# Journal of Nonlinear Mathematical Physics

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
Norbert Euler
Received 5 August 2020, Accepted 13 August 2020, Available Online 4 September 2020.
DOI
10.1080/14029251.2020.1819627How 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.

Open Access

## 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),whereu¯=α1u+β1α2u+β2(1.4)
with α1β2α2β1 ≠ 0. Clearly ut/ux 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(n3)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(b1S)2(1.7b)
ut=uxS2(1.7c)
ut=ux(a1Sa12+3a2(S22a1S3a2)1/2),(1.7d)
where the constants a1, a2 and b1 are arbitrary, except for the condition that a12+3a20 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):theSchwarzianKupershmidtIequation;(1.9a)
ut=ux(Sxx+4S2):theSchwarzianKupershmidtIIequation.(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(n3)x)(1.11)
can be presented in the form of the Möbius-invariant system
ut=uxΨ(S,Sx,Sxx,,S(n3)x)(1.12a)
St=(Dx3+2SDx+Sx)Ψ(S,Sx,Sxx,,S(n3)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(n3)x+uxΨ1(S,Sx,Sxx,,S(n4)x)(1.13a)
St=Snx+2SS(n2)x+SxS(n3)x+(Dx3+2SDx+Sx)Ψ1(S,Sx,Sxx,,S(n4)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]StS(1.14)
are generalized symmetries (also known as Lie-Bäcklund symmetries) for (1.12b) for all j𝒩. Then
Zju=uxΨ(S,Sx,Sxx,,S(n3)x)u+Rj[S]StS(1.15)
are generalized symmetries for (1.12a) for all j𝒩. 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=1qηkDx1Λk.(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 ut = uxΨ(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 Ψj 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(SxxS5/254Sx2S7/2+4S1/2)(2.4a)
ut1=ux(4Sxxb1(b1S)5+10Sx2b1(b1S)6b14Sb1(b1S)4)(2.4b)
ut1=ux(2SxxS5+5Sx2S6+2S3)(2.4c)
ut1=ux(Sxx(S22a1S3a2)5/25(Sa1)Sx22(S22a1S3a2)7/2a1S+3a2(a12+3a2)(S22a1S3a2)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=S3/2S3x92S5/2SxSxx+154S7/2Sx3.(2.5b)

A recursion operator for (2.5b) is

R[S]=1SDx252SxS2Dx2SxxS2+154Sx2S312StDx11S(2.6)
where R[S]Sx = 0 and

St1=R[S]St=S5/2S5x10S7/2SxS4x+4558S9/2Sx2S3x352S7/2SxxS3x+3154S9/2SxSxx2346516S11/2Sx3Sxx+346532S13/2Sx5.(2.7)

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

(Dx3+2SDx+Sx)Ψ1(S,Sx,Sxx)=R[S]St(2.8)
with R[S]St given by (2.7). This leads to
Ψ1(S,Sx,Sxx)=S1/2(S2Sxx54S3Sx2+4),(2.9)
so that the 5th-order Schwarzian system in the hierarchy is
ut1=ux(S5/2Sxx54S7/2Sx2+4S1/2)(2.10a)
St1=R[S]St=S5/2S5x10S7/2SxS4x+4558S9/2Sx2S3x352S7/2SxxS3x+3154S9/2SxSxx2346516S11/2Sx3Sxx+346532S13/2Sx5.(2.10b)

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

ut=ux(b1S)2(2.11a)
St=2S3x(b1S)3+18SxSxx(b1S)4+24Sx3(b1S)5+(3S+b1)Sx(b1S)3,(2.11b)
where b1 ≠ 0. A recursion operator for (2.11b) is
R[S]=1b12(b1S)2Dx2+1b110Sx(b1S)3Dx+8b1(Sxx(b1S)3+3Sx2(b1S)4+S2(b1S)2)+1b1StDx11+1b1SxDx11(b1S)2,(2.12)
whereby R[S] maps the x-translation symmetry to the t-translation symmetry. That is
R[S]Sx=2Sxxx(b1S)3+18SxSxx(b1S)4+24Sx3(b1S)5+(3S+b1)Sx(b1S)3=St.(2.13)

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

ut1=ux(4Sxxb1(b1S)5+10Sx2b1(b1S)6b14Sb1(b1S)4)(2.14a)
St1=R[S]St=4S5xb1(b1S)5+80SxS4xb1(b1S)6+140SxxS3xb1(b1S)6+780Sx2S3xb1(b1S)7+20SS3xb1(b1S)5+1080SxSxx2b1(b1S)7+4620Sx3Sxxb1(b1S)8+4(55S+b1)SxSxxb1(b1S)6+3360Sx5b1(b1S)9+10(35S+13b1)Sx3b1(b1S)7+(2S2+5b1Sb12)Sxb1(b1S)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(S3xS39SxSxxS4+12Sx3S5+3Sx2S2).(2.15b)

A recursion operator for (2.15b) is

R[S]=1S2Dx25SxS3Dx-4SxxS3+12Sx2S4+2S+St2Dx-11+Sx2Dx-11S2,(2.16)
whereby R[S] maps the x-translation symmetry to zero. Calculating R[S]St 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+70SxxS3xS6390Sx2S3xS710S3xS4540SxSxx2S7+2310Sx3SxxS8+110SxSxxS5175Sx3S61680Sx5S910SxS3.(2.17b)

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

ut=ux(a1S(a12+3a2)(S22a1S3a2)1/2)(2.18a)
St=S3x(S22a1S3a2)3/29(Sa1)SxSxx(S22a1S3a2)5/2+3(4S28a1S+5a12+3a2)Sx3(S22a1S3a2)7/2(S33a1S29a2S+3a1a2)Sx(a12+3a2)(S22a1S3a1)3/2,(2.18b)
where a12+3a20. Note that the case a12+3a2=0 is given by Case 2 above. A recursion operator for (2.18b) is
R[S]=1S22a1S3a2Dx2+5Sx(a1S)(S22a1S3a2)2Dx+4Sxx(a1S)(S22a1S3a2)2+3Sx2(4S28a1S+5a12+3a2)(S22a1S3a2)3+2SS22a1S3a2+a1a12+3a2+StDx1a1S(S22a1S3a2)1/2Sxa12+3a2Dx11,(2.19)
whereby R[S] maps the x-translation symmetry to zero. Calculating R[S]St and using Proposition 2 to determine Ψ1, we obtain the following 5th-order Schwarzian system for this hierarchy:
ut1=ux(Sxx(S22a1S3a2)5/25(Sa1)Sx22(S22a1S3a2)7/2a1S+3a2(a12+3a2)(S22a1S3a2)3/2)(2.20a)
St1=S5x(S22a1S3a2)5/220(Sa1)SxS4x(S22a1S3a2)7/235(Sa1)SxxS3x(S22a1S3a2)7/2+65(6S212a1S+3a2+7a12)Sx2S3x2(S22a1S3a2)9/2+(2a1S2+a12S+15a2S6a1a2)S3x(a123a2)(S22a1S3a2)5/2+45(6S212a1S+3a2+7a12)SxSxx2(S22a1S3a2)9/21155(Sa1)(2S24a1S+3a2+3a12)Sx3Sxx2(S22a1S3a2)11/2(18a1S3+a12S2+165a2S29a13S189a1a2S+84a12a2+90a22)SxSxx(a12+3a2)(S22a1S3a2)7/2+105(16S464a1S3+112a12S2+48a2S296a13S96a1a2S)Sx52(S22a1S3a2)13/2+105(33a14+54a12a2+9a22)Sx52(S22a1S3a2)13/2+(945a2S+30a14S693a1a22375a13a2)Sx32(a12+3a2)(S22a1S3a2)9/2+(48a1S4+525a2S317a12S333a13S2963a1a2S2+981a12a2S)Sx32(a12+3a2)(S22a1S3a2)9/2+3(a1S3+5a2S2a1a2S+3a22)Sx(a12+3a2)(S22a1S3a2)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.

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
10.1080/14029251.2020.1819627How to use a DOI?
Open Access

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  -