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Soliton, rational and special solutions of the Korteweg–de Vries hierarchy
Applied Mathematics and Computation 217 (2010) 1774–1779
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Soliton, rational and special solutions of the Korteweg–de Vries hierarchy Nikolai A. Kudryashov 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
a b s t r a c t Soliton, rational and special solutions for any member of the Korteweg–de Vries hierarchy are presented. It is shown that the soliton solutions are found by means of the Hirota formulae with change of the dispersion relation, special solutions for the Korteweg–de Vries hierarchy are expressed via the transcendents of the first and the second Painlevé hierarchies and rational solutions of members for the Korteveg–de Vries hierarchy can be presented using the Yablonski–Vorobév polynomials. Ó 2009 Elsevier Inc. All rights reserved.
Keywords: Nonlinear evolution equation Korteweg–de Vries equation Korteweg–de Vries hierarchy Soliton Self-similar solutions Rational solutions First Painlevé hierarchy Second Painlevé hierarchy
1. Introduction Any member of the Korteweg–de Vries hierarchy can be written in the form [1–3]
@u @ þ Lnþ1 ½u ¼ 0; @t @x
n ¼ 1; 2; . . . ;
ð1:1Þ
where Lnþ1 ½u is the Lenard operator that is determined by the following recursion formula [1–3]
! @ @u þ2 Ln ½u; þ 4u @x @x @3x @3
@Lnþ1 ½u ¼ @x
L1 ½u ¼ u:
ð1:2Þ
Assuming n ¼ 1; n ¼ 2 and n ¼ 3 in Eq. (1.1) we have the following formulae
L2 ½u ¼ uxx þ 3 u2 ;
ð1:3Þ
L3 ½u ¼ uxxxx þ 10uuxx þ 5 u2x þ 10 u3 ; L4 ½u ¼ uxxxxxx þ 14uuxxxx þ 28 ux uxxx þ
ð1:4Þ 21 u2xx
2
þ 70 u uxx þ
70 uu2x
4
þ 35 u
ð1:5Þ
and so on. Substituting (1.3) into (1.1) we have the famous Korteweg–de Vries equation [4–6]
ut þ 6 u ux þ uxxx ¼ 0:
ð1:6Þ
Expressions (1.4) and (1.5) allow us to obtain the Korteveg–de Vries equations of the fifth-order and the seventh-order
ut þ 10 u uxxx þ 20 ux uxxx þ 30 u2 ux þ uxxxxx ¼ 0; 3
ut þ 140 u ux þ
70 u u2x
þ
70 u3x
ð1:7Þ 2
þ 280 u ux uxx þ 70 u uxxx þ 70 uxx uxxx þ 42 ux uxxxx þ 14 u uxxxxx þ uxxxxxxx ¼ 0:
ð1:8Þ
Eq. (1.1) was firstly introduced in [1]. It is well known that the Cauchy problem for all members of this hierarchy can be solved by means of the inverse scattering transform. E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.11.024
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N.A. Kudryashov / Applied Mathematics and Computation 217 (2010) 1774–1779
The aim of this paper is to present soliton, rational and special solutions for any member of the Korteweg–de Vries hierarchy. The paper is organized as follows. In Section 2 we present soliton solutions for any member of the Korteweg–de Vries hierarchy. In Section 3 we consider the self-similar solutions of the Korteweg–de Vries hierarchy and demonstrate that rational solutions of the Korteweg–de Vries hierarchy are expressed via the Yablonski–Vorob’ev polynomials and special solutions can be found taking into account the first and the second Painlevé hierarchies.
2. Soliton solutions of any member for the Korteweg–de Vries hierarchy It is well known that the soliton of the Korteweg–de Vries Eq. (1.6) takes the form [4–6] 2
3
2
u1 ðx; tÞ ¼ 2 k cosh ðk x 4 k t þ u0 Þ;
ð2:1Þ
where k and u0 are arbitrary constants. Let us show that the soliton for any member of the Korteweg–de Vries hierarchy is expressed by formula 2
2nþ1
2
un ðx; tÞ ¼ 2 k cosh ðk x 22n k
t þ u0 Þ;
ð2:2Þ
We can see that solution (2.2) is the traveling wave solution of Eq. (1.1). So we can look for the solitary wave solutions of Eq. (1.1) in the form
n ¼ x xn t
un ðx; tÞ ¼ U n ðnÞ;
ð2:3Þ
Substituting (2.3) into Eq. (1.1) we have the nonlinear ordinary differential equation
Lnþ1 ½U n xn U n þ C 1 ¼ 0;
n ¼ 1; 2; . . .
ð2:4Þ
where C 1 is constant of integration. We are interested in solitary wave of the Korteweg–de Vries hierarchy and we assume C 1 ¼ 0. Substituting expression (2.2) into (2.4) we obtain that soliton solution (2.2) satisfy Eq. (1.1). Therefore soliton solutions for any member of the Korteweg–de Vries hierarchy is expressed by formula (2.2). To look for multi soliton solutions for any member of the Korteweg–de Vries hierarchy we can use the Hirota method [7]. We have found that multi soliton solutions for any member of the Korteweg–de Vries hierarchy is expressed by formula ðnÞ
un ¼ 2
@ 2 ln F N ðx; tÞ ; @x2
N ¼ 1; 2; . . .
ð2:5Þ 2nþ1
ðnÞ
where function F N ðx; tÞ is determined by the Hirota formulae in the case xi ¼ ki ðnÞ
F 1 ðx; tÞ ¼ 1 þ expðh1 Þ;
2nþ1
h1 ¼ k1 x k1
ðnÞ
. Using F N ðx; tÞ in the form
ð0Þ
t þ u1
ð2:6Þ
we obtain one soliton solutions (2.2) for any member of the Korteweg–de Vries hierarchy. In the case N ¼ 2 we take ðnÞ
F 2 ðx; tÞ ¼ 1 þ expðh1 Þ þ expðh2 Þ þ hi ¼ ki x
2nþ1 ki t;
ðk1 k2 Þ2 ðk1 þ k2 Þ2
expðh1 þ h2 Þ;
ð2:7Þ
ði ¼ 1; 2Þ:
and we have two soliton solution. At N ¼ 3 we have three soliton solutions by formula (2.5) for any member of the Korteweg–de Vries hierarchy if we take ðnÞ F 3 ðx; tÞ in the form ðnÞ
F 3 ðx; tÞ ¼ 1 þ expðh1 Þ þ expðh2 Þ þ expðh3 Þ þ þ
ðk2 k3 Þ2 ðk2 þ k3 Þ
2nþ1
hi ¼ ki x ki
t;
2
expðh2 þ h3 Þ þ
ðk1 k2 Þ2 ðk1 þ k2 Þ
2
expðh1 þ h2 Þ þ
ðk1 k2 Þ2 ðk1 k3 Þ2 ðk2 k3 Þ2 2
2
ðk1 þ k2 Þ ðk1 þ k3 Þ ðk2 þ k3 Þ
2
ðk1 k3 Þ2 ðk1 þ k3 Þ2
expðh1 þ h3 Þ
expðh1 þ h2 þ h3 Þ;
ð2:8Þ
ði ¼ 1; 2; 3Þ:
We do not give the proof these formulae for the soliton solutions but we have checked these formulae for the first, the second, the third and the fourth members of the Korteweg–de Vries hierarchy and we obtain that the solutions satisfy members of the Korteveg–de Vries hierarchy. These formulae are really the soliton solutions of the Korteweg–de Vries hierarchy. Taking into account the above mentioned formulae for one, two and three soliton solutions we can write soliton solutions for other members of the Korteweg–de Vries hierarchy. We note that the wave velocity of solitons is increased at ki > 1 with growth of the order of members for the Korteweg–de Vries hierarchy. In the case ki < 1 we obtain another case. However the phase shift in the case of interaction of solitons for all members of the Korteweg–de Vries hierarchy is the same.
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Taking into account the above mentioned soliton solutions for members of the Korteweg–de Vries hierarchy we can write soliton solutions for the Korteweg–de Vries hierarchy in the form M @u @ X þ an Lnþ1 ½u ¼ 0; @t @x n¼1
n ¼ 1; 2; . . . :
ð2:9Þ
Soliton solutions of hierarchy (2.9) are also determined by formulae (2.5)–(2.8) where variable hi is given by formula
hi ¼ ki x
M X
an ki2nþ1 t; i ¼ 1; 2; 3; . . .
ð2:10Þ
n¼1
We checked some of these solutions and we observe that these solutions satisfy the Korteweg–de Vries hierarchy. 3. Rational and special solutions for any member of the Korteweg–de Vries hierarchy Let us look for self-similar solutions for members of the Korteweg–de Vries hierarchy taking into account the following variables [8]
uðx; tÞ ¼
1 2 2nþ1
ð2nt þ tÞ
FðzÞ;
z¼
x 1
ð2nt þ tÞ2nþ1
ð3:1Þ
:
Substituting (3.1) into Eq. (1.1) we obtain equation with respect to FðzÞ in the form
d ðLnþ1 ½FÞ 2 F z F z ¼ 0: dz
ð3:2Þ
Assuming
F ¼ vz v2;
ð3:3Þ
where v ðzÞ is a new variable, we obtain the following equalities
d d d 1 2v þ 2v L n ½ v z v 2 z v dz dz dz 2 h d d d zi ¼ 2v þ 2v Ln ½F : dz dz dz 2
d ðLnþ1 ½FÞ 2 F z F z ¼ dz
ð3:4Þ
From Eq. (3.4) we can see that if FðzÞ is the solution of the hierarchy
Ln ½F
z ¼ 0; 2
n ¼ 1; 2; . . .
ð3:5Þ
then we obtain special solutions for any member of the Korteweg–de Vries hierarchy (1.1) by formula (3.1). This solution for the KdV equation takes the trivial form u ¼ x=ð3 tÞ but the special solution of the fifth-order Korteweg–de Vries equation is expressed via the transcendent of the first Painlevé equation. Hierarchy (3.5) was introduced in [8] and was called as the first Painleve hierarchy. Indeed assuming n ¼ 2 in (3.5) we have the first Painleve equation
F zz þ 3 F 2
z ¼ 0: 2
ð3:6Þ
In the case n ¼ 3 from Eq. (3.5) we obtain
F zzzz þ 10F F zz þ 5 F 2z þ 10 F 3
z ¼ 0: 2
ð3:7Þ
Taking into account the operator L4 ½u in (3.5) we have the sixth-order nonlinear ordinary differential equation
F zzzzzz þ 14 F F zzzz þ 28 F z F zzz þ 21 F 2zz þ 70 F 2 F zz þ 70 F F 2z þ 35 F 4
z ¼ 0: 2
ð3:8Þ
It was proved that equations of hierarchy (3.5) did not have the first integrals in polynomial form [9,10] and apparently all members of this hierarchy determine new transcendental functions similar to Painleve transcendents. However there are not the proof of irreducibility for these equations. Equations of hierarchy (3.5) were considered in many works (see Refs. [11–16]). Consider the first equality in Eq. (3.4). Let v ðzÞ be solution of hierarchy
d þ 2v Ln ½v z v 2 z v bn ¼ 0; dz
ð3:9Þ
where bn is arbitrary constants. Hierarchy (3.9) is the second Painlevé hierarchy [17–22]. Assuming n ¼ 1 in (3.9) we obtain the second Painlevé equation
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v zz 2 v 3 z v b1 ¼ 0;
ð3:10Þ
Solutions of Eq. (3.10) are the Painlevé transcenedents. However this equation has the rational and special solutions at integers and at half integers values of b1 . In the case n ¼ 2 and n ¼ 3 we have the fourth-order and sixth-order nonlinear differential equations
v zzzz 10 v zz v 2 10 v 2z v þ 6 v 5 z v b2 ¼ 0; v zzzzzz 56 v v z v zzz 42 v v 2zz þ 70 v 4 v zz 70 v 2z v zz þ 140 v 2z v 3 20 v 7 14 v 2 v zzzz z v b3 ¼ 0:
ð3:11Þ ð3:12Þ
Apparently the members of the second Painlevé hierarchy determines the transcendental functions similar to the Painlevé equations because equations of these hierarcies do not have the first integrals in the polynomial forms [9]. Special solutions of the second Painleve hierarchy can be found taking into account Eq. (3.4) and the transcendents by the first Painleve hierarchy. However solutions FðzÞ in (3.1) can also be found using formula (3.3) and taking into account the solutions of the second Painlevé hierarchy. Let us look for the rational solutions for any member of the Korteveg–de Vries hierarchy Eq. (1.1) using the formula 2
FðzÞ ¼ 2
d ln yðzÞ 2
dz
;
z¼
x 1
ð2nt þ tÞ2nþ1
ð3:13Þ
:
From Eq. (3.2) we have the hierarchy
" # 2 d d ln y 1 d ln y : z ¼ Lnþ1 2 2 dz dz 2 dz
ð3:14Þ
From hierarchy (3.14) at n ¼ 1 we obtain the nonlinear differential equations for the Yablonskii–Vorobév polynomials: y ¼ Q ð1Þ m ðzÞ [23–25]
y yzzzz 4yz yzzz þ 3 y2zz yyz zyzz þ zy2z ¼ 0: At n ¼ 2 and n ¼ 3 we also have the equations for the Yablonskii–Vorobév polynomials yðzÞ ¼ form
ð3:15Þ Q ð2Þ m
and yðzÞ ¼
Q ð3Þ m
y2 yzzzzzz 6yyz yzzzz þ 5yyzz yzzzz þ 10y2z yzzzz 20yz yzz yzzz þ 10y3zz yz y2 zyzz y2 þ zy2z y ¼ 0;
in the
ð3:16Þ
y4 yzzzzzzzz 8y3 yz yzzzzzzz þ 28y2 y2z yzzzzzz þ 7y3 y2zzzz 56yy3z yzzzzz þ 112yy2z y2zzz þ 28y2 y2zz yzzzz þ 28yy4zz þ 56y2z y3zz þ 56y4z yzzzz 112y3z yzz yzzz þ 28yy2z yzz yzzzz 56y2 yz yzzz yzzzz 112yyz y2zz yzzz yz y4 zyzz y4 þ zy2z y3 ¼ 0:
ð3:17Þ
These special polynomials are associated with the rational solutions of the second Painlevé hierarchy [26,27]. ln y Assuming w ¼ d dz we have that hierarchy (3.14) is reduced to equation
dðz wÞ dw : ¼ Lnþ1 dz dz
ð3:18Þ
All members of hierarchies (3.14) and (3.18) are integrable, because they are obtained from the Korteweg–de Vries hierarchy. Assuming n ¼ 1, n ¼ 2 and n ¼ 3 from (3.18) we obtain the integrable nonlinear differential equations
wzzz þ 3 w2z w z wz ¼ 0 wzzzzz þ 10wz wzzz þ
5w2zz
þ
ð3:19Þ 10w3z
w zwz ¼ 0
wzzzzzzz þ 14wz wzzzzz þ 28wzz wzzzz þ 21w3zzz þ 70w2z wzzz þ 70wz w2zz þ 35w4z w zwz ¼ 0
ð3:20Þ ð3:21Þ
Special polynomials Q ðnÞ m ðzÞ can be found taking into account the differential–difference hierarchy [27]
Table 3.1 The Yablonskii–Vorobev polynomials Q ð1Þ m ðzÞ. ð1Þ
Q 0 ¼ 1, ð1Þ
Q 1 ¼ z, ð1Þ
Q 2 ¼ z3 þ 4, ð1Þ
Q 3 ¼ z6 þ 20z3 80, ð1Þ
Q 4 ¼ ðz9 þ 60z6 þ 11200Þz, ð1Þ
Q 5 ¼ z15 þ 140z12 þ 2800z9 þ 78400z6 313600z3 6272000, ð1Þ
Q 6 ¼ z21 þ 280z18 þ 1848z15 þ 627200z12 17248000z9 þ 1448832000z6 þ 19317760000z3 38635520000, ð1Þ
Q 7 ¼ ðz27 þ 504z24 þ 75600z21 þ 5174400z18 þ 62092800z15 þ 13039488000z12 828731904000z9 49723914240000z6 3093932441600000Þz.
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Table 3.2 ð2Þ The Yablonskii–Vorobev polynomials Q m ðzÞ. ð2Þ
Q 0 ¼ 1, ð2Þ
Q 1 ¼ z, ð2Þ
Q 2 ¼ z3 , ð2Þ
Q 3 ¼ zðz5 144Þ, ð2Þ
Q 4 ¼ z10 1008z5 48384, ð2Þ
Q 5 ¼ z15 4032z10 3048192z5 þ 146313216, ð2Þ
Q 6 ¼ zðz20 12096z15 21337344z10 33798352896z5 4866962817024Þ, ð2Þ
Q 7 ¼ z3 ðz25 30240z20 55883520z15 1182942351360z10 þ 701543488297107456Þ.
Table 3.3 ð3Þ The Yablonskii–Vorobev polynomials Q m ðzÞ. ð3Þ
Q 0 ¼ 1, ð3Þ
Q 1 ¼ z, ð3Þ
Q 2 ¼ z3 , ð3Þ
Q 3 ¼ z6 , ð3Þ
Q 4 ¼ z3 ðz7 þ 14400Þ, ð3Þ
Q 5 ¼ zðz14 þ 129600z7 373248000Þ, ð3Þ
Q 6 ¼ z21 þ 648000z14 24634368000z7 35473489920000, ð3Þ
Q 7 ¼ z28 þ 2376000z21 825251328000z14 30436254351360000z7 þ 43828206265958400000.
" ðnÞ ðnÞ Q mþ1 Q m1
¼
ðnÞ 2 zðQ m Þ
ðnÞ
2 2ðQ ðnÞ m Þ Ln
2
#
2
d
ðln Q ðnÞ m Þ 2
dz
ð3:22Þ
:
ðnÞ
Assuming Q 0 ¼ 1 and Q 1 ¼ z from (3.22) we have special polynomials Q ðnÞ m at m > 1. Some of few special polynomials ð2Þ ð3Þ Q ð1Þ m ; Q m and Q m are given in Tables 3.1, 3.2 and 3.3. Using the Yablonskii–Vorobev polynomials we can find the rational solutions of the Korteweg–de Vries hierarchy (1.2) by the formula
u¼
2
ðnÞ d ln Q m ðzÞ
2 ðð2n þ 1ÞtÞ
2 2nþ1
2
dz
;
z¼
x 1
ð3:23Þ
:
ðð2n þ 1ÞtÞ2nþ1
The four rational solutions of the Korteweg–de Vries equation (hierarchy (1.2) at n ¼ 1) take the form [28]
2 ; x2 xðx3 þ 24tÞ
u1 ðx; tÞ ¼ u2 ðx; tÞ ¼ 6
ðx3 þ 12tÞ2
u3 ðx; tÞ ¼ 12 u4 ðx; tÞ ¼
ð3:24Þ ð3:25Þ
;
xðx9 þ 43200 t3 þ 5400 x3 t2 Þ ðx6 60x3 t þ 720t 2 Þ2 20
ð3:26Þ
;
x2 ðx9 þ 180 x6 t þ 302400 t3 Þ2 ½x18 þ 144 x15 t 2116800 x9 t3 þ 22680 x12 t 2 152409600 x6 t 4 þ 9144576000 t 6 :
ð3:27Þ
The four rational solutions of the fifth-order Korteweg–de Vries equation (hierarchy (1.2) at n ¼ 2) can be written in the form
2 ; x2 6 u2 ðx; tÞ ¼ 2 ; x 12ðx10 þ 2160 x5 t þ 86400 t 2 Þ u3 ðx; tÞ ¼ ; ðx5 þ 720tÞ2 x2
u1 ðx; tÞ ¼
u4 ðx; tÞ ¼
20 x3 ðx15 þ 5040 x10 t þ 23587200 x5 t 2 12192768000 t 3 Þ ðx10 5040 x5 t 1209600 t 2 Þ2
ð3:28Þ ð3:29Þ ð3:30Þ :
ð3:31Þ
N.A. Kudryashov / Applied Mathematics and Computation 217 (2010) 1774–1779
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The four rational solutions of the seventh-order Korteweg–de Vries equation (hierarchy (1.2) at n ¼ 3) can be given as the following
2 ; x2 6 u2 ðx; tÞ ¼ 2 ; x 12 u3 ðx; tÞ ¼ 2 ; x 20ðx14 362880 x7 t þ 3048192000 t2 Þ : u4 ðx; tÞ ¼ x2 ðx7 þ 100800 tÞ2
u1 ðx; tÞ ¼
ð3:32Þ ð3:33Þ ð3:34Þ ð3:35Þ
The rational solutions of the Korteweg–de Vries hierarchy can also be presented if we use the polynomial with respect to x and t by the formula ðnÞ Pm ðx; tÞ ¼ ðð2n þ 1ÞtÞ
m2 þm 4nþ2
Q ðnÞ m ðzÞ;
z¼
x 1
:
ðð2n þ 1ÞtÞ2nþ1
ð3:36Þ
Using these polynomials we obtain rational solutions of the Korteweg–de Vries hierarchy (1.1) by formula ðnÞ um ðx; tÞ ¼ 2
@ 2 ln PðnÞ m ðx; tÞ : @x2
ð3:37Þ
Rational solutions of the Korteweg–de Vries hierarchy (2.9) can be found by formula (3.23), if we take into account the generalized Yablonski–Vorobév polynomials [29]. 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]
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Soliton, rational and special solutions of the Korteweg–de Vries hierarchy Nikolai A. Kudryashov 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
a b s t r a c t Soliton, rational and special solutions for any member of the Korteweg–de Vries hierarchy are presented. It is shown that the soliton solutions are found by means of the Hirota formulae with change of the dispersion relation, special solutions for the Korteweg–de Vries hierarchy are expressed via the transcendents of the first and the second Painlevé hierarchies and rational solutions of members for the Korteveg–de Vries hierarchy can be presented using the Yablonski–Vorobév polynomials. Ó 2009 Elsevier Inc. All rights reserved.
Keywords: Nonlinear evolution equation Korteweg–de Vries equation Korteweg–de Vries hierarchy Soliton Self-similar solutions Rational solutions First Painlevé hierarchy Second Painlevé hierarchy
1. Introduction Any member of the Korteweg–de Vries hierarchy can be written in the form [1–3]
@u @ þ Lnþ1 ½u ¼ 0; @t @x
n ¼ 1; 2; . . . ;
ð1:1Þ
where Lnþ1 ½u is the Lenard operator that is determined by the following recursion formula [1–3]
! @ @u þ2 Ln ½u; þ 4u @x @x @3x @3
@Lnþ1 ½u ¼ @x
L1 ½u ¼ u:
ð1:2Þ
Assuming n ¼ 1; n ¼ 2 and n ¼ 3 in Eq. (1.1) we have the following formulae
L2 ½u ¼ uxx þ 3 u2 ;
ð1:3Þ
L3 ½u ¼ uxxxx þ 10uuxx þ 5 u2x þ 10 u3 ; L4 ½u ¼ uxxxxxx þ 14uuxxxx þ 28 ux uxxx þ
ð1:4Þ 21 u2xx
2
þ 70 u uxx þ
70 uu2x
4
þ 35 u
ð1:5Þ
and so on. Substituting (1.3) into (1.1) we have the famous Korteweg–de Vries equation [4–6]
ut þ 6 u ux þ uxxx ¼ 0:
ð1:6Þ
Expressions (1.4) and (1.5) allow us to obtain the Korteveg–de Vries equations of the fifth-order and the seventh-order
ut þ 10 u uxxx þ 20 ux uxxx þ 30 u2 ux þ uxxxxx ¼ 0; 3
ut þ 140 u ux þ
70 u u2x
þ
70 u3x
ð1:7Þ 2
þ 280 u ux uxx þ 70 u uxxx þ 70 uxx uxxx þ 42 ux uxxxx þ 14 u uxxxxx þ uxxxxxxx ¼ 0:
ð1:8Þ
Eq. (1.1) was firstly introduced in [1]. It is well known that the Cauchy problem for all members of this hierarchy can be solved by means of the inverse scattering transform. E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.11.024
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The aim of this paper is to present soliton, rational and special solutions for any member of the Korteweg–de Vries hierarchy. The paper is organized as follows. In Section 2 we present soliton solutions for any member of the Korteweg–de Vries hierarchy. In Section 3 we consider the self-similar solutions of the Korteweg–de Vries hierarchy and demonstrate that rational solutions of the Korteweg–de Vries hierarchy are expressed via the Yablonski–Vorob’ev polynomials and special solutions can be found taking into account the first and the second Painlevé hierarchies.
2. Soliton solutions of any member for the Korteweg–de Vries hierarchy It is well known that the soliton of the Korteweg–de Vries Eq. (1.6) takes the form [4–6] 2
3
2
u1 ðx; tÞ ¼ 2 k cosh ðk x 4 k t þ u0 Þ;
ð2:1Þ
where k and u0 are arbitrary constants. Let us show that the soliton for any member of the Korteweg–de Vries hierarchy is expressed by formula 2
2nþ1
2
un ðx; tÞ ¼ 2 k cosh ðk x 22n k
t þ u0 Þ;
ð2:2Þ
We can see that solution (2.2) is the traveling wave solution of Eq. (1.1). So we can look for the solitary wave solutions of Eq. (1.1) in the form
n ¼ x xn t
un ðx; tÞ ¼ U n ðnÞ;
ð2:3Þ
Substituting (2.3) into Eq. (1.1) we have the nonlinear ordinary differential equation
Lnþ1 ½U n xn U n þ C 1 ¼ 0;
n ¼ 1; 2; . . .
ð2:4Þ
where C 1 is constant of integration. We are interested in solitary wave of the Korteweg–de Vries hierarchy and we assume C 1 ¼ 0. Substituting expression (2.2) into (2.4) we obtain that soliton solution (2.2) satisfy Eq. (1.1). Therefore soliton solutions for any member of the Korteweg–de Vries hierarchy is expressed by formula (2.2). To look for multi soliton solutions for any member of the Korteweg–de Vries hierarchy we can use the Hirota method [7]. We have found that multi soliton solutions for any member of the Korteweg–de Vries hierarchy is expressed by formula ðnÞ
un ¼ 2
@ 2 ln F N ðx; tÞ ; @x2
N ¼ 1; 2; . . .
ð2:5Þ 2nþ1
ðnÞ
where function F N ðx; tÞ is determined by the Hirota formulae in the case xi ¼ ki ðnÞ
F 1 ðx; tÞ ¼ 1 þ expðh1 Þ;
2nþ1
h1 ¼ k1 x k1
ðnÞ
. Using F N ðx; tÞ in the form
ð0Þ
t þ u1
ð2:6Þ
we obtain one soliton solutions (2.2) for any member of the Korteweg–de Vries hierarchy. In the case N ¼ 2 we take ðnÞ
F 2 ðx; tÞ ¼ 1 þ expðh1 Þ þ expðh2 Þ þ hi ¼ ki x
2nþ1 ki t;
ðk1 k2 Þ2 ðk1 þ k2 Þ2
expðh1 þ h2 Þ;
ð2:7Þ
ði ¼ 1; 2Þ:
and we have two soliton solution. At N ¼ 3 we have three soliton solutions by formula (2.5) for any member of the Korteweg–de Vries hierarchy if we take ðnÞ F 3 ðx; tÞ in the form ðnÞ
F 3 ðx; tÞ ¼ 1 þ expðh1 Þ þ expðh2 Þ þ expðh3 Þ þ þ
ðk2 k3 Þ2 ðk2 þ k3 Þ
2nþ1
hi ¼ ki x ki
t;
2
expðh2 þ h3 Þ þ
ðk1 k2 Þ2 ðk1 þ k2 Þ
2
expðh1 þ h2 Þ þ
ðk1 k2 Þ2 ðk1 k3 Þ2 ðk2 k3 Þ2 2
2
ðk1 þ k2 Þ ðk1 þ k3 Þ ðk2 þ k3 Þ
2
ðk1 k3 Þ2 ðk1 þ k3 Þ2
expðh1 þ h3 Þ
expðh1 þ h2 þ h3 Þ;
ð2:8Þ
ði ¼ 1; 2; 3Þ:
We do not give the proof these formulae for the soliton solutions but we have checked these formulae for the first, the second, the third and the fourth members of the Korteweg–de Vries hierarchy and we obtain that the solutions satisfy members of the Korteveg–de Vries hierarchy. These formulae are really the soliton solutions of the Korteweg–de Vries hierarchy. Taking into account the above mentioned formulae for one, two and three soliton solutions we can write soliton solutions for other members of the Korteweg–de Vries hierarchy. We note that the wave velocity of solitons is increased at ki > 1 with growth of the order of members for the Korteweg–de Vries hierarchy. In the case ki < 1 we obtain another case. However the phase shift in the case of interaction of solitons for all members of the Korteweg–de Vries hierarchy is the same.
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Taking into account the above mentioned soliton solutions for members of the Korteweg–de Vries hierarchy we can write soliton solutions for the Korteweg–de Vries hierarchy in the form M @u @ X þ an Lnþ1 ½u ¼ 0; @t @x n¼1
n ¼ 1; 2; . . . :
ð2:9Þ
Soliton solutions of hierarchy (2.9) are also determined by formulae (2.5)–(2.8) where variable hi is given by formula
hi ¼ ki x
M X
an ki2nþ1 t; i ¼ 1; 2; 3; . . .
ð2:10Þ
n¼1
We checked some of these solutions and we observe that these solutions satisfy the Korteweg–de Vries hierarchy. 3. Rational and special solutions for any member of the Korteweg–de Vries hierarchy Let us look for self-similar solutions for members of the Korteweg–de Vries hierarchy taking into account the following variables [8]
uðx; tÞ ¼
1 2 2nþ1
ð2nt þ tÞ
FðzÞ;
z¼
x 1
ð2nt þ tÞ2nþ1
ð3:1Þ
:
Substituting (3.1) into Eq. (1.1) we obtain equation with respect to FðzÞ in the form
d ðLnþ1 ½FÞ 2 F z F z ¼ 0: dz
ð3:2Þ
Assuming
F ¼ vz v2;
ð3:3Þ
where v ðzÞ is a new variable, we obtain the following equalities
d d d 1 2v þ 2v L n ½ v z v 2 z v dz dz dz 2 h d d d zi ¼ 2v þ 2v Ln ½F : dz dz dz 2
d ðLnþ1 ½FÞ 2 F z F z ¼ dz
ð3:4Þ
From Eq. (3.4) we can see that if FðzÞ is the solution of the hierarchy
Ln ½F
z ¼ 0; 2
n ¼ 1; 2; . . .
ð3:5Þ
then we obtain special solutions for any member of the Korteweg–de Vries hierarchy (1.1) by formula (3.1). This solution for the KdV equation takes the trivial form u ¼ x=ð3 tÞ but the special solution of the fifth-order Korteweg–de Vries equation is expressed via the transcendent of the first Painlevé equation. Hierarchy (3.5) was introduced in [8] and was called as the first Painleve hierarchy. Indeed assuming n ¼ 2 in (3.5) we have the first Painleve equation
F zz þ 3 F 2
z ¼ 0: 2
ð3:6Þ
In the case n ¼ 3 from Eq. (3.5) we obtain
F zzzz þ 10F F zz þ 5 F 2z þ 10 F 3
z ¼ 0: 2
ð3:7Þ
Taking into account the operator L4 ½u in (3.5) we have the sixth-order nonlinear ordinary differential equation
F zzzzzz þ 14 F F zzzz þ 28 F z F zzz þ 21 F 2zz þ 70 F 2 F zz þ 70 F F 2z þ 35 F 4
z ¼ 0: 2
ð3:8Þ
It was proved that equations of hierarchy (3.5) did not have the first integrals in polynomial form [9,10] and apparently all members of this hierarchy determine new transcendental functions similar to Painleve transcendents. However there are not the proof of irreducibility for these equations. Equations of hierarchy (3.5) were considered in many works (see Refs. [11–16]). Consider the first equality in Eq. (3.4). Let v ðzÞ be solution of hierarchy
d þ 2v Ln ½v z v 2 z v bn ¼ 0; dz
ð3:9Þ
where bn is arbitrary constants. Hierarchy (3.9) is the second Painlevé hierarchy [17–22]. Assuming n ¼ 1 in (3.9) we obtain the second Painlevé equation
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v zz 2 v 3 z v b1 ¼ 0;
ð3:10Þ
Solutions of Eq. (3.10) are the Painlevé transcenedents. However this equation has the rational and special solutions at integers and at half integers values of b1 . In the case n ¼ 2 and n ¼ 3 we have the fourth-order and sixth-order nonlinear differential equations
v zzzz 10 v zz v 2 10 v 2z v þ 6 v 5 z v b2 ¼ 0; v zzzzzz 56 v v z v zzz 42 v v 2zz þ 70 v 4 v zz 70 v 2z v zz þ 140 v 2z v 3 20 v 7 14 v 2 v zzzz z v b3 ¼ 0:
ð3:11Þ ð3:12Þ
Apparently the members of the second Painlevé hierarchy determines the transcendental functions similar to the Painlevé equations because equations of these hierarcies do not have the first integrals in the polynomial forms [9]. Special solutions of the second Painleve hierarchy can be found taking into account Eq. (3.4) and the transcendents by the first Painleve hierarchy. However solutions FðzÞ in (3.1) can also be found using formula (3.3) and taking into account the solutions of the second Painlevé hierarchy. Let us look for the rational solutions for any member of the Korteveg–de Vries hierarchy Eq. (1.1) using the formula 2
FðzÞ ¼ 2
d ln yðzÞ 2
dz
;
z¼
x 1
ð2nt þ tÞ2nþ1
ð3:13Þ
:
From Eq. (3.2) we have the hierarchy
" # 2 d d ln y 1 d ln y : z ¼ Lnþ1 2 2 dz dz 2 dz
ð3:14Þ
From hierarchy (3.14) at n ¼ 1 we obtain the nonlinear differential equations for the Yablonskii–Vorobév polynomials: y ¼ Q ð1Þ m ðzÞ [23–25]
y yzzzz 4yz yzzz þ 3 y2zz yyz zyzz þ zy2z ¼ 0: At n ¼ 2 and n ¼ 3 we also have the equations for the Yablonskii–Vorobév polynomials yðzÞ ¼ form
ð3:15Þ Q ð2Þ m
and yðzÞ ¼
Q ð3Þ m
y2 yzzzzzz 6yyz yzzzz þ 5yyzz yzzzz þ 10y2z yzzzz 20yz yzz yzzz þ 10y3zz yz y2 zyzz y2 þ zy2z y ¼ 0;
in the
ð3:16Þ
y4 yzzzzzzzz 8y3 yz yzzzzzzz þ 28y2 y2z yzzzzzz þ 7y3 y2zzzz 56yy3z yzzzzz þ 112yy2z y2zzz þ 28y2 y2zz yzzzz þ 28yy4zz þ 56y2z y3zz þ 56y4z yzzzz 112y3z yzz yzzz þ 28yy2z yzz yzzzz 56y2 yz yzzz yzzzz 112yyz y2zz yzzz yz y4 zyzz y4 þ zy2z y3 ¼ 0:
ð3:17Þ
These special polynomials are associated with the rational solutions of the second Painlevé hierarchy [26,27]. ln y Assuming w ¼ d dz we have that hierarchy (3.14) is reduced to equation
dðz wÞ dw : ¼ Lnþ1 dz dz
ð3:18Þ
All members of hierarchies (3.14) and (3.18) are integrable, because they are obtained from the Korteweg–de Vries hierarchy. Assuming n ¼ 1, n ¼ 2 and n ¼ 3 from (3.18) we obtain the integrable nonlinear differential equations
wzzz þ 3 w2z w z wz ¼ 0 wzzzzz þ 10wz wzzz þ
5w2zz
þ
ð3:19Þ 10w3z
w zwz ¼ 0
wzzzzzzz þ 14wz wzzzzz þ 28wzz wzzzz þ 21w3zzz þ 70w2z wzzz þ 70wz w2zz þ 35w4z w zwz ¼ 0
ð3:20Þ ð3:21Þ
Special polynomials Q ðnÞ m ðzÞ can be found taking into account the differential–difference hierarchy [27]
Table 3.1 The Yablonskii–Vorobev polynomials Q ð1Þ m ðzÞ. ð1Þ
Q 0 ¼ 1, ð1Þ
Q 1 ¼ z, ð1Þ
Q 2 ¼ z3 þ 4, ð1Þ
Q 3 ¼ z6 þ 20z3 80, ð1Þ
Q 4 ¼ ðz9 þ 60z6 þ 11200Þz, ð1Þ
Q 5 ¼ z15 þ 140z12 þ 2800z9 þ 78400z6 313600z3 6272000, ð1Þ
Q 6 ¼ z21 þ 280z18 þ 1848z15 þ 627200z12 17248000z9 þ 1448832000z6 þ 19317760000z3 38635520000, ð1Þ
Q 7 ¼ ðz27 þ 504z24 þ 75600z21 þ 5174400z18 þ 62092800z15 þ 13039488000z12 828731904000z9 49723914240000z6 3093932441600000Þz.
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Table 3.2 ð2Þ The Yablonskii–Vorobev polynomials Q m ðzÞ. ð2Þ
Q 0 ¼ 1, ð2Þ
Q 1 ¼ z, ð2Þ
Q 2 ¼ z3 , ð2Þ
Q 3 ¼ zðz5 144Þ, ð2Þ
Q 4 ¼ z10 1008z5 48384, ð2Þ
Q 5 ¼ z15 4032z10 3048192z5 þ 146313216, ð2Þ
Q 6 ¼ zðz20 12096z15 21337344z10 33798352896z5 4866962817024Þ, ð2Þ
Q 7 ¼ z3 ðz25 30240z20 55883520z15 1182942351360z10 þ 701543488297107456Þ.
Table 3.3 ð3Þ The Yablonskii–Vorobev polynomials Q m ðzÞ. ð3Þ
Q 0 ¼ 1, ð3Þ
Q 1 ¼ z, ð3Þ
Q 2 ¼ z3 , ð3Þ
Q 3 ¼ z6 , ð3Þ
Q 4 ¼ z3 ðz7 þ 14400Þ, ð3Þ
Q 5 ¼ zðz14 þ 129600z7 373248000Þ, ð3Þ
Q 6 ¼ z21 þ 648000z14 24634368000z7 35473489920000, ð3Þ
Q 7 ¼ z28 þ 2376000z21 825251328000z14 30436254351360000z7 þ 43828206265958400000.
" ðnÞ ðnÞ Q mþ1 Q m1
¼
ðnÞ 2 zðQ m Þ
ðnÞ
2 2ðQ ðnÞ m Þ Ln
2
#
2
d
ðln Q ðnÞ m Þ 2
dz
ð3:22Þ
:
ðnÞ
Assuming Q 0 ¼ 1 and Q 1 ¼ z from (3.22) we have special polynomials Q ðnÞ m at m > 1. Some of few special polynomials ð2Þ ð3Þ Q ð1Þ m ; Q m and Q m are given in Tables 3.1, 3.2 and 3.3. Using the Yablonskii–Vorobev polynomials we can find the rational solutions of the Korteweg–de Vries hierarchy (1.2) by the formula
u¼
2
ðnÞ d ln Q m ðzÞ
2 ðð2n þ 1ÞtÞ
2 2nþ1
2
dz
;
z¼
x 1
ð3:23Þ
:
ðð2n þ 1ÞtÞ2nþ1
The four rational solutions of the Korteweg–de Vries equation (hierarchy (1.2) at n ¼ 1) take the form [28]
2 ; x2 xðx3 þ 24tÞ
u1 ðx; tÞ ¼ u2 ðx; tÞ ¼ 6
ðx3 þ 12tÞ2
u3 ðx; tÞ ¼ 12 u4 ðx; tÞ ¼
ð3:24Þ ð3:25Þ
;
xðx9 þ 43200 t3 þ 5400 x3 t2 Þ ðx6 60x3 t þ 720t 2 Þ2 20
ð3:26Þ
;
x2 ðx9 þ 180 x6 t þ 302400 t3 Þ2 ½x18 þ 144 x15 t 2116800 x9 t3 þ 22680 x12 t 2 152409600 x6 t 4 þ 9144576000 t 6 :
ð3:27Þ
The four rational solutions of the fifth-order Korteweg–de Vries equation (hierarchy (1.2) at n ¼ 2) can be written in the form
2 ; x2 6 u2 ðx; tÞ ¼ 2 ; x 12ðx10 þ 2160 x5 t þ 86400 t 2 Þ u3 ðx; tÞ ¼ ; ðx5 þ 720tÞ2 x2
u1 ðx; tÞ ¼
u4 ðx; tÞ ¼
20 x3 ðx15 þ 5040 x10 t þ 23587200 x5 t 2 12192768000 t 3 Þ ðx10 5040 x5 t 1209600 t 2 Þ2
ð3:28Þ ð3:29Þ ð3:30Þ :
ð3:31Þ
N.A. Kudryashov / Applied Mathematics and Computation 217 (2010) 1774–1779
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The four rational solutions of the seventh-order Korteweg–de Vries equation (hierarchy (1.2) at n ¼ 3) can be given as the following
2 ; x2 6 u2 ðx; tÞ ¼ 2 ; x 12 u3 ðx; tÞ ¼ 2 ; x 20ðx14 362880 x7 t þ 3048192000 t2 Þ : u4 ðx; tÞ ¼ x2 ðx7 þ 100800 tÞ2
u1 ðx; tÞ ¼
ð3:32Þ ð3:33Þ ð3:34Þ ð3:35Þ
The rational solutions of the Korteweg–de Vries hierarchy can also be presented if we use the polynomial with respect to x and t by the formula ðnÞ Pm ðx; tÞ ¼ ðð2n þ 1ÞtÞ
m2 þm 4nþ2
Q ðnÞ m ðzÞ;
z¼
x 1
:
ðð2n þ 1ÞtÞ2nþ1
ð3:36Þ
Using these polynomials we obtain rational solutions of the Korteweg–de Vries hierarchy (1.1) by formula ðnÞ um ðx; tÞ ¼ 2
@ 2 ln PðnÞ m ðx; tÞ : @x2
ð3:37Þ
Rational solutions of the Korteweg–de Vries hierarchy (2.9) can be found by formula (3.23), if we take into account the generalized Yablonski–Vorobév polynomials [29]. 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]
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