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Elliptic traveling waves of the Olver equation
Commun Nonlinear Sci Numer Simulat 17 (2012) 4104–4114
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Elliptic traveling waves of the Olver equation Nikolai A. Kudryashov ⇑, Mikhail B. Soukharev, Maria V. Demina Department of Applied Mathematics, National Research Nuclear University ‘‘MEPhI’’, 31 Kashirskoe Shosse, 115409 Moscow, Russian Federation
a r t i c l e
i n f o
Article history: Received 19 July 2011 Received in revised form 11 November 2011 Accepted 26 January 2012 Available online 15 February 2012
a b s t r a c t Nonlinear waves on water are studied. The method recently developed by Demina and Kudryashov is applied to the Olver water wave equation. New solutions of this equation are found. These solutions are expressed in terms of the Weierstrass elliptic function. Ó 2012 Elsevier B.V. All rights reserved.
Keywords: Nonlinear water wave Elliptic solution Periodic solution Meromorphic solution Olver equation Nonlinear ordinary differential equation
1. Introduction In 1984, Olver [1] derived an unidirectional model for describing long, small amplitude waves in shallow water. This model can take the wave velocity or, alternatively, the surface elevation as the principal variable. Exact solutions for the first case were obtained in the work [2]. The second case leads to the equation
gt þ gz þ q1 gzzzzz þ q2 g2 gz þ q3 ggzzz þ q4 gz gzz þ q5 gzzz þ q6 ggz ¼ 0:
ð1:1Þ
Here dependent variable g gives a surface elevation, z is the horizontal coordinate, and coefficients qi, i = 1, . . ., 6 are real constants depending on surface tension. These coefficients are
19 s s2 2 3 5 s e ; q2 ¼ ,2 ; q3 ¼ ,e; 360 12 8 8 12 4 23 5s 1 s 3 ,e; q5 ¼ e; q6 ¼ ,: þ q4 ¼ 24 8 6 2 2
q1 ¼
ð1:2Þ
Here s represents a dimensionless surface tension coefficient, , is the ratio of wave amplitude to undisturbed fluid depth, and e is the square of the ratio of fluid depth to wave length (note that in the original work [1] coefficient q3 contains a misprint). Small parameters , and e assumed to be of the same order of smallness. In the case of no surface tension all these coefficients are nonzero, otherwise some of them can take zero values. Some solitary waves of this equation with certain values of coefficients were recently obtained by Bagderina [3]. The aim of this paper is to find exact solutions expressible in terms of elliptic functions. We apply the method developed by Kudryashov [4] and Demina and Kudryashov [5,6] for this purpose. ⇑ Corresponding author. Tel./fax: +7 4993241181. E-mail address: [email protected] (N.A. Kudryashov). 1007-5704/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2012.01.033
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Cnoidal wave appeared in the theory of water waves at the close of the 19th century in the famous work of Korteweg and de Vries [7]. Now elliptic functions are widely used to construct exact solutions of nonlinear differential equations. For example, in the works [8–11] elliptic functions were used to obtain solutions of nonlinear equations integrable by the inverse scattering transform. In the papers [12–26] exact elliptic solutions of nonlinear nonintegrable equations were constructed. However, in most cases the ansatz technique is used to find elliptic solutions. Method developed in the works [5,6] is stripped of this disadvantage. It allows us to find all possible elliptic solutions of nonlinear nonintegrable differential equation. This approach was also used in the works [27–29]. This paper is organized as follows. In Section 2 we obtain the traveling wave reductions of Eq. (1.1). In section 3 we describe the method of looking for elliptic solutions. In Sections 4 and 5 we give the complete list of elliptic traveling waves obtained. 2. Traveling wave reduction of the Olver equation Eq. (1.1) contains six arbitrary constant coefficients and four nonlinear terms. But in the original model [1] coefficient q2 is q6 always nonzero. Therefore we can simplify this equation taking new dependent variable gðz; tÞ ¼ v ðz; tÞ 2q . This substitu2 tion allows us to remove the nonlinear term ggz and gives the equation
vt þ
q2 q q 1 6 v z þ q1 v zzzzz þ q2 v 2 v z þ q3 vv zzz þ q4 v z v zz þ q5 3 6 v zzz ¼ 0: 4q2 2q2
ð2:1Þ
This equation admits the traveling wave reduction v(x) = v(z, t) with x = z ct. The traveling wave reduction takes the form
q2 q q 1 c 6 v x þ q1 v xxxxx þ q2 v 2 v x þ q3 vv xxx þ q4 v x v xx þ q5 3 6 v xxx ¼ 0: 4q2 2q2
ð2:2Þ
Here c is an arbitrary constant (wave velocity). Integrating Eq. (2.2) once we obtain the ODE
q2 q q q3 2 q q 1 c 6 v þ q1 v xxxx þ 2 v 3 þ q3 vv xx þ 4 v x þ q5 3 6 v xx þ d ¼ 0; 4q2 3 2 2q2
ð2:3Þ
where d is an integration constant. Assuming q1 – 0 we can further simplify the ODE (2.3). We will distinguish two cases: q1q2 > 0 and q1q2 < 0. In the case q1q2 > 0qsurface tension dominates gravity, it takes place for water depths less than about 3–5 mm. In this case substituting ffiffiffiffiffiffi v ðxÞ ¼ 6qq21 uðxÞ into the ODE (2.3) gives the equation
uxxxx þ 2u3 þ
2c1 c3 uuxx þ c2 u2x þ uxx þ c4 u þ c5 ¼ 0 3 3
ð2:4Þ
with
c1 ¼ 3
qffiffi
q3 3 pffiffiffiffiffiffiffi ; 2 q1 q2 q2
c4 ¼ 1c 4q16q2 ; q1
qffiffi
q4 q3 3p ffiffiffiffiffiffiffi ; 2 q1 q2
c2 ¼ c5 ¼
qffiffiffiffiffiffi
q2 d 6q1 q1
c3 ¼ 3
q5 q1
q3 q6 2q ; q 1 2
ð2:5Þ
:
Eq. (2.4) describes capillary waves in very thin layers of water (when effects of gravity are negligible). The case q1q2 < 0 corresponds to water depths more than 3–5 mm (gravity dominates surface tension). In this case we use qffiffiffiffiffiffiffiffiffiffi 1 uðxÞ and obtain the substitution v ðxÞ ¼ 6q q 2
uxxxx 2u3 þ
2c1 c3 uuxx þ c2 u2x þ uxx þ c4 u þ c5 ¼ 0: 3 3
ð2:6Þ
Here
c1 ¼ 3 c4 ¼
qffiffi
1c q1
q3 3 pffiffiffiffiffiffiffiffiffi ffi; 2 q1 q2
q26 4q1 q2
;
qffiffi 4 q3 ffi c2 ¼ 32 pqffiffiffiffiffiffiffiffiffi ; q1 q2 qffiffiffiffiffiffi d 2 c5 ¼ q : 6q q 1
c3 ¼ 3
q5 q1
q3 q6 2q ; q 1 2
ð2:7Þ
1
Eq. (2.6) describes gravitational waves (when we can neglect the effects of surface tension). Further we suppose all coefficients in Eqs. (2.4) and (2.6) to be real. In fact coefficients c4 and c5 depend on arbitrary constants c and d, so c4 and c5 may be complex. But we do not consider this situation. Let us note some obvious properties of Eqs. (2.4) and (2.6). First, in the case c1 = 3c2 Eqs. (2.4) and (2.6) have first integrals. Therefore we can expect the existence of additional solutions if c1 = 3c2. Second, both ODEs (2.4) and (2.6) are invariant under the transformations
ðu; c1 ; c2 ; c5 Þ ! ðu; c1 ; c2 ; c5 Þ;
ðx; u; c3 ; c5 Þ ! ðix; u; c3 ; c5 Þ:
Therefore we can look for solutions with c1 P 0 only and reduce the number of distinct elliptic solutions.
ð2:8Þ
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3. Method applied The main idea of the method [5,6] is as follows. First, consider an ordinary differential equation. Find all Laurent series representing its formal solution in the neighborhood of movable pole x = x0. Second, choose elliptic function with unknown constant parameters that can be a solution of the ODE under study. Note that this choice may be not unique. The form of such a function depends on number and order of movable poles in formal solution of ODE. Then we find the Laurent expansions of this elliptic function in the neighborhood of all its poles inside a parallelogram of periods. Third, compare the series obtained in the first and the second steps and form a system of algebraic equations for unknown parameters. The fourth step is to solve this system. If this system has a solution then an ODE can have an elliptic solution. To prove this one should substitute the elliptic function with parameters just found into the ODE. This is the last step. It may lead to some constraints on the ODE’s coefficients for the elliptic solution to exist. Details of this algorithm (not only for elliptic solutions, but also for rational solutions and solutions expressed in terms of trigonometric or hyperbolic functions) can be found in Refs. [4–6,27–29]. In this section we consider the ODE (2.4) for capillary waves. Analysis of the ODE (2.6) is quite similar. The ODE (2.4) has constant coefficients, so it is autonomous. For any solution f(x) of an autonomous ODE the function u(x) = f(x x0) with arbitrary complex constant x0 also satisfies the equation. Further we will often omit x0 for simplicity. If a solution of an ODE has a movable pole at position x0 then this solution can be presented in the form of local Laurent series
uðxÞ ¼ ðx x0 Þp
1 X
uj ðx x0 Þj
ð3:1Þ
j¼0
with p > 0 and uj being constant coefficients. Here p is the order of movable pole x0. Substitution of expression (3.1) into the qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ODE (2.4) gives p = 2 and u0 ¼ c1 c2 ðc1 þ c2 Þ2 60. Therefore the special solution of ODE can have exactly one pole, or exactly two poles, or countable number of poles (of only one type or both types simultaneously) [5,6]. In the present work we will look for special solutions of elliptic type only. So we concentrate on the case of infinitely many poles forming a regular doubly periodic structure in the complex plane. This structure consists of identical parallelograms (called parallelograms of periods in the theory of elliptic functions). Every parallelogram contains not more than M poles, where M is the number of different Laurent series of the form (3.1). This information helps to choose the form of elliptic function at the second step of algorithm. It is well known that any elliptic function can be represented in terms of the Weierstrass Zeta-function f(x; g2, g3) (or f(x) for short) with invariants g2 and g3 [4–6,28–32]. This representation may be written as
uðxÞ ¼ h þ
X
bk fðx bk Þ þ
k
pk X
! ak;r fðr1Þ ðx bk Þ :
ð3:2Þ
r¼2
Here h is a constant, bk are the distinct poles of function u(x). Outer sum is the sum over all poles inside a parallelogram of periods. This sum obviously vanishes in the case of only one pole in a parallelogram. Constant coefficients bk are the residues P of u(x) at the poles bk. Due to the residue theorem the following correlation k bk ¼ 0 holds for u(x) to be an elliptic function. The upper limit pk of inner sum represents the order of pole bk (note that the last term with sum is absent in the case of simple poles, i.e. when pk = 1). Coefficients ak,r are some constants, f(r1)(x; g2, g3) is the (r 1)th derivative of Zeta-function. For the ODE (2.4) we take exactly two possibilities. An elliptic special solution may have either two second order poles in a parallelogram of periods or exactly one second order pole. In the case of two distinct second order poles we choose k = {1, 2} and p1 = p2 = 2 in expression (3.2). Taking into account formula f0 (x) = }(x) we can rewrite expression (3.2) in the form
uðxÞ ¼ b1 fðxÞ þ b2 fðx bÞ þ a1 }ðxÞ þ a2 }ðx bÞ þ h
ð3:3Þ
with the condition b1 + b2 = 0 for u(x) to be elliptic. Here }(x) }(x; g2, g3) is the Weierstrass elliptic function. Due to the invariance of an autonomous ODE under the transformation x ´ x x0 we always can assume that one of the poles has the position x = 0 (instead of b1) without loss of generality. Then b – 0 represents the position of another pole in a parallelogram of periods. Elliptic function (3.3) contains eight unknown parameters: a1, a2, b1, b2, h, g2, g3, and b. In the case of one pole we take k = {1} and the outer sum in (3.2) vanishes. Then due to the residue theorem for elliptic functions b1 = 0. Therefore this elliptic function takes the form
uðxÞ ¼ a}ðxÞ þ h:
ð3:4Þ
We again use the arbitrariness of x0 and take b1 = 0 for simplicity. Note that there are only 4 unknown parameters: a,h, g2 and g3. One can obtain similar results for the Eq. (2.6). In this work we consider the case when all the series of the form (3.1) do not contain arbitrary coefficient or otherwise we suppose that they are fixed.
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4. Elliptic special solutions of the ODE (2.4) 4.1. Solutions containing two poles in a parallelogram of periods In this section we are looking for special solutions of Eq. (2.4) in the form (3.3). At the first step we find all the Laurent series representing the formal solution of Eq. (2.4). Let x = 0 and x = b be the positions of poles in a parallelogram of periods. Then the Laurent series in the neighborhood of these poles are as follows
uðxÞ ¼ uðxÞ ¼
u0 u1 þ þ u2 þ u3 x þ u4 x2 þ ; x2 x
v0
ðx bÞ2
þ
v1
xb
þ v 2 þ v 3 ðx bÞ þ v 4 ðx bÞ2 þ
ð4:1Þ ð4:2Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with u0 ¼ c1 c2 þ ðc1 þ c2 Þ2 60 and v 0 ¼ c1 c2 ðc1 þ c2 Þ2 60. Additional condition is u0 – v0 for two distinct poles to exist. Due to the residue theorem for elliptic functions u1 + v1 = 0. In order to satisfy this condition we set u1 = v1 = 0. The subsequent coefficients are not uniquely determined. This is due to the series (4.1) and (4.2) can contain an arbitrary coefficient at certain values of ODE parameters c1, . . ., c5. For example if we take c3 = 0 then either u2 is arbitrary, v2 = 0, or v2 is arbitrary, u2 = 0. If we force c3 – 0 then both u2 and v2 have certain nonzero values. An analogous situation takes place while determining the values of coefficients with higher indexes. Therefore the number of distinct series (4.1) and (4.2) is infinite. Fortunately it is enough to compute coefficients up to u4 and v4 to find all the unknown parameters of the expression (3.3), so the list of (truncated) series becomes finite. We have constructed all possible pairs of series (4.1) and (4.2) (about thirty pairs) with different coefficients. We do not present here all these series because only a few of them lead to distinct elliptic solutions of Eq. (2.4). At the second step we find the Laurent series for the function (3.3) in the neighborhood of poles x = 0 and x = b. These series have the form
uðxÞ ¼
a1 b1 a1 5a2 b2 þ þ ðh þ a2 }ðbÞ b2 fðbÞÞ ða2 }0 ðbÞ þ b2 }ðbÞÞx þ g 2 þ 3a2 }2 ðbÞ þ }0 ðbÞ x2 þ 2 x x 20 2
ð4:3Þ
and
uðxÞ ¼
a2 b2 þ þ ðh þ a1 }ðbÞ þ b1 fðbÞÞ þ ða1 }0 ðbÞ b1 }ðbÞÞðx bÞ ðx bÞ2 x b a2 5a1 b1 g 2 þ 3a1 }2 ðbÞ }0 ðbÞ ðx bÞ2 þ þ 20 2
ð4:4Þ
The third step is to compare series (4.1) and (4.3), then (4.2) and (4.4). This leads us to the system of algebraic equations for determining unknown parameters of function (3.3). Note that there is an additional algebraic equation connecting the values of }0 (b) and }(b)
ð}0 ðbÞÞ2 ¼ 4}3 ðbÞ g 2 }ðbÞ g 3 :
ð4:5Þ
It follows from the Weierstrass differential equation for the function }(x). Of course, at this step we obtain the number of algebraic systems equal to the number of distinct series obtained at the first step. At the fourth step we solve these algebraic systems and find all the unknown parameters in (3.3). For all cases b1 = u1 = 0 and b2 = v2 = 0 so the expression (3.3) can be simplified
uðxÞ ¼ a1 }ðxÞ þ a2 }ðx bÞ þ h:
ð4:6Þ
Note that comparing coefficients of the Laurent series is a crucial point of the method used. The main advantage of this scheme is that we obtain linear algebraic equations for the most of parameters of solution. The reverse of the medal is that we need to compute several dozens of Laurent series representing the solution of ODE (2.4) (i.e. all possible Laurent series). It is necessary to guarantee that no elliptic solution is missed. So in the framework of this method we solve a large number (slightly more than number of series multiplied by number of unknown parameters) of simple (at most quadratic in the case of ODE (2.4)) algebraic equations. If we use a standard technique (i.e. if we substitute the formal elliptic solution (3.3) directly in the ODE (2.4)) the situation is dramatically changed. We obtain a small number of algebraic equations (namely, eighteen equations) for solution parameters. But these equations are very complicated and include high degrees of unknown parameters. It seems unreal to obtain solution of these equations. So this way (omitting the Laurent series) gives no result. The last step is to substitute expression (4.6) with already known parameters into the ODE (2.4) and to find constraints on the ODE coefficients. Note that at previous step we have found the value of }(b), not the value of b itself. Therefore we cannot use expression (4.6) for this substitution. But using the addition theorem for the Weierstrass elliptic function }(x)
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}ðx bÞ ¼ }ðxÞ }ðbÞ þ
2 1 }0 ðxÞ þ }0 ðbÞ 4 }ðxÞ }ðbÞ
ð4:7Þ
we can rewrite (4.6) as
uðxÞ ¼
2 a2 }0 ðxÞ þ }0 ðbÞ þ ða1 a2 Þ}ðxÞ þ h a2 }ðbÞ: 4 }ðxÞ }ðbÞ
ð4:8Þ
Moreover, at the fourth step we have found that for all cases }0 (b) = 0. Therefore the expression (4.8) can be further simplified using Eq. (4.5) with zero left hand side and the Weierstrass equation itself. So the final form of expression (3.3) in our case is
uðxÞ ¼
a2 12}2 ðbÞ g 2 þ a1 }ðxÞ þ a2 }ðbÞ þ h: 4 }ðxÞ }ðbÞ
ð4:9Þ
Note that the number of special solutions obtained in this way is rather large. But for some of them the equality g 32 27g 23 ¼ 0 takes place. We omit these solutions because they are degenerate elliptic and can be expressed via the hyperbolic or rational functions. The number of remaining distinct elliptic solutions can be dramatically reduced, if we take into account the invariance of the ODE under the transformations mentioned in Section 2. Namely, the first transformation allows us to consider only the special solutions with c1 P 0. The second transformation allows us to change the sign of g3 due to the homogeneity relation }(x; g2, g3) = }(ix; g2, g3). Another reduction can be done, if we set x0 = b and take into account the evenness of the Weierstrass elliptic function }(x). Performing all five steps of the method described above we obtain the following results. If c1 = 3c2 and c22 – 15=4, then the Eq. (2.4) has elliptic special solutions with two poles in a parallelogram of period. There are only three solutions of this type. The first solution takes the form
uðxÞ ¼
a2 12}2 ðbÞ g 2 c2 c3 þ a1 }ðx x0 Þ þ a2 }ðbÞ þ 2 4 }ðx x0 Þ }ðbÞ 18 c2 5
ð4:10Þ
where
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a1 ¼ 4c2 þ 2 4c22 15;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 ¼ 4c2 2 4c22 15; 2 c2 30 þ c22 þ 54c4 c22 5 c3 ; g2 ¼ 3 }ðbÞ ¼ 2 2 ; 36 c2 5 216 c22 5 4c22 27 2 c3 c23 36 þ 11c22 þ 162c4 c22 5 g3 ¼ : 2 3 23328 c2 5 4c22 27
ð4:11Þ ð4:12Þ
ð4:13Þ
Additional constraints are
2 c2 c3 c23 4c42 þ 39c22 176 þ 18c4 c22 5 4c22 5 ; 108 ðc22 5Þ3 ð4c22 27Þ 2 2 c2 5 4c2 27 – 0; c1 ¼ 3c2 : c5 ¼
ð4:14Þ ð4:15Þ
The coefficients c2, c3 and c4 can take any values. The second solution takes the form
uðxÞ ¼
pffiffiffi pffiffiffi 5 3 1323g 2 4c23 c3 2 3}ðx x0 Þ pffiffiffi 42 63}ðx x0 Þ c3 3 3
ð4:16Þ
where
g3 ¼
4c33 c3 g 2 : 250047 63
ð4:17Þ
The invariant g2 is an arbitrary constant. Additional constraints are
pffiffiffi 9 3 ; c1 ¼ 2
pffiffiffi 3 3 c2 ¼ ; 2
2c2 c4 ¼ 3 ; 9
44c3 5c23 2646g 2 pffiffiffi : c5 ¼ 3087 3
ð4:18Þ
The coefficient c3 may take any value. The third solution takes the form 2
uðxÞ ¼
pffiffiffi 3 126h þ 5c4 pffiffiffi 2 5}ðx x0 Þ 2h 28 2 5}ðx x0 Þ h
ð4:19Þ
N.A. Kudryashov et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 4104–4114
4109
where 2
g2 ¼
3h c4 þ ; 2 28
3
g3 ¼
13h hc pffiffiffi p4ffiffiffi : 20 5 56 5
ð4:20Þ
The constant h is arbitrary. Additional constraints are
pffiffiffi c 1 ¼ 3 5;
c2 ¼
pffiffiffi 5;
3
c3 ¼ 0;
c5 ¼
918h 45hc4 : 5 7
ð4:21Þ
The coefficient c4 can take any value. Note that all three solutions contain poles on the real axis if one takes x0 = 0. But these solutions can be made bounded under some conditions if we take into account the arbitrariness of x0. Namely, }0 (b) = 0 for all solutions. Therefore }(b) is one of the roots of the cubic equation 4y3 g2y g3 = 0. These roots are denoted as e1, e2, e3 in the theory of elliptic functions [32]. All these roots are real and distinct if condition D ¼ g 32 27g 23 > 0 holds. Then in usual notation e1 > e2 > e3. In the case D > 0 the function }(z) has one real half-period x1 with }(x1) = e1 and one pure imaginary half-period x3 with }(x3) = e3. The function }(z) is bounded on the lines z = x + x3 and z = iy + x1 in the complex plane. Let }(b) = e1, then both poles (inside one parallelogram of periods) of the functions (4.6) and (4.9) are on the real axis, so these functions are bounded on the line z = x + x3. Using the arbitrariness of x0 we can take x0 = x3, then all our solutions become bounded. There is no need to calculate x3 explicitly for plotting these functions. Instead of this we can use the special case of the addition theorem (4.7) in the form [31]
}ðx þ x3 Þ ¼ e3 þ
ðe3 e1 Þðe3 e2 Þ : }ðxÞ e3
ð4:22Þ
Then, using the identities e1 þ e2 þ e3 ¼ 0; g 2 ¼ 2 e21 þ e22 þ e23 ; g 3 ¼ 4e1 e2 e3 we can rewrite the expression (4.9) in the form
uðxÞ ¼ a1
ðe1 e3 Þðe2 e3 Þ ðe1 e2 Þðe3 e2 Þ þ a2 þ a1 e3 þ a2 e2 þ h: }ðxÞ e3 }ðxÞ e2
ð4:23Þ
For example let us plot the third solution using the representation (4.23). In this case we have
pffiffiffi a1 ¼ 2 5;
pffiffiffi a2 ¼ 6 5;
h e1 ¼ }ðbÞ ¼ pffiffiffi : 2 5
ð4:24Þ
The other two roots of the Equation 4y3 g2y g3 = 0 are
e1 1 e2 ¼ þ 2 2
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c4 27e21 þ ; 28
e1 1 e3 ¼ 2 2
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c4 27e21 þ : 28
ð4:25Þ
The free parameters h and c4 must be taken in such a way that the conditions D > 0 and e1 > e2 > e3 hold. For example we take c4 = 1 and h = 0.199. The resulting plot is presented in Fig. 1. 4.2. Solutions containing one pole in a parallelogram of periods In this section we are looking for the special solution of Eq. (2.4) in the form (3.4). At the first step we need to construct the Laurent series (3.1) in the neighborhood of the second order pole x = 0. This expansion has the form
Fig. 1. Solution (4.19) with c4 = 1 and h = 0.199.
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uðxÞ ¼
a c1 c2 x2
þ
u1 þ u2 þ u3 x þ u4 x2 þ u5 x3 þ u6 x4 þ x
ð4:26Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with a ¼ ðc1 þ c2 Þ2 60. In this case we need to compute coefficients up to u6 to find all the unknown parameters of the expression (3.4). Note that we can simplify the subsequent calculations if we take into account the evenness of the function (3.4). It follows that u1 = u3 = u5 = 0 in the expansion (4.26). As in the previous section, the series (4.26) may contain arbitrary coefficients. Therefore the expansion (4.26) is not unique. At the first step of the algorithm we have obtained rather large list of all possible series (4.26) with distinct coefficients u2, u4 and u6, so we do not present it here. At the second step we find the Laurent series for the function (3.4) in the neighborhood of the pole x = 0. It takes the form
uðxÞ ¼
a ag ag þ h þ 2 x2 þ 3 x4 þ x2 20 28
ð4:27Þ
At the third step we compare the series (4.26) and (4.27). For every expansion (4.26) found at the first step we have obtained a system of four algebraic equations. Solving this equations at the fourth step we find parameters a, h, g2 and g3 of the expression (3.4). At the last step we substitute the expression (3.4) with known parameters into the ODE (2.4) and find the constraints on the ODE coefficients. After that we reduce the number of solutions obtained. This was done in the same way as in the previous section. The results are as follows. Eq. (2.4) has six distinct elliptic solutions with one pole in a parallelogram of periods. The first solution takes the form
uðxÞ ¼ ða c1 c2 Þ}ðx x0 Þ þ
c3 c1 þ 3c2 3a
ð4:28Þ
where
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a ¼ ðc1 þ c2 Þ2 60; g2 ¼ g3 ¼
ð4:29Þ
180c23 þ 30c4 ð3a c1 3c2 Þ2 ðc1 þ c2 aÞð9a c1 21c2 Þð3a c1 3c2 Þ2 Ac33 þ Bc3 c4 þ Fc5
Gða c1 c2 Þð9a c1 21c2 Þð3a c1 3c2 Þ3
A ¼ 8ð11c1 þ 6c2 9aÞ;
ð4:30Þ
; ;
B ¼ 2ð3a c1 3c2 Þ2 ð7c1 3c2 3aÞ;
F ¼ ðc1 þ 21c2 9aÞð3a c1 3c2 Þ3 ;
G ¼ c2 ðc1 þ c2 aÞ 12:
ð4:31Þ ð4:32Þ ð4:33Þ
Constraints on the ODE’s coefficients are
Gða c1 c2 Þð9a c1 21c2 Þð3a c1 3c2 Þ – 0:
ð4:34Þ
The coefficients c1, c2, c3, c4, and c5 can take any values. The second solution takes the form
uðxÞ ¼
12 c2 c3 }ðx x0 Þ þ c2 2ð18 c1 c2 Þ
ð4:35Þ
where
g2 ¼
15c42 c23 þ 10c22 c4 ðc1 c2 18Þ2 16ðc1 c2 18Þ2 ð2c1 c2 3c22 27Þ
:
ð4:36Þ
The invariant g3 is an arbitrary constant. Constraints on the ODE’s coefficients are
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 c1 c21 36 ; 3 2 c2 c3 c22 c23 2c1 c2 3c22 þ 108 þ 2c4 ðc1 c2 18Þ 2c1 c2 3c22 þ 18 ; c5 ¼ 4 ðc1 c2 18Þ3 ð2c1 c2 3c22 27Þ
c2 ¼
ðc1 c2 18Þð2c1 c2 3c22 27Þ – 0:
ð4:37Þ ð4:38Þ ð4:39Þ
The coefficients c1, c3 and c4 may take any values. The third solution takes the form
uðxÞ ¼
4c2 8c1 3c3 }ðx x0 Þ þ 3 9 2ðc1 6c2 Þ
ð4:40Þ
N.A. Kudryashov et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 4104–4114
4111
where
g3 ¼
63 729c33 54c5 ðc1 6c2 Þ3 8c3 g 2 ðc1 6c2 Þ2 ð2c1 3c2 Þ2 : 16 ðc1 6c2 Þ3 ð2c1 3c2 Þð20c21 54c1 c2 þ 36c22 81Þ
ð4:41Þ
The invariant g2 is an arbitrary constant. Constraints on the ODE’s coefficients are
c2 ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 c1 3 c21 54 ; 6
c4 ¼
27c23 2ðc1 6c2 Þ2
ð4:42Þ
;
ðc1 6c2 Þð2c1 3c2 Þð20c21 54c1 c2 þ 36c22 81Þ – 0:
ð4:43Þ
The coefficients c1, c3 and c5 may take any values. The fourth solution takes the form
uðxÞ ¼
2c1 }ðx x0 Þ þ h 3
ð4:44Þ
where 2
9 6h þ c4 ; 2 c21 54 3 27 4h 2c21 27 þ 2hc4 c21 27 þ c5 c21 54 : g3 ¼ 4c1 ðc21 54Þð81 2c21 Þ
g2 ¼
ð4:45Þ ð4:46Þ
The constant h is arbitrary. Constraints on the ODE’s coefficients are
c2 ¼
45 2c1 ; c1 3
ðc21 54Þð2c21 81Þ – 0;
c3 ¼ 0;
c1 – 0:
ð4:47Þ
The coefficients c1, c4 and c5 may take any values. The fifth solution takes the form
pffiffiffi uðxÞ ¼ 3 2}ðx x0 Þ þ h
ð4:48Þ
where 2
g2 ¼
6h þ c4 : 3
ð4:49Þ
The invariant g3 and the constant h are arbitrary. Constraints on the ODE’s coefficients are
9 c1 ¼ pffiffiffi ; 2
pffiffiffi c 2 ¼ 2 2;
c3 ¼ 0;
2
c5 ¼ 2hð8h þ c4 Þ:
ð4:50Þ
The coefficient c4 can take any value. The sixth solution takes the form
pffiffiffi uðxÞ ¼ 2 6}ðx x0 Þ þ
rffiffiffiffiffiffiffiffiffi c4 6
ð4:51Þ
where
g3 ¼
pffiffiffiffiffiffiffiffiffi c5 c4 pffiffiffi ðc4 þ 18g 2 Þ: 108 12 6
ð4:52Þ
The invariant g2 is an arbitrary constant. Constraints on the ODE’s coefficients are
pffiffiffi c 1 ¼ 3 6;
c2 ¼
rffiffiffi 3 ; 2
c3 ¼ 0:
ð4:53Þ
The coefficients c4 and c5 may take any values. These solutions contain poles on the real axis if one takes x0 = 0. To make them bounded one should set x0 = x3 as in the previous section. Then using the formula (4.22) we can rewrite the expression (3.4) in the form
uðxÞ ¼ a
ðe3 e1 Þðe3 e2 Þ þ ae3 þ h: }ðxÞ e3
ð4:54Þ
This function will be bounded if we force D ¼ g 32 27g 23 > 0 and e1 > e2 > e3. pffiffiffi For example let us plot the fifth solution using representation (4.54). In this case a ¼ 3 2. Constants e1, e2, and e3 are the roots of the cubic equation
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Fig. 2. Solution (4.48) with h = 1, c4 = 38.52 and g3 = 6.84.
2
4y3 þ
6h þ c4 y g 3 ¼ 0: 3
ð4:55Þ
For example we take h = 1, c4 = 38.52, g3 = 6.84. The resulting plot is presented in Fig. 2. 5. Elliptic special solutions of the ODE (2.6) 5.1. Solution containing two poles in a parallelogram of periods Eq. (2.6) differs from ODE (2.4) only in the sign of nonlinear term u3. Therefore all computations are quite similar to the ones in Section 4. But the results are different. We obtain only one elliptic special solution of the form
uðxÞ ¼
a2 12}2 ðbÞ g 2 c2 c3 þ a1 }ðx x0 Þ þ a2 }ðbÞ þ 2 4 }ðx x0 Þ }ðbÞ 18 c2 þ 5
ð5:1Þ
where
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a1 ¼ 4c2 þ 2 4c22 þ 15;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 ¼ 4c2 2 4c22 þ 15; 2 54c4 c22 þ 5 þ c23 30 c22 c3 ; g2 ¼ }ðbÞ ¼ 2 ; 36 c22 þ 5 216 c22 þ 5 4c22 þ 27 2 c3 c23 36 11c22 þ 162c4 c22 þ 5 : g3 ¼ 2 3 2 23328 c2 þ 5 4c2 þ 27
ð5:2Þ ð5:3Þ
ð5:4Þ
Constraints on the ODE’s coefficients are
c1 ¼ 3c2 ;
c5 ¼
2 c2 c3 c23 176 þ 39c22 4c42 þ 18c4 c22 þ 5 4c22 þ 5 : 2 3 2 108 c þ 5 4c þ 27 2
ð5:5Þ
2
The coefficients c2, c3 and c4 may take any values. 5.2. Solutions containing one pole in a parallelogram of periods There are four elliptic special solutions of the form (3.4) that satisfy the Eq. (2.6). These are as follows. The first solution takes the form
uðxÞ ¼ ða þ c1 þ c2 Þ}ðx x0 Þ þ
c3 3a þ c1 þ 3c2
ð5:6Þ
where
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a ¼ ðc1 þ c2 Þ2 þ 60;
ð5:7Þ
N.A. Kudryashov et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 4104–4114
g2 ¼ g3 ¼
180c23 þ 30c4 ð3a þ c1 þ 3c2 Þ2 ða þ c1 þ c2 Þð9a þ c1 þ 21c2 Þð3a þ c1 þ 3c2 Þ2
Gða þ c1 þ c2 Þð9a þ c1 þ 21c2 Þð3a þ c1 þ 3c2 Þ3
A ¼ 8ð9a þ 11c1 þ 6c2 Þ;
ð5:8Þ
;
Ac33 þ Bc3 c4 þ Fc5
;
B ¼ 2ð3a þ c1 þ 3c2 Þ2 ð3c2 3a 7c1 Þ; 3
F ¼ ð9a þ c1 þ 21c2 Þð3a þ c1 þ 3c2 Þ ;
4113
G ¼ c2 ðc1 þ c2 þ aÞ þ 12:
ð5:9Þ ð5:10Þ ð5:11Þ
Constraints on the ODE’s coefficients are
Gða þ c1 þ c2 Þð9a þ c1 þ 21c2 Þð3a þ c1 þ 3c2 Þ – 0:
ð5:12Þ
The coefficients c1, c2, c3, c4, and c5 may take any values. The second solution takes the form
uðxÞ ¼
12 c 2 c3 }ðx x0 Þ c2 2ð18 þ c1 c2 Þ
ð5:13Þ
where
g2 ¼
15c42 c23 þ 10c22 c4 ðc1 c2 þ 18Þ2 16ðc1 c2 þ 18Þ2 ð2c1 c2 3c22 þ 27Þ
:
ð5:14Þ
The invariant g3 is arbitrary. Constraints on the ODE’s coefficients are
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 c1 c21 þ 36 ; 3 2 c2 c3 c22 c23 3c22 2c1 c2 þ 108 2c4 ðc1 c2 þ 18Þ 3c22 2c1 c2 þ 18 ; c5 ¼ 4 ðc1 c2 þ 18Þ3 ð2c1 c2 3c22 þ 27Þ
c2 ¼
ðc1 c2 þ 18Þð2c1 c2 3c22 þ 27Þ – 0:
ð5:15Þ ð5:16Þ ð5:17Þ
The coefficients c1, c3 and c4 can take any values. The third solution takes the form
uðxÞ ¼
8c1 4c2 3c3 }ðx x0 Þ þ 9 3 2ðc1 6c2 Þ
ð5:18Þ
where
g3 ¼
63 729c33 54c5 ðc1 6c2 Þ3 þ 8c3 g 2 ðc1 6c2 Þ2 ð2c1 3c2 Þ2 : 16 ðc1 6c2 Þ3 ð2c1 3c2 Þ 20c21 54c1 c2 þ 36c22 þ 81
ð5:19Þ
The invariant g2 is an arbitrary constant. Constraints on the ODE’s coefficients are
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 27c23 c1 3 c21 þ 54 ; c4 ¼ ; 6 2ðc1 6c2 Þ2 ðc1 6c2 Þð2c1 3c2 Þ 20c21 54c1 c2 þ 36c22 þ 81 – 0:
c2 ¼
ð5:20Þ ð5:21Þ
The coefficients c1, c3 and c5 may take any values. The fourth solution takes the form
uðxÞ ¼
2c1 }ðx x0 Þ þ h 3
ð5:22Þ
where 2
9 6h c4 ; 2 c21 þ 54 3 27 4h 2c21 þ 27 2hc4 c21 þ 27 c5 c21 þ 54 g3 ¼ : 4c1 ðc21 þ 54Þð2c21 þ 81Þ
g2 ¼
ð5:23Þ ð5:24Þ
The constant h is arbitrary. Constraints on the ODE’s coefficients are
c2 ¼
45 2c1 ; c1 3
c3 ¼ 0;
c1 – 0:
The coefficients c1, c4 and c5 can take any values.
ð5:25Þ
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N.A. Kudryashov et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 4104–4114
6. Conclusion In this paper we have presented elliptic traveling waves of the Olver equation. As far as we know all these solutions are completely new. To achieve this goal we have used the powerful method by Demina and Kudryashov. In the framework of this method no a priori assumptions on the form of solution are made. Therefore it allows one to find all possible elliptic solutions contrary to other methods (with the only exception made for Musette–Conte method [20], but their method involves a huge amount of computations and cannot be recommended to deal with complicated equations like the Olver equation). Moreover, almost all calculations done with Demina–Kudryashov method consist in solving linear equations to obtain constant parameters of solution. This is a great advantage of this method. Acknowledgement This research was partially supported by Federal Target Program ‘‘Research and Scientific-Pedagogical Personnel of Innovation in Russian Federation’’ on 2009–2013. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]
Olver PJ. Hamiltonian and non-Hamiltonian models for water waves. Lecture notes in physics, vol. 195. New York: Springer-Verlag; 1984. p. 273–90. Kudryashov NA, Sukharev MB. Exact solutions of a non-linear fifth-order equation for describing waves on water. J Appl Math Mech 2001;65:855–65. Bagderina YuYu. A new family of evolution water-wave equations possessing two-soliton solutions. Phys Lett A 2009;373:4322–7. Kudryashov NA. Meromorphic solutions of nonlinear ordinary differential equations. Commun Nonlinear Sci Numer Simulat 2010;15:2778–90. Demina MV, Kudryashov NA. Explicit expressions for meromorphic solutions of autonomous nonlinear ordinary differential equations. Commun Nonlinear Sci Numer Simulat 2011;16:1127–34. Demina MV, Kudryashov NA. From Laurent series to exact meromorphic solutions: the Kawahara equation. Phys Lett A 2010;374:4023–9. Korteweg DJ, de Vries G. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Philos Mag Ser 5 1995;39:422–43. Airault H, Mckean HP, Moser J. Rational and elliptic solutions of the Korteweg-de Vries equation and a related many–body problem. Commun Pure Appl Math 1977;30:95–148. Nakamura A. A direct method of calculating periodic wave solutions to nonlinear evolution equationsI. Exact two-periodic wave solution. J Phys Soc Japan 1979;47:1701–5. Krichever IM. Elliptic solutions of the Kadomtsev–Petviashvili equation and integrable systems of particles. Func Anal Appl 1980;14:282–90. Chelnokov VE, Zeitlin MG. The elliptic solution of the sinh-Gordon equation. Phys Lett A 1983;99(4):147–9. Kudryashov NA. Exact solutions of the generalized Kuramoto–Sivashinsky equation. Phys Lett A 1990;147:287–91. Kudryashov NA. Exact solutions of the non-linear wave equations arising in mechanics. J Appl Math Mech 1990;54:372–5. Kudryashov NA. On types of nonlinear nonintegrable equations with exact solutions. Phys Lett A 1991;155:269–75. Porubov AV. Exact travelling wave solutions nonlinear evolution equation of surface waves in a convecting fluid. J Phys A 1993;26:797–800. Liu SK, Fu ZT, Liu SD, Zhao Q. Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations. Phys Lett A 2001;289(1–2):69–74. Fu Z, Liu S, Liu S, Zhao Q. New Jacoby elliptic function expansion and new periodic solutions of nonlinear wave equations. Phys Lett A 2001;290:72–6. Parkes EJ, Duffy BR, Abbott PC. The Jacoby elliptic–function method for finding periodic-wave solutions to nonlinear evolution equations. Phys Lett A 2002;295:280–6. Fan E, Zhang J. Applications of the Jacobi elliptic function method to special type nonlinear equations. Phys Lett A 2002;305(6):383–92. Musette M, Conte R. Analytic solitary waves of nonintegrable equations. Physica D 2003;181:70–9. Kudryashov NA. Simplest equation method to look for exact solutions of nonlinear differential equations. Chaos Solitons Fract 2005;24:1217–31. Fan EG, Hon YC. Quasiperiodic waves and asymptotic bahavior for Bogoyavlenskii’s breaking soliton equation in (2+1) dimensions. Phys Rev E 2008;78:036607. Conte R, Musette M. Elliptic general analytic solutions. Stud Appl Math 2009;123:63–81. Ma WX, Zhou RG, Gao L. Exact one-periodic and two-periodic wave solutions to Hirota bilinear equations in (2+1) dimesions. Mod Phys Lett A 2009;24:1677–88. Tian SF, Zhang HQ. Riemann theta–functions periodic wave solutions and rational characteristics for the nonlinear equations. J Math Anal Appl 2010;371:585–608. Tian SF, Zhang HQ. A kind of explicit Riemann Theta functions periodic waves solutions for discrete soliton equations. Commun Nonlinear Sci Numer Simulat 2011;16:173–86. Kudryashov NA, Sinelshchikov DI, Demina MV. Exact solutions of the generalized Bretherton equation. Phys Lett A 2011;375:1074–9. Kudryashov NA, Sinelshchikov DI. Exact solutions of the Swift–Hohenberg equation with dispersion. Commun Nonlinear Sci Numer Simulat 2012;17:26–34. Demina MV, Kudryashov NA. On elliptic solutions of nonlinear ordinary differential equations. Appl Math Comput 2011;217:9849–53. Akhiezer NI. Elements of the theory of elliptic functions. Providence, RI: American Mathematical Society; 1990. Lawden DF. Elliptic functions and applications. Applied mathematical sciences, vol. 80. New York: Springer-Verlag; 1989. Abramowitz M, Stegun IA. Handbook of mathematical functions: with formulas, graphs, and mathematical tables. Dover Publications; 1965.
Contents lists available at SciVerse ScienceDirect
Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Elliptic traveling waves of the Olver equation Nikolai A. Kudryashov ⇑, Mikhail B. Soukharev, Maria V. Demina Department of Applied Mathematics, National Research Nuclear University ‘‘MEPhI’’, 31 Kashirskoe Shosse, 115409 Moscow, Russian Federation
a r t i c l e
i n f o
Article history: Received 19 July 2011 Received in revised form 11 November 2011 Accepted 26 January 2012 Available online 15 February 2012
a b s t r a c t Nonlinear waves on water are studied. The method recently developed by Demina and Kudryashov is applied to the Olver water wave equation. New solutions of this equation are found. These solutions are expressed in terms of the Weierstrass elliptic function. Ó 2012 Elsevier B.V. All rights reserved.
Keywords: Nonlinear water wave Elliptic solution Periodic solution Meromorphic solution Olver equation Nonlinear ordinary differential equation
1. Introduction In 1984, Olver [1] derived an unidirectional model for describing long, small amplitude waves in shallow water. This model can take the wave velocity or, alternatively, the surface elevation as the principal variable. Exact solutions for the first case were obtained in the work [2]. The second case leads to the equation
gt þ gz þ q1 gzzzzz þ q2 g2 gz þ q3 ggzzz þ q4 gz gzz þ q5 gzzz þ q6 ggz ¼ 0:
ð1:1Þ
Here dependent variable g gives a surface elevation, z is the horizontal coordinate, and coefficients qi, i = 1, . . ., 6 are real constants depending on surface tension. These coefficients are
19 s s2 2 3 5 s e ; q2 ¼ ,2 ; q3 ¼ ,e; 360 12 8 8 12 4 23 5s 1 s 3 ,e; q5 ¼ e; q6 ¼ ,: þ q4 ¼ 24 8 6 2 2
q1 ¼
ð1:2Þ
Here s represents a dimensionless surface tension coefficient, , is the ratio of wave amplitude to undisturbed fluid depth, and e is the square of the ratio of fluid depth to wave length (note that in the original work [1] coefficient q3 contains a misprint). Small parameters , and e assumed to be of the same order of smallness. In the case of no surface tension all these coefficients are nonzero, otherwise some of them can take zero values. Some solitary waves of this equation with certain values of coefficients were recently obtained by Bagderina [3]. The aim of this paper is to find exact solutions expressible in terms of elliptic functions. We apply the method developed by Kudryashov [4] and Demina and Kudryashov [5,6] for this purpose. ⇑ Corresponding author. Tel./fax: +7 4993241181. E-mail address: [email protected] (N.A. Kudryashov). 1007-5704/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2012.01.033
N.A. Kudryashov et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 4104–4114
4105
Cnoidal wave appeared in the theory of water waves at the close of the 19th century in the famous work of Korteweg and de Vries [7]. Now elliptic functions are widely used to construct exact solutions of nonlinear differential equations. For example, in the works [8–11] elliptic functions were used to obtain solutions of nonlinear equations integrable by the inverse scattering transform. In the papers [12–26] exact elliptic solutions of nonlinear nonintegrable equations were constructed. However, in most cases the ansatz technique is used to find elliptic solutions. Method developed in the works [5,6] is stripped of this disadvantage. It allows us to find all possible elliptic solutions of nonlinear nonintegrable differential equation. This approach was also used in the works [27–29]. This paper is organized as follows. In Section 2 we obtain the traveling wave reductions of Eq. (1.1). In section 3 we describe the method of looking for elliptic solutions. In Sections 4 and 5 we give the complete list of elliptic traveling waves obtained. 2. Traveling wave reduction of the Olver equation Eq. (1.1) contains six arbitrary constant coefficients and four nonlinear terms. But in the original model [1] coefficient q2 is q6 always nonzero. Therefore we can simplify this equation taking new dependent variable gðz; tÞ ¼ v ðz; tÞ 2q . This substitu2 tion allows us to remove the nonlinear term ggz and gives the equation
vt þ
q2 q q 1 6 v z þ q1 v zzzzz þ q2 v 2 v z þ q3 vv zzz þ q4 v z v zz þ q5 3 6 v zzz ¼ 0: 4q2 2q2
ð2:1Þ
This equation admits the traveling wave reduction v(x) = v(z, t) with x = z ct. The traveling wave reduction takes the form
q2 q q 1 c 6 v x þ q1 v xxxxx þ q2 v 2 v x þ q3 vv xxx þ q4 v x v xx þ q5 3 6 v xxx ¼ 0: 4q2 2q2
ð2:2Þ
Here c is an arbitrary constant (wave velocity). Integrating Eq. (2.2) once we obtain the ODE
q2 q q q3 2 q q 1 c 6 v þ q1 v xxxx þ 2 v 3 þ q3 vv xx þ 4 v x þ q5 3 6 v xx þ d ¼ 0; 4q2 3 2 2q2
ð2:3Þ
where d is an integration constant. Assuming q1 – 0 we can further simplify the ODE (2.3). We will distinguish two cases: q1q2 > 0 and q1q2 < 0. In the case q1q2 > 0qsurface tension dominates gravity, it takes place for water depths less than about 3–5 mm. In this case substituting ffiffiffiffiffiffi v ðxÞ ¼ 6qq21 uðxÞ into the ODE (2.3) gives the equation
uxxxx þ 2u3 þ
2c1 c3 uuxx þ c2 u2x þ uxx þ c4 u þ c5 ¼ 0 3 3
ð2:4Þ
with
c1 ¼ 3
qffiffi
q3 3 pffiffiffiffiffiffiffi ; 2 q1 q2 q2
c4 ¼ 1c 4q16q2 ; q1
qffiffi
q4 q3 3p ffiffiffiffiffiffiffi ; 2 q1 q2
c2 ¼ c5 ¼
qffiffiffiffiffiffi
q2 d 6q1 q1
c3 ¼ 3
q5 q1
q3 q6 2q ; q 1 2
ð2:5Þ
:
Eq. (2.4) describes capillary waves in very thin layers of water (when effects of gravity are negligible). The case q1q2 < 0 corresponds to water depths more than 3–5 mm (gravity dominates surface tension). In this case we use qffiffiffiffiffiffiffiffiffiffi 1 uðxÞ and obtain the substitution v ðxÞ ¼ 6q q 2
uxxxx 2u3 þ
2c1 c3 uuxx þ c2 u2x þ uxx þ c4 u þ c5 ¼ 0: 3 3
ð2:6Þ
Here
c1 ¼ 3 c4 ¼
qffiffi
1c q1
q3 3 pffiffiffiffiffiffiffiffiffi ffi; 2 q1 q2
q26 4q1 q2
;
qffiffi 4 q3 ffi c2 ¼ 32 pqffiffiffiffiffiffiffiffiffi ; q1 q2 qffiffiffiffiffiffi d 2 c5 ¼ q : 6q q 1
c3 ¼ 3
q5 q1
q3 q6 2q ; q 1 2
ð2:7Þ
1
Eq. (2.6) describes gravitational waves (when we can neglect the effects of surface tension). Further we suppose all coefficients in Eqs. (2.4) and (2.6) to be real. In fact coefficients c4 and c5 depend on arbitrary constants c and d, so c4 and c5 may be complex. But we do not consider this situation. Let us note some obvious properties of Eqs. (2.4) and (2.6). First, in the case c1 = 3c2 Eqs. (2.4) and (2.6) have first integrals. Therefore we can expect the existence of additional solutions if c1 = 3c2. Second, both ODEs (2.4) and (2.6) are invariant under the transformations
ðu; c1 ; c2 ; c5 Þ ! ðu; c1 ; c2 ; c5 Þ;
ðx; u; c3 ; c5 Þ ! ðix; u; c3 ; c5 Þ:
Therefore we can look for solutions with c1 P 0 only and reduce the number of distinct elliptic solutions.
ð2:8Þ
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3. Method applied The main idea of the method [5,6] is as follows. First, consider an ordinary differential equation. Find all Laurent series representing its formal solution in the neighborhood of movable pole x = x0. Second, choose elliptic function with unknown constant parameters that can be a solution of the ODE under study. Note that this choice may be not unique. The form of such a function depends on number and order of movable poles in formal solution of ODE. Then we find the Laurent expansions of this elliptic function in the neighborhood of all its poles inside a parallelogram of periods. Third, compare the series obtained in the first and the second steps and form a system of algebraic equations for unknown parameters. The fourth step is to solve this system. If this system has a solution then an ODE can have an elliptic solution. To prove this one should substitute the elliptic function with parameters just found into the ODE. This is the last step. It may lead to some constraints on the ODE’s coefficients for the elliptic solution to exist. Details of this algorithm (not only for elliptic solutions, but also for rational solutions and solutions expressed in terms of trigonometric or hyperbolic functions) can be found in Refs. [4–6,27–29]. In this section we consider the ODE (2.4) for capillary waves. Analysis of the ODE (2.6) is quite similar. The ODE (2.4) has constant coefficients, so it is autonomous. For any solution f(x) of an autonomous ODE the function u(x) = f(x x0) with arbitrary complex constant x0 also satisfies the equation. Further we will often omit x0 for simplicity. If a solution of an ODE has a movable pole at position x0 then this solution can be presented in the form of local Laurent series
uðxÞ ¼ ðx x0 Þp
1 X
uj ðx x0 Þj
ð3:1Þ
j¼0
with p > 0 and uj being constant coefficients. Here p is the order of movable pole x0. Substitution of expression (3.1) into the qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ODE (2.4) gives p = 2 and u0 ¼ c1 c2 ðc1 þ c2 Þ2 60. Therefore the special solution of ODE can have exactly one pole, or exactly two poles, or countable number of poles (of only one type or both types simultaneously) [5,6]. In the present work we will look for special solutions of elliptic type only. So we concentrate on the case of infinitely many poles forming a regular doubly periodic structure in the complex plane. This structure consists of identical parallelograms (called parallelograms of periods in the theory of elliptic functions). Every parallelogram contains not more than M poles, where M is the number of different Laurent series of the form (3.1). This information helps to choose the form of elliptic function at the second step of algorithm. It is well known that any elliptic function can be represented in terms of the Weierstrass Zeta-function f(x; g2, g3) (or f(x) for short) with invariants g2 and g3 [4–6,28–32]. This representation may be written as
uðxÞ ¼ h þ
X
bk fðx bk Þ þ
k
pk X
! ak;r fðr1Þ ðx bk Þ :
ð3:2Þ
r¼2
Here h is a constant, bk are the distinct poles of function u(x). Outer sum is the sum over all poles inside a parallelogram of periods. This sum obviously vanishes in the case of only one pole in a parallelogram. Constant coefficients bk are the residues P of u(x) at the poles bk. Due to the residue theorem the following correlation k bk ¼ 0 holds for u(x) to be an elliptic function. The upper limit pk of inner sum represents the order of pole bk (note that the last term with sum is absent in the case of simple poles, i.e. when pk = 1). Coefficients ak,r are some constants, f(r1)(x; g2, g3) is the (r 1)th derivative of Zeta-function. For the ODE (2.4) we take exactly two possibilities. An elliptic special solution may have either two second order poles in a parallelogram of periods or exactly one second order pole. In the case of two distinct second order poles we choose k = {1, 2} and p1 = p2 = 2 in expression (3.2). Taking into account formula f0 (x) = }(x) we can rewrite expression (3.2) in the form
uðxÞ ¼ b1 fðxÞ þ b2 fðx bÞ þ a1 }ðxÞ þ a2 }ðx bÞ þ h
ð3:3Þ
with the condition b1 + b2 = 0 for u(x) to be elliptic. Here }(x) }(x; g2, g3) is the Weierstrass elliptic function. Due to the invariance of an autonomous ODE under the transformation x ´ x x0 we always can assume that one of the poles has the position x = 0 (instead of b1) without loss of generality. Then b – 0 represents the position of another pole in a parallelogram of periods. Elliptic function (3.3) contains eight unknown parameters: a1, a2, b1, b2, h, g2, g3, and b. In the case of one pole we take k = {1} and the outer sum in (3.2) vanishes. Then due to the residue theorem for elliptic functions b1 = 0. Therefore this elliptic function takes the form
uðxÞ ¼ a}ðxÞ þ h:
ð3:4Þ
We again use the arbitrariness of x0 and take b1 = 0 for simplicity. Note that there are only 4 unknown parameters: a,h, g2 and g3. One can obtain similar results for the Eq. (2.6). In this work we consider the case when all the series of the form (3.1) do not contain arbitrary coefficient or otherwise we suppose that they are fixed.
N.A. Kudryashov et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 4104–4114
4107
4. Elliptic special solutions of the ODE (2.4) 4.1. Solutions containing two poles in a parallelogram of periods In this section we are looking for special solutions of Eq. (2.4) in the form (3.3). At the first step we find all the Laurent series representing the formal solution of Eq. (2.4). Let x = 0 and x = b be the positions of poles in a parallelogram of periods. Then the Laurent series in the neighborhood of these poles are as follows
uðxÞ ¼ uðxÞ ¼
u0 u1 þ þ u2 þ u3 x þ u4 x2 þ ; x2 x
v0
ðx bÞ2
þ
v1
xb
þ v 2 þ v 3 ðx bÞ þ v 4 ðx bÞ2 þ
ð4:1Þ ð4:2Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with u0 ¼ c1 c2 þ ðc1 þ c2 Þ2 60 and v 0 ¼ c1 c2 ðc1 þ c2 Þ2 60. Additional condition is u0 – v0 for two distinct poles to exist. Due to the residue theorem for elliptic functions u1 + v1 = 0. In order to satisfy this condition we set u1 = v1 = 0. The subsequent coefficients are not uniquely determined. This is due to the series (4.1) and (4.2) can contain an arbitrary coefficient at certain values of ODE parameters c1, . . ., c5. For example if we take c3 = 0 then either u2 is arbitrary, v2 = 0, or v2 is arbitrary, u2 = 0. If we force c3 – 0 then both u2 and v2 have certain nonzero values. An analogous situation takes place while determining the values of coefficients with higher indexes. Therefore the number of distinct series (4.1) and (4.2) is infinite. Fortunately it is enough to compute coefficients up to u4 and v4 to find all the unknown parameters of the expression (3.3), so the list of (truncated) series becomes finite. We have constructed all possible pairs of series (4.1) and (4.2) (about thirty pairs) with different coefficients. We do not present here all these series because only a few of them lead to distinct elliptic solutions of Eq. (2.4). At the second step we find the Laurent series for the function (3.3) in the neighborhood of poles x = 0 and x = b. These series have the form
uðxÞ ¼
a1 b1 a1 5a2 b2 þ þ ðh þ a2 }ðbÞ b2 fðbÞÞ ða2 }0 ðbÞ þ b2 }ðbÞÞx þ g 2 þ 3a2 }2 ðbÞ þ }0 ðbÞ x2 þ 2 x x 20 2
ð4:3Þ
and
uðxÞ ¼
a2 b2 þ þ ðh þ a1 }ðbÞ þ b1 fðbÞÞ þ ða1 }0 ðbÞ b1 }ðbÞÞðx bÞ ðx bÞ2 x b a2 5a1 b1 g 2 þ 3a1 }2 ðbÞ }0 ðbÞ ðx bÞ2 þ þ 20 2
ð4:4Þ
The third step is to compare series (4.1) and (4.3), then (4.2) and (4.4). This leads us to the system of algebraic equations for determining unknown parameters of function (3.3). Note that there is an additional algebraic equation connecting the values of }0 (b) and }(b)
ð}0 ðbÞÞ2 ¼ 4}3 ðbÞ g 2 }ðbÞ g 3 :
ð4:5Þ
It follows from the Weierstrass differential equation for the function }(x). Of course, at this step we obtain the number of algebraic systems equal to the number of distinct series obtained at the first step. At the fourth step we solve these algebraic systems and find all the unknown parameters in (3.3). For all cases b1 = u1 = 0 and b2 = v2 = 0 so the expression (3.3) can be simplified
uðxÞ ¼ a1 }ðxÞ þ a2 }ðx bÞ þ h:
ð4:6Þ
Note that comparing coefficients of the Laurent series is a crucial point of the method used. The main advantage of this scheme is that we obtain linear algebraic equations for the most of parameters of solution. The reverse of the medal is that we need to compute several dozens of Laurent series representing the solution of ODE (2.4) (i.e. all possible Laurent series). It is necessary to guarantee that no elliptic solution is missed. So in the framework of this method we solve a large number (slightly more than number of series multiplied by number of unknown parameters) of simple (at most quadratic in the case of ODE (2.4)) algebraic equations. If we use a standard technique (i.e. if we substitute the formal elliptic solution (3.3) directly in the ODE (2.4)) the situation is dramatically changed. We obtain a small number of algebraic equations (namely, eighteen equations) for solution parameters. But these equations are very complicated and include high degrees of unknown parameters. It seems unreal to obtain solution of these equations. So this way (omitting the Laurent series) gives no result. The last step is to substitute expression (4.6) with already known parameters into the ODE (2.4) and to find constraints on the ODE coefficients. Note that at previous step we have found the value of }(b), not the value of b itself. Therefore we cannot use expression (4.6) for this substitution. But using the addition theorem for the Weierstrass elliptic function }(x)
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}ðx bÞ ¼ }ðxÞ }ðbÞ þ
2 1 }0 ðxÞ þ }0 ðbÞ 4 }ðxÞ }ðbÞ
ð4:7Þ
we can rewrite (4.6) as
uðxÞ ¼
2 a2 }0 ðxÞ þ }0 ðbÞ þ ða1 a2 Þ}ðxÞ þ h a2 }ðbÞ: 4 }ðxÞ }ðbÞ
ð4:8Þ
Moreover, at the fourth step we have found that for all cases }0 (b) = 0. Therefore the expression (4.8) can be further simplified using Eq. (4.5) with zero left hand side and the Weierstrass equation itself. So the final form of expression (3.3) in our case is
uðxÞ ¼
a2 12}2 ðbÞ g 2 þ a1 }ðxÞ þ a2 }ðbÞ þ h: 4 }ðxÞ }ðbÞ
ð4:9Þ
Note that the number of special solutions obtained in this way is rather large. But for some of them the equality g 32 27g 23 ¼ 0 takes place. We omit these solutions because they are degenerate elliptic and can be expressed via the hyperbolic or rational functions. The number of remaining distinct elliptic solutions can be dramatically reduced, if we take into account the invariance of the ODE under the transformations mentioned in Section 2. Namely, the first transformation allows us to consider only the special solutions with c1 P 0. The second transformation allows us to change the sign of g3 due to the homogeneity relation }(x; g2, g3) = }(ix; g2, g3). Another reduction can be done, if we set x0 = b and take into account the evenness of the Weierstrass elliptic function }(x). Performing all five steps of the method described above we obtain the following results. If c1 = 3c2 and c22 – 15=4, then the Eq. (2.4) has elliptic special solutions with two poles in a parallelogram of period. There are only three solutions of this type. The first solution takes the form
uðxÞ ¼
a2 12}2 ðbÞ g 2 c2 c3 þ a1 }ðx x0 Þ þ a2 }ðbÞ þ 2 4 }ðx x0 Þ }ðbÞ 18 c2 5
ð4:10Þ
where
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a1 ¼ 4c2 þ 2 4c22 15;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 ¼ 4c2 2 4c22 15; 2 c2 30 þ c22 þ 54c4 c22 5 c3 ; g2 ¼ 3 }ðbÞ ¼ 2 2 ; 36 c2 5 216 c22 5 4c22 27 2 c3 c23 36 þ 11c22 þ 162c4 c22 5 g3 ¼ : 2 3 23328 c2 5 4c22 27
ð4:11Þ ð4:12Þ
ð4:13Þ
Additional constraints are
2 c2 c3 c23 4c42 þ 39c22 176 þ 18c4 c22 5 4c22 5 ; 108 ðc22 5Þ3 ð4c22 27Þ 2 2 c2 5 4c2 27 – 0; c1 ¼ 3c2 : c5 ¼
ð4:14Þ ð4:15Þ
The coefficients c2, c3 and c4 can take any values. The second solution takes the form
uðxÞ ¼
pffiffiffi pffiffiffi 5 3 1323g 2 4c23 c3 2 3}ðx x0 Þ pffiffiffi 42 63}ðx x0 Þ c3 3 3
ð4:16Þ
where
g3 ¼
4c33 c3 g 2 : 250047 63
ð4:17Þ
The invariant g2 is an arbitrary constant. Additional constraints are
pffiffiffi 9 3 ; c1 ¼ 2
pffiffiffi 3 3 c2 ¼ ; 2
2c2 c4 ¼ 3 ; 9
44c3 5c23 2646g 2 pffiffiffi : c5 ¼ 3087 3
ð4:18Þ
The coefficient c3 may take any value. The third solution takes the form 2
uðxÞ ¼
pffiffiffi 3 126h þ 5c4 pffiffiffi 2 5}ðx x0 Þ 2h 28 2 5}ðx x0 Þ h
ð4:19Þ
N.A. Kudryashov et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 4104–4114
4109
where 2
g2 ¼
3h c4 þ ; 2 28
3
g3 ¼
13h hc pffiffiffi p4ffiffiffi : 20 5 56 5
ð4:20Þ
The constant h is arbitrary. Additional constraints are
pffiffiffi c 1 ¼ 3 5;
c2 ¼
pffiffiffi 5;
3
c3 ¼ 0;
c5 ¼
918h 45hc4 : 5 7
ð4:21Þ
The coefficient c4 can take any value. Note that all three solutions contain poles on the real axis if one takes x0 = 0. But these solutions can be made bounded under some conditions if we take into account the arbitrariness of x0. Namely, }0 (b) = 0 for all solutions. Therefore }(b) is one of the roots of the cubic equation 4y3 g2y g3 = 0. These roots are denoted as e1, e2, e3 in the theory of elliptic functions [32]. All these roots are real and distinct if condition D ¼ g 32 27g 23 > 0 holds. Then in usual notation e1 > e2 > e3. In the case D > 0 the function }(z) has one real half-period x1 with }(x1) = e1 and one pure imaginary half-period x3 with }(x3) = e3. The function }(z) is bounded on the lines z = x + x3 and z = iy + x1 in the complex plane. Let }(b) = e1, then both poles (inside one parallelogram of periods) of the functions (4.6) and (4.9) are on the real axis, so these functions are bounded on the line z = x + x3. Using the arbitrariness of x0 we can take x0 = x3, then all our solutions become bounded. There is no need to calculate x3 explicitly for plotting these functions. Instead of this we can use the special case of the addition theorem (4.7) in the form [31]
}ðx þ x3 Þ ¼ e3 þ
ðe3 e1 Þðe3 e2 Þ : }ðxÞ e3
ð4:22Þ
Then, using the identities e1 þ e2 þ e3 ¼ 0; g 2 ¼ 2 e21 þ e22 þ e23 ; g 3 ¼ 4e1 e2 e3 we can rewrite the expression (4.9) in the form
uðxÞ ¼ a1
ðe1 e3 Þðe2 e3 Þ ðe1 e2 Þðe3 e2 Þ þ a2 þ a1 e3 þ a2 e2 þ h: }ðxÞ e3 }ðxÞ e2
ð4:23Þ
For example let us plot the third solution using the representation (4.23). In this case we have
pffiffiffi a1 ¼ 2 5;
pffiffiffi a2 ¼ 6 5;
h e1 ¼ }ðbÞ ¼ pffiffiffi : 2 5
ð4:24Þ
The other two roots of the Equation 4y3 g2y g3 = 0 are
e1 1 e2 ¼ þ 2 2
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c4 27e21 þ ; 28
e1 1 e3 ¼ 2 2
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c4 27e21 þ : 28
ð4:25Þ
The free parameters h and c4 must be taken in such a way that the conditions D > 0 and e1 > e2 > e3 hold. For example we take c4 = 1 and h = 0.199. The resulting plot is presented in Fig. 1. 4.2. Solutions containing one pole in a parallelogram of periods In this section we are looking for the special solution of Eq. (2.4) in the form (3.4). At the first step we need to construct the Laurent series (3.1) in the neighborhood of the second order pole x = 0. This expansion has the form
Fig. 1. Solution (4.19) with c4 = 1 and h = 0.199.
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N.A. Kudryashov et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 4104–4114
uðxÞ ¼
a c1 c2 x2
þ
u1 þ u2 þ u3 x þ u4 x2 þ u5 x3 þ u6 x4 þ x
ð4:26Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with a ¼ ðc1 þ c2 Þ2 60. In this case we need to compute coefficients up to u6 to find all the unknown parameters of the expression (3.4). Note that we can simplify the subsequent calculations if we take into account the evenness of the function (3.4). It follows that u1 = u3 = u5 = 0 in the expansion (4.26). As in the previous section, the series (4.26) may contain arbitrary coefficients. Therefore the expansion (4.26) is not unique. At the first step of the algorithm we have obtained rather large list of all possible series (4.26) with distinct coefficients u2, u4 and u6, so we do not present it here. At the second step we find the Laurent series for the function (3.4) in the neighborhood of the pole x = 0. It takes the form
uðxÞ ¼
a ag ag þ h þ 2 x2 þ 3 x4 þ x2 20 28
ð4:27Þ
At the third step we compare the series (4.26) and (4.27). For every expansion (4.26) found at the first step we have obtained a system of four algebraic equations. Solving this equations at the fourth step we find parameters a, h, g2 and g3 of the expression (3.4). At the last step we substitute the expression (3.4) with known parameters into the ODE (2.4) and find the constraints on the ODE coefficients. After that we reduce the number of solutions obtained. This was done in the same way as in the previous section. The results are as follows. Eq. (2.4) has six distinct elliptic solutions with one pole in a parallelogram of periods. The first solution takes the form
uðxÞ ¼ ða c1 c2 Þ}ðx x0 Þ þ
c3 c1 þ 3c2 3a
ð4:28Þ
where
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a ¼ ðc1 þ c2 Þ2 60; g2 ¼ g3 ¼
ð4:29Þ
180c23 þ 30c4 ð3a c1 3c2 Þ2 ðc1 þ c2 aÞð9a c1 21c2 Þð3a c1 3c2 Þ2 Ac33 þ Bc3 c4 þ Fc5
Gða c1 c2 Þð9a c1 21c2 Þð3a c1 3c2 Þ3
A ¼ 8ð11c1 þ 6c2 9aÞ;
ð4:30Þ
; ;
B ¼ 2ð3a c1 3c2 Þ2 ð7c1 3c2 3aÞ;
F ¼ ðc1 þ 21c2 9aÞð3a c1 3c2 Þ3 ;
G ¼ c2 ðc1 þ c2 aÞ 12:
ð4:31Þ ð4:32Þ ð4:33Þ
Constraints on the ODE’s coefficients are
Gða c1 c2 Þð9a c1 21c2 Þð3a c1 3c2 Þ – 0:
ð4:34Þ
The coefficients c1, c2, c3, c4, and c5 can take any values. The second solution takes the form
uðxÞ ¼
12 c2 c3 }ðx x0 Þ þ c2 2ð18 c1 c2 Þ
ð4:35Þ
where
g2 ¼
15c42 c23 þ 10c22 c4 ðc1 c2 18Þ2 16ðc1 c2 18Þ2 ð2c1 c2 3c22 27Þ
:
ð4:36Þ
The invariant g3 is an arbitrary constant. Constraints on the ODE’s coefficients are
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 c1 c21 36 ; 3 2 c2 c3 c22 c23 2c1 c2 3c22 þ 108 þ 2c4 ðc1 c2 18Þ 2c1 c2 3c22 þ 18 ; c5 ¼ 4 ðc1 c2 18Þ3 ð2c1 c2 3c22 27Þ
c2 ¼
ðc1 c2 18Þð2c1 c2 3c22 27Þ – 0:
ð4:37Þ ð4:38Þ ð4:39Þ
The coefficients c1, c3 and c4 may take any values. The third solution takes the form
uðxÞ ¼
4c2 8c1 3c3 }ðx x0 Þ þ 3 9 2ðc1 6c2 Þ
ð4:40Þ
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4111
where
g3 ¼
63 729c33 54c5 ðc1 6c2 Þ3 8c3 g 2 ðc1 6c2 Þ2 ð2c1 3c2 Þ2 : 16 ðc1 6c2 Þ3 ð2c1 3c2 Þð20c21 54c1 c2 þ 36c22 81Þ
ð4:41Þ
The invariant g2 is an arbitrary constant. Constraints on the ODE’s coefficients are
c2 ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 c1 3 c21 54 ; 6
c4 ¼
27c23 2ðc1 6c2 Þ2
ð4:42Þ
;
ðc1 6c2 Þð2c1 3c2 Þð20c21 54c1 c2 þ 36c22 81Þ – 0:
ð4:43Þ
The coefficients c1, c3 and c5 may take any values. The fourth solution takes the form
uðxÞ ¼
2c1 }ðx x0 Þ þ h 3
ð4:44Þ
where 2
9 6h þ c4 ; 2 c21 54 3 27 4h 2c21 27 þ 2hc4 c21 27 þ c5 c21 54 : g3 ¼ 4c1 ðc21 54Þð81 2c21 Þ
g2 ¼
ð4:45Þ ð4:46Þ
The constant h is arbitrary. Constraints on the ODE’s coefficients are
c2 ¼
45 2c1 ; c1 3
ðc21 54Þð2c21 81Þ – 0;
c3 ¼ 0;
c1 – 0:
ð4:47Þ
The coefficients c1, c4 and c5 may take any values. The fifth solution takes the form
pffiffiffi uðxÞ ¼ 3 2}ðx x0 Þ þ h
ð4:48Þ
where 2
g2 ¼
6h þ c4 : 3
ð4:49Þ
The invariant g3 and the constant h are arbitrary. Constraints on the ODE’s coefficients are
9 c1 ¼ pffiffiffi ; 2
pffiffiffi c 2 ¼ 2 2;
c3 ¼ 0;
2
c5 ¼ 2hð8h þ c4 Þ:
ð4:50Þ
The coefficient c4 can take any value. The sixth solution takes the form
pffiffiffi uðxÞ ¼ 2 6}ðx x0 Þ þ
rffiffiffiffiffiffiffiffiffi c4 6
ð4:51Þ
where
g3 ¼
pffiffiffiffiffiffiffiffiffi c5 c4 pffiffiffi ðc4 þ 18g 2 Þ: 108 12 6
ð4:52Þ
The invariant g2 is an arbitrary constant. Constraints on the ODE’s coefficients are
pffiffiffi c 1 ¼ 3 6;
c2 ¼
rffiffiffi 3 ; 2
c3 ¼ 0:
ð4:53Þ
The coefficients c4 and c5 may take any values. These solutions contain poles on the real axis if one takes x0 = 0. To make them bounded one should set x0 = x3 as in the previous section. Then using the formula (4.22) we can rewrite the expression (3.4) in the form
uðxÞ ¼ a
ðe3 e1 Þðe3 e2 Þ þ ae3 þ h: }ðxÞ e3
ð4:54Þ
This function will be bounded if we force D ¼ g 32 27g 23 > 0 and e1 > e2 > e3. pffiffiffi For example let us plot the fifth solution using representation (4.54). In this case a ¼ 3 2. Constants e1, e2, and e3 are the roots of the cubic equation
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Fig. 2. Solution (4.48) with h = 1, c4 = 38.52 and g3 = 6.84.
2
4y3 þ
6h þ c4 y g 3 ¼ 0: 3
ð4:55Þ
For example we take h = 1, c4 = 38.52, g3 = 6.84. The resulting plot is presented in Fig. 2. 5. Elliptic special solutions of the ODE (2.6) 5.1. Solution containing two poles in a parallelogram of periods Eq. (2.6) differs from ODE (2.4) only in the sign of nonlinear term u3. Therefore all computations are quite similar to the ones in Section 4. But the results are different. We obtain only one elliptic special solution of the form
uðxÞ ¼
a2 12}2 ðbÞ g 2 c2 c3 þ a1 }ðx x0 Þ þ a2 }ðbÞ þ 2 4 }ðx x0 Þ }ðbÞ 18 c2 þ 5
ð5:1Þ
where
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a1 ¼ 4c2 þ 2 4c22 þ 15;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 ¼ 4c2 2 4c22 þ 15; 2 54c4 c22 þ 5 þ c23 30 c22 c3 ; g2 ¼ }ðbÞ ¼ 2 ; 36 c22 þ 5 216 c22 þ 5 4c22 þ 27 2 c3 c23 36 11c22 þ 162c4 c22 þ 5 : g3 ¼ 2 3 2 23328 c2 þ 5 4c2 þ 27
ð5:2Þ ð5:3Þ
ð5:4Þ
Constraints on the ODE’s coefficients are
c1 ¼ 3c2 ;
c5 ¼
2 c2 c3 c23 176 þ 39c22 4c42 þ 18c4 c22 þ 5 4c22 þ 5 : 2 3 2 108 c þ 5 4c þ 27 2
ð5:5Þ
2
The coefficients c2, c3 and c4 may take any values. 5.2. Solutions containing one pole in a parallelogram of periods There are four elliptic special solutions of the form (3.4) that satisfy the Eq. (2.6). These are as follows. The first solution takes the form
uðxÞ ¼ ða þ c1 þ c2 Þ}ðx x0 Þ þ
c3 3a þ c1 þ 3c2
ð5:6Þ
where
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a ¼ ðc1 þ c2 Þ2 þ 60;
ð5:7Þ
N.A. Kudryashov et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 4104–4114
g2 ¼ g3 ¼
180c23 þ 30c4 ð3a þ c1 þ 3c2 Þ2 ða þ c1 þ c2 Þð9a þ c1 þ 21c2 Þð3a þ c1 þ 3c2 Þ2
Gða þ c1 þ c2 Þð9a þ c1 þ 21c2 Þð3a þ c1 þ 3c2 Þ3
A ¼ 8ð9a þ 11c1 þ 6c2 Þ;
ð5:8Þ
;
Ac33 þ Bc3 c4 þ Fc5
;
B ¼ 2ð3a þ c1 þ 3c2 Þ2 ð3c2 3a 7c1 Þ; 3
F ¼ ð9a þ c1 þ 21c2 Þð3a þ c1 þ 3c2 Þ ;
4113
G ¼ c2 ðc1 þ c2 þ aÞ þ 12:
ð5:9Þ ð5:10Þ ð5:11Þ
Constraints on the ODE’s coefficients are
Gða þ c1 þ c2 Þð9a þ c1 þ 21c2 Þð3a þ c1 þ 3c2 Þ – 0:
ð5:12Þ
The coefficients c1, c2, c3, c4, and c5 may take any values. The second solution takes the form
uðxÞ ¼
12 c 2 c3 }ðx x0 Þ c2 2ð18 þ c1 c2 Þ
ð5:13Þ
where
g2 ¼
15c42 c23 þ 10c22 c4 ðc1 c2 þ 18Þ2 16ðc1 c2 þ 18Þ2 ð2c1 c2 3c22 þ 27Þ
:
ð5:14Þ
The invariant g3 is arbitrary. Constraints on the ODE’s coefficients are
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 c1 c21 þ 36 ; 3 2 c2 c3 c22 c23 3c22 2c1 c2 þ 108 2c4 ðc1 c2 þ 18Þ 3c22 2c1 c2 þ 18 ; c5 ¼ 4 ðc1 c2 þ 18Þ3 ð2c1 c2 3c22 þ 27Þ
c2 ¼
ðc1 c2 þ 18Þð2c1 c2 3c22 þ 27Þ – 0:
ð5:15Þ ð5:16Þ ð5:17Þ
The coefficients c1, c3 and c4 can take any values. The third solution takes the form
uðxÞ ¼
8c1 4c2 3c3 }ðx x0 Þ þ 9 3 2ðc1 6c2 Þ
ð5:18Þ
where
g3 ¼
63 729c33 54c5 ðc1 6c2 Þ3 þ 8c3 g 2 ðc1 6c2 Þ2 ð2c1 3c2 Þ2 : 16 ðc1 6c2 Þ3 ð2c1 3c2 Þ 20c21 54c1 c2 þ 36c22 þ 81
ð5:19Þ
The invariant g2 is an arbitrary constant. Constraints on the ODE’s coefficients are
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 27c23 c1 3 c21 þ 54 ; c4 ¼ ; 6 2ðc1 6c2 Þ2 ðc1 6c2 Þð2c1 3c2 Þ 20c21 54c1 c2 þ 36c22 þ 81 – 0:
c2 ¼
ð5:20Þ ð5:21Þ
The coefficients c1, c3 and c5 may take any values. The fourth solution takes the form
uðxÞ ¼
2c1 }ðx x0 Þ þ h 3
ð5:22Þ
where 2
9 6h c4 ; 2 c21 þ 54 3 27 4h 2c21 þ 27 2hc4 c21 þ 27 c5 c21 þ 54 g3 ¼ : 4c1 ðc21 þ 54Þð2c21 þ 81Þ
g2 ¼
ð5:23Þ ð5:24Þ
The constant h is arbitrary. Constraints on the ODE’s coefficients are
c2 ¼
45 2c1 ; c1 3
c3 ¼ 0;
c1 – 0:
The coefficients c1, c4 and c5 can take any values.
ð5:25Þ
4114
N.A. Kudryashov et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 4104–4114
6. Conclusion In this paper we have presented elliptic traveling waves of the Olver equation. As far as we know all these solutions are completely new. To achieve this goal we have used the powerful method by Demina and Kudryashov. In the framework of this method no a priori assumptions on the form of solution are made. Therefore it allows one to find all possible elliptic solutions contrary to other methods (with the only exception made for Musette–Conte method [20], but their method involves a huge amount of computations and cannot be recommended to deal with complicated equations like the Olver equation). Moreover, almost all calculations done with Demina–Kudryashov method consist in solving linear equations to obtain constant parameters of solution. This is a great advantage of this method. Acknowledgement This research was partially supported by Federal Target Program ‘‘Research and Scientific-Pedagogical Personnel of Innovation in Russian Federation’’ on 2009–2013. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]
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