Bäcklund Transformations and Non-Abelian Nonlinear Evolution Equations

Bäcklund Transformations and Non-Abelian
Nonlinear Evolution Equations:
a Novel Bäcklund Chart\star\star$\star$This paper is a contribution to the
Special Issue on Analytical Mechanics and Differential Geometry in honour of
Sergio Benenti. The full collection is available at
http://www.emis.de/journals/SIGMA/Benenti.html

Sandra CARILLO ${}^{{\dagger}^{1}{\dagger}^{2}}$, Mauro LO SCHIAVO
${}^{{\dagger}^{1}}$ and Cornelia SCHIEBOLD ${}^{{\dagger}^{3}{\dagger}^{4}}$

S. Carillo, M. Lo Schiavo and C. Schiebold

${}^{{\dagger}^{1}}$ Dipartimento “Scienze di Base e Applicate per
l’Ingegneria”, Sapienza - Università di Roma,
${}^{{\dagger}^{1}}$ 16, Via A. Scarpa, 00161 Rome, Italy
<EMAIL_ADDRESS><EMAIL_ADDRESS>http://www.sbai.uniroma1.it/~sandra.carillo/
http://www.sbai.uniroma1.it/~mauro.loschiavo/

${}^{{\dagger}^{2}}$ I.N.F.N. - Sez. Roma1, Gr. IV - Mathematical Methods in
NonLinear Physics, Rome, Italy

${}^{{\dagger}^{3}}$ Department of Science Education and Mathematics, Mid
Sweden University,
${}^{{\dagger}^{3}}$ S-851 70 Sundsvall, Sweden<EMAIL_ADDRESS>http://www.miun.se/personal/corneliaschiebold

${}^{{\dagger}^{4}}$ Instytut Matematyki, Uniwersytet Jana Kochanowskiego w
Kielcach, Poland

Received December 08, 2015, in final form August 24, 2016; Published online
August 30, 2016

Classes of third order non-Abelian evolution equations linked to that of
Korteweg–de Vries-type are investigated and their connections represented in a
non-commutative Bäcklund chart, generalizing results in [Fuchssteiner B.,
Carillo S., Phys. A 154 (1989), 467–510]. The recursion operators are shown to
be hereditary, thereby allowing the results to be extended to hierarchies. The
present study is devoted to operator nonlinear evolution equations: general
results are presented. The implied applications referring to finite-
dimensional cases will be considered separately.

Dedicated to Sergio Benenti on his 70th birthday

##  1 Introduction

Bäcklund transformations and their associated nonlinear superposition
principles, known as permutability theorems, play a central role in modern
soliton theory as described in [29, 48, 49], see also [46] for further
applications. The importance of the combination of gauge and reciprocal
transformations to link hierarchies of integrable systems is well-established
(see, e.g., [42] and literature cited therein). Here, the concern is with the
application of Bäcklund transformations to connect various non-Abelian
equations to a canonical KdV-type equation and to construct their recursion
operators. Hierarchies of such nonlinear equations are considered wherein the
unknown is an operator on a Banach space. Such operator equations were
originally introduced by Marchenko [39] and further investigated and
developed, in the framework of Banach space geometry, in [1, 15, 16].

The results comprised in this article represent the continuation of the study
in [9, 11, 12, 13, 14]. In particular, in [11, 13] the focus is on the
operator potential Korteweg–de Vries, Korteweg–de Vries and modified
Korteweg–de Vries equations, their connections via Bäcklund transformations
and the construction of the recursion operators they admit.

One of the main advantages of connecting non-Abelian equations via Bäcklund
transformations is that solutions can be transferred from one equation to
another. In [11] operator-valued solutions (which can be interpreted as
operator analogs of solitons) to the pKdV, KdV and mKdV hierarchies are
constructed. In [13] suitable projection techniques are exploited to derive
solution formulae to the corresponding scalar and matrix hierarchies. Note
that the study of non-Abelian nonlinear evolution equations found its original
interest in the case of matrix equations [6, 36]. The results in the present
article are valid for operator-valued functions. This level of generality
permits to construct solutions to scalar and matrix equations that can be
viewed as countable superposition of solitons, see also [50, 53] for the
connection between countable nonlinear superposition and Banach space
geometry, and [51, 52] for further applications.

In [10], the term Bäcklund chart was introduced to indicate the net of
Bäcklund transformations connecting different evolution equations. In [23] a
wide Bäcklund chart which includes scalar 3rd order Abelian evolution
equations is constructed. It connects, further to the KdV and the modified KdV
equations, in particular the KdV singularity manifold equation, also known as
as UrKdV or Schwarz-KdV [18, 55, 57, 58], and the KdV interacting soliton
equation [22]. The connections among the equations in this Abelian Bäcklund
chart are applied to find, or recover, the recursion operators admitted by all
these nonlinear evolution equations. Moreover, other structural properties
such as the Hamiltonian, and bi-Hamiltonian, structure of these equations are
also obtained via the Bäcklund chart [23]. The present study is concerned
about the extension of the Bäcklund chart in [11] to obtain the non-Abelian
analog of the links established in the scalar case [23]. Precisely, in
addition to the pKdV, KdV and mKdV equations already connected in [11], a
different version of the non-Abelian modified KdV and two non-Abelian
equations, respectively, analogs of the KdV singularity manifold and of the
KdV interacting soliton equation, are all linked together via Bäcklund
transformations. New, in the present Bäcklund chart, are the inclusion of a
second modified KdV equation, denoted as amKdV, for alternative mKdV equation
[34, 44], new are also the non-Abelian KdV singularity manifold equation and
the non-Abelian KdV interacting soliton equation.

All the non-Abelian nonlinear evolution equations in the Bäcklund chart admit
a recursion operator. The recursion operators of some of them, such as the
pKdV, KdV and mKdV [11, 44, 54] are known. We recover, via the established
connections, the recursion operators admitted by the non-Abelian amKdV, given
in [44]. Then, the recursion operators of the non-Abelian KdV singularity
manifold and of the KdV interacting soliton equations, both new, are
constructed. Furthermore, the hereditariness of all the obtained recursion
operators is proved combining links via Bäcklund transformations [19, 20],
with the hereditariness of non-Abelian KdV recursion operator [54]. Finally,
since all the nonlinear third order non-Abelian evolution equations admit
hereditary recursion operators, according to [19, 20], all the links in the
Bäcklund chart can be extended to the corresponding whole hierarchies.

The material is organized as follows. The opening Section 2 is devoted to the
Bäcklund charts connecting nonlinear evolution equations which generalize, to
the operator level, the pKdV, KdV and mKdV equations [11, 13]. Sections 3 and
4 are devoted to the construction of a novel Bäcklund chart. In Section 3, the
Bäcklund chart is extended to incorporate also the non-Abelian amKdV equation;
its recursion operator is constructed from the known recursion operators of
the mKdV equations; finally, the connection between the two different non-
Abelian modified KdV equations is provided.

In Section 4, the novel Bäcklund chart is further enlarged. Then, the
recursion operators of the KdV interacting soliton and of the KdV singularity
manifold equations are constructed. In addition, in Section 4, a Möbius type
invariance exhibited by the non-Abelian KdV singularity manifold equation is
established.

The subsequent Section 5 is devoted to the proof of hereditariness of all the
recursion operators in the previous sections. The hereditariness of all the
recursion operators in the Bäcklund chart guarantees that the links can be
transferred to whole hierarchies, relating their corresponding members. In the
closing Section 6, further to some remarks on the interest of the present
study, open problems and perspectives suggested by our new results are briefly
outlined. The article is complemented with an Appendix where some needed
definitions, such as the definition of Bäcklund transformation [19], adopted
throughout, as well as known results obtained in the case of Abelian nonlinear
evolution equations [23] are comprised.

##  2 Non-Abelian Bäcklund charts

In [11], the recursion operators admitted by the non-Abelian potential KdV,
KdV and modified KdV equations were shown to be related by Bäcklund
transformations. Consider first the operator potential KdV equation (pKdV)

$\displaystyle W_{t}=W_{xxx}+{3}W_{x}^{2},$ (2.1)

where the unknown $W$ is a function whose values are bounded linear operators
on some Banach space111Here capital letters are used to emphasize that the
unknown is an operator acting on a Banach space.. It admits the recursion
operator

$\displaystyle\hat{\Psi}(W)=D^{2}+A_{W_{x}}+D^{-1}A_{W_{x}}D+D^{-1}C_{W_{x}}D^{-1}C_{W_{x}},$
(2.2)

where $D$ denotes the derivative with respect to $x$, and $C_{T}$, $A_{T}$
denote the commutator and anti-commutator with respect to $T$, namely,

$\displaystyle C_{T}S:=[T,S]\equiv TS-ST,\qquad A_{T}S:=\\{T,S\\}\equiv
TS+ST.$

For earlier occurrences of recursion operators in the non-Abelian setting we
refer to [5, 25, 32, 44]. Consider next the operator Korteweg–de Vries
equation (KdV)

$\displaystyle U_{t}=U_{xxx}+3\\{U,U_{x}\\},$ (2.3)

which admits the recursion operator

$\displaystyle\Phi(U)=D^{2}+2A_{U}+A_{U_{x}}D^{-1}+C_{U}D^{-1}C_{U}D^{-1}.$
(2.4)

The recursion operators (2.2) and (2.4) are linked via the Bäcklund
transformation

$\displaystyle B_{1}\colon\ U-W_{x}=0.$

In fact, the transformation operator is $\Pi_{B_{1}}=D^{-1}$, and
$\hat{\Psi}(W)=D^{-1}\Phi(U)D$. Hence, the non-Abelian pKdV (2.1) and KdV
(2.3) equations can also be written as

$\displaystyle W_{t}=\hat{\Phi}(W)W_{x}\qquad\text{and}\qquad
U_{t}=\Phi(U)U_{x}.$

Finally, the KdV equation (2.3) is related to the operator modified KdV
equation (mKdV)

$\displaystyle V_{t}=V_{xxx}-3\big{\\{}V^{2},V_{x}\big{\\}}$ (2.5)

via the Miura transformation

$\displaystyle M\colon\ U+V_{x}+V^{2}=0.$ (2.6)

Hence, the Miura transformation (2.6) allows to obtain the recursion operator
$\Psi(V)$ of the mKdV (2.5) from the recursion operator $\Phi(U)$ of the KdV
(2.4) via

$\displaystyle\Psi(V)=\Pi_{M}\Phi(U)\Pi_{M}^{-1},\qquad\text{where}\quad\Pi_{M}=-(D+A_{V})^{-1}.$

The latter implies

$\displaystyle\Psi(V)=\big{(}D-C_{V}D^{-1}C_{V}\big{)}\big{(}D-A_{V}D^{-1}A_{V}\big{)}$

(see [11] for details), which can be equivalently written as

$\displaystyle\Psi(V)=(D-C_{V})D^{-1}(D+C_{V})(D-A_{V})D^{-1}(D+A_{V}).$ (2.7)

The following Bäcklund chart summarizes the links among the non-Abelian pKdV
(2.1), KdV (2.3), and mKdV (2.5) equations:

$\displaystyle\boxed{W_{t}=W_{xxx}+3W^{2}_{x}}\buildrel
B_{1}\over{\text{\textendash\textendash}}\boxed{U_{t}=U_{xxx}+3\\{U,U_{x}\\}}\buildrel
M\over{\text{\textendash\textendash}}\boxed{V_{t}=V_{xxx}-3\big{\\{}V^{2},V_{x}\big{\\}}}\,.$

When the respective recursion operators are applied to the above equations
iteratively, the Bäcklund chart can be extended to the corresponding
hierarchies, and the connections can be summarized in

$\displaystyle\boxed{W_{t}=[\hat{\Phi}(W)]^{n}W_{x}}\buildrel
B_{1}\over{\text{\textendash\textendash}}\boxed{U_{t}=[\Phi(U)]^{n}U_{x}}\buildrel
M\over{\text{\textendash\textendash}}\boxed{V_{t}=[\Psi(V)]^{n}V_{x}}\,.$
(2.8)

Note that (2.8) is the natural non-Abelian counterpart of the corresponding
part of the Bäcklund chart (A.4) introduced in [23].

##  3 On the non-Abelian mKdV equations

A distinguished feature of the Bäcklund chart studied in this article is that
it proceeds via two versions of the non-Abelian mKdV equation. The link
between those versions is considered in this section. The alternative non-
Abelian mKdV equation (amKdV)

$\displaystyle\widetilde{V}_{t}=\widetilde{V}_{xxx}+3[\widetilde{V},\widetilde{V}_{xx}]-6\widetilde{V}\widetilde{V}_{x}\widetilde{V}$
(3.1)

was first described in [34], where the Lax pair formulation and the inverse
scattering problem were studied. In contrast to the non-Abelian mKdV (2.5)
studied in the previous section, (3.1) does not admit a Miura transformation
[44]. Note also that, in the matrix case, (2.5) is invariant under both
$V\rightarrow V^{*}$ and $V\rightarrow-V$, whereas (3.1) is only invariant
under $\widetilde{V}\rightarrow-\widetilde{V}^{*}$ ($V^{*}$ denoting the
transpose of $V$).

As already pointed out in [37], these two versions of the mKdV equation are
linked by the gauge transformation $\widetilde{V}=G^{-1}VG$ with $G_{x}=VG$.
Consider the Bäcklund transformations

$\displaystyle B_{2}\colon\ G_{x}-VG=0,$ (3.2) $\displaystyle B_{3}\colon\
G_{x}-G\widetilde{V}=0.$ (3.3)

Obviously, subsequent application of these Bäcklund transformations links
(2.5) to (3.1).

The recursion operator $\widetilde{\Psi}(\widetilde{V})$ of (3.1) is

$\displaystyle\widetilde{\Psi}\big{(}\widetilde{V}\big{)}=\big{(}D+2C_{\widetilde{V}}\big{)}\big{(}D-2R_{\widetilde{V}}\big{)}\big{(}D+C_{\widetilde{V}}\big{)}^{-1}\big{(}D+2L_{\widetilde{V}}\big{)}D\big{(}D+C_{\widetilde{V}}\big{)}^{-1}.$
(3.4)

It was first given in [32], where it was derived using Lax representation.
Here we use the Bäcklund link between the mKdV and the amKdV equation to give
an alternative derivation together with a more conceptual formulation of the
operator itself.

To this end, we introduce the derivation

$\displaystyle\mathbb{D}:=D+C_{\widetilde{V}}.$ (3.5)

######  Theorem 3.1.

On use of the derivation (3.5), the recursion operator (3.4) of the amKdV
equation can be written as

$\displaystyle\widetilde{\Psi}\big{(}\widetilde{V}\big{)}=\big{(}\mathbb{D}+C_{\widetilde{V}}\big{)}\big{(}\mathbb{D}-A_{\widetilde{V}}\big{)}\mathbb{D}^{-1}\big{(}\mathbb{D}+A_{\widetilde{V}}\big{)}\big{(}\mathbb{D}-C_{\widetilde{V}}\big{)}\mathbb{D}^{-1}.$

######  Remark 3.2.

Recall that two operators $T$ and $S$ are called _related_ if they are of the
form $T=AB$ and $S=BA$. In [11], relatedness of the non-Abelian KdV and mKdV
recursion operators on the image of the Miura transform was exploited to
derive solutions of the non-Abelian mKdV hierarchy from solutions of the non-
Abelian KdV hierarchy.

Here we observe that $\Psi(V)$ and $\widetilde{\Psi}(\widetilde{V})$ are
related in a generalized sense: If $P(D,V)=(D-C_{V})D^{-1}$ and
$Q(D,V)=(D+C_{V})(D-A_{V})D^{-1}(D+A_{V})$, then

$\displaystyle\Psi(V)=P(D,V)Q(D,V),\qquad\widetilde{\Psi}\big{(}\widetilde{V}\big{)}=Q\big{(}\mathbb{D},\widetilde{V}\big{)}P\big{(}\mathbb{D},\widetilde{V}\big{)}.$

Before proving Theorem 3.1, we observe some crucial properties of the
derivation (3.5).

######  Proposition 3.3.

Let $V$ and $\widetilde{V}$ be related to each other via the Bäcklund
transformations $B_{2}$ and $B_{3}$ with some intermediate invertible function
$G$. Then the derivation (3.5) is the result of conjugation of $D$ with
$K_{G}^{-1}$, i.e.,

$\displaystyle\mathbb{D}=K_{G}DK_{G}^{-1},$

where $K_{G}$ denotes the conjugation operator, $K_{G}T=G^{-1}TG$.

Moreover, it holds:

  * $a)$

$K_{G}(D\pm C_{V})K_{G}^{-1}=\mathbb{D}\pm C_{\widetilde{V}}$,

  * $b)$

$K_{G}(D\pm A_{V})K_{G}^{-1}=\mathbb{D}\pm A_{\widetilde{V}}$.

###### Proof.

As an example, we verify $K_{G}DK_{G}^{-1}=\mathbb{D}$ and
$K_{G}C_{V}K_{G}^{-1}=C_{\widetilde{V}}$.

Indeed,

$\displaystyle
K_{G}DK_{G}^{-1}T=G^{-1}\big{(}GTG^{-1}\big{)}_{x}G=G^{-1}\big{(}G_{x}TG^{-1}+GT_{x}G^{-1}-GTG^{-1}G_{x}G^{-1}\big{)}G$
$\displaystyle\hphantom{K_{G}DK_{G}^{-1}T}{}=G^{-1}G_{x}T+T_{x}-TG^{-1}G_{x}=\widetilde{V}T+T_{x}-T\widetilde{V}=\big{(}D+C_{\widetilde{V}}\big{)}T=\mathbb{D}T.$

Analogously,

$\displaystyle
K_{G}C_{V}K_{G}^{-1}T=G^{-1}\big{[}V,GTG^{-1}\big{]}G=G^{-1}\big{(}VGTG^{-1}-GTG^{-1}V\big{)}G$
$\displaystyle\hphantom{K_{G}C_{V}K_{G}^{-1}T}{}=G^{-1}VGT-
TG^{-1}VG=\big{[}G^{-1}VG,T\big{]}=\big{[}\widetilde{V},T\big{]}=C_{\widetilde{V}}T.$
∎

###### Proof of Theorem 3.1.

On use of the Bäcklund transformations $B_{2}$, $B_{3}$, the amKdV recursion
operator $\widetilde{\Psi}(\widetilde{V})$ is related to the mKdV recursion
operator $\Psi(V)$ via

$\displaystyle\widetilde{\Psi}(\widetilde{V})=\Pi\Psi(V)\Pi^{-1}$ (3.6)

with $\Pi=\Pi_{3}\Pi_{2}$, where

$\displaystyle\Pi_{2}=-(B_{2})_{\vphantom{\widetilde{V}}G}^{-1}(B_{2})_{\vphantom{\widetilde{V}}V}^{\vphantom{-1}},\qquad\Pi_{3}=-(B_{3})_{\widetilde{V}}^{-1}(B_{3})_{\vphantom{\widetilde{V}}G}^{\vphantom{-1}}.$

Let $L_{V}$ and $R_{V}$ denote left and right multiplication by $V$, then
$C_{V}T=(L_{V}-R_{V})T$ and $A_{V}T=(L_{V}+R_{V})T$. Now it is straightforward
to verify $(B_{2})_{G}=(D-L_{V})$, $(B_{2})_{V}=-R_{G}$. This leads to the
transformation operator $\Pi_{2}=(D-L_{V})^{-1}R_{G}$. Similarly,
$(B_{3})_{G}=(D-R_{\widetilde{V}})$, $(B_{3})_{\widetilde{V}}=-L_{G}$ leading
to $\Pi_{3}=L_{G^{-1}}(D-R_{\widetilde{V}})$. Hence,

$\displaystyle\Pi=L_{G^{-1}}\big{(}D-R_{\widetilde{V}}\big{)}(D-L_{V})^{-1}R_{G}.$

Next, using the identities $(D-L_{V})R_{G}=R_{G}(D-C_{V})$ and
$(D-R_{\widetilde{V}})R_{G}=R_{G}D$, which are straightforward to check, it
follows

$\displaystyle\Pi=L_{G^{-1}}\big{(}D-R_{\widetilde{V}}\big{)}\big{(}(D-L_{V})^{-1}R_{G}\big{)}=L_{G^{-1}}\big{(}\big{(}D-R_{\widetilde{V}}\big{)}R_{G}\big{)}(D-C_{V})^{-1}$
$\displaystyle\hphantom{\Pi}{}=L_{G^{-1}}R_{G}D(D-C_{V})^{-1}=K_{G}D(D-C_{V})^{-1}.$
(3.7)

Hence, the amKdV recursion operator $\widetilde{\Psi}(\widetilde{V})$ is
obtained from the recursion operator $\Psi(V)$ of the mKdV equation as stated
in (2.7) via

$\displaystyle\widetilde{\Psi}\big{(}\widetilde{V}\big{)}\stackrel{{\scriptstyle\eqref{rec
with
pi},\eqref{pi}}}{{=}}\big{(}K_{G}D(D-C_{V})^{-1}\big{)}(D-C_{V})D^{-1}(D+C_{V})(D-A_{V})$
$\displaystyle\hphantom{\widetilde{\Psi}\big{(}\widetilde{V}\big{)}\stackrel{{\scriptstyle\eqref{rec
with pi},\eqref{pi}}}{{=}}}{}\times
D^{-1}(D+A_{V})\big{(}(D-C_{V})D^{-1}K_{G^{-1}}\big{)}$
$\displaystyle\hphantom{\widetilde{\Psi}\big{(}\widetilde{V}\big{)}}{}=K_{G^{-1}}(D+C_{V})(D-A_{V})D^{-1}(D+A_{V})(D-C_{V})D^{-1}K_{G}$
$\displaystyle\hphantom{\widetilde{\Psi}\big{(}\widetilde{V}\big{)}}{}=(\mathbb{D}+C_{\widetilde{V}})(\mathbb{D}-A_{\widetilde{V}})\mathbb{D}^{-1}(\mathbb{D}+A_{\widetilde{V}})(\mathbb{D}-C_{\widetilde{V}})\mathbb{D}^{-1},$

where the last step requires Proposition 3.3. ∎

To conclude this section, for the reader’s convenience we give a direct
verification that the mKdV and the amKdV equation are related via
$\widetilde{V}=G^{-1}G_{x}$ and $V=G_{x}G^{-1}$. More precisely, we show

  * a)

$V_{t}=\Psi(V)V_{x}\Longleftrightarrow\widetilde{V}_{t}=\widetilde{\Psi}(\widetilde{V})\widetilde{V}_{x}$,

  * b)

$\widetilde{\Psi}(\widetilde{V})\widetilde{V}_{x}$ constitutes the right-hand
side of (3.1).

To prove a), recall that $\widetilde{\Psi}(\widetilde{V})=\Pi\Psi(V)\Pi^{-1}$
where $\Pi=K_{G}D(D-C_{V})^{-1}$, compare (3.6), (3.7) in the proof of Theorem
3.1. Noting that $\widetilde{V}=K_{G}V$ implies
$\widetilde{V}_{x}=K_{G}V_{x}$, since
$(G^{-1}G_{x})_{x}=G^{-1}(G_{x}G^{-1})_{x}G$ (observe that
$\widetilde{V}_{xx}=K_{G}V_{xx}$ does not hold). Hence

$\displaystyle\Pi^{-1}\widetilde{V}_{x}\stackrel{{\scriptstyle\eqref{pi}}}{{=}}(D-C_{V})D^{-1}K_{G}^{-1}\widetilde{V}_{x}=(D-C_{V})D^{-1}V_{x}=(D-C_{V})V=V_{x}.$
(3.8)

Furthermore,
$\widetilde{V}_{t}=(G^{-1}G_{x})_{t}=G^{-1}G_{xt}-G^{-1}G_{t}G^{-1}G_{x}=G^{-1}(G_{xt}G^{-1}-G_{t}G^{-1}G_{x}G^{-1})G=K_{G}D(G_{t}G^{-1})$.
This shows

$\displaystyle\Pi^{-1}\widetilde{V}_{t}=(D-C_{V})D^{-1}K_{G}^{-1}\widetilde{V}_{t}=(D-C_{V})\big{(}G_{t}G^{-1}\big{)}=V_{t}.$
(3.9)

Now a) follows from
$\widetilde{\Psi}(\widetilde{V})\widetilde{V}_{x}=\Pi\Psi(V)\Pi^{-1}\widetilde{V}_{x}\stackrel{{\scriptstyle\eqref{h1}}}{{=}}\Pi\Psi(V)V_{x}=\Pi
V_{t}\stackrel{{\scriptstyle\eqref{h2}}}{{=}}\widetilde{V}_{t}$.

To prove b), note first that
$\widetilde{V}_{x}=D\widetilde{V}=(D+C_{\widetilde{V}})\widetilde{V}=\mathbb{D}\widetilde{V}$.
Similarly,
$(\mathbb{D}-C_{\widetilde{V}})\widetilde{V}=\mathbb{D}\widetilde{V}$, and
then
$(\mathbb{D}+A_{\widetilde{V}})\mathbb{D}\widetilde{V}=\mathbb{D}^{2}\widetilde{V}+\\{\widetilde{V},\mathbb{D}\widetilde{V}\\}=\mathbb{D}(\widetilde{V}^{2}+\mathbb{D}\widetilde{V})$.
Using Theorem 3.1, we hence get

$\displaystyle\widetilde{\Psi}\big{(}\widetilde{V}\big{)}\widetilde{V}_{x}=\big{(}\mathbb{D}+C_{\widetilde{V}}\big{)}\big{(}\mathbb{D}-A_{\widetilde{V}}\big{)}\mathbb{D}^{-1}\big{(}\mathbb{D}+A_{\widetilde{V}}\big{)}\big{(}\mathbb{D}-C_{\widetilde{V}}\big{)}\mathbb{D}^{-1}\mathbb{D}\widetilde{V}$
$\displaystyle\hphantom{\widetilde{\Psi}\big{(}\widetilde{V}\big{)}\widetilde{V}_{x}}{}=\big{(}\mathbb{D}+C_{\widetilde{V}}\big{)}\big{(}\mathbb{D}-A_{\widetilde{V}}\big{)}\mathbb{D}^{-1}\big{(}\mathbb{D}+A_{\widetilde{V}}\big{)}\mathbb{D}\widetilde{V}=\big{(}\mathbb{D}+C_{\widetilde{V}}\big{)}\big{(}\mathbb{D}-A_{\widetilde{V}}\big{)}\big{(}\mathbb{D}\widetilde{V}+\widetilde{V}^{2}\big{)}$
$\displaystyle\hphantom{\widetilde{\Psi}\big{(}\widetilde{V}\big{)}\widetilde{V}_{x}}{}=\big{(}\mathbb{D}+C_{\widetilde{V}}\big{)}\big{(}\mathbb{D}^{2}\widetilde{V}-2\widetilde{V}^{3}\big{)}=\mathbb{D}^{3}\widetilde{V}+\big{[}\widetilde{V},\mathbb{D}^{2}\widetilde{V}\big{]}-2\mathbb{D}\big{(}\widetilde{V}^{3}\big{)}.$

Finally, evaluation of $\mathbb{D}=D+C_{\widetilde{V}}$ gives the right-hand
side of the amKdV equation.

##  4 A new non-Abelian Bäcklund chart

In the last two sections a non-Abelian generalization of the Bäcklund chart
linking pKdV, KdV and mKdV equations was presented, and extended to include
the amKdV equation (3.1). The aim of the present section is a further
extension to also include non-Abelian analogues of the KdV singularity
manifold equation and the KdV interacting soliton equation. To the best of the
authors’ knowledge, the non-Abelian equations presented in this sections are
new. Together with the links presented in the previous section, the resulting
non-Abelian Bäcklund chart222To facilitate the comparison with the Abelian
case, a brief overview of the results in [23] is provided in the Appendix. we
obtain can be summarized as follows:

$\boxed{\mbox{pKdV}(W)}\,{\buildrel
B_{1}\over{\text{\textendash\textendash}}}\,\boxed{\mbox{KdV}(U)}\,{\buildrel
M\over{\text{\textendash\textendash}}}\,\boxed{\mbox{mKdV}(V)}\,{\buildrel
B_{2},B_{3}\over{\text{\textendash\textendash}}}\,\boxed{\mbox{amKdV}(\widetilde{V})}{\buildrel
B_{4}\over{\text{\textendash\textendash}}}\,\boxed{\mbox{KdV
sing.}(\phi)}\,{\buildrel B_{5}\over{\text{\textendash\textendash}}}\
\boxed{\mbox{int.\, soliton KdV}(S)}\,.$ (4.1)

Moreover, the corresponding recursion operators are derived. Note that this
chart extends to the respective hierarchies. In conclusion, it is shown that
all recursion operators are hereditary. Finally, it is well-known that the KdV
singularity manifold equation is invariant under the full Möbius group, and it
is shown that a generalized property holds for its non-Abelian counterpart.

To begin with, we introduce a non-Abelian analogue of the KdV singularity
manifold equation (KdV sing.)

$\displaystyle\phi_{t}=\phi_{x}\\{\phi;x\\},$ (4.2)

where $\\{\phi;x\\}$ denotes the following non-Abelian version of the
Schwarzian derivative

$\displaystyle\\{\phi;x\\}:=\big{(}\phi_{x}^{-1}\phi_{xx}\big{)}_{x}-\frac{1}{2}\big{(}\phi_{x}^{-1}\phi_{xx}\big{)}^{2}.$

######  Remark 4.1.

  1. a)

To the best of the authors’ knowledge, equation (4.2) is new. For earlier non-
Abelian versions of (4.2) we refer to [56], see also [2]. The difference to
our approach is the non-Abelian interpretation of terms of the form
$1/u_{x}^{2}$.

  2. b)

In the scalar case there is a very satisfying group theoretic explanation of
the relation between the KdV sing. (also called UrKdV), mKdV and KdV equations
[58], see also [18, 55]. It seems to be an interesting problem to find an
appropriate extension of these ideas to the non-Abelian setting. Departing
from the present results, a first major difficulty is that our link between
(2.5) and (3.1) (which reduce to the same scalar equation) is not realized by
an explicit mapping $V=F(\widetilde{V})$.

The KdV sing. equation (4.2) exhibits invariances analogous to its scalar
counterpart. Note that the following theorem implies invariance under the full
Möbius group in the scalar case.

######  Theorem 4.2.

Let $A$, $B$, $C$, $D$ be constant operators. Then the non-Abelian KdV
singularity manifold equation (4.2) is invariant under the transformations

$\displaystyle\phi\mapsto(\phi A+B)^{-1}(\phi
C+D),\qquad\phi\mapsto(C\phi+D)(A\phi+B)^{-1},$

provided that $A$ and $D-BA^{-1}C$ are invertible.

In other words, if $\phi$ is an invertible solution of (4.2), then also $(\phi
A+B)^{-1}(\phi C+D)$ and $(C\phi+D)(A\phi+B)^{-1}$ are solutions of (4.2).

######  Remark 4.3.

The condition that $D-BA^{-1}C$ is invertible can be understood as a
generalization of the scalar condition $ad-bc\not=0$ in the case $a\not=0$.
Note however, that this is not the only possibility to guarantee the
invariances of Theorem 4.2. For example, the assumption that $B$ and
$D-AB^{-1}C$ are invertible is also sufficient.

The main ingredient of the proof of Theorem 4.2 is the following observation.

######  Proposition 4.4.

The non-Abelian KdV singularity manifold equation (4.2) is invariant under the
transformation $\phi\mapsto\phi^{-1}$.

###### Proof.

To start with we calculate the Schwarzian derivative of $\psi:=\phi^{-1}$.

Since $\psi_{x}=-\phi^{-1}\phi_{x}\phi^{-1}$, we get
$\psi_{xx}=-\phi^{-1}\phi_{xx}\phi^{-1}+2\phi^{-1}\phi_{x}\phi^{-1}\phi_{x}\phi^{-1}$.
Hence

$\displaystyle\psi_{x}^{-1}\psi_{xx}=\phi\phi_{x}^{-1}\phi_{xx}\phi^{-1}-2\phi_{x}\phi^{-1},$

and therefore

$\displaystyle\\{\psi;x\\}=\big{(}\psi_{x}^{-1}\psi_{xx}\big{)}_{x}-\frac{1}{2}\big{(}\psi_{x}^{-1}\psi_{xx}\big{)}^{2}$
$\displaystyle\hphantom{\\{\psi;x\\}}{}=\phi_{xx}\phi^{-1}+\boxed{\phi\big{(}\phi_{x}^{-1}\phi_{xx}\big{)}_{x}\phi^{-1}}-\phi\phi_{x}^{-1}\phi_{xx}\phi^{-1}\phi_{x}\phi^{-1}-2\phi_{xx}\phi^{-1}+2\phi_{x}\phi^{-1}\phi_{x}\phi^{-1}$
$\displaystyle\hphantom{\\{\psi;x\\}=}{}\boxed{-\frac{1}{2}\phi\big{(}\phi_{x}^{-1}\phi_{xx}\big{)}^{2}\phi^{-1}}+\phi\phi_{x}^{-1}\phi_{xx}\phi^{-1}\phi_{x}\phi^{-1}+\phi_{xx}\phi^{-1}-2\phi_{x}\phi^{-1}\phi_{x}\phi^{-1}.$

All terms with exception of the boxed ones cancel out, hence
$\\{\psi;x\\}=\phi\\{\phi,x\\}\phi^{-1}$. As a result,

$\displaystyle\psi_{t}=-\phi^{-1}\phi_{t}\phi^{-1}\stackrel{{\scriptstyle\eqref{nc-
KdVsing}}}{{=}}-\phi^{-1}\big{(}\phi_{x}\\{\phi;x\\}\big{)}\phi^{-1}=\big{(}{-}\phi^{-1}\phi_{x}\phi^{-1}\big{)}\big{(}\phi\\{\phi;x\\}\phi^{-1}\big{)}=\psi_{x}\\{\psi;x\\},$

which completes the proof. ∎

The following invariance is straightforward to verify.

######  Proposition 4.5.

The non-Abelian KdV singularity manifold equation (4.2) is invariant under the
transformation $\phi\mapsto A\phi B+C$ where $A$, $B$, $C$ are constant
operators, provided that $A$ and $B$ are invertible.

###### Proof of Theorem 4.2.

Let $\phi$ be a solution of (4.2). By Proposition 4.5, also $\phi A+B$ solves
(4.2) since $A$ is invertible. Then, by Proposition 4.4, the inverse
$\psi=(\phi A+B)^{-1}$ is a solution of (4.2). Applying Proposition 4.5,
$\psi(D-BA^{-1}C)+A^{-1}C=(\phi A+B)^{-1}(D-BA^{-1}C+(\phi A+B)A^{-1}C)=(\phi
A+B)^{-1}(\phi C+D)$ is a solution of (4.2). ∎

Finally, we introduce the non-Abelian KdV interacting soliton equation (int.
soliton KdV)

$\displaystyle
S_{t}=\left(S\left(\big{(}S^{-1}S_{x}\big{)}_{x}-\frac{1}{2}\big{(}S^{-1}S_{x}\big{)}^{2}\right)\right)_{x}.$
(4.3)

Again, to the best of the authors’ knowledge, this equation is new.

Analogously as in Section 3, it can be verified that the amKdV (3.1), KdV
sing. (4.2) and int. soliton KdV (4.3) equations are related by the Bäcklund
transformations

$\displaystyle B_{4}\colon\ 2\phi_{x}{\widetilde{V}}-\phi_{xx}=0,$ (4.4)
$\displaystyle B_{5}\colon\ \phi_{x}-S=0.$ (4.5)

######  Remark 4.6.

Note that the counterpart of the transformation
$\widetilde{V}=\frac{1}{2}\phi_{x}^{-1}\phi_{xx}$ from (4.4), namely,
$\widehat{V}=-\frac{1}{2}\phi_{xx}\phi_{x}^{-1}$, also maps solutions $\phi$
of the KdV sing. equation (4.2) to solutions $\widehat{V}$ of the amKdV
equation (3.1).

Both the recursion operators $\Upsilon(\phi)$ of (4.2) as well as $\chi(S)$ of
(4.3) can be constructed on use of the Bäcklund links (4.4), (4.5).

######  Theorem 4.7.

Define $N(T)=\frac{1}{2}T^{-1}T_{x}$. Then the recursion operators
$\Upsilon(\phi)$ of KdV sing. (4.2) and $\chi(S)$ of (4.3) are given by

$\displaystyle\Upsilon(\phi)=L_{\phi_{x}}\mathbb{D}^{-1}\big{(}\mathbb{D}-A_{N(\phi_{x})}\big{)}\big{(}\mathbb{D}-C_{N(\phi_{x})}\mathbb{D}^{-1}C_{N(\phi_{x})}\big{)}\big{(}\mathbb{D}+A_{N(\phi_{x})}\big{)}L_{\phi_{x}^{-1}},$
(4.6) $\displaystyle\chi(S)=\textstyle
L_{S}\big{(}\mathbb{D}-A_{N(S)}\mathbb{D}^{-1}A_{N(S)}\big{)}\big{(}\mathbb{D}-C_{N(S)}\mathbb{D}^{-1}C_{N(S)}\big{)}L_{S}^{-1}.$
(4.7)

Note that $\mathbb{D}=D+C_{N(\phi_{x})}=D+C_{N(S)}$.

###### Proof.

Indeed, from the Bäcklund links we have that the recursion operators satisfy

$\displaystyle\Upsilon(\phi)=\Pi_{4}\widetilde{\Psi}\big{(}\widetilde{V}\big{)}\Pi_{4}^{-1},\qquad$
$\displaystyle\Pi_{4}=-(B_{4})_{\phi}^{-1}(B_{4})_{\widetilde{V}},$
$\displaystyle\chi(S)=\Pi_{5}\Upsilon(\phi)\Pi_{5}^{-1},\qquad$
$\displaystyle\Pi_{5}=-(B_{5})_{S}^{-1}(B_{5})_{\phi}.$

It is straightforward to calculate $(B_{4})_{\widetilde{V}}=2L_{\phi_{x}}$,
$(B_{4})_{\phi}=-(D-2R_{\widetilde{V}})D$. Therefore, we get

$\displaystyle\Pi_{4}=2D^{-1}\big{(}D-2R_{\widetilde{V}}\big{)}^{-1}L_{\phi_{x}}.$

Similarly $(B_{5})_{S}=-I$, $(B_{5})_{\phi}=D$, and hence $\Pi_{5}=D$.
Consequently,

$\displaystyle\Pi_{5}\Pi_{4}=2\big{(}D-2R_{\widetilde{V}}\big{)}^{-1}L_{\phi_{x}}=2\big{(}\mathbb{D}-A_{\widetilde{V}}\big{)}^{-1}L_{S}.$

Using the identity
$(\mathbb{D}-A_{\widetilde{V}})L_{S}=L_{S}(\mathbb{D}+C_{\widetilde{V}})$,
which can be checked directly using only the product rule and
$\widetilde{V}=\frac{1}{2}S^{-1}S_{x}$, we find

$\displaystyle\Pi_{5}\Pi_{4}=2L_{S}\big{(}\mathbb{D}+C_{\widetilde{V}}\big{)}^{-1}.$

From this and Theorem 3.1, we get

$\displaystyle\chi(S)=(\Pi_{5}\Pi_{4})\widetilde{\Psi}\big{(}\widetilde{V}\big{)}(\Pi_{5}\Pi_{4})^{-1}=L_{S}\big{(}\mathbb{D}+C_{\widetilde{V}}\big{)}^{-1}\widetilde{\Psi}\big{(}\widetilde{V}\big{)}\big{(}\mathbb{D}+C_{\widetilde{V}}\big{)}L_{S}^{-1}$
$\displaystyle\hphantom{\chi(S)}{}=L_{S}\big{(}\mathbb{D}-A_{\widetilde{V}}\big{)}\mathbb{D}^{-1}\big{(}\mathbb{D}+A_{\widetilde{V}}\big{)}\big{(}\mathbb{D}-C_{\widetilde{V}}\big{)}\mathbb{D}^{-1}\big{(}\mathbb{D}+C_{\widetilde{V}}\big{)}L_{S^{-1}}.$

The claim for $\chi(S)$ follows from observing that
$(\mathbb{D}-A_{\widetilde{V}})\mathbb{D}^{-1}(\mathbb{D}+A_{\widetilde{V}})=\mathbb{D}-A_{\widetilde{V}}\mathbb{D}^{-1}A_{\widetilde{V}}=\mathbb{D}-A_{N(S)}\mathbb{D}^{-1}A_{N(S)}$
and
$(\mathbb{D}-C_{\widetilde{V}})\mathbb{D}^{-1}(\mathbb{D}+C_{\widetilde{V}})=\mathbb{D}-C_{\widetilde{V}}\mathbb{D}^{-1}C_{\widetilde{V}}=\mathbb{D}-C_{N(S)}\mathbb{D}^{-1}C_{N(S)}$.
Finally,

$\displaystyle\Upsilon(\phi)=\Pi_{5}^{-1}\chi(S)\Pi_{5}=D^{-1}L_{S}\big{(}\mathbb{D}-A_{N(S)}\mathbb{D}^{-1}A_{N(S)}\big{)}\big{(}\mathbb{D}-C_{N(S)}\mathbb{D}^{-1}C_{N(S)}\big{)}L_{S}^{-1}D.$

Since $DL_{S}^{-1}=L_{S}^{-1}D-L_{S}^{-1}L_{S_{x}}L_{S}^{-1}$, the identity
$L_{S}^{-1}D=(D+L_{S^{-1}S_{x}})L_{S}^{-1}=(D+2L_{N(S)})L_{S}^{-1}=(\mathbb{D}+A_{N(S)})L_{S}^{-1}$
holds, and thus

$\displaystyle\Upsilon(\phi)=L_{S}\big{(}\mathbb{D}+A_{N(S)}\big{)}^{-1}\big{(}\mathbb{D}-A_{N(S)}\mathbb{D}^{-1}A_{N(S)}\big{)}\big{(}\mathbb{D}-C_{N(S)}\mathbb{D}^{-1}C_{N(S)}\big{)}\big{(}\mathbb{D}+A_{N(S)}\big{)}L_{S}^{-1}$
$\displaystyle\hphantom{\Upsilon(\phi)}{}=L_{S}\mathbb{D}^{-1}\big{(}\mathbb{D}-A_{N(S)}\big{)}\big{(}\mathbb{D}-C_{N(S)}\mathbb{D}^{-1}C_{N(S)}\big{)}\big{(}\mathbb{D}+A_{N(S)}\big{)}L_{S}^{-1},$

which is the claim for $\Upsilon(\phi)$ upon substituting $S=\phi_{x}$. ∎

In the scalar case, the recursion operator for the KdV sing. equation (4.2) is
well-known, see, e.g., [23], where it appears in the form

$\displaystyle\Upsilon(\phi)=\phi_{x}D^{-1}\big{(}D^{3}+2D{\sf U}+2{\sf
U}D\big{)}\phi_{x}^{-1}.$

with ${\sf
U}=\frac{1}{2}(\phi_{xx}/\phi_{x})_{x}-\frac{1}{4}(\phi_{xx}/\phi_{x})^{2}$.
To facilitate comparison to the non-Abelian case, we state the following
reformulation of (4.6).

######  Corollary 4.8.

The recursion operator $\Upsilon(\phi)$ of the KdV sing. equation (4.2) in
Theorem 4.7 can be written as

$\displaystyle\Upsilon(\phi)=L_{\phi_{x}}\mathbb{D}^{-1}\big{(}\mathbb{D}^{3}+\mathbb{D}A_{\sf
U}+A_{\sf U}\mathbb{D}+C_{\sf U}\mathbb{D}^{-1}C_{\sf
U}\big{)}L_{\phi_{x}^{-1}},$

where ${\sf U}=(N(\phi_{x}))_{x}-(N(\phi_{x}))^{2}$.

###### Proof.

The corollary is an immediate consequence of Proposition 4.10, the fact that
$N(\phi_{x})=\widetilde{V}$ and $\widetilde{V}_{x}=\mathbb{D}\widetilde{V}$. ∎

######  Remark 4.9.

Note that in the scalar case ${\sf U}=v_{x}-v^{2}$, i.e., $\sf U$ stems from a
Miura transformation of a solution $v$ to the mKdV equation. In the non-
Abelian setting, we have ${\sf U}=\widetilde{V}_{x}-\widetilde{V}^{2}$, which
means that $\sf U$ stems from a (non-Abelian) Miura transformation of a
solution $\widetilde{V}$ to the amKdV equation (3.1).

######  Proposition 4.10.

Let $\sf D$ be a derivation, and $\sf U$, $\sf V$ related by the Miura
transform ${\sf U}={\sf D}{\sf V}-{\sf V}^{2}$. Then the following
factorization holds:

$\displaystyle{\sf D}^{3}+A_{\sf U}{\sf D}+{\sf D}A_{\sf U}+C_{\sf U}{\sf
D}^{-1}C_{\sf U}=({\sf D}-A_{\sf V})\big{(}{\sf D}-C_{\sf V}{\sf D}^{-1}C_{\sf
V}\big{)}({\sf D}+A_{\sf V})$ $\displaystyle\hphantom{{\sf D}^{3}+A_{\sf
U}{\sf D}+{\sf D}A_{\sf U}+C_{\sf U}{\sf D}^{-1}C_{\sf U}}{}=({\sf D}-A_{\sf
V})({\sf D}\pm C_{\sf V}){\sf D}^{-1}({\sf D}\mp C_{\sf V})({\sf D}+A_{\sf
V}).$

###### Proof.

The proof of the proposition is completely analogous to the proof of the
factorization of the non-Abelian KdV recursion operator on the image of a
Miura transformation in [11, Proposition 14]. The crucial observation is that
due to the product rule the identities

$\displaystyle{\sf D}A_{T}=A_{T}{{\sf D}}+A_{{\sf D}T},\qquad{\sf
D}C_{T}=C_{T}{{\sf D}}+C_{{\sf D}T}$

hold for any derivation ${\sf D}$. ∎

There are close algebraic relations between the non-Abelian recursion
operators of the amKdV, KdV sing. and int. soliton KdV equations.

######  Corollary 4.11.

For $\widetilde{V}=\frac{1}{2}\phi_{x}^{-1}\phi_{xx}$ and $S=\phi_{x}$, the
recursion operators in Theorem 3.1 and the recursion operators in (4.6), (4.7)
are pairwise related. More precisely, it holds

$\displaystyle\widetilde{\Psi}\big{(}\widetilde{V}\big{)}=P_{1}P_{2}=Q_{1}Q_{2},\qquad\chi(S)=Q_{2}Q_{1}=R_{1}R_{2},\qquad\Upsilon(\phi)=P_{2}P_{1}=R_{2}R_{1},$

where

$\displaystyle
P_{1}=\big{(}\mathbb{D}+C_{\widetilde{V}}\big{)}\big{(}\mathbb{D}+A_{\widetilde{V}}\big{)}L_{\phi_{x}}^{-1},\qquad$
$\displaystyle
P_{2}=L_{\phi_{x}}\mathbb{D}^{-1}\big{(}\mathbb{D}-A_{\widetilde{V}}\big{)}(\mathbb{D}-C_{\widetilde{V}})\mathbb{D}^{-1},$
$\displaystyle
Q_{1}=\big{(}\mathbb{D}+C_{\widetilde{V}}\big{)}L_{\phi_{x}}^{-1},\qquad$
$\displaystyle
Q_{2}=L_{\phi_{x}}\big{(}\mathbb{D}-A_{\widetilde{V}}\mathbb{D}^{-1}A_{\widetilde{V}}\big{)}\big{(}\mathbb{D}-C_{\widetilde{V}}\big{)}\mathbb{D}^{-1},$
$\displaystyle
R_{1}=L_{\phi_{x}}\big{(}\mathbb{D}+A_{\widetilde{V}}\big{)}L_{\phi_{x}}^{-1},\qquad$
$\displaystyle
R_{2}=L_{\phi_{x}}\mathbb{D}^{-1}\big{(}\mathbb{D}-A_{\widetilde{V}}\big{)}\big{(}\mathbb{D}-C_{\widetilde{V}}\mathbb{D}^{-1}C_{\widetilde{V}}\big{)}L_{\phi_{x}}^{-1}.$

##  5 Hereditariness and hierarchies

The links obtained in the last two sections are summarized in

$\displaystyle\boxed{\mbox{mKdV}(V)}\ {\buildrel
B_{2},B_{3}\over{\text{\textendash\textendash}}}\
\boxed{\mbox{amKdV}(\widetilde{V})}\ {\buildrel
B_{4}\over{\text{\textendash\textendash}}}\ \boxed{\mbox{KdV sing.}(\phi)}\
{\buildrel B_{5}\over{\text{\textendash\textendash}}}\ \boxed{\mbox{int.\
soliton KdV}(S)}\,,$

where the Bäcklund transformations $B_{2}$, $B_{3}$, $B_{4}$ and $B_{5}$ are
given in (3.2), (3.3), (4.4) and (4.5), respectively. Applying the respective
recursion operators, the Bäcklund chart extends to the whole hierarchies as
follows

$\displaystyle\boxed{V_{t}=[\Psi(V)]^{n}V_{x}}\buildrel
B_{2},B_{3}\over{\text{\textendash\textendash}}\boxed{\widetilde{V}_{t}=[\widetilde{\Psi}(\widetilde{V})]^{n}\widetilde{V}_{x}}\buildrel
B_{4}\over{\text{\textendash\textendash}}\boxed{\phi_{t}=[\Upsilon(\phi)]^{n}\phi_{x}}\buildrel
B_{5}\over{\text{\textendash\textendash}}\boxed{S_{t}=[\chi(S)]^{n}S_{x}}\,.$
(5.1)

As known from the scalar case, hereditariness is a crucial property of
recursion operators [20, 38]. Unfortunately its direct verification often
requires involved computations, in particular in the non-Abelian case. The
following proposition uses the Bäcklund links established in the present
article to avoid computations by reducing the proof of hereditariness to the
hereditariness of the non-Abelian KdV recursion operator, which is proved in
[54].

######  Proposition 5.1.

Each of the recursion operators $\Psi$ from the Bäcklund chart (4.1) has the
following properties:

  1. $a)$

$\Psi$ is hereditary,

  2. $b)$

$\Psi$ is a strong symmetry for all equations of the hierarchy generated by
$\Psi$.

###### Proof.

In [54, (32)] it is verified that the non-Abelian KdV recursion operator (2.4)
satisfies the identity

$\displaystyle[D,\Phi(U)]=\Phi^{\prime}(U)[U_{x}],$

implying that $\Phi(U)$ is a strong symmetry (recursion operator in the sense
of [43]) for the trivial member $U_{t}=U_{x}$ of the non-Abelian KdV hierarchy
[20]. Moreover, the main result in [54] is that $\Phi(U)$ is hereditary. Hence
$\Phi(U)$ is a strong symmetry for all equations of the non-Abelian KdV
hierarchy [20]. As shown in [26], the properties of a) and b) are preserved
under Bäcklund transformations. ∎

Each of the hierarchies in (5.1) is of the form

$\displaystyle U_{t}=[\Phi(U)]^{n}U_{x}.$ (5.2)

We may rewrite the right-hand side of (5.2) as

$\displaystyle[\Phi(U)]^{n}U_{x}=X_{n}(U),$

where $X_{n}$ is a vector field on the space of $x$-dependent operator-valued
functions (see [54] for details). The main consequence of Proposition 5.1 is

######  Corollary 5.2.

Let $X_{1},X_{2},\ldots$ be the vector fields of one of the hierarchies in
(5.2). Then we have

$\displaystyle[X_{m},X_{n}]=0$

for all $m,n=1,2,\ldots$.

The proof uses arguments explained in [54, Section VI].

##  6 Remarks, perspectives and open problems

This section collects some remarks on the results previously presented
together with some perspectives study and open problems. The chain of Bäcklund
transformations we obtained, represents a not at all trivial generalization to
the operator level of the corresponding one [23] which links the scalar pKdV,
KdV, mKdV, KdV interacting soliton and KdV singularity manifold hierarchies.
Furthermore, it generalizes the non-Abelian Bäcklund chart in [11] since it
connects further non-Abelian hierarchies. Specifically, new non-Abelian
equations, and, then, the corresponding hierarchies, arise, such as the amKdV,
in (3.1).

Remarks

  * •

The two Bäcklund charts, respectively, in the Abelian [23] (see Appendix) and
the non Abelian case, connect the pKdV, KdV, mKdV, KdV-singularity manifold
and Interacting Soliton KdV equations or their non-Abelian analogs.

  * •

Known the recursion operator of a nonlinear evolution equation then all the
other nonlinear evolution equations linked to it via a Bäcklund chart admit a
recursion operator. The latter can be constructed in the Abelian as well as in
the non-Abelian case.

  * •

All nonlinear evolution equations in the same Bäcklund chart share all the
structural properties which are preserved under Bäcklund transformations as
soon as a single equation, in it, enjoys them. Remarkable is the
hereditariness of the recursion operators [54].

  * •

Also in the non-Abelian operator case, given the hereditary recursion
operators, the Bäcklund chart can be extended to the corresponding generated
hierarchies. Again, the Bäcklund chart relates the corresponding members of
each one of the involved hierarchies of nonlinear evolution equations.

  * •

Even if there are similarities between the Abelian scalar case and the non-
Abelian operator case, in the second case the structure is richer. Indeed, two
distinct non-Abelian mKdV equations appear: the non-Abelian mKdV and an
alternative mKdV equations which do coincide in the Abelian case.
Correspondingly, when commutativity is assumed, combination of the Bäcklund
transformations $B_{2}$ and $B_{3}$ produces the identity transformation and,
hence, the Abelian Bäcklund chart [23] is recovered.

  * •

A similar behavior can be observed also when the Cole–Hopf link connecting
Burgers equation to linear heat equation is extended to the non-Abelian case
[12, 14]. The heat equation is connected to two different Burgers equations,
termed Burgers and mirror Burgers equations in [9, 35]. Then, recursion
operators and the corresponding hierarchies follow from the Cole–Hopf link,
which can be regarded as a special case of Bäcklund transformation.

Perspectives and open problems

  * •

We expect a similar situation when the 5th order nonlinear evolution equations
which appear in the Bäcklund chart in [8, 47] are extended to the non-Abelian
case. This study is currently under investigation and we are devising also
computer aided routines to check the algebraic properties of the recursion
operators. Indeed, already in the Abelian case, the computations involved are
very long and complicated.

  * •

Furthermore, if a nonlinear evolution equation admits a Hamiltonian and bi-
Hamiltonian structure, related to the recursion operator [21, 24, 27, 38],
then all nonlinear evolution equations in the same Bäcklund chart admit a
Hamiltonian and bi-Hamiltonian structure.

  * •

The approach can be extended also when nonlinear evolution equations in
$(2+1)$ dimensions are considered, namely, the unknown function is supposed to
depend on two space variables rather than on a single space variable. Thus, in
[40], the Kadomtsev–Petviashvili (KP), modified Kadomtsev–Petviashvili (mKP)
and $(2+1)$-dimensional Harry Dym equations, which represent, in turn, the
$(2+1)$-dimensional analog of KdV, mKdV, and Harry Dym equations are all
connected via Bäcklund transformations. Notably, the connection among their
$(1+1)$-dimensional corresponding equations [41] follows on imposing suitable
constraints to the $(2+1)$-dimensional ones.

##  Appendix A Appendix

In this Appendix some background definitions which are of use throughout the
whole article are briefly recalled in the opening subsection: this choice is
due to the lack of uniqueness in many definitions in the literature. In the
following subsection, some results strictly connected to the present
investigation are retrieved.

###  A.1 Background notions

Here, to improve readability, some fundamental definitions are provided. First
of all, we recall the notion of Bäcklund transformation, according to the
definition of Fokas and Fuchssteiner [19] (see also the book by Rogers and
Shadwick [49]). Consider non linear evolution equations of the type

$\displaystyle u_{t}=K(u),$ (A.1)

where the unknown function $u$ depends on the independent variables $x$, $t$
and, for fixed $t$, $u(x,t)\in M$, a manifold modeled on a linear topological
space so that the generic fiber $T_{u}M$, at $u\in M$, can be identified with
$M$ itself333It is generally assumed that $M$ is the space of functions
$u(x,t)$ which, for each fixed $t$, belong to the Schwartz space $S$ of
rapidly decreasing functions on ${\mathbb{R}}^{n}$, i.e.,
$S({\mathbb{R}}^{n}):=\\{f\in
C^{\infty}({\mathbb{R}}^{n})\colon\|f\|_{\alpha,\beta}<\infty,\forall\alpha,\beta\\}$,
where
$\|f\|_{\alpha,\beta}:=\sup\limits_{x\in{\mathbb{R}}^{n}}\left|x^{\alpha}D^{\beta}f(x)\right|$,
and $D^{\beta}:=\partial^{\beta}/{\partial x}^{\beta}$. In the present section
$n=1$., and $K\colon M\rightarrow TM$, is an appropriate $C^{\infty}$ vector
field on a manifold $M$ to its tangent manifold $TM$.

Let, now

$\displaystyle v_{t}=G(v)$ (A.2)

denote a second nonlinear evolution equation. Accordingly, in turn, when
$u(x,t)\in M_{1}$, $v(x,t)\in M_{2}$ and $M_{1}$, $M_{2}$ represent manifolds
modeled on a linear topological space then, $K\colon M_{1}\rightarrow TM_{1}$
and $G\colon M_{2}\rightarrow TM_{2}$ denote appropriate $C^{\infty}$-vector
fields on the manifolds $M_{i}$, $i=1,2$,

$\displaystyle u_{t}=K(u),\qquad$ $\displaystyle K\colon\ M_{1}\rightarrow
TM_{1},\qquad$ $\displaystyle u\colon\
(x,t)\in{\mathbb{R}}\times{\mathbb{R}}\to u(x,t)\in M_{1},$ $\displaystyle
v_{t}=G(v),\qquad$ $\displaystyle G\colon\ M_{2}\rightarrow TM_{2},\qquad$
$\displaystyle v\colon\ (x,t)\in{\mathbb{R}}\times{\mathbb{R}}\to v(x,t)\in
M_{2}.$

As usual, when soliton solutions are considered, the further assumption
$M:=M_{1}\equiv M_{2}$ is adopted. Then, the definition of Bäcklund
transformation, according to [19], can be stated as follows.

######  Definition A.1.

Given two evolution equations, $u_{t}=K(u)$ and $v_{t}=G(v)$, then $B(u,v)=0$
represents a Bäcklund transformation between them if, whenever two solutions
of these equations are given, let denote them $u(x,t)$ and $v(x,t)$
respectively, such that $B(u,v)=0$ at the initial time $t=0$, then this holds
true for all times, namely,

$\displaystyle B(u(x,t),v(x,t))|_{t=0}=0\quad\Longrightarrow\quad
B(u(x,t),v(x,t))|_{t=\bar{t}}=0,\qquad\forall\,\bar{t}>0,\quad\forall\,x\in{\mathbb{R}}.$

Therefore, the Bäcklund transformation $B$ establishes a correspondence
between solutions of the evolution equations it connects. This link is
graphically represented as

$\displaystyle\boxed{u_{t}=K(u)}\ {\buildrel
B\over{\text{\textendash\textendash}}}\ \boxed{v_{t}=G(v)}$

and, then, if the nonlinear evolution equation $u_{t}=K(u)$ admits a
hereditary recursion operator $\Phi(u)$ [19], namely,

$\displaystyle u_{t}=\Phi(u)u_{x},\qquad\text{where}\quad K(u)=\Phi(u)u_{x},$

also the equation $v_{t}=G(v)$ admits a hereditary recursion operator, say
$\Psi(v)$. The Bäcklund transformation $B$ allows to find $\Psi(v)$ via

$\displaystyle\Psi(v)=\Pi\Phi(u)\Pi^{-1},\qquad\text{where}\quad\Pi:=-B_{v}^{-1}B_{u},\qquad\Pi\colon\
TM_{1}\rightarrow TM_{2},$ (A.3)

where $B_{u}$ and $B_{v}$ denote the Frechet derivatives of the Bäcklund
transformation $B(u,v)$. The Bäcklund transformation $B$ [19], via relation
(A.3), guarantees that $\Psi(v)$ represents the hereditary recursion operator
admitted by the equation $v_{t}=G(v)$. Hence, the two hierarchies
$u_{t}=[\Phi(u)]^{n}u_{x}$ and $v_{t}=[\Psi(v)]^{n}v_{x}$, $n\geq 0$, of
nonlinear evolution equations can be constructed [20] and their base members
equations are, in turn, (A.1) and (A.2). Each equation in these two
hierarchies, parametrized444Here and throught
${\mathbb{N}}_{0}:={\mathbb{N}}\cup\\{0\\}$. by $n\in{\mathbb{N}}_{0}$, is
connected, via the same Bäcklund transformation $B$, to the corresponding
equation, parametrized by the same $n$, in the other hierarchy. This extension
to the whole hierarchies is depicted in the following Bäcklund chart

$\displaystyle\boxed{u=[\Phi(u)]^{n}u_{x}}\ {\buildrel
B\over{\text{\textendash\textendash}}}\ \boxed{v_{t}=[\Psi(v)]^{n}v_{x}}\,,$

which emphasizes that the link between the two equations (A.1) and (A.2) is
inherited555These recursion operators are termed hereditary in [20] to stress
that the property to be recursion operator, established referring to the base
member equation, is inherited by all the other members in the hierarchy of
nonlinear evolution equations. by each member of the two hierarchies
generated, respectively, by the recursion operators $\Phi$ and $\Psi$.

###  A.2 Abelian KdV connected Bäcklund chart

Here some results, obtained in [23], concerning the links among nonlinear
evolution equations whose base member is a 3-rd order equation, are recalled.
An overview on further previous result is given in [7]. Thus, the Bäcklund
chart here below, (A.4), allows to find solutions of Harry Dym equation [28].
This method to find solutions is not restricted to third order nonlinear
evolution equations; indeed, for instance, also solutions of Burgers equation
are obtained via Bäcklund transformations (see [17, 30, 33, 36]) to mention a
few of them.

The following Bäcklund chart summarizes links relating 3rd order nonlinear
evolution equations, when the links in [23] are combined with the link between
the KdV and potential KdV (pKdV),

$\boxed{\\!\text{pKdV}(q)\\!}\,{\buildrel(a)\over{\text{\textendash\textendash}}}\,\boxed{\\!\text{KdV}(u)\\!}\,{\buildrel(b)\over{\text{\textendash\textendash}}}\,\boxed{\\!\text{mKdV}(v)\\!}\,{\buildrel(c)\over{\text{\textendash\textendash}}}\,\boxed{\\!\text{KdV~{}sing.}(\varphi)\\!}{\buildrel(d)\over{\text{\textendash\textendash}}}\,\boxed{\\!\text{int.
soliton
KdV}(s)\\!}\,{\buildrel(e)\over{\text{\textendash\textendash}}}\,\boxed{\\!\text{Dym}(\rho)\\!}\,,$
(A.4)

wherein the third order nonlinear evolution equations are, in turn, given by

$\displaystyle q_{t}=q_{xxx}+3q^{2}_{x}\qquad$ $\displaystyle\text{(pKdV)},$
$\displaystyle u_{t}=u_{xxx}+6uu_{x}\qquad$ $\displaystyle\text{(KdV)},$
$\displaystyle v_{t}=v_{xxx}-6v^{2}v_{x}\qquad$ $\displaystyle\text{(mKdV)},$
$\displaystyle\varphi_{t}=\varphi_{x}\\{\varphi;x\\},\quad\text{where}\ \
\\{\varphi;x\\}:=\left({\varphi_{xx}\over\varphi_{x}}\right)_{x}-{1\over
2}\left({\varphi_{xx}\over\varphi_{x}}\right)^{2}\qquad$
$\displaystyle\text{(KdV~{}sing.)},$ $\displaystyle
s^{2}s_{t}=s^{2}s_{xxx}-3ss_{x}s_{xx}+{3\over 2}{s_{x}}^{3}\qquad$
$\displaystyle\text{(int.\ soliton~{}KdV)},$
$\displaystyle\rho_{t}=\rho^{3}\rho_{\xi\xi\xi}\qquad$
$\displaystyle\text{(Dym)},$

in addition, the Bäcklund transformations which link them, are, respectively:

$\displaystyle(a)\ \ u-q_{x}=0,\qquad(b)\ \ u+v_{x}+v^{2}=0,\qquad(c)\ \
v-{{1\over 2}{\varphi_{xx}\over\varphi_{x}}}=0,$ $\displaystyle(d)\ \
s-\varphi_{x}=0,\qquad(e)\ \ {\bar{x}}:=D^{-1}s(x)\qquad\text{where}\quad
D^{-1}:=\int_{-\infty}^{x}d\xi.$

The transformation $(e)$ is a reciprocal666Reciprocal-type transformations
were originally introduced in $(2+0)$-dimensional isentropic gasdynamics by
Bateman [3, 4] and in $(1+1)$-dimensional anisentropic gasdynamics by Rogers
[45]. Connection with invariance of these gasdynamic systems under Bäcklund
transformations is described in [49]. transformation [4, 49]: according to [8,
23], it represents a Bäcklund transformation whose transformation operator
$\Pi$ can be constructed, provided it is considered an extended manifold which
comprises also the independent space variable $x$, further to the dependent
variable $u$. That is, while the Bäcklund transformations $(a)$, $(b)$ and
$(c)$ admit operators $\Pi$

$\displaystyle\Pi_{a}\colon\ TM_{u}\rightarrow TM_{v},\qquad\Pi_{b}\colon\
TM_{v}\rightarrow TM_{\varphi},\qquad\Pi_{c}\colon\ TM_{\varphi}\rightarrow
TM_{s},$

when the reciprocal transformation $(e)$ is considered, the corresponding
operator is

$\displaystyle\Pi_{e}\colon\ TM_{(x,s)}\rightarrow TM_{(\bar{x},\rho)}.$

The Bäcklund chart (A.4) allows to find solutions to initial boundary value
problems of the Dym equation [23, 28]. Furthermore, the links depicted in the
Bäcklund chart relate each member of the involved hierarchies to the
corresponding member of all the other hierarchies in the Bäcklund chart.
Composition of Bäcklund transformations in (A.4) allows to retrieve and
generalize to the whole hierarchies, the direct Bäcklund transformation
between the KdV and Dym equations in [31].

The Bäcklund chart (A.4) turned out to be very helpful in revealing new
invariances enjoyed by any of the equations which appear in the Bäcklund chart
itself. Indeed, let

$\displaystyle\\{\varphi;x\\}=\left({\varphi_{xx}\over\varphi_{x}}\right)_{x}-{1\over
2}\left({\varphi_{xx}\over\varphi_{x}}\right)^{2}$

denote the Schwarzian derivative, then, the KdV Singularity equation
$\varphi_{t}=\varphi_{x}\\{\varphi;x\\}$ is invariant under the Möbius group
of transformations

$\displaystyle\widetilde{\varphi}={{a\varphi+b}\over{c\varphi+d}},\qquad
a,b,c,d\in\mathbb{C}\qquad\text{such that}\quad ad-bc\neq 0,$

since the Schwarzian derivative of $\varphi$ with respect to $x$,
$\\{\varphi;x\\}$, enjoys such a property.

The invariance, under the Möbius group of transformations of all the members
of the KdV Singularity hierarchy allowed to recover the Invariance $I$,
enjoyed by the Dym hierarchy [23] and to find a new invariance enjoyed by the
Kawamoto equation [47] in the case of the 5th order Bäcklund chart.

### Acknowledgements

The financial support of G.N.F.M.-I.N.d.A.M., I.N.F.N. and Sapienza University
of Rome, Italy are gratefully acknowledged. C. Schiebold wishes also to thank
S.B.A.I. Dept. and Sapienza University of Rome for the kind hospitality. The
authors wish to thank the referees who carefully read this work: their
comments were of help in improving its presentation.

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