On the hierarchies of the fully nonlinear Möbius-invariant and symmetry-integrable evolution equations of order three

Marianna Euler; Norbert Euler

doi:10.1080/14029251.2020.1819627

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Volume 27, Issue 4, September 2020, Pages 521 - 528

On the hierarchies of the fully nonlinear Möbius-invariant and symmetry-integrable evolution equations of order three

Authors

Marianna Euler, Norbert Euler^*

Department of Mathematics, Jinan University, 601 Huangpu W Ave, 510632 Guangzhou, People’s Republic of China and Centro Internacional de Ciencias, Av. Universidad s/n, Colonia Chamilpa, 62210 Cuernavaca, Morelos, Mexico

^*Corresponding author’s email address: euler199@gmail.com

Corresponding Author

Norbert Euler

Received 5 August 2020, Accepted 13 August 2020, Available Online 4 September 2020.

DOI: 10.1080/14029251.2020.1819627 How to use a DOI?
Keywords: Symmetry-Integrable Nonlinear Evolution Equations; Fully Nonlinear PDEs; Möbius transformations
Abstract: This is a follow-up paper to the results published in Studies in Applied Mathematics 143, 139–156 (2019), where we reported a classification of 3rd- and 5th-order semi-linear symmetry-integrable evolution equations that are invariant under the Möbius transformation, which includes a list of fully nonlinear 3rd-order equations that admit these properties. In the current paper we propose a simple method to compute the higher-order equations in the hierarchies for the fully nonlinear 3rd-order equations. We apply the proposed method to compute the 5th-order members of the hierarchies explicitly.
Copyright: © 2020 The Authors. Published by Atlantis Press and Taylor & Francis
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. Introduction

In an earlier work [2] we derived all (1 + 1)-dimensional semi-linear evolution equations of order three and order five which are both Möbius-invariant and symmetry-integrable. The classification was done for semi-linear and fully nonlinear 3rd-order equations (meaning nonlinear in the highest derivative) of the form

ut=uxΨ(S)(1.1)

and for semi-linear 5th-order equations of the form

ut=uxΨ(S,Sx,Sxx).(1.2)

Here and throughout this paper, S denotes the Schwarzian derivative in terms of u, namely

S:=uxxxux=32(uxxux)2.(1.3)

We remark that S, itself, is invariant under the Möbius transformation.That is

S(u¯)=S(u), where u¯=α1u+β1α2u+β2(1.4)

with α₁β₂ − α₂β₁ ≠ 0. Clearly u_t/u_x is also Möbius invariant, as well as the x-derivatives of S, so that the nth-order equation in u,

u=uxΨ(S,Sx,Sxx,S3x,…,S(n−3)x),(1.5)

is Möbius invariant for any smooth function Ψ with n ⩾ 3.

It follows [2] that the only semi-linear equation of the form (1.1) which is symmetry-integrable is the Schwarzian Korteweg-de Vries equation

ut=uxS.(1.6)

When (1.1) is not required to be semi-linear, four additional symmetry-integrable fully nonlinear equations follow, namely [2]

ut=−2uxS(1.7a)

ut=ux(b1−S)2(1.7b)

ut=uxS2(1.7c)

ut=ux(a1−Sa12+3a2(S2−2a1S−3a2)1/2),(1.7d)

where the constants a₁, a₂ and b₁ are arbitrary, except for the condition that a12+3a2≠0 and b₁ ≠ 0.

For the 5th-order semi-linear equations

ut=uxSxx+uxΦ1(S,Sx,Sxx)(1.8)

we obtained two equations, namely [2]

ut=ux(Sxx+14S2): the Schwarzian Kupershmidt I equation;(1.9a)

ut=ux(Sxx+4S2): the Schwarzian Kupershmidt II equation.(1.9b)

In addition, the 5th-order Möbius-invariant equation

ut=ux(Sxx+32S2)(1.10)

follows from the 2nd member of the Schwarzian Korteweg-de Vries hierarchy of (1.6).

The following statement is essential for this classification:

Lemma 1 ([2]).

The nth-order Möbius-invariant equation

ut=uxΨ(S,Sx,Sxx,…,S(n−3)x)(1.11)

can be presented in the form of the Möbius-invariant system

ut=uxΨ(S,Sx,Sxx,…,S(n−3)x)(1.12a)

St=(Dx3+2SDx+Sx)Ψ(S,Sx,Sxx,…,S(n−3)x),(1.12b)

known as the Schwarzian system, where S denotes the Schwarzian derivative in terms of u and n ⩾ 3. For semi-linear evolution equations with n > 3, system (1.12a)–(1.12b) takes the following form:

ut=uxS(n−3)x+uxΨ1(S,Sx,Sxx,…,S(n−4)x)(1.13a)

St=Snx+2SS(n−2)x+SxS(n−3)x+(Dx3+2SDx+Sx)Ψ1(S,Sx,Sxx,…,S(n−4)x).(1.13b)

The Möbius-invariant and symmetry-integrable equations listed above then follow from

Proposition 1 ([2]).

Let R[S] be a recursion operator for (1.12b), such that

ZjS=Rj[S]St∂∂S(1.14)

are generalized symmetries (also known as Lie-Bäcklund symmetries) for (1.12b) for all j ∈ 𝒩. Then

Zju=uxΨ(S,Sx,Sxx,…,S(n−3)x)∂∂u+Rj[S]St∂∂S(1.15)

are generalized symmetries for (1.12a) for all j ∈ 𝒩. Therefore, (1.12a) is symmetry-integrable if (1.12b) is symmetry-integrable.

The hierarchies of higher-order members of the Möbius-invariant and symmetry-integrable equations (1.6), (1.9a) and (1.9b) are well known and are best presented in terms of their recursion operators ([1], [2], [4]). However, for the fully nonlinear equations (1.7a)–(1.7d) one encounters a problem as we have found that these equations do not admit recursion operators of the usual linear form

R[u]=∑j=0pGjDxj+∑k=1qηkDx−1∘Λk.(1.16)

In this paper we propose and alternate approach to compute and present the higher-order members of the fully nonlinear hierarchies.

Motivation. The results for 3rd-order and 5th-order semi-linear equations reported in [2] show that the Möbius-invariant systems that are identified by Proposition 1 are exactly those equations that play a central role in the construction of nonlocal and auto-B¨acklund transformations by multipotentialisation, namely the Schwarzian KdV equation (1.6), the Schwarzian Kupershmidt I equation (1.9a) and the Schwarzian Kupershmidt II equation (1.9b) (see [1] for more details). We expect that the Möbius-invariant and symmetry-integrable fully nonlinear equations of 3rd and higher order are of similar importance in the study of fully nonlinear evolution equations.

2. Hierarchies of the fully nonlinear evolution equations of order three

Lemma 1 directly leads to the following proposition by which it is relatively easy to compute the higher-order members of the Möbius-invariant and symmetry-integrable hierarchies of the 3rd-order equation u_t = u_xΨ(S):

Proposition 2.

Let

ut=uxΨ(S)(2.1a)

St=(Dx3+2SDx+Sx)Ψ(S)(2.1b)

be a Möbius-invariant and symmetry-integrable system for some given function Ψ = Ψ(S), where S is the Schwarzian derivative in u. Let R[S] be a 2nd-order recursion operator for (2.1b). Then the higher-order equations in the hierarchy of the symmetry-integrable equation (2.1a) are of the form

utj=uxΨj(S,Sx,Sxx,S(2j)x), j=0,1,2,…,(2.2)

where Ψ_j is to be solved for every j > 0 from the relation

(Dx3+2SDx+Sx)Ψj(S,Sx,…,S(2j)x)=Rj[S](Dx3+2SDx+Sx)Ψ(S)(2.3)

and Ψ₀ ≡ Ψ.

Remark 1.

Note that (2.1b) is never fully nonlinear (in the highest derivative of S), so that the existence of a recursion operator R[S] of the form (1.16) for the symmetry-integrable equation (2.1b) can be assumed.

Result. Applying Proposition 2 we obtain the following 5th-order equations that belong to the hierarchies of fully nonlinear 3rd-order equations (1.7a), (1.7b), (1.7c) and (1.7d), respectively:

ut1=ux(SxxS5/2−54Sx2S7/2+4S1/2)(2.4a)

ut1=ux(4Sxxb1(b1−S)5+10Sx2b1(b1−S)6−b1−4Sb1(b1−S)4)(2.4b)

ut1=ux(−2SxxS5+5Sx2S6+2S3)(2.4c)

ut1=ux(Sxx(S2−2a1S−3a2)5/2−5(S−a1)Sx22(S2−2a1S−3a2)7/2−a1S+3a2(a12+3a2)(S2−2a1S−3a2)3/2).(2.4d)

To discuss the derivation of (2.4a)–(2.4d), we consider the equations (1.7a)–(1.7d) in four separate cases, where we also provide the recursion operators for each S-equation associated to (1.7a)–(1.7d):

Case 1: We consider the 3rd-order Schwarzian system that is associated with the equation (1.7a), namely

ut=−2uxS(2.5a)

St=S−3/2S3x−92S−5/2SxSxx+154S−7/2Sx3.(2.5b)

A recursion operator for (2.5b) is

R[S]=1SDx2−52SxS2Dx−2SxxS2+154Sx2S3−12StDx−1∘1S(2.6)

where R[S]S_x = 0 and

St1=R[S]St=S−5/2S5x−10S−7/2SxS4x+4558S−9/2Sx2S3x−352S−7/2SxxS3x+3154S−9/2SxSxx2−346516S−11/2Sx3Sxx+346532S−13/2Sx5.(2.7)

Applying Proposition 2 we need to find the general solution for Ψ₁(S, S_x, S_xx) from the relation

(Dx3+2SDx+Sx)Ψ1(S,Sx,Sxx)=R[S]St(2.8)

with R[S]S_t given by (2.7). This leads to

Ψ1(S,Sx,Sxx)=S−1/2(S−2Sxx−54S−3Sx2+4),(2.9)

so that the 5th-order Schwarzian system in the hierarchy is

ut1=ux(S−5/2Sxx−54S−7/2Sx2+4S−1/2)(2.10a)

St1=R[S]St=S−5/2S5x−10S−7/2SxS4x+4558S−9/2Sx2S3x−352S−7/2SxxS3x+3154S−9/2SxSxx2−346516S−11/2Sx3Sxx+346532S−13/2Sx5.(2.10b)

Case 2: We consider the 3rd-order Schwarzian system that is associated with equation (1.7b), namely

ut=ux(b1−S)2(2.11a)

St=2S3x(b1−S)3+18SxSxx(b1−S)4+24Sx3(b1−S)5+(3S+b1)Sx(b1−S)3,(2.11b)

where b₁ ≠ 0. A recursion operator for (2.11b) is

R[S]=1b12(b1−S)2Dx2+1b110Sx(b1−S)3Dx+8b1(Sxx(b1−S)3+3Sx2(b1−S)4+S2(b1−S)2)+1b1StDx−1∘1+1b1SxDx−1∘1(b1−S)2,(2.12)

whereby R[S] maps the x-translation symmetry to the t-translation symmetry. That is

R[S]Sx=2Sxxx(b1−S)3+18SxSxx(b1−S)4+24Sx3(b1−S)5+(3S+b1)Sx(b1−S)3=St.(2.13)

Calculating R[S]S_t and using Proposition 2 to determine Ψ₁, we obtain the following 5th-order Schwarzian system for this hierarchy:

ut1=ux(4Sxxb1(b1−S)5+10Sx2b1(b1−S)6−b1−4Sb1(b1−S)4)(2.14a)

St1=R[S]St=4S5xb1(b1−S)5+80SxS4xb1(b1−S)6+140SxxS3xb1(b1−S)6+780Sx2S3xb1(b1−S)7+20SS3xb1(b1−S)5+1080SxSxx2b1(b1−S)7+4620Sx3Sxxb1(b1−S)8+4(55S+b1)SxSxxb1(b1−S)6+3360Sx5b1(b1−S)9+10(35S+13b1)Sx3b1(b1−S)7+(2S2+5b1S−b12)Sxb1(b1−S)5.(2.14b)

Case 3: We consider the 3rd-order Schwarzian system that is associated with the equation (1.7c), namely

ut=ux(1S2)(2.15a)

St=−2(S3xS3−9SxSxxS4+12Sx3S5+3Sx2S2).(2.15b)

A recursion operator for (2.15b) is

R[S]=1S2Dx2−5SxS3Dx-4SxxS3+12Sx2S4+2S+St2Dx-1∘1+Sx2Dx-1∘1S2,(2.16)

whereby R[S] maps the x-translation symmetry to zero. Calculating R[S]S_t and using Proposition 2 to determine Ψ₁, we obtain the following 5th-order Schwarzian system for this hierarchy:

ut1=ux(−2SxxS5+5Sx2S6+2S3)(2.17a)

St1=R[S]St=−2S5xS5+40SxS4xS6+70SxxS3xS6−390Sx2S3xS7−10S3xS4−540SxSxx2S7+2310Sx3SxxS8+110SxSxxS5−175Sx3S6−1680Sx5S9−10SxS3.(2.17b)

Case 4: We consider the 3rd-order Schwarzian system that is associated with equation (1.7d), namely

ut=ux(a1−S(a12+3a2)(S2−2a1S−3a2)1/2)(2.18a)

St=S3x(S2−2a1S−3a2)3/2−9(S−a1)SxSxx(S2−2a1S−3a2)5/2+3(4S2−8a1S+5a12+3a2)Sx3(S2−2a1S−3a2)7/2−(S3−3a1S2−9a2S+3a1a2)Sx(a12+3a2)(S2−2a1S−3a1)3/2,(2.18b)

where a12+3a2≠0. Note that the case a12+3a2=0 is given by Case 2 above. A recursion operator for (2.18b) is

R[S]=1S2−2a1S−3a2Dx2+5Sx(a1−S)(S2−2a1S−3a2)2Dx+4Sxx(a1−S)(S2−2a1S−3a2)2+3Sx2(4S2−8a1S+5a12+3a2)(S2−2a1S−3a2)3+2SS2−2a1S−3a2+a1a12+3a2+StDx−1∘a1−S(S2−2a1S−3a2)1/2−Sxa12+3a2Dx−1∘1,(2.19)

whereby R[S] maps the x-translation symmetry to zero. Calculating R[S]S_t and using Proposition 2 to determine Ψ₁, we obtain the following 5th-order Schwarzian system for this hierarchy:

ut1=ux(Sxx(S2−2a1S−3a2)5/2−5(S−a1)Sx22(S2−2a1S−3a2)7/2−a1S+3a2(a12+3a2)(S2−2a1S−3a2)3/2)(2.20a)

St1=S5x(S2−2a1S−3a2)5/2−20(S−a1)SxS4x(S2−2a1S−3a2)7/2−35(S−a1)SxxS3x(S2−2a1S−3a2)7/2+65(6S2−12a1S+3a2+7a12)Sx2S3x2(S2−2a1S−3a2)9/2+(2a1S2+a12S+15a2S−6a1a2)S3x(a12−3a2)(S2−2a1S−3a2)5/2+45(6S2−12a1S+3a2+7a12)SxSxx2(S2−2a1S−3a2)9/2−1155(S−a1)(2S2−4a1S+3a2+3a12)Sx3Sxx2(S2−2a1S−3a2)11/2−(18a1S3+a12S2+165a2S2−9a13S−189a1a2S+84a12a2+90a22)SxSxx(a12+3a2)(S2−2a1S−3a2)7/2+105(16S4−64a1S3+112a12S2+48a2S2−96a13S−96a1a2S)Sx52(S2−2a1S−3a2)13/2+105(33a14+54a12a2+9a22)Sx52(S2−2a1S−3a2)13/2+(945a2S+30a14S−693a1a22−375a13a2)Sx32(a12+3a2)(S2−2a1S−3a2)9/2+(48a1S4+525a2S3−17a12S3−33a13S2−963a1a2S2+981a12a2S)Sx32(a12+3a2)(S2−2a1S−3a2)9/2+3(a1S3+5a2S2−a1a2S+3a22)Sx(a12+3a2)(S2−2a1S−3a2)5/2.(2.20b)

3. Concluding remarks

We have introduced a simple method, given by Proposition 2, by which it is relatively easy to compute the higher-order equations for a hierarchy of Möbius-invariant and symmetry-integrable equations. This is of particular intertest for fully nonlinear equations (1.7a)–(1.7a), where it is diffi-cult to obtain the recursion operators of the equations. Proposition 2 applies to 3rd-order equations, but the extension to higher-order equations is straightforward. Our study of 5th-order quasi-linear and fully nonlinear Möbius-invariant and symmetry-integrable evolution equations is ongoing and will be published in the near future.

References

[1]M. Euler and N. Euler, Nonlocal invariance of the multipotentialisations of the Kupershmidt equation and its higher-order hierarchies, N. Euler (editor), Nonlinear Systems and Their Remarkable Mathematical Structures, CRC Press, Boca Raton, 2018, pp. 317-351.

[2]M. Euler and N. Euler, On Möbius-invariant and symmetry-integrable evolution equations and the Schwarzian derivative, Studies in Applied Mathematics, Vol. 143, 2019, pp. 139-156.

[3]M. Euler, N. Euler, and E.G. Reyes, Multipotentialisation and nonlocal symmetries: Kupershmidt, Kaup-Kupershmidt and Sawada-Kotera equations, J. Nonlinear Math. Phys., Vol. 24, No. 3, 2017, pp. 303-314.

[4]J.A. Sanders and J.P. Wang, Integrable systems and their recursion operators, Nonlinear Analysis, Vol. 47, 2001, pp. 5213-5240.

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Journal: Journal of Nonlinear Mathematical Physics
Volume-Issue: 27 - 4
Pages: 521 - 528
Publication Date: 2020/09/04
ISSN (Online): 1776-0852
ISSN (Print): 1402-9251
DOI: 10.1080/14029251.2020.1819627 How to use a DOI?
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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ris enw bib

TY  - JOUR
AU  - Marianna Euler
AU  - Norbert Euler
PY  - 2020
DA  - 2020/09/04
TI  - On the hierarchies of the fully nonlinear Möbius-invariant and symmetry-integrable evolution equations of order three
JO  - Journal of Nonlinear Mathematical Physics
SP  - 521
EP  - 528
VL  - 27
IS  - 4
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2020.1819627
DO  - 10.1080/14029251.2020.1819627
ID  - Euler2020
ER  -

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