Remarks on rational solutions for the Korteveg - de Vries hierarchy

0.09 MB·English

by  Nikolai A. Kudryashov

Remarks on rational solutions for the
Korteveg - de Vries hierarchy
Nikolai A. Kudryashov
7
0
0 Department of Applied Mathematics
2
Moscow Engineering and Physics Institute
n
a
(State University)
J

7 31 Kashirskoe Shosse, 115409, Moscow,
1
Russian Federation

]
I
S
. Abstract
n
i
l Differential equations for the special polynomials associated with
n
[ the rational solutions of the second Painlev´e hierarchy are introduced.

It is shown rational solutions of the Korteveg - de Vries hierarchy can
1
v be found taking the Yablonskii - Vorob’ev polynomials into account.
4 Specialpolynomials associated with rational solutions ofthe Korteveg
3
- de Vries hierarchy are presented.
0
1
0 Keywords: The Korteveg - de Vries hierarchy, Special polynomials, the
7
Yablonskii - Vorob’ev polynomials, Rational solutions,
0
/
n
i PACS: 02.30.Hq - Ordinary differential equations
l
n
:
v
1 Introduction
i
X
r
a In this paper our interest is in rational solutions of the Korteveg - de Vries
hierarchy [1]
∂
u + L [u] = 0 (1.1)
t N+1
∂x
Here the operator L is determined by the Lenard recursion formula [1–3]
N
d d3 d 1
L [u] = +4u +2u L [u], L [u] = (1.2)
dz N+1 (cid:18)dz3 dz z(cid:19) N 0 2
1
From (1.2) we have
L [u] = u, (1.3)
1
L [u] = u +3u2, (1.4)
2 zz
L [u] = u +10uu +5u2 +10u3, (1.5)
3 zzzz zz z
L [u] = u +14uu +28u u +21u3 +
4 zzzzzz zzzz z zzz zz
(1.6)
+70u2u +70uu2+35u4
zz z
Hierarchy (1.1) is integrable by the inverse scattering transform [4]. The
rational solutions of the Korteveg - de Vries equation (hierarchy (1.1) at
N = 1) are studied in [5–7].
The aim of this paper is to introduce the hierarchy for the special polyno-
mials associated with rational solutions of the second Painlev´e hierarchy (the
Yablonskii - Vorob’ev polynomials [8–13]). These special polynomials can be
used to find the rational solutions of the Korteveg - de Vries hierarchy. As
additional result of this paper is special polynomials associated with rational
solutions of the Korteveg - de Vries hierarchy.
2 Hierarchy for the Yablonskii - Vorob’ev poly-
nomials
Suppose u(x,t in (1.1) takes the form
∂2ln(ϕ(t)y(z)) x
u = 2 , z = (2.1)
∂x2 1
((2N +1)t)2N+1
where ϕ(t) is arbitrary function of t and y(z) is a new function of z. Then
from equation (1.1) we have the hierarchy
d dlny 1 d2lny
z = L 2 (2.2)
dz (cid:18) dz (cid:19) 2 N+1(cid:20) dz2 (cid:21)
It can be shown that special polynomials Q(N)(z) associated with the
n
rational solutions of the second Painlev´e hierarchy satisfy hierarchy (2.2).
Assuming N = 1, N = 2andN = 3in(2.2)wehave differentialequations
for y(z) ≡ Q(1)(z) in (2.3) , y(z) ≡ Q(2)(z) in (2.4) and y(z) ≡ Q(3)(z) in
n n n
(2.5)
yy −4y y +3y 2 −yy −zy y +zy 2 = 0 (2.3)
zzzz z zzz zz z zz z
2
Table 2.1: The Yablonskii - Vorob’ev polynomials Q(1)(z)
n
Q(1)(z) = 1,
0
Q(1)(z) = z,
1
Q(1)(z) = z3 +4,
2
Q(1)(z) = z6 +20z3 −80,
3
Q(1)(z) = (z9 +60z6 +11200)z,
4
Q(1)(z) = z15 +140z12 +2800z9 +78400z6 −313600z3 −6272000,
5
Q(1)(z) = z21 +280z18 +18480z15 +627200z12 −17248000z9+1448832000z6
6
[phone]z3−38635520000,
Q(1)(z) = (z27 +504z24 +75600z21 +5174400z18 [phone]z15+13039488000z12−
7
−828731904000z9−49723914240000z6−3093932441600000)z
y2y −6yy y +5yy y +10y 2y −20y y y +
zzzzzz z zzzzz zz zzzz z zzzz z zz zzz
(2.4)
+10y 3 −y y2 −zy y2 +zy 2y = 0
zz z zz z
y4y −8y3y y +28y2y 2y +7y3y 2−
zzzzzzzz z zzzzzzz z zzzzzz zzzz
−56yy 3y +112yy 2y 2 +28y2y 2y +28yy 4+
z zzzzz z zzz zz zzzz zz
(2.5)
+56y 2y 3 +56y 4y −112y 3y y +28yy 2y y −
z zz z zzzz z zz zzz z zz zzzz
−56y2yzyzzzyzzzz −112yyzyzz2yzzz −yzy4 −zyzzy4 +zyz2y3 = 0
Equation (2.3) for Q(1)(z) was found in [10] but equations (2.4) and (2.5)
n
for special polynomials Q(2)(z) and Q(3)(z) are new. Hierarchy (2.2) for the
n n
special polynomials associated with rational solutions of the second Painlev´e
hierarchy is new as well and we need to study this hierarchy in future.
Denote by w = dlny then equation (2.2) is reduced to equation
dz
d(zw) dw
= L (2.6)
N+1
dz (cid:20) dz (cid:21)
Assuming N = 1, N = 2 and N = 3 from (2.6) we obtain the differential
equations
w +3w2−w−zw = 0, (2.7)
zzz z z
3
Table 2.2: The Yablonskii - Vorob’ev polynomials Q(2)(z)
(n)
Q(2)(z) = 1,
0
Q(2)(z) = z,
1
Q(2)(z) = z3,
2
Q(2)(z) = z(z5 −144),
3
Q(2)(z) = z10 −1008z5 −48384,
4
Q(2)(z) = z15 −4032z10 −3048192z5 [phone],
5
Q(2)(z) = z(z20 −12096z15 −21337344z10−33798352896z5−4866962817024),
6
Q(2)(z) = z3(z25 −30240z20 −55883520z15 −1182942351360z10+701543488297107456)
7
w +10w w +5w2 +10w3−w−zw = 0, (2.8)
zzzzz z zzz zz z z
w +14w w +28w w +21w3 +70w2w +
zzzzzzz z zzzzz zz zzzz zzz z zzz
(2.9)
+70w w2 +35w4−w−zw = 0
z zz z z
Equation (2.7) for dln(Q(n1)(z)) was studied in [10] but equations (2.8) and (2.9)
dz
for special polynomials dln(Q(n2)(z)) and dln(Q(n3)(z)) are new.
dz dz
Multiplying (2.7) on (2w −1) we have the first integral of this equation
zz 2
in the form [10]
1 z
w2 − w +2(2w2− w) w − = C (2.10)
zz 2 zz z z 4 1
(cid:16) (cid:17)
Taking C = n(n+1) in (2.10) we have solutions of (2.10) in the form of the
1 4
Yablonskii - Vorob’ev polynomials. Using w(z) = z2 + p(z) in (2.10) we
8
obtain the equation in the form [10]
2
1 1
p2 +4 (p )3 +2z (p )2 −2pp − n+ = 0 (2.11)
zz z z z 4 (cid:18) 2(cid:19)
Hierarchy (2.6) is integrable because this is obtained from the Korteveg - de
Vries hierarchy. The study of equation (2.7) by the Painlev´e test [14] yields
the Fuchs indices: j = −1, j = 1 and j = 6. These indices correspond to
1 2
4
Table 2.3: The Yablonskii - Vorob’ev polynomials for Q(3)(z)
n
Q(3)(z) = 1
0
Q(3)(z) = z
1
Q (z) = z3
2
Q(3)(z) = z6
3
Q(3)(z) = z3(z7 +14400)
4
Q(3)(z) = z(z14 +129600z7 −373248000)
5
Q(3)(z) = z21 +648000z14 −24634368000z7−35473489920000
6
Q(3)(z) = z28 +2376000z21 −825251328000z14−30436254351360000z7
7
+43828206265958400000
the arbitrary constants z , a and a in the Laurent series
0 1 6
2 z z 2 a
w(z) = +a + 0 (z −z )− 0 + 1 (z −z 3−
(z −z ) 1 6 0 (cid:18)360 30(cid:19) 0
0
(2.12)
z z 2 a
− (z −z )4 +a (z −z )5 − 0 + 1 (z −z )6 +...
0 6 0 0
72 (cid:18)4320 360 (cid:19)
Special polynomials Q(N)(z) were found in works [11,13]. It is very im-
n
portant to remark that these polynomials can be found taking into account
the differential - difference hierarchy [15]
d2
Q(N) Q(N) = z(Q(N))2 −2(Q(N))2L 2 lnQ(N) , n ≥ 1 (2.13)
n+1 n−1 n n N (cid:20) dz2 n (cid:21)
(cid:0) (cid:1)
Assuming Q(N)(z) = 1 and Q(N)(z) = z from (2.13) we have special
0 0
polynomials Q(N)(z) at n ≥ 1. Some of few special polynomials Q(1)(z)
n n
Q(2)(z) and Q(3)(z) are given in tables (2.1), (2.2) and (2.3).
n n
3 Rational solutions of the Korteveg - de Vries
hierarchy
Using the special polynomials associated with the rational solutions of the
second Painlev´e hierarchy (the Yablonskii - Vorob’ev polynomials) we can
5
find the rational solutions of the Korteveg - de Vries hierarchy (1.1) by the
formula
∂2 lnQ(N)(z) x
n
u(x,t) = 2 , z = (3.1)
∂x2 1
((2N +1)t)2N+1
Table 3.1: Polynomials P(1)(x,t)
n
P(1)(x,t) = x,
1
P(1)(x,t) = x3 +12t,
2
P(1)(x,t) = x6 +60x3t−720t2,
3
P(1)(x,t) = (x9 +180x6t+302400t3) x,
4
P(1)(x,t) = x15 +420x12t+25200x9t2 +2116800x6t3−
5
−2540166000x3t4 −1524096000t5,
P(1)(x,t) = x21 +840x18t+166320x15t2 [phone]x12t3 −1397088000x9t4+
6
[phone]x6t5 [phone]x3t6 −84495882240000t7,
P(1)(x,t) = x(x27 +1512x24t+680400x21t2 [phone]x18t3+
7
[phone]x15t4 [phone]x12t5 −604145558016000x9t6−
−108746200442880000x6t7 −60897872248012800000t9)
The four rational solutions of the Korteveg - de Vries equation (hierarchy
(1.1) at N = 1) take the form
2
u (x,t) = − , (3.2)
1 x2
x(−x3 +24t)
u (x,t) = 6 , (3.3)
2 (x3 +12t)2
x(x9 +43200t3+5400x3t2)
u (x,t) = −12 , (3.4)
3 (−x6 −60x3t+720t2)2
20
u (x,t) = − [x18 +144x15t−
4 x2(x9 +180x6t+302400t3)2 (3.5)
−2116800x9t3 +22680x12t2 −152409600x6t4 [phone]t6]
The four rational solutions of the fifth-order Korteveg - de Vries equation
(hierarchy (1.1) at N = 2) can be written in the form
2
u (x,t) = − , (3.6)
1 x2
6
Table 3.2: Polynomials P(2)(x,t)
n
P(2)(x,t) = x,
1
P(2)(x,t) = x3,
2
P(2)(x,t) = x(x5 −720t),
3
P(2)(x,t) = x10 −5040x5t−1209600t2,
4
P(2)(x,t) = x15 −20160x10t−76204800x5t2 [phone]t3,
5
P(2)(x,t) = x(x20 −60480x15t−533433600x10t2 −4224794112000x5t3−
6
−3041851760640000t4),
P(2)(x,t) = x3(x25 −151200x20t−1397088000x15t2 −147867793920000x10t3+
7
+2192323400928460800000t5)
6
u (x,t) = − , (3.7)
2 x2
12(x10 +2160x5t+86400t2)
u (x,t) = − (3.8)
3 (−x5 +720t)2x2
20x3(x15 +5040x10t+23587200x5t2 −12192768000t3)
u (x,t) = − (3.9)
4 (x10 −5040x5t−1209600t2)2
The four rational solutions of the seventh-order Korteveg - de Vries equation
(hierarchy (1.1) at N = 3) can be given as the following
2
u (x,t) = − , (3.10)
1 x2
6
u (x,t) = − , (3.11)
2 x2
12
u (x,t) = − , (3.12)
3 x2
20(x14 −362880x7t+3048192000t2)
u (x,t) = − (3.13)
4 x2(x7 +100800t)2
The rational solutions of the Korteveg - de Vries hierarchy can be presented
in the form of special polynomials taking into account the arbitrary function
7
Table 3.3: Polynomials P(3)(x,t)
n
P(3)(x,t) = x,
1
P(3)(x,t) = x3,
2
P(3)(x,t) = x6,
3
P(3)(x,t) = x3(x7 +100800t),
4
P(3)(x,t) = x(x14 +907200x7t−18289152000t2),
5
P(3)(x,t) = x21 +4536000x14t−1207084032000x7t2 −12167407042560000t3
6
P(3)(x,t) = x28 [phone]x21t−40437315072000x14t2−
7
−10439635242516480000x7t3 +105231523244566118400000t4
ϕ(t) in (2.1). These polynomials are determined by means of formulae
n2+n x
P(N)(x,t) = ((2N +1)t)“4N+2” Q(N)(z), z = (3.14)
n n 1
((2N +1)t)2N+1
Using these polynomials we obtain rational solutions from (2.4) of the Ko-
rteveg - de Vries hierarchy (1.1) by formula
∂2 lnP(N)(x,t)
u(N)(x,t) = 2 n (3.15)
n ∂x2
Some of these polynomials P(1)(x,t), P(2)(x,t) and P(3)(x,t) are given in
n n n
tables (3.1), (3.2) and (3.3).
4 Acknowledgments
This workwassupportedbytheInternationalScience andTechnology Center
under Project B 1213.
References
[1] P.D. Lax, Integrals of nonlinear equations of evolution and solitary
waves, Comm. Pure Appl. Math., v.21, (1968), 467 - 490
8
[2] N.A. Kudryashov, The first Painleve and the second Painleve equations
of higher order and some relations between them, Physics Letters A,
v.224, (1997), 353 - 360
[3] N.A. Kudryashov, Analytical theory of nonlinear differential equations,
Institute of Computer Investigations, Moscow-Izhevsk, (2004), 360 p.
(in Russian)
[4] C.S. Gardner, J.M. Greene, M.D. Kruskal,, R.M. Miura, Method for
solving the korteveg - de Vries equation, v.19, (1967), 1095 - 1097
[5] M.J. Ablowitz, H. Segur, Solitons and the Inverse Scattering TraOn
classes of integrable systems and the painleve property, J. Math. th.
(SIAM, Philadelphia,1981)
[6] H. Airault, H.P. McKean, J. Moser, Rational and Elliptic Solutions of
the Korteveg - de Vries Equation and a related Many- Body Problem,
Comm. Pure Appl. Math. 30, (1977) 95
[7] J. Weiss, On classes of integrable systems and the painleve property, J.
Math. Phys., 25(1) (1984) 13-24
[8] A.I. Yablonskii, On rational solutions of the second Painleve equations,
Vesti Akad. Nauk BSSR, Ser. Fiz. Tkh. Nauk, 3 (1959), 30 - 35 (in
Russian)
[9] A.P. Vorob’ev, On rational solutions of the second Painleve equations,
Differential equations 1 (1965) 79 - 81 (in Russian)
[10] P.A. Clarkson, Remarks on the Yablonskii - Vorob’ev Polynomials,
Physics Letters A, 319, (2003) 137 - 144
[11] P.A. Clarkson, E.L. Mansfield, The second Painleve equation, its hier-
archy and associated special polynomials, Nonlinearity 16 (2003) R1
[12] P.A. Clarkson, Painleve equations - Nonlinear Special Functions, (2004)
Report of IMC, 15 december 2004
[13] M.V. Demina, N.A. Kudryashov,TheYablonskii-Vorob’evpolynomials
for the second Painleve hierarchy, Chaos, Solitons and Fractals, 32(2),
(2007), 526 -537
[14] R. Conte, The Painleve approach to Nonlinear Ordinary Differential
equations, In ”The Painlev´e property. One century later, CRM Series in
Mathematical Physics”, Springer, (1999), New York, 177 - 180
9
[15] M.V. Demina, N.A. Kudryashov, Special polynomials and rational so-
lutions of the second Painlev´e hierarchy, submitted to Theoretical and
Mathematical Physics, (2006)
10