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

^{*}

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

- 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.- 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

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

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

*α*

_{1}

*β*

_{2}−

*α*

_{2}

*β*

_{1}≠ 0. Clearly

*u*

_{t}*/u*is also Möbius invariant, as well as the

_{x}*x*-derivatives of

*S*, so that the

*n*th-order equation in

*u*,

*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

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

*a*

_{1},

*a*

_{2}and

*b*

_{1}are arbitrary, except for the condition that

*b*

_{1}≠ 0.

For the 5th-order semi-linear equations

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

The following statement is essential for this classification:

### Lemma 1 ([2]).

*The nth-order Möbius-invariant equation*

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

*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:*

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*

*are generalized symmetries*(

*also known as Lie-Bäcklund symmetries*)

*for*(1.12b)

*for all j*∈

*𝒩. Then*

*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

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*Ψ(

_{x}*S*):

### Proposition 2.

*Let*

*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*

*where*Ψ

_{j}*is to be solved for every j >*0

*from the relation*

*and*Ψ

_{0}≡ 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* 5*th-order equations that belong to the hierarchies of fully nonlinear* 3*rd-order equations* (1.7a), (1.7b), (1.7c) *and* (1.7d)*, respectively:*

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

A recursion operator for (2.5b) is

*R*[

*S*]

*S*= 0 and

_{x}Applying Proposition 2 we need to find the general solution for Ψ_{1}(*S*, *S _{x}*,

*S*) from the relation

_{xx}*R*[

*S*]

*S*given by (2.7). This leads to

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

*b*

_{1}≠ 0. A recursion operator for (2.11b) is

*R*[

*S*] maps the

*x*-translation symmetry to the

*t*-translation symmetry. That is

Calculating *R*[*S*]*S _{t}* and using Proposition 2 to determine Ψ

_{1}, we obtain the following 5th-order Schwarzian system for this hierarchy:

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

A recursion operator for (2.15b) is

*R*[

*S*] maps the

*x*-translation symmetry to zero. Calculating

*R*[

*S*]

*S*and using Proposition 2 to determine Ψ

_{t}_{1}, we obtain the following 5th-order Schwarzian system for this hierarchy:

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

*R*[

*S*] maps the

*x*-translation symmetry to zero. Calculating

*R*[

*S*]

*S*and using Proposition 2 to determine Ψ

_{t}_{1}, we obtain the following 5th-order Schwarzian system for this hierarchy:

## 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

### Cite this article

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 -