1 if A <
0, and for z = z2 if A > 0. The value of z2 is the
same as that found for the exact soliton solution in [10,
Eq. (28)]. In general
1440(1 − v) + Aα(1 − m) [1080 − 135 Aα(1 − m)]
This looks like a contradiction, but substitution z =
z1 = (43 − √2305)/152 in both (42) and (43) gives
the same result
A1 =
.
For z = z2 = (43 +
is
A2 =
This means that not only are Eqs. (34) and (35)
equivalent, but also (32) and (33), as well. Therefore, Eqs.
(32)–(35) supply only three independent conditions for
the coefficients of KdV2 solutions in the form (17).
Now, using z = z1 and A1 given by (44) we obtain
from (39)
vnum−(m)
v1 = vden−(m)
vnum+(m)
v2 = vden+(m) ,
and with z = z2 and A2 given by (45)
(39)
(40)
(41)
(44)
(45)
(46)
(47)
where
vnum∓(m) = 6
2912513 ∓ 58361√2305 m2
2.4 Discussion of mathematical solutions
From a strictly mathematical point of view, we found
two families of solutions determined by coefficients
A, B, v as functions of the elliptic parameter m. There
are two cases.
43 − √2305
Case 2 z = z2 =
Then
Case 1 z = z1 =
152
This case leads to B2 < 0 and has to be rejected as
in previous papers [
10,18
].
43 + √2305
As
Bs
Vs
B1
V1
0
0.2
0.8
1
0.4
0.6
m
and v2 is given by (47). Since m ∈ [
0, 1
], (m−5) < 0
then B2 is real. The solution in this case is
1
η2(x − v2t, m) = 2 A2 dn2(B2(x − v2t), m)
+√m cn(B2(x − v2t), m) dn(B2(x − v2t), m) .
(50)
Coefficients A2, B2, v2 of superposition solutions
(17) to KdV2 as functions of m are presented in Fig. 3
for α = β = 110 and compared to corresponding
solutions to KdV. Here, similarly as in Fig. 1, we assume
that AKdV = AKdV2.
Physically relevant exact superposition solutions to
KdV2 can be found for greater values of the
parameter α than 110 . In Fig. 4, the amplitude A(α, m) given
by (48) is shown as a contour plot for the region
0.0
0.1
0.2
0.3
0.4
0.5
α ∈ [0.01, 0.5], m ∈ [
0, 1
]. It is clear that reasonable
amplitudes occur in wide regions of α and m values,
similarly like in the case dn2. The amplitudes become
too big only for α → 0, but in such cases KdV works
very well.
2.5 Superposition 1 “ dn2 − √m cn dn”
Now we check the alternative superposition “ dn2 −
√m cn dn”
(51)
η−(y) = 21 A dn2(By, m) − √m cn(By, m) dn(By, m) .
In this case, the derivatives are given by formulas
similar to (18)–(21) with some signs altered. Analogous
changes occur in formulas (23)–(29). Then (22) has a
similar form like (30)
21 A B √m −√m cn + dn
2
sn × (F0 + Fcd cn dn
+Fc2 cn2 + Fc3d cn3 dn + Fc4 cn4
= 0.
(52)
Equation (52) is valid for arbitrary arguments when all
coefficients F0, Fcd , Fc2 , Fc3d , Fc4 vanish
simultaneously. This gives us a set of equations for the
coefficients v, A, B. Despite some changes in signs on
the way to (52), this set is the same as for “ dn2 +
√m cn dn” superposition (31)–(35). Then the
coefficients A, B, v for superposition “ dn2 √m cn dn”
are the same as for superposition “ dn2 −+ √m cn dn”
given above. This property for KdV2 is the same as for
KdV, see [
15
]. It follows from periodicity of the Jacobi
elliptic functions. From
cn(y + 2K (m), m) = − cn(y, m),
dn(y + 2K (m), m) = dn(y, m)
it follows that
dn2(y + 2K (m), m)
+
= dn2(x , m) −
(m) cn(x + 2K (m), m) dn(x + 2K (m), m)
(m) cn(x , m) dn(x , m).
(53)
So both superpositions η+ (17) and η− (51) represent
the same solution, but shifted by the period of the Jacobi
elliptic functions. This property is well seen in Figs. 5,
6 and 7.
3 Examples
Below, some examples of wave profiles for both KdV
and KdV2 are presented. We know from Sect. 2 that
for a given m, the coefficients A, B, v of KdV2
solutions are fixed. As we have already written, this is not
the case for A, B, v of KdV solutions. So, there is one
free parameter. In order to compare KdV2 solutions to
those of KdV for identical m, we set AKdV = AKdV2. In
Figs. 5, 6 and 7 below, KdV solutions of the forms (3),
1.4
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
2.5
2
1
1.5
0.5
0
0
2.5
2
1
1.5
0
0
0
2
4
8
10
12
6
x
sup+ KdV
sup+ KdV2
single KdV
single KdV2
sup- KdV
sup- KdV2
Fig. 5 Profiles of KdV and KdV2 waves for m = 0.1. (Color
figure online)
8
x
sup+ KdV
sup+ KdV2
single KdV
single KdV2
sup- KdV
sup- KdV2
2
4
6
10
12
14
16
18
Fig. 6 Profiles of KdV and KdV2 waves for m = 0.9. (Color
figure online)
15
x
sup+ KdV
sup+ KdV2
single KdV
single KdV2
sup- KdV
sup- KdV2
5
10
20
25
30
35
Fig. 7 Profiles of KdV and KdV2 waves for m = 0.99. (Color
figure online)
1.4
1.3
(17) and (51) are drawn with solid red, green and blue
lines, respectively. For KdV2 solutions, the same color
convention is used, but with dashed lines. In all the
presented cases, the parameters α = β = 0.1 were used.
Comparison of wave profiles for different m
suggests several observations. For small m, solutions given
by the single formula (3) differ substantially from those
given by superpositions (17) and (51). Note that (3)
is equal to the sum of both superpositions and when
m → 1 the distance between crests of η+ and η−
increases to infinity (in the m = 1 limit). All three
solutions converge to the same soliton.
In order to check whether the obtained analytic
solutions are really true solutions to KdV2 several
numerical simulations were performed. In each of them, the
numerical FDM code used with success in previous
studies [
9–11,14,18,19
] was applied. Since the
calculations concerned periodic solutions, the periodic
boundary conditions were used with an x interval equal to the
particular wavelength. In Figs. 5, 6 and 7 dashed lines
display profiles of single dn2 (3) and superposition η+
and η− (4) solutions for three values of m = 0.1, 0.9
and 0.99. Below in Figs. 6, 7 and 8 six examples of
time evolution for these solutions obtained in
numerics are presented. Profiles of solutions at time instants
t = 0, T /4, T /2, 3T /4 and T , where T = λ/v are
displayed. Open symbols represent the profiles at t = T
which overlap with those at t = 0 with numerical
deviations less than 10−11. In all the presented examples, as
well as all others not shown here, numerics confirmed
a uniform motion and a fixed shape for the considered
solutions (Figs. 8, 9, 10).
1.5
0.5
2.5
2
1
1.5
0.5
0
0
0
0
1
2 3
single KdV2
4
5 6
sup+ KdV2
7
8
9
Fig. 9 Same as in Fig. 8 but for m = 0.9. (Color figure online)
2
4
single KdV2
6
8 10
sup+ KdV2
12
14
Fig. 10 Same as in Fig. 8 but for m = 0.99. (Color figure online)
4 Conclusions
The most important results of the paper can be
summarized as follows.
It is shown that several kinds of analytic solutions of
KdV2 have the same forms as corresponding solutions
to KdV but with different coefficients. This statement
is true for our single solitonic solutions [
10
], periodic
solutions in the form of single Jacobi elliptic functions
cn2 [
18
] or dn2, and for periodic solutions in the form
of superpositions dn2 ± √m cn dn (this paper).
Coefficients A, B, v of these solutions to KdV2 are fixed by
coefficients of the equation, that is by values of α, β
parameters. This is in contradiction to the KdV case
where one coefficient (usually A) is arbitrary.
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