Nonlinear differential equations associated with the first Painlevé hierarchy

Accepted Manuscript Nonlinear differential equations associated with the first Painlev´e hierarchy

Nikolay A. Kudryashov

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S0893-9659(18)30386-0 https://doi.org/10.1016/j.aml.2018.11.013 AML 5698

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Applied Mathematics Letters

Received date : 29 October 2018 Revised date : 18 November 2018 Accepted date : 18 November 2018 Please cite this article as: N.A. Kudryashov, Nonlinear differential equations associated with the first Painlev´e hierarchy, Applied Mathematics Letters (2018), https://doi.org/10.1016/j.aml.2018.11.013 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Nonlinear differential equations associated with the first Painlev´ e hierarchy. Nikolay A. Kudryashov National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), 31 Kashirskoe Shosse, 115409 Moscow, Russian Federation

Abstract The first Painlev´e hierarchy with general solutions in the form of the Painlev´e transcendents is considered. The linear system associated with this hierarchy is given. Some new hierarchies with properties similar to the Painlev´e hierarchies are presented. It is shown that the solutions of these hierarchies are expressed via the transcendents of the first Painlev´e hierarchy. Thus, the list of nonlinear differential equations whose solutions are expressed in terms of non-classical functions are extended. Key words: First Painlev´e hierarchy, Painlev´e transcendent, Transformation.

The Painlev´e equations were discovered more than one hundred ago as equations with solutions without critical movable points on the complex plane [1,2]. For a long time, it was believed that these equations have interesting properties, but do not have any applications in the description of physical or any other processes. Behavior to Painlev´e equations changed in the seventies of the last century after the discovery of the Inverse Scattering Transform method for solving the Cauchy problem of nonlinear evolution equations with soliton solutions. In papers [3, 4] it was fixed that invariant solutions of many nonlinear evolution equations are expressed through solutions of the Painlev´e equations. In addition, almost at the same time, nonlinear evolution equations with soliton solutions appeared in the description of many physical processes. As a result, there was a great interest in the study of Painlev´e equations. The first Painlev´e hierarchy was introduced in work [5] and takes the form Ln [w] = z,

w ≡ w(z),

(n = 1, 2, . . .),

(1)

∗ Corresponding author. Email address: [email protected] (Nikolay A. Kudryashov).

Preprint submitted to Applied Mathematics Letters

18 November 2018

where z is independent variable, w(z) is dependent variable and the operator Ln [w] is determined by formula d Ln+1 [w] = dz

d dw d3 − 4w −2 dz3 dz dz

!

Ln [w],

1 L0 [w] ≡ − . 2

(2)

Taking into account the recursion relation (2) we have the first Painlev´e equation in the form L2 [w] = wzz − 3 w2 = z. (3)

In the case n = 3 we obtain the second member of the first Painlev e hierarchy in the form L3 [w] = wzzzz − 10 w wzz − 5 wz2 + 10 w3 = z. (4) Assuming n = 4 we have the third member of the first Painlev´e hierarchy 2 L4 [w] = wzzzzzz − 14 w wzzzz − 28 wz wzzz − 21 wzz +

(5)

+70 w2 wzz + 70 w wz2 − 35 w4 = z. It is obvious that n-th member of the first Painlev´e hierarchy has (2 n − 2) – th order of differential equation. The Cauchy problem for equations of the first Painlev´e hierarchy (1) can be solved taking into account the linear system of equations associated with this hierarchy. The Painlev´e hierarchy (1) can be obtained from the following linear system of equations Ψzz = U(z, λ) Ψ, ω(λ) Ψλ = 2 A(z, λ) Ψz − Az (z, λ) Ψ

(6)

Using the compatibility condition for system of equations (6) d2 d Ψzz = 2 Ψλ dλ dz

(7)

we obtain the following equation [6] Azzz − 4 U Az − 2 Uz A + ω(λ) Uλ = 0

(8)

Assuming ω(λ) = −1 in (8) we have the equation Azzz − 4 U Az − 2 Uz A − Uλ = 0.

(9)

Assuming the potential U(w, λ) in the form [6] U = w(z) − λ 2

(10)

we get from equation (9) Azzz − 4 w Az − 2 A wz + 4 λ Az + 1 = 0.

(11)

Let us look for A(w, wz , . . . , λ) in (11) in the form A(w, wz , . . . , λ) =

n−1 X

ak (w, wz , . . .) (−4 λ)n−1−k .

(12)

k=0

Substituting (12) into equation (11) and equating the expressions at various powers λ to zero, we have coefficients ak (w, wz , . . .) in the form ak = − Lk [w],

(k = 0, 1, . . . , n − 1)

(13)

and

d (14) Ln [w] = 1. dz From equation (14) we have the first Painlev´e hierarchy (1) after integration with respect to z. Taking into account (6) and (13) we obtain the linear system associated with the first Painlev´e equation (3) in the form Ψzz = (w − λ) Ψ, Ψλ = 2 (w + 2 λ) Ψz − wz Ψ.

(15)

The second member of the first Painlev´e hierarchy (5) is associated with the following linear system of equations Ψzz = (w − λ) Ψ, Ψλ = 2 (wzz − 3 w2 − 4 λ w − 8 λ2 ) Ψz − (wzzz − 6 w wz − 4 λ wz ) Ψ.

(16)

In the general case the first Painlev´e hierarchy (1) can be associated with the following system

Ψλ = 2

" n−1 X k=0

Ψzz = (w − λ) Ψ, (−4 λ)

n−1−k

#

Lk [w] Ψz −

" n−1 X

(−4 λ)

n−1−k

k=1

#

Lk,z [w] Ψ.

(17)

Linear system (17) can be used to solve the Cauchy problem for equations of the first Painlev´e hierarchy (1). Taking in (1) the Schwarzian derivative as new variable w = {y; z} = 3

2 yzzz 3 yzz − yz 2 yz2

(18)

we have the first Schwarzian Painlev´e hierarchy in the form Ln ({y; z}) = z.

(19)

This hierarchy can be obtained if we use the self-similar variable in the singular manifold equation for the family of the Korteweg-de Vriez equations. From (19) at n = 1 we obtain the third-order differential equation in the form yz yzzz −

3 2 y − z yz2 = 0. 2 zz

(20)

Assuming in (19) n = 2 we have the fifth-order differential equation in the form 2 2 yz 3 yzzzzz − 5 yz2 yzz yzzzz + 26 yz yzz yzzz − 7 yz2 yzzz −

Taking into account a new variable p(z) = 

1 2 p 2

Ln pz −

yzz yz



63 4 y − z yz4 = 0. (21) 4 zz

in (19) we obtain the hierarchy (22)

= z.

At n = 1 from (22) we have the equation pz −

1 2 p = z. 2

(23)

The general solution of equation (23) is expressed by means of the Airy functions. It follows from the linear equation ϕzz +

z ϕ = 0, 2

p=−

2 ϕz . ϕ

(24)

At n = 2 we obtain the third-order nonlinear differential equation in the form pzzz − p pzz − 4 p2z + 3 p2 pz −

3 4 p = z. 2

(25)

At known solution of the first Painlev´e hierarchy W(z) one can find solution y(z) of hierarchy (19) from the differential equation 2 yzzz 3 yzz = W. − yz 2 yz2

(26)

Equation (26) can be written in the form of the linear equation if we use the transformation yz = φ−2 (27) 4

. We obtain that φ satisfies the second-order linear equation in the form φzz +

1 W φ = 0. 2

(28)

Substituting w(z) = {y; z} into (17) we have the linear system that can be used to look for the solution of the Cauchy problem for hierarchy (19). New hierarchy can be obtained using hierarchy (19) if we take into account the Cole-Hopf transformation yz u= . (29) y Using (29) we have yz = u y, yzz = (uz + u2 ) y, (30) 3 yzzz = (uzz + 3 uz u + u ) y. Substituting (30) into hierarchy (19) we obtain the new hierarchy in the form Ln

"

uzz 3 u2z u2 − − u 2 u2 2

#

= z,

(n = 1, 2, . . .).

(31)

Assuming n = 1 in (31) we have from the second-order differential equation in the form 3 u4 (32) u uzz − u2z − − z u2 = 0. 2 2 Taking n = 2 in (31) we get the nonlinear fourth-order ordinary differential equation in the form u3 uzzzz − 5 u2 uz uzzz + 26 u u2z uzz − 7 u2 u2zz − −

3 11 4 2 u uz + 2 u5 uzz − u8 − z u4 = 0. 2 4

63 4 u − 4 z

(33)

Solutions of equations (32), (33) and hierarchy (31) can be found by means of formula (29) at the known solutions of hierarchy (19). Linear problems associated with hierarchy (31) can be found from the system (17) if we use in (17) the expression w(z) =

uzz 3 u2z u2 − − . 2 u 2u 2

(34)

Using new variable in hierarchy (31) u = eκ v 5

(35)

we have the hierarchy in the form Ln

"

#

κ2 2 1 2 κ v = z. κ vzz − v − e 2 z 2

(36)

Solutions of hierarchy (36) are found taking into account solutions of equation (31) by formula 1 (37) v = ln (u(z)). k From (36) at n = 1 we have the second-order differential equation in the form κ vzz −

κ2 2 1 2 κ v v − e = z. 2 z 2

(38)

At n = 2 from (36) we get the fourth-order differential equation in the form 2 k vzzzz − k2 vz vzzz − 4 k2 vzz + 3 k3 vz2 vzz −

+2 k vzz e2 k v −

3 4 4 k vz + 4

7 2 2 2kv 3 4kv k vz e − e = z. 2 4

(39)

Linear systems to solve the Cauchy problems for hierarchy (36) can be found from (17) using the formula w(z) = κ vzz −

κ2 2 1 2 κ v v − e . 2 z 2

(40)

Let us transform the first Painlev´e hierarchy in the form d Ln+1 dz

(

Fzzz 3 F2zz − Fz 2 F2z

)!

=1

(41)

using variables [7] Fz = q, We also have the following formulas d d =q , dz dx

x = F.

Fzz = q qx ,

Fzzz = q2 qxx + q q2x .

(42)

(43)

At n = 1 in (41) we have the third-order differential equation in the form qxxx = q−2 .

(44)

The general solution of equation (44) is given by means of the Bessel function in book [8]. At n = 2 we obtain the fifth-order equation from (41) in the form q2 qxxxxx + 5 q qx qxxxx + 7 q2x qxxx − 4 q qxx qxxx = q−2 . 6

(45)

The general solution of equation (45) can be found taking into account the solutions of the Painlev´e functions. In this Letter we have considered nonlinear differential equations which are related by some transformations with the first Painlev´e hierarchy. Since the general solutions of Painlev´e equations are non-classical functions [9, 10], then the equations presented above are found via the solutions of non-classical functions too. The Cauchy problems for these nonlinear differential equations can be found by means of the Inverse Isomonodromic Transform using the linear system of equations associated with considered above hierarchies. We have demonstrated that linear systems corresponding to new hierarchies can be obtained from the linear system for the first Painlev´e hierarchy. The reported study was funded by RFBR according to the research project 18-29-10025.

References [1] P. Painlev´e, Sur les ´equations diff´eretielles du second ordre et d’ordre sup´errieur dont l’integrate g´en´erale est uniforme, Acta Mathh., 25, (1902) 1–85. [2] B. Gambier, Sur les ´equations diff´eretielles dont l’integrate g´en´erale est uniforme, C.R. Acad. Sc. Paris, 142, (1906) 266–269, 1403–1406, 1497–1500. [3]

M.J. Ablowitz, H. Segur , Exact linearization of a Painleve transcendent, Phys. Rev. Lett., 38 (1977), 11031106

[4] M.J. Ablowitz, A. Ramani, H. Segur A connection between nonlinear evolution equations and ordinary differential equations of P-type. I, J. Math. Phys., 21 (1980), 715721 [5] N.A. Kudryashov, The First and Second Painlev´e equations of higher order and some relations between them, Physics letters A, 224, (1997) 353–360. [6] N.A.. Kudryashov, Amalgamations of the Painlev´e equations, Journal Mathematical Physics, 44(12), (2003) 6160–6178. [7] N.A. Kudryashov, From singular manifold equation to integrable evolution equations, J.Phys.A: Math.Gen, 27 (1994) 2457–2470. [8] A.D. Polyanin, V.F. Zaitsev, 2003 Handbook of Exact Solutions for Ordinary Differential Equations, (Boca Ration, Chapman and Hall/CRC). [9] N.A. Kudryashov, Transcendents defined by nonlinear fourth-order differential equations, J.Phys.A: Math.Gen, 31 (1999) 999–1013. [10] N.A. Kudryashov, Higher Painlev´e Transcensents as Special Solutions of Some Nonlinear Integrable Hierarchies, Regular and Chaotic Dynamics, 19(1), (2014) 48–63.

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