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Exact solutions of the Swift–Hohenberg equation with dispersion
Commun Nonlinear Sci Numer Simulat 17 (2012) 26–34
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Exact solutions of the Swift–Hohenberg equation with dispersion Nikolai A. Kudryashov ⇑, Dmitry I. Sinelshchikov 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 20 January 2011 Received in revised form 3 April 2011 Accepted 4 April 2011 Available online 13 April 2011
a b s t r a c t The Swift–Hohenberg equation with dispersion is considered. Traveling wave solutions of the Swift–Hohenberg equation with dispersion are presented. The classification of these solutions is given. It is shown that the Swift–Hohenberg equation without dispersion has only stationary meromorphic solution. Ó 2011 Elsevier B.V. All rights reserved.
Keywords: Swift–Hohenberg equation Meromorphic exact solutions Dispersion
1. Introduction The Swift–Hohenberg equation is one of important equations for description localized structures in the modern physics. This equation occurs in fluid dynamics, optical physics and other fields [1–4]. The Swift–Hohenberg equation with dispersion takes the form [5]
ut þ 2uxx ruxxx þ uxxxx ¼ au þ bu2 cu3 ;
ð1:1Þ
where r, a, b and c are parameters of equation. At r = 0 Eq. (1.1) is reduced to the standard Swift–Hohenberg equation. The wave breaking phenomenon for Eq. (1.1) was numerically investigated in [5]. It was shown that localized structures of Eq. (1.1) are drift in contrast of structures described by the original Swift–Hohenberg equation. The aim of this paper is to construct and to classify traveling wave solutions of the Swift–Hohenberg equation with dispersion. We present the traveling wave solutions of the Swift–Hohenberg equation with dispersion and give the classification of all meromorphic traveling wave solutions. We show that the Swift–Hohenberg equation without dispersion admits only the stationary traveling wave solutions. We also present classification of these stationary solutions. To achieve our aim we use the method introduced in [6,7]. In recent papers [6,7] two theorems were proved for possible representation of rational, one periodic and doubly periodic solutions in the complex plane of the autonomous nonlinear ordinary differential equations. These theorems allow us to present general forms of one periodic and doubly periodic solutions. At the same time this theorems lead to a new method for constructing meromorphic exact solutions of autonomous nonlinear ordinary differential equations. In the case of one branch of the general solution in the Laurent series we obtain known methods for finding exact solutions, which were developed in the last years [8–23]. In the case of two or more branches for the expansions of the general solution in the Laurent series we obtain new expressions for exact solutions of nonlinear differential equations. The essence of the approach [6,7] is that the exact solutions of nonlinear ordinary differential equations are found by comparison of the two Laurent series: one of them is the Laurent series for the general solution of ordinary differential ⇑ Corresponding author. Tel./fax: +7 4953241181. E-mail address: [email protected] (N.A. Kudryashov). 1007-5704/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2011.04.008
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equation and the other is the Laurent series for function which can be a solution to the original equation. Using this approach the meromorphic exact solutions of the third order differential equation were obtained in [6]. The exact solutions of the Kawahara and the Bretherton equations expressed via the Weierstrass elliptic function were found in [7,24]. The outline of this paper is the following. In Section 2 we study the Swift–Hohenberg equation with dispersion. We find the traveling wave solutions of this equation and obtain classification of them. In Section 3 we consider the exact solution of Eq. (1.1) at r = 0 and show that there is the stationary meromorphic solutions of this equation. We present the classification of these solutions. In the conclusion we summarize and discuss the results of this paper. 2. Traveling wave solutions of the Swift–Hohenberg equation with dispersion Let us consider the traveling wave solutions of the Swift–Hohenberg equation with dispersion. Without loss of generality we set c = 30 in Eq. (1.1). It is convenient to take this value of the coefficient c for our calculations. In fact, we can obtain 1=3 0 0 this coefficient taking the transformation u ¼ ð 30 c Þ u into account, where u is the new variable. In this case Eq. (1.1) is reduced to the form
ut þ 2uxx þ uxxxx ruxxx ¼ au þ bu2 þ 30u3 :
ð2:1Þ
Using the traveling wave solutions u(x, t) = w(z), z = x C0t in (2.1) we have the nonlinear ordinary differential equation
wzzzz rwzzz þ 2wzz C 0 wz aw bw2 30w3 ¼ 0:
ð2:2Þ
We suppose that solutions of Eq. (2.2) has the form of the Laurent series in a neighborhood of the pole z = z0
wðzÞ ¼
1 X
ak ðz z0 Þkp ;
p > 0:
ð2:3Þ
k¼0
Eq. (2.2) is autonomous and without loss of the generality we can set z0 = 0. Eq. (2.2) admits two different formal Laurent expansions in a neighborhood of the second order (p = 2) pole z = 0. Corresponding values of a0 and a1 are the following ð1;2Þ
a0
¼ 2;
ð1;2Þ
a1
¼
r 7
ð2:4Þ
:
Necessary conditions for existence of elliptic solutions are ð1Þ
1Þa1 ¼ 0;
ð2Þ
2Þa1 ¼ 0;
ð1Þ
ð2Þ
3Þa1 þ a1 ¼ 0:
ð2:5Þ
We see that at r – 0 elliptic solutions can exist only in the third case. At r = 0 the elliptic solutions can exist in all cases. So at r – 0 we can look for the elliptic solutions if and only if we take into account both Laurent series for solution of Eq. (2.2). In accordance with classification of meromorphic solutions of autonomous ordinary differential equation given in [6,7], there are different types of meromorphic exact solutions of Eq. (2.2). The first type is the elliptic solutions corresponding to the both Laurent series. The second type is the simply periodic solutions corresponding to one of the Laurent series or to the both Laurent series. The Fuchs indices corresponding to expansions of the solution in the Laurent series are the following
j1 ¼ 1;
j2 ¼ 8;
j3;4 ¼
pffiffiffiffiffiffi 1 7 i 71 : 2
ð2:6Þ
We see that the Fuchs indices j2 has the positive integer value. So two expansions of the solution can exist if a8 is arbitrary constant. ð1Þ We have the formal Laurent expansion for the solution of Eq. (2.2) corresponding to a0 ¼ 2 in the form
wðzÞ ¼
2 r 1 23r2 b 3r 3 C0 r ð1Þ z þ þ a8 z6 þ þ þ þ þ z2 7z 15 2940 90 3430 90 126
ð2:7Þ
Series (2.7) corresponds to the solution of Eq. (2.2) in the case
r 3115800r7 50774976r5 þ 54096588r4 C 0 þ 60289110r3 a 669879b2 r3 26682793200 þ 222832008r3 401581656C 0 r2 169414560ra þ 235298b3 r þ 196944426rC 20 203297472r C2 31765230rab þ 1882384b2 r þ 338829120aC 0 þ 338829120C 0 3764768b2 C 0 0 ¼ 0: 225
ð2:8Þ
The last equality is the compatibility condition for existence of the Laurent series (2.7). Series (2.7) does not exist if relation (2.8) is not satisfied.
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The Laurent expansion for the solution of Eq. (2.2) at a0 ¼ 2 can be presented in the form
2 r 23r 1 b r C 0 3r3 ð2Þ þ þ z þ þ a8 z6 þ 2 z 7z 2940 15 90 126 90 3430 2
wðzÞ ¼
ð2:9Þ
ð2Þ
The Laurent series (2.9) exists and the coefficient a8 can be arbitrary constant if there is the compatibility condition for the Laurent expansion (2.9) in the form
r 3115800r7 50774976r5 þ 54096588r4 C 0 þ 60289110r3 a 669879b2 r3 26682793200 þ 222832008r3 401581656C 0 r2 169414560ra 235298b3 r þ 196944426rC 20 203297472r C2 þ 31765230rab þ 1882384b2 r þ 338829120aC 0 þ 338829120C 0 3764768b2 C 0 þ 0 ¼ 0: 225
ð2:10Þ
In fact the compatibility conditions (2.8) and (2.10) are different. However in the case of
a¼
b2 : 135
ð2:11Þ
conditions (2.8) and (2.10) are the same. b2 from (2.8) and (2.10) we have At a ¼ 135
r
C0 ¼
18522ð93r2 56Þ
54r2 ð32522 4381r2 Þ þ 5488ðb2 270Þ 20412r2 442123r6 1593648r2
1=2 1846684r4 5883136 þ 686b2 11365704r2 þ 1118799r4 þ 43904b2 15410304 þ 351298031616 : ð2:12Þ 2
b Thus Eq. (2.2) has the elliptic traveling wave solutions if and only if a ¼ 135 and C0 is determined by expression (2.12). Let us present the approach for classification of the meromorphic exact solutions for Eq. (2.2). For this purpose we use method of constructing general meromorphic solutions for ordinary differential equations [6,7]. The outline of this method is the following.
(1) In the first step we construct the formal Laurent series for the general solution of the equation. (2) In the second step we use the general form of the possible elliptic (simple periodic) solution of the equation and find the Laurent expansions for these functions. (3) In the third step we compare the Laurent series for the general solution of the equation, that were found on the first step, with the Laurent expansions of the possible elliptic (simple periodic) solutions, that were found on the second step. (4) In the fourth step we solve the system of algebraic equations obtained on the third step and find the values of parameters of the equation and parameters of the possible elliptic (simple periodic) solutions. The formal Laurent expansions of the possible elliptic solutions can be found using textbook [25] or Maple. Theorem 2.1. The elliptic solution of Eq. (2.2) corresponding to both of the Laurent series (2.7) and (2.9) exists if there is the following relation between parameters b and r
14ð14bÞ2 ¼ 27ð941r4 13384r2 þ 43904Þ:
ð2:13Þ
Proof. In accordance with method from Refs. [6,7] we look for the possible elliptic solution of Eq. (2.2) in the form
wðzÞ ¼
2 1 }0 ðz; g 2 ; g 3 Þ þ B r }0 ðz; g 2 ; g 3 Þ þ B þ þ4}ðz; g 2 ; g 3 Þ þ h0 : 2 }ðz; g 2 ; g 3 Þ A 14 }ðz; g 2 ; g 3 Þ A
ð2:14Þ
Comparing expansion (2.7) with the Laurent expansion of expression (2.14), we have the following values of parameters and invariants
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1843968 þ 39522r4 562128r2 23r2 196 ; 2940 5880 2 4 38640r 3627r 21952 r 23r2 196 ; B ¼ 0; C 0 ¼ ; A¼ 5880 1372ð93r2 56Þ 2311r4 38024r2 þ 153664 ; g2 ¼ 6914880 2 4 23r 196 9439r 154056r2 þ 614656 ; g3 ¼ 203297472000 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 1843968 þ 39522r4 562128r2 b¼ 196 h0 ¼
ð2:15Þ
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Solving the algebraic system of equations on parameters h0, A, B, g2, g3, b, C0 we obtain values for constant a8 in the form ð1;2Þ
a8
¼
1147313r8 61991r6 1621r4 41r2 1 : 14344669624320000 18296772480000 31116960000 119070000 1215000
ð2:16Þ
Expressions (2.16) are necessary conditions for existence of the Laurent series (2.7) and (2.9). These conditions show us that the elliptic solution of Eq. (2.2) contains only one arbitrary constant corresponding to Eq. (2.2). In this case we can add the arbitrary constant z0 to variable z. We have to remember this fact for exact solutions of Eq. (2.2). Taking into account formulae (2.14) we obtain the elliptic solution of Eq. (2.2) in the form
" #2 1 }0 ðz; g 2 ; g 3 Þ r }0 ðz; g 1 ; g 2 Þ 23r2 196 wðzÞ ¼ þ 4}ðz; g 2 ; g 3 Þ þ 2 196 2 196 2 }ðz; g 2 ; g 3 Þ 23r5880 2940 14 }ðz; g 2 ; g 3 Þ 23r5880 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1843968 þ 39522r4 562128r2 ; 5880
ð2:17Þ
where invariants g2, g3 are defined by relations (2.15). This solution exists if b is defined by (2.13).This completes the proof. h Let us show that elliptic solution (2.17) can be reduced to the simple periodic solution at Weierstrass elliptic function is defined by following differential equation
pffiffiffi
pffiffiffiffiffiffiffiffi 1513 . 89
r ¼ 7 and r ¼ 7
}02 ¼ 4}3 g 2 } g 3 :
The
ð2:18Þ
where g2, g3 and b are determined by (2.15) and (2.13). If the right-hand side of Eq. (2.18) has multiple roots then the Weierstrass elliptic function is degenerated to the trigonometric or rational function. It is possible if invariants are defined by (2.15) and the values of r are the following
pffiffiffi
r ¼ 7; r ¼
pffiffiffiffiffiffiffiffiffiffiffiffi 7 1513 : 89
ð2:19Þ
pffiffiffi In the case of r ¼ 7 the simple periodic solutions of Eq. (2.2) take the form
1 2 w ¼ csch 14
pffiffiffi ! pffiffiffi ! pffiffiffi ! pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7z 7z 7z 1 1 154350 2 csch sech : 14 14 28 14 28 5880
ð2:20Þ
In this case the parameters A, B, h0, b, C0 can be written as
A¼
1 ; 168
In the case of
r¼
B ¼ 0; pffiffiffiffiffiffiffiffi 1513 7 89
h0 ¼
70 þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 154350 ; 5880
C0 ¼
pffiffiffi 17 7 ; 196
b¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 154350 : 196
ð2:21Þ
elliptic solution (2.17) is reduced to the simple periodic solution in the form
pffiffiffiffiffiffiffiffiffi ! pffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi ! pffiffiffiffiffiffiffiffiffi ! 7 7 623z 1513 623 623z 623z þ cosec2 sec2 w¼ cosec 178 356 178 15842 178 356 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffi 784980 355468050 7921 : þ 46575480
ð2:22Þ
Solution (2.22) satisfies Eq. (2.2) when the parameters are the following
7 ; A¼ 2136
B ¼ 0;
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffi 355468050 7921 7 h0 ¼ ; þ 1068 46575480
pffiffiffiffiffiffiffiffiffiffiffiffi 49 1513 C0 ¼ ; 31684
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffi 3 355468050 7921 b¼ : 1552516
ð2:23Þ
Consider simple periodic solutions of Eq. (2.2) corresponding to the Laurent series (2.7). The following theorem is valid. Theorem 2.2. The general form of the simple periodic solution of Eq. (2.2) corresponding to the Laurent series (2.7) exist if the following relations between parameters b and r are hold
7r2 ð283r4 4592r2 þ21952Þ 3 1=3 ; b ¼ 196 þ r p p1=3 pffiffiffi p ¼ r3 41r2 392 71r2 392 11r2 392 i30 3r5 51r2 392 31r2 392
ð2:24Þ
or
49r2 ð421r4 4664r2 þ12544Þ ; b ¼ 983r q1=3 þ 1=3 q pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q ¼ 30r4 3ð98 15r2 Þ 614656 189336r2 þ 14629r4 r3 239r2 1568 127r2 784 11r2 392 : ð2:25Þ
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N.A. Kudryashov, D.I. Sinelshchikov / Commun Nonlinear Sci Numer Simulat 17 (2012) 26–34
Proof. The general form of the simple periodic solution can be written in the form [6,7]
wðzÞ ¼
p r T 7
cot
npzo T
2
npzo d cot þ h0 : dz T
ð2:26Þ
Also comparing the Laurent expansion for function (2.26) with expansion (2.7) for the general solution of Eq. (2.2) we obtain
qffiffiffiffiffiffiffiffiffi T ¼ 28p r12 ;
8
r a8 ¼ 127508174438400 ;
2
r ; h0 ¼ 6b 41 90 5880
2
r 1960Þ C 0 ¼ rð2012744 :
ð2:27Þ
Solution (2.26) satisfies Eq. (2.2) if the parameter a takes the form
a¼
1 10976b2 686b 2352 1149r2 2 2963520ð343b 383r 784Þ þ 81r2 9505216 1565928r2 þ 82933r4 697019904
ð2:28Þ
and b is defined by (2.24). Comparing the Laurent series for function (2.26) with the Laurent series (2.7) for the general solution of Eq. (2.2) we obtain
14p b þ 24 127r2 rð171r2 980Þ ; T ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; h0 ¼ ; C0 ¼ 2 90 686 2940 98 15r r2 15r2 196 225r4 2940r2 þ 19208 1 þ a8 ¼ : 33205253760 5400
ð2:29Þ
In this case solution (2.26) satisfies Eq. (2.2) if the parameter a takes the form
1 686b2 21168 þ 686b 4689r2 185220ð7056 þ 343b 1563r2 Þ þ 27r2 101286528 27836424r2 þ 2136139r4 1858719744
a¼
ð2:30Þ
and b is defined by (2.25). As result we have the periodic solution of Eq. (2.2) taking formulae (2.26) into account in the following form
wðzÞ ¼
p r T 7
cot
npzo T
2
npzo 6 b 41r2 d cot þ ; dz T 90 5880
ð2:31Þ
where T and b are defined by (2.27) and (2.24) or by (2.29) and (2.25) correspondingly. This completes the proof. h The simple periodic solution corresponding to Laurent expansion (2.9) can be obtained by analogy with solution (2.31). Let us note that solution (2.31) contains one arbitrary constant corresponding to autonomous of Eq. (2.2). If we take this constant purely imaginary then all poles of solution (2.31) will be on the imaginary axis of the complex plane. The picture of solution (2.31) at different values of r on the real axis is demonstrated on Fig. (1).
3
2
1
Fig. 1. Exact solution (2.31) of Eq. (2.2) at r = 2.6, 2.65, 2.7 (curves 1, 2, 3).
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N.A. Kudryashov, D.I. Sinelshchikov / Commun Nonlinear Sci Numer Simulat 17 (2012) 26–34
Let us discuss the stability of the simple periodic solution (2.31) of Eq. (1.1). To investigate the stability of this solution we use the numerical approach that is the IFRK4 method [26,27]. With this aim we have considered the propagation of nonlinear waves described by Eq. (1.1) with periodic boundary conditions. To test our numerical approach we use exact solution (2.31). This solution is not periodic and we mirror-reflect solution (2.31) with respect to the certain point to obtain the periodic solution in the form of the superposition of exact and reflected solutions as initial conditions. We observe that the relative error of our calculations was less then 5 % during the time of calculations but the relative error was calculated only on the non-reflected part of initial data. To study the stability of solution (2.31) we perturb the initial solution using the random noise. The amplitude of the random noise was given between 1% to 5% percents of the initial data. We have obtained that the shape of perturbed solutions is not changed at numerical modeling and we believe that exact solution (2.31) is stable. 3. Traveling wave solutions of Eq. (1.1) at r = 0 Consider the Swift–Hohenberg equation Eq. 1.1 at r = 0.
ut þ 2uxx þ uxxxx ¼ au þ bu2 cu3
ð3:1Þ
Without loss of generality we take c = 30 in (3.1) again. In this case taking into account the traveling wave solutions u(x, t) = w(z), where z = x C0t in Eq. (2.2) we obtain the nonlinear ordinary differential equation in the form
wzzzz þ 2wzz C 0 wz aw bw2 30w3 ¼ 0:
ð3:2Þ
Suppose that there is the expansion of the general solution of Eq. (3.2) in the form of the Laurent series in a neighborhood of the pole z = z0
wðzÞ ¼
1 X
ak ðz z0 Þkp ;
p > 0:
ð3:3Þ
k¼0
As Eq. (3.2) is autonomous we can consider that z0 = 0 again. Substituting (3.3) into Eq. (3.2) we find that p = 2 and a0 = ±2 again. The Fuchs indices corresponding of each expansions are the following
j1 ¼ 1;
j2 ¼ 8;
j3;4 ¼
pffiffiffiffiffiffi 1 7 i 71 : 2
ð3:4Þ
Substituting (3.3) into Eq. (3.2) and computing coefficients ak we obtain that the compatibility condition for existence of the Laurent series (3.3) is the following
C 0 ¼ 0:
ð3:5Þ
As result we obtain that the Swift–Hohenberg equation has only stationary traveling wave solutions. Let us present classification of the stationary meromorphic exact solutions of Eq. (3.2). For this purpose we use method for constructing general meromorphic solutions of ordinary differential equations from Refs. [6,7] again. Expansions in the Laurent series of the solution of Eq. (3.2) in a neighborhood of the pole z = 0 are the following
wðzÞ ¼
! 2 b6 b2 90a 36 2 z þ þ a 8 z6 þ z2 90 16200
ð3:6Þ
From expansion (3.6) we see that the necessary condition for existence of the elliptic solutions is satisfied. From theorems presented in Refs, [6,7] we obtain that there are some types of meromorphic solutions of Eq. (3.2): simple periodic solutions (and elliptic solutions) corresponding to one of Laurent series (3.6) of the general solution and simple periodic solutions (and elliptic solutions) corresponding to both of Laurent series (3.6). 2
b Theorem 3.1. The elliptic solutions of Eq. (3.2) corresponding to both of the Laurent series (3.6) exist if and only if a ¼ 135 or b = 0.
Proof. In accordance with method from the references [6,7] we look for the possible elliptic solution of Eq. (3.2) in the form
wðzÞ ¼
ð}0 ðz; g 2 ; g 3 Þ þ BÞ2 2ð}ðz; g 2 ; g 3 Þ AÞ2
þ 4}ðz; g 2 ; g 3 Þ þ h0 :
ð3:7Þ
Here }(z, g2, g3) is the Weierstrass elliptic function with invariants g2, g3 and }0 (z, g2, g3) is its derivative with respect to z. Comparing one of expansions (3.6) with the Laurent expansion for expression (3.7) we obtain that elliptic solutions exist b2 only if a ¼ 135 or b = 0. b2 In the case of a ¼ 135 we get the following values of parameters A, B, h0
A¼
1 ; 30
B ¼ 0;
h0 ¼
bþ6 : 90
ð3:8Þ
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N.A. Kudryashov, D.I. Sinelshchikov / Commun Nonlinear Sci Numer Simulat 17 (2012) 26–34
The elliptic solution (3.7) of Eq. (3.2) takes the form
wðzÞ ¼
}02 ðz; g 2 ; g 3 Þ 2ð}ðz; g 2 ; g 3 Þ þ
1 2 Þ 30
þ 4}ðz; g 2 ; g 3 Þ
bþ6 ; 90
ð3:9Þ
where invariants g2, g3 are defined by formulae
g2 ¼
12 15a ; 1620
g3 ¼
8 25a : 81000
ð3:10Þ
Necessary conditions for series (3.6) to exist is the following
a8 ¼
23 11a 7a2 : 10935000 2187000 3499200
ð3:11Þ
At b = 0 parameters A, B, h0 can be found from (3.8) at b = 0. Elliptic solution of Eq. (3.2) is expressed by formula (3.9) at b = 0 as well with invariants (3.10) and values a8 from (3.11). h pffiffiffiffiffiffi Let us show that elliptic solution (3.9) is reduced to the simple periodic solution in the case of b ¼ 3 5330 i. It is known that the Weierstrass elliptic function is defined by the differential equation
}02 ¼ 4}3 g 2 } g 3 :
ð3:12Þ
We obtain that the right-hand side of Eq. (3.12) has two multiple root at b ¼ function is reduced to the hyperbolic function. pffiffiffiffiffiffi Solution (3.9) of Eq. (3.2) at b ¼ 3 5330 i takes the form
wðzÞ ¼
2 2 csch 5
pffiffiffiffiffiffi 3 5330 i.
In this case the Weierstrass elliptic
(pffiffiffi ) (pffiffiffi ) pffiffiffiffiffiffiffiffiffi 5z 5z 1 330 2 sech i: 5 10 5 150
ð3:13Þ
The compatibility condition in this case is the following
a8 ¼
127 : 27000000
ð3:14Þ
In the case of a ¼ 11 elliptic solution (3.9) is also reduced to the simple periodic solution of Eq. (3.2) in the form 25
2 2 wðzÞ ¼ csch 5
(pffiffiffi ) (pffiffiffi ) 5z 5z 1 2 sech : 5 10 5
ð3:15Þ
Let us consider the elliptic solutions of Eq. (3.2) which corresponds to one of expansions (3.6). The following theorem can be formulated. Theorem 3.2. Eq. (3.2) has the elliptic solution which corresponds to one of expansions (3.6) at any values of a and b. Proof. Following the method from the references [6,7] we use the possible elliptic exact solution of Eq. (3.2) corresponding to the first Laurent series (3.6) in the form
w ¼ 2}ðz; g 2 ; g 3 Þ þ h0 :
ð3:16Þ
Comparing the Laureant series corresponding to function (3.16) with the first of the expansion (3.6), we get
h0 ¼
6b ; 90
g2 ¼
b2 90a 36 ; 1620
g3 ¼
b3 þ 6b2 540a 135ab 432 : 291600
ð3:17Þ
The compatibility condition for the first Laurent series (3.6) leads to the equality
a8 ¼
ð36 b2 þ 90aÞ2 : 1574640000
ð3:18Þ
From (3.16) as the result of calculations we have
! b2 90a 36 b3 þ 6b2 540a 135ab 432 6b ; þ : wðzÞ ¼ 2} z; 1620 291600 90
ð3:19Þ
This completes the proof. h The elliptic solution of Eq. (3.2) corresponding to the second expansion from (3.6) can be obtained by analogy with the above presented solution. Finally let us consider the simple periodic solutions corresponding to one of series (3.6). The following theorem is hold.
N.A. Kudryashov, D.I. Sinelshchikov / Commun Nonlinear Sci Numer Simulat 17 (2012) 26–34
33
Theorem 3.3. Eq. (3.2) has the simple periodic solution corresponding to one of expansions (3.6) if
a¼
ðb 6Þðb 24Þ 225
ð3:20Þ
a¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðb 6Þðb þ 12Þ 7bðb þ 6Þ ðb þ 6Þ 15ðb þ 12Þð7b 12Þ þ 504 : pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 9 5ðb þ 12Þ 15ðb þ 12Þð7b 12Þ
ð3:21Þ
or
Proof. The general form of the simple periodic solution corresponding to the first Laurent series (3.6) can be written [6,7] as
wðzÞ ¼
npzo 2p d cot þ h0 : T T dz
ð3:22Þ
Comparing the Laurent expansion for function (3.22) with the first series (3.6) we find the following set of T, h0 and a in the form
30p T 1;2 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 15ð6 bÞ
h0 ¼ 0
ð3:23Þ
where a is defined by (3.20). We also have the values
T 3;4 ¼ T 5;6 ¼ ð3;4Þ;ð5;6Þ
h0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6ðb þ 12Þðb 6Þð60 þ 5b þ 15 ðb þ 12Þð7b 12ÞÞp
; ðb þ 12Þðb 6Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi pffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6ðb þ 12Þðb 6Þð60 þ 5b þ 15 ðb þ 12Þð7b 12ÞÞp ðb þ 12Þðb 6Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðb 6Þ 15ðb þ 12Þð7b 12Þ 15ðb þ 12Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 90ð 15ðb þ 12Þð7b 12Þ 5ðb þ 12ÞÞ
; ð3:24Þ
where a is defined by (3.21). Substituting values of T and h0 from (3.23) or (3.24) we obtain the simple periodic solutions of Eq. (3.2) at alpha given by (3.20) or (3.21) correspondingly. This completes the proof. h Note that solution (3.22) contains one arbitrary constant corresponding to autonomous of Eq. (3.2). We can take this constant imaginary then all poles of solution (3.22) will be on the imaginary axis of the complex plane. The picture of solution (3.22) on the real axis at different values of b is demonstrate on the Fig. (2) for this case.
Fig. 2. Exact solution (3.22) of Eq. (3.2) at b = 50, 70, 85.
34
N.A. Kudryashov, D.I. Sinelshchikov / Commun Nonlinear Sci Numer Simulat 17 (2012) 26–34
We have investigated the stability of solution (3.22) by the numerical method again as for solution (2.31). Solution (3.22) is periodic and we use (3.22) as initial data for our numerical calculations. We have perturbed the initial conditions using the random noise with the same amplitude as for solution (2.31). The results of the numerical simulation showed that the shape of perturbed solutions is not changed at numerical modeling. Thus we believe that exact solution (3.22) is stable as well. 4. Conclusion In this paper we have studied the traveling wave solutions of the Swift–Hohenberg equation with dispersion. We have shown that nonstationary traveling wave solutions exist in the case of the Swift–Hohenberg equation with dispersion. We have found the elliptic and the simple periodic traveling wave solutions. We have shown that the Swift–Hohenberg equation without dispersion has only the stationary meromorphic traveling wave solutions. We have obtained these exact solutions and presented their classification as well. Acknowledgements This research was supported by Federal Target Programm 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]
Swift J, Hohenberg PC. Hydrodynamic fluctuations at the convective instability. Phys Rev A 1977;15:319–28. Aranson IS, Gorshkov KA, Lomov AS, Rabinovich MI. Stable particle-like solutions of multidimensional nonlinear fields. Phys D 1990;43:435–53. Pomeau Y, Zaleski S. Dislocation motion in cellular structures. J Phys 1981;42:2710–26. Glebsky LYu, Lerman LM. On small stationary localized solutions for the generalized 1-D Swift–Hohenberg equation. CHAOS 1995;5:424–31. Burke J, Houghton SM, Knobloch E. Swift–Hohenberg equation with broken reflection symmetry. Phys Rev E 2009;80. 036202. 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. Kudryashov NA. Exact soliton solutions of the generalized evolution equation of wave dynamics. J Appl Math Mech 1988;52:361–5. 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. Partial differential equations with solutions having movable first – order singularities. Phys Lett A 1992;169:237–42. Parkes EJ, Duffy BR. An automated tanh–function method for finding solitary wave solutions to non–linear evolution equations. Comput Phys Commun 1996;98:288–300. Fan E. Extended tanh-function method and its applications to nonlinear equations. Phys Lett A 2000;277:212–8. Fu Z, Liu S, Liu S. New transformations and new approach to find exact solutions to nonlinear equations. Phys Lett A 2002;299:507–12. Parkes EJ, Duffy BR, Abbott PC. The Jacobi elliptic–function method for finding periodic–wave solutions to nonlinear evolution equations. Phys Lett A 2002;295:280–6. Kudryashov NA. Simplest equation method to look for exact solutions of nonlinear differential equations. Chaos Soliton Fract 2005;24:1217–31. Polyanin AD, Zaitsev VF, Zhyrov AI. Methods of nonlinear equations of mathematical physics and mechanics. Moscow: Fizmatlit; 2005. 260 P. Vernov SYu. Proof of the absence of elliptic solutions of the cubic complex Ginzburg–Landau equation. Theor Math Phys 2006;146:131–9. Biswas A. Solitary wave solution for the generalized Kawahara equation. Appl Math Lett 2009;22:208–10. Kudryashov NA, Loguinova NB. Extended simplest equation method for nonlinear differential equations. Appl Math Comput 2008;205:396–402. Kudryashov NA. Seven common errors in finding exact solutions of nonlinear differential equations. Commun Nonlinear Sci Numer Simulat 2009;14:3507–29. Vitanov NK. Application of simplest equations of Bernoulli and Riccati kind for obtaining exact traveling-wave solutions for a class of PDEs with polynomial nonlinearity. Commun Nonlinear Sci Numer Simulat 2010;15:2050–60. Kudryashov NA. Meromorphic solutions of nonlinear ordinary differential equations. Commun Nonlinear Sci Numer Simulat 2010;15:2778–90. Kudryashov NA, Sinelshchikov DI, Demina MV. Exact solutions of the generalized Bretherton equation. Phys Lett A 2011;375:1074–9. Abramowitz M, Stegun IA. Handbook of mathematical functions: with formulas, graphs and mathematical tables. Dover Publications; 1965. 1046 P. Cox SM, Matthews PC. Exponential time differencing for stiff systems. J Comput Phys 2002;176:430–55. Kassam A, Trefethen LN. Fourth-order time stepping for stiff PDEs. SIAM J Sci Comput 2005;26:1214–33.
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Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Exact solutions of the Swift–Hohenberg equation with dispersion Nikolai A. Kudryashov ⇑, Dmitry I. Sinelshchikov 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 20 January 2011 Received in revised form 3 April 2011 Accepted 4 April 2011 Available online 13 April 2011
a b s t r a c t The Swift–Hohenberg equation with dispersion is considered. Traveling wave solutions of the Swift–Hohenberg equation with dispersion are presented. The classification of these solutions is given. It is shown that the Swift–Hohenberg equation without dispersion has only stationary meromorphic solution. Ó 2011 Elsevier B.V. All rights reserved.
Keywords: Swift–Hohenberg equation Meromorphic exact solutions Dispersion
1. Introduction The Swift–Hohenberg equation is one of important equations for description localized structures in the modern physics. This equation occurs in fluid dynamics, optical physics and other fields [1–4]. The Swift–Hohenberg equation with dispersion takes the form [5]
ut þ 2uxx ruxxx þ uxxxx ¼ au þ bu2 cu3 ;
ð1:1Þ
where r, a, b and c are parameters of equation. At r = 0 Eq. (1.1) is reduced to the standard Swift–Hohenberg equation. The wave breaking phenomenon for Eq. (1.1) was numerically investigated in [5]. It was shown that localized structures of Eq. (1.1) are drift in contrast of structures described by the original Swift–Hohenberg equation. The aim of this paper is to construct and to classify traveling wave solutions of the Swift–Hohenberg equation with dispersion. We present the traveling wave solutions of the Swift–Hohenberg equation with dispersion and give the classification of all meromorphic traveling wave solutions. We show that the Swift–Hohenberg equation without dispersion admits only the stationary traveling wave solutions. We also present classification of these stationary solutions. To achieve our aim we use the method introduced in [6,7]. In recent papers [6,7] two theorems were proved for possible representation of rational, one periodic and doubly periodic solutions in the complex plane of the autonomous nonlinear ordinary differential equations. These theorems allow us to present general forms of one periodic and doubly periodic solutions. At the same time this theorems lead to a new method for constructing meromorphic exact solutions of autonomous nonlinear ordinary differential equations. In the case of one branch of the general solution in the Laurent series we obtain known methods for finding exact solutions, which were developed in the last years [8–23]. In the case of two or more branches for the expansions of the general solution in the Laurent series we obtain new expressions for exact solutions of nonlinear differential equations. The essence of the approach [6,7] is that the exact solutions of nonlinear ordinary differential equations are found by comparison of the two Laurent series: one of them is the Laurent series for the general solution of ordinary differential ⇑ Corresponding author. Tel./fax: +7 4953241181. E-mail address: [email protected] (N.A. Kudryashov). 1007-5704/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2011.04.008
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N.A. Kudryashov, D.I. Sinelshchikov / Commun Nonlinear Sci Numer Simulat 17 (2012) 26–34
equation and the other is the Laurent series for function which can be a solution to the original equation. Using this approach the meromorphic exact solutions of the third order differential equation were obtained in [6]. The exact solutions of the Kawahara and the Bretherton equations expressed via the Weierstrass elliptic function were found in [7,24]. The outline of this paper is the following. In Section 2 we study the Swift–Hohenberg equation with dispersion. We find the traveling wave solutions of this equation and obtain classification of them. In Section 3 we consider the exact solution of Eq. (1.1) at r = 0 and show that there is the stationary meromorphic solutions of this equation. We present the classification of these solutions. In the conclusion we summarize and discuss the results of this paper. 2. Traveling wave solutions of the Swift–Hohenberg equation with dispersion Let us consider the traveling wave solutions of the Swift–Hohenberg equation with dispersion. Without loss of generality we set c = 30 in Eq. (1.1). It is convenient to take this value of the coefficient c for our calculations. In fact, we can obtain 1=3 0 0 this coefficient taking the transformation u ¼ ð 30 c Þ u into account, where u is the new variable. In this case Eq. (1.1) is reduced to the form
ut þ 2uxx þ uxxxx ruxxx ¼ au þ bu2 þ 30u3 :
ð2:1Þ
Using the traveling wave solutions u(x, t) = w(z), z = x C0t in (2.1) we have the nonlinear ordinary differential equation
wzzzz rwzzz þ 2wzz C 0 wz aw bw2 30w3 ¼ 0:
ð2:2Þ
We suppose that solutions of Eq. (2.2) has the form of the Laurent series in a neighborhood of the pole z = z0
wðzÞ ¼
1 X
ak ðz z0 Þkp ;
p > 0:
ð2:3Þ
k¼0
Eq. (2.2) is autonomous and without loss of the generality we can set z0 = 0. Eq. (2.2) admits two different formal Laurent expansions in a neighborhood of the second order (p = 2) pole z = 0. Corresponding values of a0 and a1 are the following ð1;2Þ
a0
¼ 2;
ð1;2Þ
a1
¼
r 7
ð2:4Þ
:
Necessary conditions for existence of elliptic solutions are ð1Þ
1Þa1 ¼ 0;
ð2Þ
2Þa1 ¼ 0;
ð1Þ
ð2Þ
3Þa1 þ a1 ¼ 0:
ð2:5Þ
We see that at r – 0 elliptic solutions can exist only in the third case. At r = 0 the elliptic solutions can exist in all cases. So at r – 0 we can look for the elliptic solutions if and only if we take into account both Laurent series for solution of Eq. (2.2). In accordance with classification of meromorphic solutions of autonomous ordinary differential equation given in [6,7], there are different types of meromorphic exact solutions of Eq. (2.2). The first type is the elliptic solutions corresponding to the both Laurent series. The second type is the simply periodic solutions corresponding to one of the Laurent series or to the both Laurent series. The Fuchs indices corresponding to expansions of the solution in the Laurent series are the following
j1 ¼ 1;
j2 ¼ 8;
j3;4 ¼
pffiffiffiffiffiffi 1 7 i 71 : 2
ð2:6Þ
We see that the Fuchs indices j2 has the positive integer value. So two expansions of the solution can exist if a8 is arbitrary constant. ð1Þ We have the formal Laurent expansion for the solution of Eq. (2.2) corresponding to a0 ¼ 2 in the form
wðzÞ ¼
2 r 1 23r2 b 3r 3 C0 r ð1Þ z þ þ a8 z6 þ þ þ þ þ z2 7z 15 2940 90 3430 90 126
ð2:7Þ
Series (2.7) corresponds to the solution of Eq. (2.2) in the case
r 3115800r7 50774976r5 þ 54096588r4 C 0 þ 60289110r3 a 669879b2 r3 26682793200 þ 222832008r3 401581656C 0 r2 169414560ra þ 235298b3 r þ 196944426rC 20 203297472r C2 31765230rab þ 1882384b2 r þ 338829120aC 0 þ 338829120C 0 3764768b2 C 0 0 ¼ 0: 225
ð2:8Þ
The last equality is the compatibility condition for existence of the Laurent series (2.7). Series (2.7) does not exist if relation (2.8) is not satisfied.
28
N.A. Kudryashov, D.I. Sinelshchikov / Commun Nonlinear Sci Numer Simulat 17 (2012) 26–34 ð1Þ
The Laurent expansion for the solution of Eq. (2.2) at a0 ¼ 2 can be presented in the form
2 r 23r 1 b r C 0 3r3 ð2Þ þ þ z þ þ a8 z6 þ 2 z 7z 2940 15 90 126 90 3430 2
wðzÞ ¼
ð2:9Þ
ð2Þ
The Laurent series (2.9) exists and the coefficient a8 can be arbitrary constant if there is the compatibility condition for the Laurent expansion (2.9) in the form
r 3115800r7 50774976r5 þ 54096588r4 C 0 þ 60289110r3 a 669879b2 r3 26682793200 þ 222832008r3 401581656C 0 r2 169414560ra 235298b3 r þ 196944426rC 20 203297472r C2 þ 31765230rab þ 1882384b2 r þ 338829120aC 0 þ 338829120C 0 3764768b2 C 0 þ 0 ¼ 0: 225
ð2:10Þ
In fact the compatibility conditions (2.8) and (2.10) are different. However in the case of
a¼
b2 : 135
ð2:11Þ
conditions (2.8) and (2.10) are the same. b2 from (2.8) and (2.10) we have At a ¼ 135
r
C0 ¼
18522ð93r2 56Þ
54r2 ð32522 4381r2 Þ þ 5488ðb2 270Þ 20412r2 442123r6 1593648r2
1=2 1846684r4 5883136 þ 686b2 11365704r2 þ 1118799r4 þ 43904b2 15410304 þ 351298031616 : ð2:12Þ 2
b Thus Eq. (2.2) has the elliptic traveling wave solutions if and only if a ¼ 135 and C0 is determined by expression (2.12). Let us present the approach for classification of the meromorphic exact solutions for Eq. (2.2). For this purpose we use method of constructing general meromorphic solutions for ordinary differential equations [6,7]. The outline of this method is the following.
(1) In the first step we construct the formal Laurent series for the general solution of the equation. (2) In the second step we use the general form of the possible elliptic (simple periodic) solution of the equation and find the Laurent expansions for these functions. (3) In the third step we compare the Laurent series for the general solution of the equation, that were found on the first step, with the Laurent expansions of the possible elliptic (simple periodic) solutions, that were found on the second step. (4) In the fourth step we solve the system of algebraic equations obtained on the third step and find the values of parameters of the equation and parameters of the possible elliptic (simple periodic) solutions. The formal Laurent expansions of the possible elliptic solutions can be found using textbook [25] or Maple. Theorem 2.1. The elliptic solution of Eq. (2.2) corresponding to both of the Laurent series (2.7) and (2.9) exists if there is the following relation between parameters b and r
14ð14bÞ2 ¼ 27ð941r4 13384r2 þ 43904Þ:
ð2:13Þ
Proof. In accordance with method from Refs. [6,7] we look for the possible elliptic solution of Eq. (2.2) in the form
wðzÞ ¼
2 1 }0 ðz; g 2 ; g 3 Þ þ B r }0 ðz; g 2 ; g 3 Þ þ B þ þ4}ðz; g 2 ; g 3 Þ þ h0 : 2 }ðz; g 2 ; g 3 Þ A 14 }ðz; g 2 ; g 3 Þ A
ð2:14Þ
Comparing expansion (2.7) with the Laurent expansion of expression (2.14), we have the following values of parameters and invariants
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1843968 þ 39522r4 562128r2 23r2 196 ; 2940 5880 2 4 38640r 3627r 21952 r 23r2 196 ; B ¼ 0; C 0 ¼ ; A¼ 5880 1372ð93r2 56Þ 2311r4 38024r2 þ 153664 ; g2 ¼ 6914880 2 4 23r 196 9439r 154056r2 þ 614656 ; g3 ¼ 203297472000 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 1843968 þ 39522r4 562128r2 b¼ 196 h0 ¼
ð2:15Þ
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N.A. Kudryashov, D.I. Sinelshchikov / Commun Nonlinear Sci Numer Simulat 17 (2012) 26–34
Solving the algebraic system of equations on parameters h0, A, B, g2, g3, b, C0 we obtain values for constant a8 in the form ð1;2Þ
a8
¼
1147313r8 61991r6 1621r4 41r2 1 : 14344669624320000 18296772480000 31116960000 119070000 1215000
ð2:16Þ
Expressions (2.16) are necessary conditions for existence of the Laurent series (2.7) and (2.9). These conditions show us that the elliptic solution of Eq. (2.2) contains only one arbitrary constant corresponding to Eq. (2.2). In this case we can add the arbitrary constant z0 to variable z. We have to remember this fact for exact solutions of Eq. (2.2). Taking into account formulae (2.14) we obtain the elliptic solution of Eq. (2.2) in the form
" #2 1 }0 ðz; g 2 ; g 3 Þ r }0 ðz; g 1 ; g 2 Þ 23r2 196 wðzÞ ¼ þ 4}ðz; g 2 ; g 3 Þ þ 2 196 2 196 2 }ðz; g 2 ; g 3 Þ 23r5880 2940 14 }ðz; g 2 ; g 3 Þ 23r5880 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1843968 þ 39522r4 562128r2 ; 5880
ð2:17Þ
where invariants g2, g3 are defined by relations (2.15). This solution exists if b is defined by (2.13).This completes the proof. h Let us show that elliptic solution (2.17) can be reduced to the simple periodic solution at Weierstrass elliptic function is defined by following differential equation
pffiffiffi
pffiffiffiffiffiffiffiffi 1513 . 89
r ¼ 7 and r ¼ 7
}02 ¼ 4}3 g 2 } g 3 :
The
ð2:18Þ
where g2, g3 and b are determined by (2.15) and (2.13). If the right-hand side of Eq. (2.18) has multiple roots then the Weierstrass elliptic function is degenerated to the trigonometric or rational function. It is possible if invariants are defined by (2.15) and the values of r are the following
pffiffiffi
r ¼ 7; r ¼
pffiffiffiffiffiffiffiffiffiffiffiffi 7 1513 : 89
ð2:19Þ
pffiffiffi In the case of r ¼ 7 the simple periodic solutions of Eq. (2.2) take the form
1 2 w ¼ csch 14
pffiffiffi ! pffiffiffi ! pffiffiffi ! pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7z 7z 7z 1 1 154350 2 csch sech : 14 14 28 14 28 5880
ð2:20Þ
In this case the parameters A, B, h0, b, C0 can be written as
A¼
1 ; 168
In the case of
r¼
B ¼ 0; pffiffiffiffiffiffiffiffi 1513 7 89
h0 ¼
70 þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 154350 ; 5880
C0 ¼
pffiffiffi 17 7 ; 196
b¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 154350 : 196
ð2:21Þ
elliptic solution (2.17) is reduced to the simple periodic solution in the form
pffiffiffiffiffiffiffiffiffi ! pffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi ! pffiffiffiffiffiffiffiffiffi ! 7 7 623z 1513 623 623z 623z þ cosec2 sec2 w¼ cosec 178 356 178 15842 178 356 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffi 784980 355468050 7921 : þ 46575480
ð2:22Þ
Solution (2.22) satisfies Eq. (2.2) when the parameters are the following
7 ; A¼ 2136
B ¼ 0;
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffi 355468050 7921 7 h0 ¼ ; þ 1068 46575480
pffiffiffiffiffiffiffiffiffiffiffiffi 49 1513 C0 ¼ ; 31684
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffi 3 355468050 7921 b¼ : 1552516
ð2:23Þ
Consider simple periodic solutions of Eq. (2.2) corresponding to the Laurent series (2.7). The following theorem is valid. Theorem 2.2. The general form of the simple periodic solution of Eq. (2.2) corresponding to the Laurent series (2.7) exist if the following relations between parameters b and r are hold
7r2 ð283r4 4592r2 þ21952Þ 3 1=3 ; b ¼ 196 þ r p p1=3 pffiffiffi p ¼ r3 41r2 392 71r2 392 11r2 392 i30 3r5 51r2 392 31r2 392
ð2:24Þ
or
49r2 ð421r4 4664r2 þ12544Þ ; b ¼ 983r q1=3 þ 1=3 q pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q ¼ 30r4 3ð98 15r2 Þ 614656 189336r2 þ 14629r4 r3 239r2 1568 127r2 784 11r2 392 : ð2:25Þ
30
N.A. Kudryashov, D.I. Sinelshchikov / Commun Nonlinear Sci Numer Simulat 17 (2012) 26–34
Proof. The general form of the simple periodic solution can be written in the form [6,7]
wðzÞ ¼
p r T 7
cot
npzo T
2
npzo d cot þ h0 : dz T
ð2:26Þ
Also comparing the Laurent expansion for function (2.26) with expansion (2.7) for the general solution of Eq. (2.2) we obtain
qffiffiffiffiffiffiffiffiffi T ¼ 28p r12 ;
8
r a8 ¼ 127508174438400 ;
2
r ; h0 ¼ 6b 41 90 5880
2
r 1960Þ C 0 ¼ rð2012744 :
ð2:27Þ
Solution (2.26) satisfies Eq. (2.2) if the parameter a takes the form
a¼
1 10976b2 686b 2352 1149r2 2 2963520ð343b 383r 784Þ þ 81r2 9505216 1565928r2 þ 82933r4 697019904
ð2:28Þ
and b is defined by (2.24). Comparing the Laurent series for function (2.26) with the Laurent series (2.7) for the general solution of Eq. (2.2) we obtain
14p b þ 24 127r2 rð171r2 980Þ ; T ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; h0 ¼ ; C0 ¼ 2 90 686 2940 98 15r r2 15r2 196 225r4 2940r2 þ 19208 1 þ a8 ¼ : 33205253760 5400
ð2:29Þ
In this case solution (2.26) satisfies Eq. (2.2) if the parameter a takes the form
1 686b2 21168 þ 686b 4689r2 185220ð7056 þ 343b 1563r2 Þ þ 27r2 101286528 27836424r2 þ 2136139r4 1858719744
a¼
ð2:30Þ
and b is defined by (2.25). As result we have the periodic solution of Eq. (2.2) taking formulae (2.26) into account in the following form
wðzÞ ¼
p r T 7
cot
npzo T
2
npzo 6 b 41r2 d cot þ ; dz T 90 5880
ð2:31Þ
where T and b are defined by (2.27) and (2.24) or by (2.29) and (2.25) correspondingly. This completes the proof. h The simple periodic solution corresponding to Laurent expansion (2.9) can be obtained by analogy with solution (2.31). Let us note that solution (2.31) contains one arbitrary constant corresponding to autonomous of Eq. (2.2). If we take this constant purely imaginary then all poles of solution (2.31) will be on the imaginary axis of the complex plane. The picture of solution (2.31) at different values of r on the real axis is demonstrated on Fig. (1).
3
2
1
Fig. 1. Exact solution (2.31) of Eq. (2.2) at r = 2.6, 2.65, 2.7 (curves 1, 2, 3).
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N.A. Kudryashov, D.I. Sinelshchikov / Commun Nonlinear Sci Numer Simulat 17 (2012) 26–34
Let us discuss the stability of the simple periodic solution (2.31) of Eq. (1.1). To investigate the stability of this solution we use the numerical approach that is the IFRK4 method [26,27]. With this aim we have considered the propagation of nonlinear waves described by Eq. (1.1) with periodic boundary conditions. To test our numerical approach we use exact solution (2.31). This solution is not periodic and we mirror-reflect solution (2.31) with respect to the certain point to obtain the periodic solution in the form of the superposition of exact and reflected solutions as initial conditions. We observe that the relative error of our calculations was less then 5 % during the time of calculations but the relative error was calculated only on the non-reflected part of initial data. To study the stability of solution (2.31) we perturb the initial solution using the random noise. The amplitude of the random noise was given between 1% to 5% percents of the initial data. We have obtained that the shape of perturbed solutions is not changed at numerical modeling and we believe that exact solution (2.31) is stable. 3. Traveling wave solutions of Eq. (1.1) at r = 0 Consider the Swift–Hohenberg equation Eq. 1.1 at r = 0.
ut þ 2uxx þ uxxxx ¼ au þ bu2 cu3
ð3:1Þ
Without loss of generality we take c = 30 in (3.1) again. In this case taking into account the traveling wave solutions u(x, t) = w(z), where z = x C0t in Eq. (2.2) we obtain the nonlinear ordinary differential equation in the form
wzzzz þ 2wzz C 0 wz aw bw2 30w3 ¼ 0:
ð3:2Þ
Suppose that there is the expansion of the general solution of Eq. (3.2) in the form of the Laurent series in a neighborhood of the pole z = z0
wðzÞ ¼
1 X
ak ðz z0 Þkp ;
p > 0:
ð3:3Þ
k¼0
As Eq. (3.2) is autonomous we can consider that z0 = 0 again. Substituting (3.3) into Eq. (3.2) we find that p = 2 and a0 = ±2 again. The Fuchs indices corresponding of each expansions are the following
j1 ¼ 1;
j2 ¼ 8;
j3;4 ¼
pffiffiffiffiffiffi 1 7 i 71 : 2
ð3:4Þ
Substituting (3.3) into Eq. (3.2) and computing coefficients ak we obtain that the compatibility condition for existence of the Laurent series (3.3) is the following
C 0 ¼ 0:
ð3:5Þ
As result we obtain that the Swift–Hohenberg equation has only stationary traveling wave solutions. Let us present classification of the stationary meromorphic exact solutions of Eq. (3.2). For this purpose we use method for constructing general meromorphic solutions of ordinary differential equations from Refs. [6,7] again. Expansions in the Laurent series of the solution of Eq. (3.2) in a neighborhood of the pole z = 0 are the following
wðzÞ ¼
! 2 b6 b2 90a 36 2 z þ þ a 8 z6 þ z2 90 16200
ð3:6Þ
From expansion (3.6) we see that the necessary condition for existence of the elliptic solutions is satisfied. From theorems presented in Refs, [6,7] we obtain that there are some types of meromorphic solutions of Eq. (3.2): simple periodic solutions (and elliptic solutions) corresponding to one of Laurent series (3.6) of the general solution and simple periodic solutions (and elliptic solutions) corresponding to both of Laurent series (3.6). 2
b Theorem 3.1. The elliptic solutions of Eq. (3.2) corresponding to both of the Laurent series (3.6) exist if and only if a ¼ 135 or b = 0.
Proof. In accordance with method from the references [6,7] we look for the possible elliptic solution of Eq. (3.2) in the form
wðzÞ ¼
ð}0 ðz; g 2 ; g 3 Þ þ BÞ2 2ð}ðz; g 2 ; g 3 Þ AÞ2
þ 4}ðz; g 2 ; g 3 Þ þ h0 :
ð3:7Þ
Here }(z, g2, g3) is the Weierstrass elliptic function with invariants g2, g3 and }0 (z, g2, g3) is its derivative with respect to z. Comparing one of expansions (3.6) with the Laurent expansion for expression (3.7) we obtain that elliptic solutions exist b2 only if a ¼ 135 or b = 0. b2 In the case of a ¼ 135 we get the following values of parameters A, B, h0
A¼
1 ; 30
B ¼ 0;
h0 ¼
bþ6 : 90
ð3:8Þ
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N.A. Kudryashov, D.I. Sinelshchikov / Commun Nonlinear Sci Numer Simulat 17 (2012) 26–34
The elliptic solution (3.7) of Eq. (3.2) takes the form
wðzÞ ¼
}02 ðz; g 2 ; g 3 Þ 2ð}ðz; g 2 ; g 3 Þ þ
1 2 Þ 30
þ 4}ðz; g 2 ; g 3 Þ
bþ6 ; 90
ð3:9Þ
where invariants g2, g3 are defined by formulae
g2 ¼
12 15a ; 1620
g3 ¼
8 25a : 81000
ð3:10Þ
Necessary conditions for series (3.6) to exist is the following
a8 ¼
23 11a 7a2 : 10935000 2187000 3499200
ð3:11Þ
At b = 0 parameters A, B, h0 can be found from (3.8) at b = 0. Elliptic solution of Eq. (3.2) is expressed by formula (3.9) at b = 0 as well with invariants (3.10) and values a8 from (3.11). h pffiffiffiffiffiffi Let us show that elliptic solution (3.9) is reduced to the simple periodic solution in the case of b ¼ 3 5330 i. It is known that the Weierstrass elliptic function is defined by the differential equation
}02 ¼ 4}3 g 2 } g 3 :
ð3:12Þ
We obtain that the right-hand side of Eq. (3.12) has two multiple root at b ¼ function is reduced to the hyperbolic function. pffiffiffiffiffiffi Solution (3.9) of Eq. (3.2) at b ¼ 3 5330 i takes the form
wðzÞ ¼
2 2 csch 5
pffiffiffiffiffiffi 3 5330 i.
In this case the Weierstrass elliptic
(pffiffiffi ) (pffiffiffi ) pffiffiffiffiffiffiffiffiffi 5z 5z 1 330 2 sech i: 5 10 5 150
ð3:13Þ
The compatibility condition in this case is the following
a8 ¼
127 : 27000000
ð3:14Þ
In the case of a ¼ 11 elliptic solution (3.9) is also reduced to the simple periodic solution of Eq. (3.2) in the form 25
2 2 wðzÞ ¼ csch 5
(pffiffiffi ) (pffiffiffi ) 5z 5z 1 2 sech : 5 10 5
ð3:15Þ
Let us consider the elliptic solutions of Eq. (3.2) which corresponds to one of expansions (3.6). The following theorem can be formulated. Theorem 3.2. Eq. (3.2) has the elliptic solution which corresponds to one of expansions (3.6) at any values of a and b. Proof. Following the method from the references [6,7] we use the possible elliptic exact solution of Eq. (3.2) corresponding to the first Laurent series (3.6) in the form
w ¼ 2}ðz; g 2 ; g 3 Þ þ h0 :
ð3:16Þ
Comparing the Laureant series corresponding to function (3.16) with the first of the expansion (3.6), we get
h0 ¼
6b ; 90
g2 ¼
b2 90a 36 ; 1620
g3 ¼
b3 þ 6b2 540a 135ab 432 : 291600
ð3:17Þ
The compatibility condition for the first Laurent series (3.6) leads to the equality
a8 ¼
ð36 b2 þ 90aÞ2 : 1574640000
ð3:18Þ
From (3.16) as the result of calculations we have
! b2 90a 36 b3 þ 6b2 540a 135ab 432 6b ; þ : wðzÞ ¼ 2} z; 1620 291600 90
ð3:19Þ
This completes the proof. h The elliptic solution of Eq. (3.2) corresponding to the second expansion from (3.6) can be obtained by analogy with the above presented solution. Finally let us consider the simple periodic solutions corresponding to one of series (3.6). The following theorem is hold.
N.A. Kudryashov, D.I. Sinelshchikov / Commun Nonlinear Sci Numer Simulat 17 (2012) 26–34
33
Theorem 3.3. Eq. (3.2) has the simple periodic solution corresponding to one of expansions (3.6) if
a¼
ðb 6Þðb 24Þ 225
ð3:20Þ
a¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðb 6Þðb þ 12Þ 7bðb þ 6Þ ðb þ 6Þ 15ðb þ 12Þð7b 12Þ þ 504 : pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 9 5ðb þ 12Þ 15ðb þ 12Þð7b 12Þ
ð3:21Þ
or
Proof. The general form of the simple periodic solution corresponding to the first Laurent series (3.6) can be written [6,7] as
wðzÞ ¼
npzo 2p d cot þ h0 : T T dz
ð3:22Þ
Comparing the Laurent expansion for function (3.22) with the first series (3.6) we find the following set of T, h0 and a in the form
30p T 1;2 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 15ð6 bÞ
h0 ¼ 0
ð3:23Þ
where a is defined by (3.20). We also have the values
T 3;4 ¼ T 5;6 ¼ ð3;4Þ;ð5;6Þ
h0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6ðb þ 12Þðb 6Þð60 þ 5b þ 15 ðb þ 12Þð7b 12ÞÞp
; ðb þ 12Þðb 6Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi pffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6ðb þ 12Þðb 6Þð60 þ 5b þ 15 ðb þ 12Þð7b 12ÞÞp ðb þ 12Þðb 6Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðb 6Þ 15ðb þ 12Þð7b 12Þ 15ðb þ 12Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 90ð 15ðb þ 12Þð7b 12Þ 5ðb þ 12ÞÞ
; ð3:24Þ
where a is defined by (3.21). Substituting values of T and h0 from (3.23) or (3.24) we obtain the simple periodic solutions of Eq. (3.2) at alpha given by (3.20) or (3.21) correspondingly. This completes the proof. h Note that solution (3.22) contains one arbitrary constant corresponding to autonomous of Eq. (3.2). We can take this constant imaginary then all poles of solution (3.22) will be on the imaginary axis of the complex plane. The picture of solution (3.22) on the real axis at different values of b is demonstrate on the Fig. (2) for this case.
Fig. 2. Exact solution (3.22) of Eq. (3.2) at b = 50, 70, 85.
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We have investigated the stability of solution (3.22) by the numerical method again as for solution (2.31). Solution (3.22) is periodic and we use (3.22) as initial data for our numerical calculations. We have perturbed the initial conditions using the random noise with the same amplitude as for solution (2.31). The results of the numerical simulation showed that the shape of perturbed solutions is not changed at numerical modeling. Thus we believe that exact solution (3.22) is stable as well. 4. Conclusion In this paper we have studied the traveling wave solutions of the Swift–Hohenberg equation with dispersion. We have shown that nonstationary traveling wave solutions exist in the case of the Swift–Hohenberg equation with dispersion. We have found the elliptic and the simple periodic traveling wave solutions. We have shown that the Swift–Hohenberg equation without dispersion has only the stationary meromorphic traveling wave solutions. We have obtained these exact solutions and presented their classification as well. Acknowledgements This research was supported by Federal Target Programm 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]
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