Modified AKNS model, Riccati-type pseudo-potential approach and infinite
towers of quasi-conservation laws

H. Blas(a), M. Cerna Maguiña(b) and L.F. dos Santos(c)

(a)Instituto de Física

Universidade Federal de Mato Grosso

Av. Fernando Correa, $N^{0}$ 2367

Bairro Boa Esperança, Cep 78060-900, Cuiabá - MT - Brazil.

(b) Departamento de Matemática

Universidad Nacional Santiago Antúnez de Mayolo

Campus Shancayán, Av. Centenario 200, Huaraz - Perú

(c) Centro Federado de Educação Tecnologica-CEFET-RJ

Campus Angra dos Reis, Rua do Areal, 522, Angra dos Reis- RJ -Brazil

A dual Riccati-type pseudo-potential formulation is introduced for a modified
AKNS system (MAKNS) and infinite towers of novel anomalous conservation laws
are uncovered. In addition, infinite towers of exact non-local conservation
laws are uncovered in a linear formulation of the system. It is shown that
certain modifications of the non-linear Schrödinger model (MNLS) can be
obtained through a reduction process starting from the MAKNS model. So, the
novel infinite sets of quasi-conservation laws and related anomalous charges
are constructed by an unified and rigorous approach based on the Riccati-type
pseudo-potential method, for the standard NLS and modified MNLS cases,
respectively. The non-local properties, the complete list of towers of
infinite number of anomalous charges and the (non-local) exact conservation
laws of the quasi-integrable systems, such as the deformed Bullough-Dodd,
Toda, KdV and SUSY sine-Gordon systems can be studied in the framework
presented in this paper. Our results may find many applications since the
AKNS-type system arises in several branches of non-linear physics, such as
Bose-Einstein condensation, superconductivity and soliton turbulence.

##  1 Introduction

Certain non-linear field theory models with relevant physical applications and
modeling solitary waves are not integrable. Recently, the concept of quasi-
integrability for some deformations of integrable models has been put forward.
In that context, some properties of the deformations of the soliton models,
such as SG, NLS, Toda, KdV, Boullogh-Dodd and SUSY-SG have been examined in
the frameworks of the anomalous zero-curvature formulations [1, 2, 3, 4, 5, 6,
7, 8, 9, 10] and the deformations of the Riccati-type pseudo-potential
approach [11, 12].

The quasi-integrable models set forward in the literature [1, 2, 4, 8, 9, 3,
7, 12] possess important structures, such as infinite sets of non-local
conserved charges, and some types of linear formulations. In this context, the
Riccati-type representations have recently been presented for the deformed KdV
and sine-Gordon models [12, 11]. Moreover, in the context of the Riccati-type
pseudo-potential approach to quasi-integrability, it has been shown that the
deformed SG and KdV models [11, 12] can be formulated as the compatibility
condition of certain linear systems of equations and that they possess
infinite towers of exact non-local conservation laws.

In this paper we will tackle the problem of extending the Riccati-type pseudo-
potential formalism, which has been used for a variety of well known
integrable systems, to deformed AKNS models. The new properties mentioned
above have been examined for the deformations of the relativistic invariant
sine-Gordon model with topological solitons and the non-relativistic KdV model
with non-topological and unidirectional solitons, respectively.
Conventionally, both of them are defined for real scalar fields. So, it would
be desirable to examine those properties for NLS-type models defined for a
complex field with envelope solitons. Some of the mentioned properties have
recently been examined for a deformed NLS model in [13] by direct construction
of novel quasi-conservation laws starting from the eqs. of motion.

In general, those integrable models can be formulated in the framework of the
AKNS system, from which they can be obtained through relevant reduction
processes. The NLS-type model stands in the same level of importance as the
KdV-type and SG-type models in their potential applications, since they are
ubiquitous in all areas of nonlinear physics, such as Bose-Einsten
condensation and superconductivity [14, 15, 16], soliton gas and soliton
turbulence in fluid dynamics [17, 18, 19, 20], the Alice-Bob physics [21], and
the understanding of a kind of triality among the gauge theories, integrable
models and gravity theories (see [22] and references therein).

We perform a particular deformation of the Riccati-type pseudo-potential
approach related to the AKNS system [23], from which the NLS-like modified
model is obtained through a particular reduction process. A modified AKNS
system (MAKNS) is defined by introducing a deformed potential $V$ and some
auxiliary fields into the pair of Riccati-type system of equations and another
system of equations for the set of auxiliary fields, such that the
compatibility condition applied to the extended system gives rise to the
modified AKNS model equations of motions. Then, it is constructed a set of
infinite number of quasi-conservation laws order by order in powers of the
spectral parameter.

Remarkably, a new dual system of the Riccati-type pseudo-potential approach is
introduced which provides infinite sets of novel anomalous conservation laws.
Those quasi-conservation laws encompass, as a subset, the ones obtained by a
direct constructive method for the MNLS model in the companion paper [13]. So,
in this paper it will be shown unified and rigorous constructions of the novel
anomalous charges for the standard NLS and modified MNLS cases, respectively.

In addition, in the framework of the pseudo-potential approach, it is proposed
a linear system of equations whose compatibility condition gives rise to the
MAKNS equations of motion. As an application of the linear system formulation
of the modified AKNS model, it is obtained a pair of infinite towers of exact
non-local conservation laws. A particular reduction $MAKNS\rightarrow MNLS$
allows one to reproduce the relevant quantities of the MNLS model out of the
ones constructed for the MAKNS system.

This paper is organized as follows: The section 2 considers a particular
deformation of the $sl(2)$ AKNS model in the context of the Riccati-type
pseudo-potential approach, and it discusses a particular reduction to the
modified NLS model. An infinite set of quasi-conservation laws are
constructed. In sec. 3 a dual Riccati-type formulation and novel anomalous
charges are presented. In sec. 4 it is found a linear system formulation of
the deformed AKNS model and constructed an infinite set of non-local
conservation laws. In sec. 5 we present our conclusions and discussions. The
appendices A , B and C present the components of the expansions in power of
$\zeta$ of the Riccati-type pseudo-potentials.

##  2 Riccati-type pseudo-potential and modified $sl(2)$ AKNS model

The standard NLS model can be obtained as a special reduction of the AKNS
system; so, in the next sections we consider a convenient deformation of the
usual pseudo-potential approach to the AKNS integrable field theory.
Subsequently, we will discuss its reduction process leading to the modified
NLS model. In [23] it has been generated the both Lax equations and Bäcklund
transformations for well-known non-linear evolution equations using the
concept of pseudo-potentials and the related properties of the Riccati
equation. These applications have been done in the context of a variety of
integrable systems (sine-Gordon, KdV, NLS, etc), and allow the Lax pair
formulation, the construction of conservation laws and the Bäcklund
transformations for them [23, 24].

So, let us consider the system of Riccati-type equations

$\displaystyle\partial_{x}u$ $\displaystyle=$
$\displaystyle-2i\zeta\,u+q+\,\bar{q}\,\,u^{2},$ (2.1)
$\displaystyle\partial_{t}u$ $\displaystyle=$ $\displaystyle
2A\,u-C\,u^{2}+B+r-u\,s,$ (2.2)

where $u$ is the Riccati-type pseudo-potential, $r$ and $s$ are auxiliary
fields, and $q$ and $\bar{q}$ are the fields of the model. Let us assume

$\displaystyle A$ $\displaystyle\equiv$
$\displaystyle-2i\zeta^{2}-\frac{1}{2}i\,V^{(1)},$ (2.3) $\displaystyle B$
$\displaystyle\equiv$ $\displaystyle 2\zeta q+i\partial_{x}q,$ (2.4)
$\displaystyle C$ $\displaystyle\equiv$
$\displaystyle-2\zeta\bar{q}+i\partial_{x}\bar{q},\,\,\,\,\,V^{(1)}\equiv\frac{dV[\rho]}{d\rho},\,\,\,\rho\equiv\bar{q}q,$
(2.5)

where $V(\bar{q}q)$ is the potential of the modified AKNS equation (MAKNS) and
$\zeta$ is the spectral parameter. In fact, the form of (2.1) is similar to
the first Riccati equation for the standard AKNS model. The functions $B$ and
$C$ are the same as in the usual second Riccati equation; whereas, the funtion
$A$ has been modified to contain a generalized potential as compared to the
standard AKNS model [24].

We consider the following equations for the auxiliary fields $r(x,t)$ and
$s(x,t)$

$\displaystyle\partial_{x}r$ $\displaystyle=$ $\displaystyle
q\,s+(-2i\zeta+Q\,u)\,r,$ (2.6) $\displaystyle\partial_{x}s$ $\displaystyle=$
$\displaystyle Q\,r-2\,\bar{q}\,r+u\bar{q}\,s+2iX,$ (2.7) $\displaystyle X$
$\displaystyle\equiv$
$\displaystyle-\partial_{x}\left(\frac{1}{2}V^{(1)}+\bar{q}q\right),$ (2.8)

where $Q$ is an arbitrary field. So, one has a set of two deformed Riccati-
type equations for the pseudo-potential $u$ (2.1)-(2.2) and a system of
equations (2.6)-(2.7) for the auxiliary fields $r$ and $s$.

Notice that, for the integrable AKNS model one has the potential

$\displaystyle
V_{NLS}(\bar{q}q)=-\left(\bar{q}q\right)^{2}\,\rightarrow\,V_{NLS}^{(1)}(\bar{q}q)=-2(\bar{q}q),$
(2.9)

and, therefore, $X=0$ in (2.8), and so the auxiliary system of eqs.
(2.6)-(2.7) possesses the trivial solution $r=s=0$. Inserting this trivial
solution into the system (2.1)-(2.2) and considering the potential (2.9), one
has a set of two Riccati equations for the standard AKNS model and they play
an important role in order to study its properties, such as the derivation of
the infinite number of conserved charges and the Bäcklund transformations,
relating the fields $(q,\bar{q})$ with another set of solutions
$(q^{\prime},\bar{q}^{\prime})$ [24].

Note that only the $t-$component $\partial_{t}u$ of the Riccati equation
associated to the ordinary AKNS equation has been deformed away from the AKNS
potential (2.9), and it carries all the information regarding the deformation
of the model which are encoded in the potential $V(\bar{q}q)$ and the
auxiliary fields $r(x,t)$ and $s(x,t)$. The form of the $x-$component
$\partial_{x}u$ remains the same as the usual Riccati equation associated to
the AKNS model.

We have computed the compatibility condition
$[\partial_{t}\partial_{x}u-\partial_{x}\partial_{t}u]=0$ for the Riccati-type
equations (2.1)-(2.2), taking into account the auxiliary system of equations
(2.6)-(2.7) and then derived the eqs. of motion for the fields $q$ and
$\bar{q}$

$\displaystyle i\partial_{t}q+\partial^{2}_{x}q-V^{(1)}q$ $\displaystyle=$
$\displaystyle 0,$ (2.10)
$\displaystyle-i\partial_{t}\bar{q}+\partial^{2}_{x}\bar{q}-V^{(1)}\bar{q}$
$\displaystyle=$ $\displaystyle 0.$ (2.11)

This is a modified $sl(2)$ AKNS system (MAKNS) for arbitrary potential of type
$V(\bar{q}q)$. An important observation in the constructions above is that
$\frac{\partial}{\partial t}\zeta=0$, as it can be checked by direct
computation using the system (2.1)-(2.2) and (2.6)-(2.7), provided that the
system of eqs. (2.10)-(2.11) is satisfied. So, the modified system MAKNS
possesses an isospectral parameter $\zeta$.

Next, providing that the identifications

$\displaystyle q\equiv
i(-\eta)^{1/2}\,\psi,\,\,\,\bar{q}\equiv-i(-\eta)^{1/2}\,\bar{\psi},\,\,\,\,\eta\in\leavevmode\hbox{\rm
I\kern-1.79993ptR}$ (2.12)

are performed in the system (2.10)-(2.11), where $\bar{\psi}$ stands for
complex conjugation of the field $\psi$, one can obtain the modified NLS model

$\displaystyle i\frac{\partial}{\partial
t}\psi(x,t)+\frac{\partial^{2}}{\partial
x^{2}}\psi(x,t)-V^{(1)}(|\psi(x,t)|^{2})\psi(x,t)$ $\displaystyle=$
$\displaystyle 0,$ (2.13) $\displaystyle V^{(n)}(I)$ $\displaystyle\equiv$
$\displaystyle\frac{d^{n}}{dI^{n}}V(I),\,\,\,\,I\equiv\bar{\psi}\psi,$ (2.14)

where $\psi$ is a complex scalar field. If the potential and its derivatives
are taken as

$\displaystyle V(I)=-\eta I^{2},\,\,\,V^{(1)}(I)=-2\eta
I\,\,\,\mbox{and}\,\,\,\,V^{(2)}(I)=-2\eta.$ (2.15)

the reduction (2.12) defines the standard NLS model. This is a process, we
have just mentioned above, through which the standard NLS model is obtained as
a special reduction of the AKNS system.

Let us emphasize that for the standard NLS model we have the trivial solution
of the system (2.6)-(2.7), i.e. $X=0\rightarrow r=s=0$, and the existence of
the Lax pair of de ordinary NLS model reflects in its equivalent Riccati-type
representation, provided by the system (2.1)-(2.2) with the well known
potential (2.15) [23, 24].

Next, let us consider a special space-time symmetry related to soliton-type
solutions of the model. So, consider a reflection around a fixed point
$(x_{\Delta},t_{\Delta})$

$\displaystyle\widetilde{{\cal
P}}:(\widetilde{x},\widetilde{t})\rightarrow(-\widetilde{x},-\widetilde{t});\,\,\,\,\,\,\,\,\widetilde{x}=x-x_{\Delta},\,\,\widetilde{t}=t-t_{\Delta}.$
(2.16)

The transformation $\widetilde{{\cal P}}$ defines a shifted parity ${\cal
P}_{s}$ for the spatial variable $x$ and a delayed time reversal ${\cal
T}_{d}$ for the time variable $t$. Notice that, when $x_{\Delta}=0$
($t_{\Delta}=0$), ${\cal P}_{s}$ (${\cal T}_{d}$) is reduced to the pure
parity ${\cal P}$ (pure time reversal ${\cal T}$).

We define the quasi-integrable MAKNS model for field configurations $q$ and
$\bar{q}$ satisfying (2.10)-(2.11) such that the fields and the deformed
potential transform under the space-time transformation (2.16) as

$\displaystyle\widetilde{{\cal P}}(q)=\bar{q},\,\,\,\,\,\,\widetilde{{\cal
P}}(\bar{q})=q,\,\,\,\,\,\mbox{and }\,\,\,\,\,\widetilde{{\cal
P}}[V(\rho)]=V(\rho),\,\,\,\rho\equiv\bar{q}q.$ (2.17)

Under this transformation one has that $X$ from (2.8) becomes an odd function

$\displaystyle\widetilde{{\cal P}}(X)=-X.$ (2.18)

Next, let us discuss the relevant conservation laws in the context of the
Riccati-type system (2.1)-(2.2) and the auxiliary equations (2.6)-(2.7). So,
substituting the expression for $u^{2}$ from (2.1) into (2.2) and considering
(2.6)-(2.7), one gets the following relationship

$\displaystyle\partial_{t}[i\bar{q}\,u]-\partial_{x}\Big{[}2i\zeta\bar{q}\,u-\bar{q}q+u\,\partial_{x}\bar{q}\Big{]}$
$\displaystyle=$ $\displaystyle i\bar{q}(r-s\,u).$ (2.19)

Defining the r.h.s. of (2.19) as

$\displaystyle\chi\equiv i\bar{q}(r-s\,u),$ (2.20)

and using the system (2.1)-(2.2) and the auxiliary equations (2.6)-(2.7) one
can write a first order differential equation for the auxiliary field $\chi$

$\displaystyle\partial_{x}\chi$ $\displaystyle=$
$\displaystyle\left(-2i\zeta+2u\bar{q}+\frac{\partial_{x}\bar{q}}{\bar{q}}\right)\chi+2\bar{q}\,uX.$
(2.21)

The eqs. (2.19) and (2.21) will be used below in order to uncover an infinite
tower of quasi-conservation laws associated to the modified AKNS model
(2.10)-(2.11). We will construct the relevant charges order by order in powers
of the parameter $\zeta$. So, let us consider the expansions

$\displaystyle
u=\sum_{n=1}^{\infty}u_{n}\,\zeta^{-n},\,\,\,\,\chi=\sum_{n=1}^{\infty}\chi_{n}\zeta^{-n-1}.$
(2.22)

The coefficients $u_{n}$ of the expansion above can be determined order by
order in powers of $\zeta$ from the Riccati equation (2.1). In appendix A we
provide the recursion relation for the $u_{n}\,^{\prime}s$ and the expressions
for the first $u_{n}$. Likewise, using the results for the $u_{n}\,^{\prime}s$
we get the relevant expressions for the $\chi_{n}\,^{\prime}s$ from (2.21).
The first components $\chi_{n}$ are provided in appendix B.

Then, making use of the $u_{n}$ and $\chi_{n}$ components of the expansions of
$u$ and $\chi$, respectively, provided in (2.22), one can find the
conservation laws, order by order in powers of $\zeta$. So, by inserting those
expansions into the eq. (2.19) one has that the coefficient of the
$n^{\prime}$th order term becomes

$\displaystyle\partial_{t}a_{x}^{(n)}$ $\displaystyle+$
$\displaystyle\partial_{x}a_{t}^{(n)}=\chi_{n-1},\,\,\,\,\,n=0,1,2,3,....;\,\chi_{0}\equiv
0$ (2.23) $\displaystyle a_{x}^{(n)}$ $\displaystyle\equiv$ $\displaystyle
i\bar{q}\,u_{n},\,\,\,\,a_{t}^{(n)}\equiv-\left(2i\bar{q}\,u_{n+1}-\bar{q}q\,\delta_{0,n}+\partial_{x}\bar{q}\,u_{n}\right),\,\,\,\,\,u_{0}\equiv
0.$ (2.24)

Notice that making the substitution $\chi_{n-1}\equiv 0$ into the eq. (2.23)
one can get the tower of exact conservation laws of the usual AKNS system. A
truly conservation law character of this equation, at each order $n$, remains
to be clarified, since the field components $\chi_{n-1}$ in the r.h.s. of
(2.23), as they can be seen in the appendix B, do not present the adequate
forms to be directly incorporated into the l.h.s. of the conservation laws. We
will tackle this construction order by order for each $\chi_{n-1}$ component.
Notice that analogous quasi-conservation laws have been obtained in the
context of the anomalous zero-curvature formulation of the modified NLS model
and its associated anomalous Lax pair in [4].

We will show below that the r.h.s. of (2.23) for $\chi_{1},\chi_{2}$ and
$\chi_{3}$ can be written in the form
$\chi_{j}\equiv\partial_{x}\chi^{x}_{j}+\partial_{t}\chi^{t}_{j}$, with
$\chi^{x}_{j}$ and $\chi^{t}_{j}$ being certain local functions of
$\\{\bar{q},q,V\\}$ and their $x$ and $t-$derivatives; i.e. there exist local
expressions for some $\chi_{j}\,(j=1,2,3)$, such that the eq. (2.23) provides
a proper local conservation law.

Let us compute the charges order by order in $n$ using the eq. (2.23) and the
relevant expressions presented in the appendices A and B.

The zero’th order provides a trivial identity.

The order $n=1$ and the field normalization

In this case the anomaly is trivial $\chi_{0}=0$. So, one has

$\displaystyle\partial_{t}\left(\frac{1}{2}\bar{q}q\right)-\partial_{x}\left(\frac{1}{2}i\bar{q}\partial_{x}q-\frac{1}{2}iq\partial_{x}\bar{q}\right)=0.$
(2.25)

It provides the conserved charge

$\displaystyle N=\int dx\,\bar{q}q$ (2.26)

The order $n=2$ and momentum conservation

At this order one has

$\displaystyle\partial_{t}\left(\frac{1}{4}i\bar{q}\partial_{x}q\right)+\partial_{x}\left(\frac{1}{4}[(\bar{q}q)^{2}+\bar{q}\partial_{x}^{2}q-\partial_{x}\bar{q}\partial_{x}q]\right)$
$\displaystyle=$ $\displaystyle\chi_{1}.$ (2.27)

The function $\chi_{1}$ can be rewritten as

$\displaystyle\chi_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\partial_{x}[F(\rho)],\,\,\,\,\rho\equiv\bar{q}q.$
(2.28) $\displaystyle F(I)$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}\rho\frac{d}{d\rho}V(\rho)-\frac{1}{2}V(\rho)+\frac{1}{2}\rho^{2}.$
(2.29)

So, from (2.27), taking into account (2.28), one can write the conserved
charge

$\displaystyle P=i\int
dx\,\left(\bar{q}\partial_{x}q-q\partial_{x}\bar{q}\right)$ (2.30)

The order $n=3$ and energy conservation

One has the conservation law

$\displaystyle\partial_{t}[-\frac{1}{8}(\bar{q}q)^{2}-\frac{1}{8}\bar{q}\partial^{2}_{x}q]-\partial_{x}\left(2i\bar{q}u_{4}+\partial_{x}\bar{q}u_{3}\right)$
$\displaystyle=$ $\displaystyle\chi_{2}.$ (2.31)

The function $\chi_{2}$ can be rewritten as

$\displaystyle\chi_{2}\equiv-\frac{1}{8}\partial_{t}V-\frac{1}{8}\partial_{t}(\bar{q}q)^{2}-\frac{1}{4}i\partial_{x}\Big{[}X\bar{q}q-X(q\partial_{x}\bar{q}-\bar{q}\partial_{x}q)\Big{]}.$
(2.32)

So, (2.31) provides the conserved charge

$\displaystyle H_{MNLS}=\int
dx\,[\,\partial_{x}\bar{q}\partial_{x}q+V(\bar{q}q)\,].$ (2.33)

Notice that in order to get the identity (2.32) we have used the eqs. of
motion (2.10)-(2.11). Since we have considered $\chi_{2}\neq 0$ in the r.h.s.
of (2.31), which carries the effect of the modified potential, the expression
of the energy (2.33) is valid for the general MNLS model. In particular, for
the ordinary AKNS the energy follows directly from the l.h.s. of (2.31)
(provided that $\chi_{2}=0$ in the r.h.s. of that eq.), i.e. $H_{NLS}=\int
dx\,[\,\partial_{x}\bar{q}\partial_{x}q+V_{NLS}(\bar{q}q)\,]$, where
$V_{NLS}=-(\bar{q}q)^{2}$ as in (2.9).

The order $n=4$: A first trivial charge and its associated anomalous charge

One has

$\displaystyle\partial_{t}\left(-\frac{3}{16}i\bar{q}q\bar{q}\partial_{x}q-\frac{1}{32}i\partial_{x}(\bar{q}q)^{2}-\frac{1}{16}i\bar{q}\partial^{3}_{x}q\right)-\partial_{x}[2i\bar{q}\,u_{5}+\partial_{x}\bar{q}\,u_{4}]$
$\displaystyle=$ $\displaystyle\chi_{3}.$ (2.34)

Remarkably, the expression for $\chi_{3}$ can be written as

$\displaystyle\chi_{3}$ $\displaystyle\equiv$
$\displaystyle\partial_{x}[\chi^{(3)}_{x}]+\partial_{t}[\chi^{(3)}_{t}],$
(2.35) $\displaystyle\chi^{(3)}_{t}$ $\displaystyle=$
$\displaystyle-\frac{3}{16}i\bar{q}q\bar{q}\partial_{x}q-\frac{1}{16}i\bar{q}\partial^{3}_{x}q,$
(2.36) $\displaystyle\chi^{(3)}_{x}$ $\displaystyle=$
$\displaystyle\frac{3}{8}X\bar{q}\partial_{x}q+\frac{3}{8}H_{1}(\rho)+\frac{1}{8}\bar{q}q\partial_{x}X-\frac{1}{8}X\partial_{x}(\bar{q}q)-\frac{1}{16}[\partial_{x}(\bar{q}q)]^{2}+\frac{3}{8}\bar{q}q\partial_{x}\bar{q}\partial_{x}q-\frac{3}{8}H_{2}(\rho)+$
(2.37)
$\displaystyle\frac{3}{32}i\bar{q}\partial_{t}q^{2}-\frac{1}{16}V^{(1)}\left(q\partial^{2}_{x}\bar{q}+\bar{q}\partial^{2}_{x}q\right)+\frac{1}{16}V^{(1)}\partial_{x}q\partial_{x}\bar{q}+$
$\displaystyle\frac{1}{16}i[\partial_{t}q\partial^{2}_{x}\bar{q}-\partial_{x}\partial_{t}q\partial_{x}\bar{q}+\bar{q}\partial_{t}\partial^{2}_{x}q].$
$\displaystyle\frac{d}{d\rho}H_{1}(\rho)$ $\displaystyle\equiv$
$\displaystyle-(V^{(2)}/2+1)\rho^{2},\,\,\,\,\,\frac{d}{d\rho}H_{2}(\rho)\equiv\rho
V^{(1)},\,\,\,\,\,\rho\equiv\bar{q}q.$ (2.38)

Therefore, at this order, the eq. (2.34) can be written as an exact
conservation law. However, taking into account the term
$\partial_{t}\chi^{(3)}_{t}$ of $\chi_{3}$ in (2.35)-(2.36) and the relevant
terms in the l.h.s. of (2.34) with partial $t-$derivatives one gets a fourth
order trivial charge $Q^{(4)}=0$, provided that the surface term
$\sim\partial_{x}(\bar{q}q)^{2}$ is dropped, since upon integration in $x$ in
order to define the charge it vanishes for suitable boundary conditions. So,
at this order of the above formulation, the charge $Q^{(4)}$ trivially
vanishes.

However, at this order and in the higher order ones, one can define an
asymptotically conserved charge for the MAKNS model

$\displaystyle Q^{(4)}_{a}=\frac{i}{2}\int
dx\,\Big{[}3\bar{q}q\left(\bar{q}\partial_{x}q-q\partial_{x}\bar{q}\right)+\bar{q}\partial^{3}_{x}q-q\partial^{3}_{x}\bar{q}\Big{]},$
(2.39)

such that

$\displaystyle\frac{d}{dt}Q^{(4)}_{a}$ $\displaystyle=$
$\displaystyle\hat{\tau}$ (2.40) $\displaystyle\hat{\tau}$ $\displaystyle=$
$\displaystyle-8\int\,dx\,\chi_{3},$ $\displaystyle=$
$\displaystyle-8\int\,dx\,\Big{\\{}\frac{1}{8}\Big{[}3\partial_{x}(\bar{q}\partial_{x}qX)-3\partial_{x}(\bar{q}\partial_{x}q)X+3\bar{q}(\bar{q}q^{2}+\partial^{2}_{x}q)X+\bar{q}q\partial^{2}_{x}X\Big{]}\Big{\\}}$
$\displaystyle=$
$\displaystyle-\int\,dx\,[3(\bar{q}q)^{2}X+\bar{q}q\partial^{2}_{x}X-3\partial_{x}\bar{q}\partial_{x}qX],$
(2.42)

where in (2) the expression of $\chi_{3}$ from (B.7) must be inserted and the
final form of the anomaly density in (2.42 ) is obtained by dropping a surface
term. Notice that the anomaly density in (2.42) possesses an odd parity under
(2.16) and (2.17) taking into account that $X$ is an odd function according to
(2.18). Therefore, one has $\int dt\int dx\,\chi_{3}=0$ implying the
asymptotically conservation of the charge $Q^{(4)}_{a}$.

The charge $Q^{(4)}_{a}$ in (2.39) takes the same form as the fourth order
charge in the standard AKNS model. In fact, when the r.h.s. of (2.34)
vanishes, i.e. $\chi_{3}=0$, one has a charge similar in form to the one in
(2.39), conveniently rewritten by discarding surface terms. Taking into
account the reduction process (2.12) one can get a similar anomalous charge
for the MNLS model, as presented in [4, 8, 9]. In fact, upon the reduction
(2.12) the anomalous charge $Q^{(4)}_{a}$ in (2.39) corresponds to the one for
the MNLS model in sec. 3.5 of [13].

The order $n=5$ and the quasi-conserved charge

At this order one has

$\displaystyle\frac{1}{32}\partial_{t}\Big{[}2(\bar{q}q)^{3}+5\bar{q}^{2}(\partial_{x}q)^{2}+6\bar{q}q\left(\partial_{x}q\partial_{x}\bar{q}+\bar{q}\partial^{2}_{x}q\right)+\bar{q}q^{2}\partial^{2}_{x}\bar{q}+\bar{q}\partial^{4}_{x}q\Big{]}-\partial_{x}[2i\bar{q}\,u_{6}+\partial_{x}\bar{q}\,u_{5}]=\chi_{4}.$
(2.43)

Likewise, the expression for $\chi_{4}$ can be written as

$\displaystyle\chi_{4}$ $\displaystyle\equiv$
$\displaystyle\partial_{x}[\chi^{(4)}_{x}]+\frac{1}{16}\partial_{t}[Z(\rho)]+\beta_{1},$
(2.44) $\displaystyle\chi^{(4)}_{x}$ $\displaystyle=$
$\displaystyle\frac{i}{16}\Big{\\{}Z^{(1)}(\rho)(q\partial_{x}\bar{q}-\bar{q}\partial_{x}q)+6\bar{q}\partial^{2}_{x}qX+4\bar{q}\partial_{x}q\partial_{x}X-4\partial_{x}(\bar{q}\partial_{x}q)X-\partial_{x}(\bar{q}q)\partial_{x}X+\partial^{2}_{x}(\bar{q}q)X+$
$\displaystyle\bar{q}q\partial^{2}_{x}X-(\frac{1}{2}V^{(1)}+\bar{q}q)[-6\partial_{x}(\bar{q}\partial^{2}_{x}q)+4\partial^{2}_{x}(\bar{q}\partial_{x}q)+4\bar{q}\partial^{3}_{x}q-\partial^{3}_{x}(\bar{q}q)]\Big{\\}}$
$\displaystyle\beta_{1}$ $\displaystyle=$
$\displaystyle\frac{i}{32}\left(\frac{2\bar{q}q}{V^{(1)}}+1\right)\left(\bar{q}\partial_{x}^{4}q-q\partial^{4}_{x}\bar{q}\right)V^{(1)}$
(2.45) $\displaystyle\frac{d}{d\rho}Z(\rho)$ $\displaystyle\equiv$
$\displaystyle
6\int_{\rho_{0}}^{\rho}\hat{\rho}[\frac{1}{2}V^{(2)}(\hat{\rho})+1]\,d\hat{\rho},$
(2.46)

where the function $\beta_{1}$ defines the anomaly associated with the quasi-
conservation law (2.43). Let us write the next identity

$\displaystyle\left(\bar{q}\partial_{x}^{4}q-q\partial^{4}_{x}\bar{q}\right)V^{(1)}$
$\displaystyle=$
$\displaystyle\Big{[}(-i\partial_{t}\bar{q}+\partial_{x}^{2}\bar{q})\partial_{x}^{4}q-(i\partial_{t}q+\partial_{x}^{2}q)\partial^{4}_{x}\bar{q}\Big{]}$
(2.47) $\displaystyle=$ $\displaystyle\Big{[}\partial_{x}{\cal
M}-i\partial_{t}(\bar{q}\partial^{4}_{x}q)\Big{]}$ (2.48) $\displaystyle{\cal
M}$ $\displaystyle\equiv$
$\displaystyle\partial^{2}_{x}\bar{q}\partial^{3}_{x}q-\partial^{3}_{x}\bar{q}\partial^{2}_{x}q-i\partial^{3}_{x}\bar{q}\partial_{t}q+i\partial^{2}_{x}\bar{q}\partial_{x}\partial_{t}q-i\partial_{x}\bar{q}\partial^{2}_{x}\partial_{t}q+i\bar{q}\partial^{3}_{x}\partial_{t}q,$
(2.49)

which is derived by using the eqs. of motion (2.10)-(2.11). Therefore, using
(2.48) the anomaly $\beta_{1}$ can be written as

$\displaystyle\beta_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{32}\left(\frac{2\bar{q}q}{V^{(1)}}+1\right)\Big{[}\partial_{x}{\cal
M}-i\partial_{t}(\bar{q}\partial^{4}_{x}q)\Big{]}.$ (2.50)

Next, taking into account the relevant terms of $\chi_{4}$ in (2.44) and the
terms in the l.h.s. of (2.43) with partial $t-$derivatives and discarding the
boundary terms with partial $x-$derivatives one can define the fifth order
quasi-conserved charge

$\displaystyle\frac{d}{dt}Q^{(5)}_{a}$ $\displaystyle=$ $\displaystyle\int
dx\,\beta_{1},$ (2.51) $\displaystyle Q^{(5)}_{a}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{32}\int
dx\,\Big{[}2(\bar{q}q)^{3}-8\bar{q}q\partial_{x}q\partial_{x}\bar{q}-\bar{q}^{2}(\partial_{x}q)^{2}-q^{2}(\partial_{x}\bar{q})^{2}+\partial_{x}^{2}\bar{q}\partial_{x}^{2}q-2Z(\rho)\Big{]},$
(2.52)

where the anomaly $\beta_{1}$ can take the form (2.45) or, alternatively, the
form (2.50). Notice that the form of the anomaly in (2.45) possesses an odd
parity under (2.16) and (2.17). Therefore, one has $\int dt\int
dx\,\beta_{1}=0$ implying the asymptotically conservation of the charge
$Q^{(5)}_{a}$.

Therefore, the fifth order eq. (2.43) has been written as a quasi-conservation
law. Through the reduction process (2.12) one can get an anomalous charge and
its relevant anomaly $\beta_{1}$ at this order for the MNLS model, as
presented in [4, 8, 9]. In fact, upon the reduction (2.12) the anomalous
charge $Q^{(5)}_{a}$ in (2.51)-(2.52) can be identified, dropping surface
terms, to the one for the MNLS model discussed in sec. 3.6 of [13].

Notice that, in the usual AKNS limit, i.e. when $V^{(1)}=-2\bar{q}q$ and
$V^{(2)}=-2$ for the AKNS potential as in (2.9), the factor
$\left(\frac{2\bar{q}q}{V^{(1)}}+1\right)$ of the anomaly $\beta_{1}$ in
(2.45) vanishes, and the term $Z(\rho)$ in the density of the charge (2.52)
can be set to zero (see (2.46)). Therefore, the quasi-conserved charge
$Q^{(5)}_{a}$ in (2.51) becomes the fifth order charge $Q^{(5)}$ of the usual
AKNS model. Actually, in this limit one has that $\chi_{4}=0$ for $X=0$ (see
B.7), then the r.h.s. of (2.43) vanishes, and so, this eq. can be written as
an exact conservation law.

So, we have constructed the set of (quasi-)conservation laws of type (2.23)
using the Riccati-type approach of the modified AKNS model. By a suitable
reduction process these charges can be identified to the ones of the MNLS
model, as discussed above. In ref. [4] in the context of the anomalous Lax
pair formulation of modified NLS models and through the abelianization
procedure it has been constructed an infinite set of asymptotically conserved
charges, which are similar in form to the exact conserved charges of the
standard NLS model.

##  3 Dual Riccati-type formulation and novel anomalous charges

In this section we will derive the novel anomalous conservation laws through
the dual formulation of the Riccati-type pseudo-potential approach. So, in
order to discuss a dual formulation, let us rewrite the Riccati-type system
(2.1)-(2.2) and the auxiliary eq. (2.21) as

$\displaystyle\partial_{x}u$ $\displaystyle=$
$\displaystyle-2i\zeta\,u+q+\,\bar{q}\,\,u^{2},$ (3.1)
$\displaystyle\partial_{t}u$ $\displaystyle=$ $\displaystyle
2A\,u-C\,u^{2}+B-i\frac{\chi}{\bar{q}},$ (3.2) $\displaystyle\partial_{x}\chi$
$\displaystyle=$
$\displaystyle\left(-2i\zeta+2u\bar{q}+\frac{\partial_{x}\bar{q}}{\bar{q}}\right)\chi+2\bar{q}\,uX,$
(3.3)

where the $A,B$ and $C$ functions are defined in (2.3)-(2.5).

Notice that the system of differential eqs. (2.10)-(2.11) is invariant under
the transformations: $q\leftrightarrow\bar{q}$ and $i\leftrightarrow-i$. So, a
dual formulation of the Ricati-type system (3.1)-(3.3) is achieved by
performing the changes $q\leftrightarrow\bar{q}$ and $i\leftrightarrow-i$,
$u\rightarrow\bar{u}$ and $\chi\rightarrow\bar{\chi}$ into the system above.
So, one has

$\displaystyle\partial_{x}\bar{u}$ $\displaystyle=$ $\displaystyle
2i\zeta\,\bar{u}+\bar{q}+\,q\,\,\bar{u}^{2},$ (3.4)
$\displaystyle\partial_{t}\bar{u}$ $\displaystyle=$ $\displaystyle
2\bar{A}\,\bar{u}-\bar{C}\,\bar{u}^{2}+\bar{B}+i\frac{\bar{\chi}}{q},$ (3.5)
$\displaystyle\partial_{x}\bar{\chi}$ $\displaystyle=$
$\displaystyle\left(2i\zeta+2\bar{u}q+\frac{\partial_{x}q}{q}\right)\bar{\chi}+2q\,\bar{u}X.$
(3.6)

where $\bar{u}$ is a new Riccati-type pseudo-potential and $\bar{\chi}$ is a
new auxiliary field. Notice that the function $X$ defined in (2.8) remains the
same. The functions $\bar{A},\bar{B}$ and $\bar{C}$ become

$\displaystyle\bar{A}$ $\displaystyle\equiv$ $\displaystyle
2i\zeta^{2}+\frac{1}{2}i\,V^{(1)},$ (3.7) $\displaystyle\bar{B}$
$\displaystyle\equiv$ $\displaystyle 2\zeta\bar{q}-i\partial_{x}\bar{q},$
(3.8) $\displaystyle\bar{C}$ $\displaystyle\equiv$ $\displaystyle-2\zeta
q-i\partial_{x}q.$ (3.9)

It is a simple calculation to verify that this dual Riccati-type system
(3.4)-(3.6) reproduces the eqs. (2.10)-(2.11).

Considering the expansions

$\displaystyle\bar{u}=\sum_{n=1}^{\infty}\bar{u}_{n}\,\zeta^{-n},\,\,\,\,\bar{\chi}=\sum_{n=1}^{\infty}\bar{\chi}_{n}\zeta^{-n-1},$
(3.10)

the coefficients $\bar{u}_{n}$ and $\bar{\chi}_{n}\,^{\prime}s$ can be
obtained from (3.4) and (3.6), respectively. The first components are provided
in appendix C.

For the fields $q,\bar{q}$ and $X$ satisfying the transformation laws (2.17)
and (2.18), respectively, one can verify from the system of dual equations
(3.1)-(3.3) and (3.4)-(3.6) the following symmetry transformations

$\displaystyle\widetilde{{\cal P}}(u)$ $\displaystyle=$
$\displaystyle-\bar{u},\,\,\,\,\,\,\,\widetilde{{\cal P}}(\bar{u})=-u,$ (3.11)
$\displaystyle\widetilde{{\cal P}}(\chi)$ $\displaystyle=$
$\displaystyle-\bar{\chi},\,\,\,\,\,\,\,\widetilde{{\cal
P}}(\bar{\chi})=-\chi.$ (3.12)

In fact, a careful inspection of the first six and five lowest order
components for the expressions of $\\{u,\bar{u}\\}$ and
$\\{\chi,\bar{\chi}\\}$, respectively, provided in the appendices A, B and C,
are in accordance, order by order in $n$, with the symmetries above, i.e.

$\displaystyle\widetilde{{\cal P}}(u_{n})$ $\displaystyle=$
$\displaystyle-\bar{u}_{n},\,\,\,\,\,\,\,\widetilde{{\cal
P}}(\bar{u}_{n})=-u_{n},\,\,\,\,\,\,n=1,2,...,6;$ (3.13)
$\displaystyle\widetilde{{\cal P}}(\chi_{n}\pm\bar{\chi}_{n})$
$\displaystyle=$
$\displaystyle\mp(\chi_{n}\pm\bar{\chi}_{n}),\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,n=1,2,...,5;$
(3.14)

From the both dual systems of Riccati-type eqs. (3.1)-(3.3) and (3.4)-(3.6)
one can write the next equations, respectively

$\displaystyle\partial_{t}(i\bar{q}u)-\partial_{x}(2i\zeta\bar{q}u-\bar{q}q+u\partial_{x}\bar{q})=\chi$
(3.15)

and

$\displaystyle\partial_{t}(iq\bar{u})-\partial_{x}(2i\zeta
q\bar{u}+\bar{q}q-\bar{u}\partial_{x}q)=-\bar{\chi},$ (3.16)

where (3.15) has already been considered in (2.19) with $\chi$ defined in
(2.20). Subtracting the b.h.s. of (3.15) and (3.16) one has

$\displaystyle\partial_{t}[i\bar{q}u-iq\bar{u}]-\partial_{x}[2i\zeta(\bar{q}u-q\bar{u})-2\bar{q}q+u\partial_{x}\bar{q}+\bar{u}\partial_{x}q]=\chi+\bar{\chi}.$
(3.17)

Notice that the r.h.s. of the last equation is an odd expression under the
special space-time operator; i.e. taking into account (3.12) one has
$\widetilde{{\cal P}}(\chi+\bar{\chi})=-(\chi+\bar{\chi})$. So, the equation
(3.17) defines a quasi-conservation law. In fact, the first five lowest order
components are indeed odd functions as written in (3.14). A usual computation
shows that the components of the expansion in powers of $\zeta^{-n}$ of the
quasi-conservation law (3.17) give rise to the normalization $(n=1)$, momentum
$(n=2)$ and energy $(n=3)$ conserved charges; whereas, the higher order ones
provide the same anomalous charges as the ones discussed in sec. 2. An
important observation is that the density charges of the (quasi-)conservation
eq. (3.17) are even functions, since the expression $[i\bar{q}u-iq\bar{u}]$
inside the partial time derivative in the l.h.s. of (3.17) is an even parity
function.

However, the summation of the b.h.s. of (3.15) and (3.16) will not reproduce a
quasi-conservation law, since the anomaly $(\chi-\bar{\chi})$ is an even
function according to (3.12). In addition, in this case the expression
$[i\bar{q}u+iq\bar{u}]$ of the charge density will be an odd parity function,
furnishing a trivial charge.

In the following we construct new towers of quasi-conservation laws with true
anomalies, i.e. expressions with odd parities under the symmetry
transformation (2.16). Let us consider even parity expressions of the types:
$\bar{u}u$, $i(\bar{u}\partial_{x}u-u\partial_{x}\bar{u})$,
$\partial_{x}\bar{u}\partial_{x}u$,
$i(\bar{u}\partial_{x}^{3}u-u\partial_{x}^{3}\bar{u})$,
$i\bar{u}u(\bar{u}\partial_{x}u-u\partial_{x}\bar{u}),...$ So, one can write
an infinite tower of quasi-conservation laws on top of every monomial or
polynomial of these types. Next, we show the first examples of this novel set
of infinite number of quasi-conservation laws.

The next equation follows from the both dual systems of eqs. (3.1)-(3.3) and
(3.4)-(3.6)

$\displaystyle\partial_{t}[\frac{1}{k}(\bar{u}u)^{k}]-\partial_{x}[i(\bar{u}u)^{k-1}(\bar{u}\partial_{x}u-u\partial_{x}\bar{u})]$
$\displaystyle=$ $\displaystyle{\cal A}^{(k)},\,\,\,\,\,\,\,\,\,\,k=1,2,3,...$
(3.18) $\displaystyle{\cal A}^{(k)}$ $\displaystyle=$
$\displaystyle(\bar{u}u)^{k-1}{\cal
A}-i(k-1)(\bar{u}u)^{k-2}[(\bar{u}\partial_{x}u)^{2}-(u\partial_{x}\bar{u})^{2}],$
(3.19) $\displaystyle{\cal A}$ $\displaystyle\equiv$ $\displaystyle
4\zeta\bar{u}u[\bar{q}u+q\bar{u}]+2i\bar{u}u[\bar{u}\partial_{x}q-u\partial_{x}\bar{q}]-2i\bar{u}u[\bar{q}^{2}u^{2}-q^{2}\bar{u}^{2}]+$
(3.20) $\displaystyle i[u\frac{\bar{\chi}}{q}-\bar{u}\frac{\chi}{\bar{q}}].$

Notice that ${\cal A}$ is an odd function, and so is the general function
${\cal A}^{(k)}$ for any positive integer $k$. Remarkably, the anomaly ${\cal
A}^{(k)}$ encompasses two types of anomalies. In fact, the last terms of
${\cal A}$ in (3.20) show the auxiliary potentials $\bar{\chi}$ and $\chi$
which incorporate the information of the modification of the AKNS model. The
remaining terms of ${\cal A}^{(k)}$ do not depend explicitly on those fields;
and so, they will be present even in the standard AKNS model, i.e. for
$\bar{\chi}=\chi=0$. This is our first description, in the pseudo-potential
approach, of the presence of these type of quasi-conservation laws even for a
truly integrable system.

Let us examine the first three lowest order equations $(n=2,3,4)$, for the
case $k=1$ in (3.18). The first two orders $n=2,3$ correspond, up to overall
constant factors, to the exact conservation laws for the normalization and
momentum charges, (2.26) and (2.30), respectively. The order $n=4$ can be
written as

$\displaystyle\partial_{t}\\{\partial_{x}\bar{q}\partial_{x}q-\bar{q}\partial^{2}_{x}q-q\partial^{2}_{x}\bar{q}-2(\bar{q}q)^{2}\\}$
$\displaystyle-$
$\displaystyle\partial_{x}\\{i[2(\partial_{x}\bar{q}\partial^{2}_{x}q-\partial_{x}q\partial^{2}_{x}\bar{q})-(\bar{q}\partial^{3}_{x}q-q\partial^{3}_{x}\bar{q})-$
(3.21) $\displaystyle
2\bar{q}q(\bar{q}\partial_{x}q-q\partial_{x}\bar{q})]\\}={\cal A}_{0}+{\cal
A}_{X}$ $\displaystyle{\cal A}_{0}$ $\displaystyle\equiv$
$\displaystyle-2i\bar{q}q(\bar{q}\partial^{2}_{x}q-q\partial^{2}_{x}\bar{q})-4i[(\bar{q}\partial_{x}q)^{2}-(q\partial_{x}\bar{q})^{2}]$
(3.22) $\displaystyle{\cal A}_{X}$ $\displaystyle\equiv$
$\displaystyle-6i(\bar{q}\partial_{x}q-q\partial_{x}\bar{q})X,$ (3.23)

with the odd functions ${\cal A}_{0}$ and ${\cal A}_{X}$ defining the relevant
anomaly $({\cal A}_{0}+{\cal A}_{X})$. Notice that ${\cal A}_{X}$ contains the
deformation variable, $X$, and it will vanish for the standard AKNS model,
i.e. ${\cal A}_{X}=0$. Whereas, the term ${\cal A}_{0}$ will be present as an
anomaly even for the usual AKNS model.

Next, let us examine the first two lowest order conservation laws for $k=2$ in
(3.18). The first non-trivial quasi-conservation law is for the order $n=4$

$\displaystyle\partial_{t}[(\bar{q}q)^{2}]-\partial_{x}[i\bar{q}q(\bar{q}\partial_{x}q-q\partial_{x}\bar{q})]=-i\bar{q}q[(\bar{q}\partial_{x}q)^{2}-(q\partial_{x}\bar{q})^{2}],$
(3.24)

where the r.h.s. is an odd function which does not depend explicitly on the
deformation variable $X$. This quasi-conservation law corresponds, up to an
overall factor $\frac{1}{4}$, to the eqs. in sec. 3.2 (with $n=2$) of the
companion paper [13] for the (modified) NLS model provided that the field
identifications (2.12) are performed.

The next order $n=5$ provides

$\displaystyle\partial_{t}[i\bar{q}q(\bar{q}\partial_{x}q-q\partial_{x}\bar{q})]$
$\displaystyle-$
$\displaystyle\partial_{x}\\{\frac{1}{2}[\partial_{x}q^{2}\partial_{x}\bar{q}^{2}-((\bar{q}\partial_{x}q)^{2}-(q\partial_{x}\bar{q})^{2})-\bar{q}q(\bar{q}\partial^{2}_{x}q+q\partial^{2}_{x}\bar{q})-$
(3.25)
$\displaystyle\frac{4}{3}(\bar{q}q)^{3}]\\}=[\bar{q}^{2}\partial_{x}q\partial^{2}_{x}q+q^{2}\partial_{x}\bar{q}\partial^{2}_{x}\bar{q}]-[\bar{q}(\partial_{x}q)^{2}\partial_{x}\bar{q}+q\partial_{x}q(\partial_{x}\bar{q})^{2}]-$
$\displaystyle 2(\bar{q}q)^{2}X.$

In this case the anomaly contains a term with $X$ and another terms which will
be present in the usual AKNS model for $X=0$.

In the following, let us consider the quasi-conservation law

$\displaystyle\partial_{t}[\partial_{x}\bar{u}\partial_{x}u]-\partial_{x}[i(\partial_{x}\bar{u}\partial^{2}_{x}u-\partial_{x}u\partial^{2}_{x}\bar{u})]$
$\displaystyle=$ $\displaystyle
2\zeta(\partial_{x}\bar{u}\partial_{x}q_{+}\partial_{x}u\partial_{x}\bar{q})+2\zeta(\bar{u}^{2}\partial_{x}u\partial_{x}q+u^{2}\partial_{x}\bar{u}\partial_{x}\bar{q})+$
$\displaystyle
2\zeta(\bar{q}\partial_{x}\bar{u}\partial_{x}u^{2}+q\partial_{x}u\partial_{x}\bar{u}^{2})-i(\partial_{x}\bar{q}\partial_{x}\bar{u}\partial_{x}u^{2}-\partial_{x}q\partial_{x}u\partial_{x}\bar{u}^{2})-$
$\displaystyle
i(\partial^{2}_{x}\bar{q}\partial_{x}u-\partial^{2}_{x}q\partial_{x}\bar{u})-i(\partial^{2}_{x}\bar{q}\,u^{2}\partial_{x}\bar{u}-\partial^{2}_{x}q\,\bar{u}^{2}\partial_{x}u)-$
$\displaystyle
i(u\partial_{x}\bar{u}-\bar{u}\partial_{x}u)\partial_{x}V^{(1)}+i(\partial_{x}\bar{q}\partial_{x}\bar{u}\frac{\chi}{\bar{q}^{2}}-\partial_{x}q\partial_{x}u\frac{\bar{\chi}}{q^{2}})+$
$\displaystyle
i(\partial_{x}u\frac{\partial_{x}\bar{\chi}}{q}-\partial_{x}\bar{u}\frac{\partial_{x}\chi}{\bar{q}})-i(\partial_{x}\bar{u}\partial^{3}_{x}u-\partial_{x}u\partial^{3}_{x}\bar{u}).$

The lowest order of (3 ) is for $n=2$ and it becomes the quasi-conservation
law

$\displaystyle\partial_{t}[\frac{1}{4}\partial_{x}\bar{q}\partial_{x}q]-\partial_{x}[\frac{i}{4}(\partial_{x}\bar{q}\partial^{2}_{x}q-\partial_{x}q\partial^{2}_{x}\bar{q})]=\frac{i}{4}[(\bar{q}\partial_{x}q)^{2}-(q\partial_{x}\bar{q})^{2}]V^{(2)}.$
(3.27)

Similarly, this quasi-conservation law corresponds, up to an overall factor
$\frac{1}{4}$, to the eq. 3.26 (with $n=1$) in the companion paper [13] for
the modified NLS model provided that the field identifications (2.12) are
performed.

Analogous construction process for the polynomial
$i(\bar{u}\partial_{x}u-u\partial_{x}\bar{u})$ will provide the momentum
charge, at the lowest order conservation law ($n=2$), and in top of it an
infinite set of anomalous charges as in the tower of charges of the sec. 3.3
of the companion paper [13].

We have presented an unified and rigorous constructions of the first
representatives of the novel type of quasi-conservation laws in the pseudo-
potential approach. Let us emphasize that those quasi-conservation laws are
present in the deformations of AKNS model of the type (2.10)-(2.11), as well
as in the standard AKNS model for potential (2.9). Certainly, they deserve a
more careful consideration in the context of (quasi-)integrability phenomena
and, in general, the dynamics of soliton collisions, which we will address in
a future work.

Since the early days of integrable models the presence of an infinite number
of conservation laws is among the most important features of integrability,
and the presence of anomalous conservation laws, in the N-soliton sector of
them, is a novelty. The anomalous charges of deformations of the SG and KdV
models have also been performed in the Ricatti-type pseudo-potential approach
[11, 12]), reproducing the results of [7, 1] obtained in the zero-curvature
method. In fact, this method reproduces the anomalous charges which possess
the same form as the ones of the standard integrable models. In sec. 3 of the
companion paper [13] we have obtained infinite number of anomalous charges of
the modified NLS by a direct construction method, and in this section we have
uncovered novel anomalous charges in the pseudo-potential approach.

The partial differential equations have been regarded as infinite-dimensional
manifolds and the so-called differential coverings have been introduced, which
have been used to construct some structures such as Lax pairs and Bäcklund
transformations (see e.g. [25, 26]). In particular, an auto-Bäcklund
transformation is associated to an automorphism of the covering. Then, it
would be interesting to study the properties of the dual system (3.1)-(3.3)
and (3.4)-(3.6) as some types of differential coverings of the MAKNS system
(2.10)-(2.11).

Recently, several analytical and numerical techniques have been set forward in
order to study non-linear equations possessing soliton type solutions (see
e.g. [27] and references therein).

##  4 Linear system formulation of modified AKNS system

In the next steps we will pursue a linear system of equations associated to
the deformed AKNS system (2.10)-(2.11). In order to tackle this problem we
will resort to the Riccati-type formulation of the model presented above; so,
in this context we will make use of three of the eqs. presented above, the
Riccati eq. (2.1), the quasi-conservation law (2.19) and the eq. for the
auxiliary field $\chi$ in (2.21). Next, let us consider the transformation

$\displaystyle u=-\frac{1}{\bar{q}}\partial_{x}\log{\phi},$ (4.1)

where $\phi$ represents a new pseudo-potential.

With the substitution (4.1) the Riccati eq. (2.1) becomes

$\displaystyle\phi_{xx}=-[2i\zeta-\frac{\partial_{x}\bar{q}}{\bar{q}}]\phi_{x}-\bar{q}q\phi.$
(4.2)

Next, consider the quasi-conservation law (2.19) and integrate that eq. once
in $x$. Then one gets

$\displaystyle
s(x,t)=\bar{q}q-\frac{1}{\phi}\Big{[}i\phi_{t}-2i\zeta\phi_{x}-\frac{\partial_{x}\bar{q}}{\bar{q}}\phi_{x}\Big{]},$
(4.3)

where

$\displaystyle s(x,t)\equiv\int^{x}dx^{\prime}\chi.$ (4.4)

The auxiliary eq. (2.21) upon substitution of (4.1) becomes

$\displaystyle
s_{xx}=-2X\partial_{x}\log{\phi}-[2i\zeta-\frac{\partial_{x}\bar{q}}{\bar{q}}+2\partial_{x}\log{\phi}]s_{x}.$
(4.5)

The compatibility condition
$\partial_{t}[\partial^{2}_{x}\phi]-\partial^{2}_{x}[\partial_{t}\phi]=0$
gives rise to the eq. of motion of the deformed AKNS (2.10)-(2.11).

Some comments are in order here. First, in the absence of deformations the
auxiliary eq. (4.5) becomes a trivial one, i.e. $\chi=X=0$ implies $s=0$.
Second, one can get the linear system of eqs. (4.2)-(4.3) as the linear
formulation of the standard AKNS model, provided that $s\equiv 0$ in the
l.h.s. of (4.3).

So, following analogous constructions presented in [11, 12] related to the
deformations of the sine-Gordon and KdV models, we look for a linear system of
eqs. associated to the deformed AKNS system. Notice that the function $s$ in
(4.3)-(4.4) will inherit from $\chi$ in (2.21) a highly nonlinear dependence
on $u$; then, through the transformation (4.1), $s$ will have in general a
nonlinear dependence on $\phi$. However, one can argue that the eq. (4.3)
would represent a linear eq. for the pseudo-potential $\phi$ provided that the
auxiliary field $s$ is written solely in terms of the fields $q,\bar{q}$ and
$X$ and their derivatives. So, let us assume the next Ansatz

$\displaystyle\partial_{x}\phi$ $\displaystyle=$ $\displaystyle{\cal
A}_{x}\phi,$ (4.6) $\displaystyle\partial_{t}\phi$ $\displaystyle=$
$\displaystyle{\cal A}_{t}\phi,$ (4.7)

where the functions ${\cal A}_{x}$ and ${\cal A}_{t}$ represent the gauge
connections and depend on the fields of the model. The compatibility condition
of this system will provide the eq. of motion

$\displaystyle\partial_{t}{\cal A}_{x}-\partial_{x}{\cal A}_{t}=0.$ (4.8)

Using (4.6) into (4.2) one gets the following Riccati eq. for ${\cal A}_{x}$

$\displaystyle\partial_{x}{\cal A}_{x}=-2i\zeta{\cal A}_{x}-({\cal
A}_{x})^{2}-\bar{q}q+\frac{\partial_{x}\bar{q}}{\bar{q}}{\cal A}_{x}.$ (4.9)

Likewise, replacing (4.6)-(4.7) into (4.3) one gets a relationship for the
quantity $s$

$\displaystyle s=\bar{q}q-i{\cal
A}_{t}+(2i\zeta+\frac{\partial_{x}\bar{q}}{\bar{q}}){\cal A}_{x}.$ (4.10)

Substituting this form of $s$ into the eq. (4.5) and using the eqs.
(4.8)-(4.9) one gets the eq. of motion of the deformed AKNS (2.10)-(2.11). So,
the form of $s$ in (4.10) is consistent with the dynamics of the deformed
model.

Notice that the system of eqs. (4.6)-(4.7) are defined up to a gauge
transformation of the type

$\displaystyle\phi$ $\displaystyle\rightarrow$ $\displaystyle e^{\Lambda}\phi$
(4.11) $\displaystyle{\cal A}_{x}$ $\displaystyle\rightarrow$
$\displaystyle{\cal A}_{x}+\partial_{x}\Lambda$ (4.12) $\displaystyle{\cal
A}_{t}$ $\displaystyle\rightarrow$ $\displaystyle{\cal
A}_{t}+\partial_{t}\Lambda,$ (4.13)

for an arbitrary function $\Lambda$. We will discuss this point below in more
detail.

In order to find a linear system formulation of the modified AKNS it is needed
a certain amount of guesswork out of the eq. (4.3). Moreover, due to the gauge
symmetry (4.11)-(4.13) it is possible to make a particular choice for the
connections ${\cal A}_{x}$ and ${\cal A}_{t}$. Let us propose the following
linear system of equations as the linear formulation of the deformed AKNS
111Below we will provide a gauge transformation between the systems
(4.14)-(4.15) and (4.6)-(4.7).

$\displaystyle\partial_{t}\Phi$ $\displaystyle=$ $\displaystyle A_{t}\Phi$
(4.14) $\displaystyle\partial_{x}\Phi$ $\displaystyle=$ $\displaystyle
A_{x}\Phi$ (4.15) $\displaystyle A_{x}$ $\displaystyle\equiv$
$\displaystyle-\zeta\partial_{x}\bar{q}+2\bar{q}\,\left(\frac{a_{0}\zeta+\zeta^{2}a_{1}}{2\zeta\bar{q}+i\partial_{x}\bar{q}}\right),$
(4.16) $\displaystyle A_{t}$ $\displaystyle\equiv$
$\displaystyle\frac{b_{0}+\zeta
b_{1}}{2i\zeta^{2}\bar{q}-\zeta\partial_{x}\bar{q}}+\zeta\,\int^{x}\,dx^{\prime}\,\frac{b_{0}+b_{1}\zeta+b_{2}\zeta^{2}}{(2\zeta\bar{q}+i\partial_{x^{\prime}}\bar{q})^{2}},$
(4.17)

such that

$\displaystyle b_{0}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}i\partial_{x}\bar{q}^{2}(2\partial_{t}a_{0}+\partial_{x}V^{(1)}\partial_{x}\bar{q})+i(\partial_{x}\bar{q})^{2}(V^{(1)}\partial_{x}\bar{q}-\partial^{3}_{x}\bar{q})+2a_{0}(\bar{q}^{2}\partial_{x}V^{(1)}+\partial_{x}\bar{q}\partial^{2}_{x}\bar{q}-\bar{q}\partial^{3}_{x}\bar{q})$
$\displaystyle b_{1}$ $\displaystyle\equiv$ $\displaystyle
2\bar{q}^{2}[2\partial_{t}a_{0}+\partial_{x}V^{(1)}(a_{1}+2\partial_{x}\bar{q})]+2a_{1}\partial_{x}\bar{q}\partial^{2}_{x}\bar{q}+\bar{q}[2i\partial_{t}a_{1}\partial_{x}\bar{q}+4V^{(1)}(\partial_{x}\bar{q})^{2}-2(a_{1}+2\partial_{x}\bar{q})\partial^{3}_{x}\bar{q}],$
$\displaystyle b_{2}$ $\displaystyle\equiv$ $\displaystyle
8\bar{q}^{2}[\partial_{t}a_{1}-i(\bar{q}\partial_{x}V^{(1)}+V^{(1)}\partial_{x}\bar{q}-\partial^{3}_{x}\bar{q})],$

where $a_{0}$ and $a_{1}$ are some nonvanishing auxiliary functions. The
compatibility condition of the system of eqs. (4.14)-(4.15); i.e.
$\partial_{t}\partial_{x}(\Phi)-\partial_{x}\partial_{t}(\Phi)=0$, furnishes
the next equation

$\displaystyle\partial_{t}A_{x}-\partial_{x}A_{t}=0.$ (4.18)

Substituting the expressions of the connection components
$\\{A_{x}\,,\,A_{t}\\}$ defined in (4.16)-(4.17) into (4.18) one gets the next
expression, which is a polynomial in powers of $\zeta$

$\displaystyle\\{8i\bar{q}^{2}\partial_{x}(V^{(1)}\bar{q}+i\partial_{t}\bar{q}-\partial^{2}_{x}\bar{q})\\}\zeta^{2}-$
(4.19)
$\displaystyle\\{2a_{1}[\partial_{x}\bar{q}(\partial^{2}_{x}\bar{q}-i\partial_{t}\bar{q})+\bar{q}(\bar{q}\partial_{x}V^{(1)}+i\partial_{x}\partial_{t}\bar{q}-\partial^{3}_{x}\bar{q})]+2\partial_{x}\bar{q}^{2}\partial_{x}[V^{(1)}\bar{q}+i\partial_{t}\bar{q}-\partial^{2}_{x}\bar{q}]\\}\zeta-$
(4.20)
$\displaystyle\\{2a_{0}[\partial_{x}\bar{q}(\partial^{2}_{x}\bar{q}-i\partial_{t}\bar{q})+\bar{q}(\bar{q}\partial_{x}V^{(1)}+i\partial_{x}\partial_{t}\bar{q}-\partial^{3}_{x}\bar{q})]+i(\partial_{x}\bar{q})^{2}\partial_{x}(V^{(1)}\bar{q}+i\partial_{t}\bar{q}-\partial^{2}_{x}\bar{q})\\}\equiv
0$ (4.21)

Therefore, equating to zero the coefficient of $\zeta^{2}$ in (4.19) provides
the identity

$\displaystyle\partial_{x}[V^{(1)}\bar{q}+i\partial_{t}\bar{q}-\partial^{2}_{x}\bar{q}]=0.$
(4.22)

Next, replacing this identity into the coefficients of $\zeta$ and $\zeta^{0}$
in the eqs. (4.20) and (4.21), respectively, one can get

$\displaystyle
a_{i}\,[-i\partial_{t}\bar{q}+\partial^{2}_{x}\bar{q}-V^{(1)}\bar{q}]=0,\,\,\,\,\,i=0,1.$
(4.23)

Since the auxiliary fields $a_{i}$ are non-vanishing arbitrary functions one
gets the second eq. (2.11) of the AKNS system. The first eq. (2.10) we will
derive below.

Next, let us consider the linear system 222Notice that the connections from
(4.24)-(4.25) and (4.14)-(4.15) can be related as $\widetilde{{\cal
A}}_{\mu}=\widetilde{{\cal P}}({\cal
A}_{\mu})\,\,(\mu=\\{x,t\\}),\,\,\,\widetilde{a}_{i}=\widetilde{{\cal
P}}(a_{i})\,\,(i=0,1)$, where $\widetilde{{\cal P}}$ is the parity
transformation defined in (2.16) and (2.17).

$\displaystyle\partial_{t}\widetilde{\Phi}$ $\displaystyle=$
$\displaystyle\widetilde{A}_{t}\widetilde{\Phi}$ (4.24)
$\displaystyle\partial_{x}\widetilde{\Phi}$ $\displaystyle=$
$\displaystyle\widetilde{A}_{x}\widetilde{\Phi}$ (4.25)
$\displaystyle\widetilde{A}_{x}$ $\displaystyle\equiv$
$\displaystyle\zeta\partial_{x}q+2q\,\left(\frac{\widetilde{a}_{0}\zeta+\zeta^{2}\widetilde{a}_{1}}{2\zeta
q-i\partial_{x}q}\right),$ (4.26) $\displaystyle\widetilde{A}_{t}$
$\displaystyle\equiv$
$\displaystyle\frac{\bar{b}_{0}+\zeta\bar{b}_{1}}{-2i\zeta^{2}q-\zeta\partial_{x}q}-\zeta\,\int^{x}\,dx^{\prime}\,\frac{\widetilde{b}_{0}+\widetilde{b}_{1}\zeta+\widetilde{b}_{2}\zeta^{2}}{(2\zeta
q-i\partial_{x^{\prime}}q)^{2}},$ (4.27)

with

$\displaystyle\widetilde{b}_{0}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}i\partial_{x}q^{2}(2\partial_{t}\widetilde{a}_{0}-\partial_{x}V^{(1)}\partial_{x}q)-i(\partial_{x}q)^{2}(V^{(1)}\partial_{x}q-\partial^{3}_{x}q)-2\widetilde{a}_{0}(q^{2}\partial_{x}V^{(1)}+\partial_{x}q\partial^{2}_{x}q-q\partial^{3}_{x}q)$
$\displaystyle\widetilde{b}_{1}$ $\displaystyle\equiv$
$\displaystyle-2q^{2}[2\partial_{t}\widetilde{a}_{0}+\partial_{x}V^{(1)}(\widetilde{a}_{1}-2\partial_{x}q)]-2\widetilde{a}_{1}\partial_{x}q\partial^{2}_{x}q+q[2i\partial_{t}\widetilde{a}_{1}\partial_{x}q+4V^{(1)}(\partial_{x}q)^{2}+2(\widetilde{a}_{1}-2\partial_{x}q)\partial^{3}_{x}q],$
$\displaystyle\widetilde{b}_{2}$ $\displaystyle\equiv$
$\displaystyle-8q^{2}[\partial_{t}\widetilde{a}_{1}-i(q\partial_{x}V^{(1)}+V^{(1)}\partial_{x}q-\partial^{3}_{x}q)].$

The compatibility condition of the system of eqs. (4.24)-(4.25); i.e.
$\partial_{t}\partial_{x}(\widetilde{\Phi})-\partial_{x}\partial_{t}(\widetilde{\Phi})=0$,
furnishes the next polynomial in powers of $\zeta$

$\displaystyle\\{8iq^{2}\partial_{x}(V^{(1)}q+i\partial_{t}q-\partial^{2}_{x}q)\\}\zeta^{2}-$
(4.28)
$\displaystyle\\{2\widetilde{a}_{1}[\partial_{x}q(\partial^{2}_{x}q-i\partial_{t}q)+\bar{q}(q\partial_{x}V^{(1)}+i\partial_{x}\partial_{t}q-\partial^{3}_{x}q)]+2\partial_{x}q^{2}\partial_{x}[V^{(1)}q+i\partial_{t}q-\partial^{2}_{x}q]\\}\zeta-$
(4.29)
$\displaystyle\\{2\widetilde{a}_{0}[\partial_{x}q(\partial^{2}_{x}q-i\partial_{t}q)+q(q\partial_{x}V^{(1)}+i\partial_{x}\partial_{t}q-\partial^{3}_{x}q)]+i(\partial_{x}q)^{2}\partial_{x}(V^{(1)}q+i\partial_{t}q-\partial^{2}_{x}q)\\}\equiv
0.$ (4.30)

Therefore, equating to zero the coefficient of $\zeta^{2}$ in (4.28) provides
the identity

$\displaystyle\partial_{x}[V^{(1)}q-i\partial_{t}q-\partial^{2}_{x}q]=0.$
(4.31)

Next, replacing this identity into the coefficients of $\zeta$ and $\zeta^{0}$
in the eqs. (4.29) and (4.30), respectively, one can get

$\displaystyle\widetilde{a}_{i}\,[i\partial_{t}q+\partial^{2}_{x}q-V^{(1)}q]=0,\,\,\,\,\,i=0,1.$
(4.32)

Since the auxiliary fields $\widetilde{a}_{i}$ are nonvanishing arbitrary
functions one gets the first eq. (2.10) of the AKNS system.

Therefore, the linear formulation (4.24)-(4.25) is related to the first
deformed AKNS eq. (2.10), whereas the linear formulation (4.14)-(4.15) is
related to the second deformed AKNS eq. (2.11). These separate formulations
can be joined together into just one linear system defined as

$\displaystyle\partial_{x}\left(\begin{array}[]{c}\Phi\\\
\widetilde{\Phi}\end{array}\right)$ $\displaystyle=$ $\displaystyle{\cal
M}\,\left(\begin{array}[]{c}\Phi\\\
\widetilde{\Phi}\end{array}\right),\,\,\,\,\,\,{\cal
M}\equiv\left(\begin{array}[]{cc}A_{x}&0\\\
0&\widetilde{A}_{x}\end{array}\right)$ (4.39)
$\displaystyle\partial_{t}\left(\begin{array}[]{c}\Phi\\\
\widetilde{\Phi}\end{array}\right)$ $\displaystyle=$ $\displaystyle{\cal
N}\,\left(\begin{array}[]{c}\Phi\\\
\widetilde{\Phi}\end{array}\right),\,\,\,\,\,\,{\cal
N}\equiv\left(\begin{array}[]{cc}A_{t}&0\\\
0&\widetilde{A}_{t}\end{array}\right)\,.$ (4.46)

So, the compatibility condition of this system provides the zero-curvature eq.

$\displaystyle\partial_{t}{\cal M}-\partial_{x}{\cal N}+\Big{[}{\cal
M},\,{\cal N}\Big{]}=0.$ (4.47)

Notice that ${\cal M}$ and ${\cal N}$ are diagonal matrices, and so, one has
$[{\cal M},\,{\cal N}]=0$; therefore, the linear formulation (4.39)-(4.46)
splits into the relevant formulations in (4.14)-(4.15) and (4.24)-(4.25),
respectively. Then, the eqs. of motion of the deformed AKNS model
(2.10)-(2.11) can be obtained from (4.47).

For completeness we provide a gauge transformation between the system
(4.6)-(4.7) and the system (4.14)-(4.15). So, the gauge transformation
(4.11)-(4.13) can be written as

$\displaystyle\phi$ $\displaystyle=$ $\displaystyle e^{-\Lambda}\Phi$ (4.48)
$\displaystyle A_{x}$ $\displaystyle=$ $\displaystyle{\cal
A}_{x}+\partial_{x}\Lambda$ (4.49) $\displaystyle A_{t}$ $\displaystyle=$
$\displaystyle{\cal A}_{t}+\partial_{t}\Lambda,$ (4.50)

where $\Omega\equiv\partial_{x}\Lambda$ satisfies the Riccati eq.

$\displaystyle\partial_{x}\Omega=2\Omega^{2}-(2i\zeta+2A_{x}-\frac{\partial_{x}\bar{q}}{\bar{q}})\,\Omega+i\zeta
A_{x}+\frac{1}{2}A^{2}_{x}+\frac{1}{2}\bar{q}q+\frac{1}{2}\partial_{x}A_{x}-\frac{1}{2}A_{x}\frac{\partial_{x}\bar{q}}{\bar{q}}.$
(4.51)

A similar construction can be performed for the gauge transformation of the
sector with the connection
$\left(\widetilde{A}_{x},\,\widetilde{A}_{t}\right)$. In fact, in order to
perform that gauge transformation for the full system (4.39)-(4.46) we can
have

$\displaystyle\left(\begin{array}[]{c}\phi\\\
\widetilde{\phi}\end{array}\right)$ $\displaystyle=$ $\displaystyle
g\,\left(\begin{array}[]{c}\Phi\\\
\widetilde{\Phi}\end{array}\right),\,\,\,\,\,\,g\equiv\left(\begin{array}[]{cc}e^{-\Lambda}&0\\\
0&e^{-\widetilde{\Lambda}}\end{array}\right)$ (4.58) $\displaystyle A_{\mu}$
$\displaystyle=$ $\displaystyle{\cal A}_{\mu}+\partial_{\mu}\Lambda$ (4.59)
$\displaystyle\widetilde{A}_{\mu}$ $\displaystyle=$
$\displaystyle\widetilde{{\cal
A}}_{\mu}+\partial_{\mu}\widetilde{\Lambda},\,\,\,\,\mu=\\{x,t\\}.$ (4.60)

Since the matrices ${\cal M}$ , ${\cal N}$ and $g$ are diagonal the connection
components from the both sectors do not couple in the process.

Actually, all of the constructions above can be reproduced for the modified
NLS model (2.13) since one can perform the reduction process of the modified
AKNS model ($MAKNS\rightarrow MNLS$) through the identifications (2.12).

###  4.1 Infinite set of non-local conserved charges

For each of the linear systems in (4.14)-(4.15) and (4.24)-(4.25) it is
possible to construct a set of non-local conserved charges. The construction
of analogous linear systems and their associated non-local charges have
recently been performed for some deformations of the sine-Gordon and KdV
models [11, 12]. In fact, following an iterative technique developed by Brézin
et.al. [28], the authors in [11, 12] have uncovered infinite set of non-local
conservation laws for the relevant linear systems associated to the
deformations of the SG and KdV models, respectively. So, in order to make this
paper self-contained we summarize the main points of the construction since
the procedure is quite similar to the ones undertaken for the SG and KdV
models. So, let us define the currents

$\displaystyle J_{\mu}^{(n)}$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial x_{\mu}}\chi^{(n)},\,\,\,x_{\mu}\equiv
x,t;\,\,\,\,n=0,1,2,...$ (4.61) $\displaystyle d\chi^{(1)}$ $\displaystyle=$
$\displaystyle A_{\mu}dx_{\mu}$ (4.62) $\displaystyle\equiv$ $\displaystyle
A_{x}dx+A_{t}dt,$ (4.63) $\displaystyle J_{\mu}^{(n+1)}$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial
x_{\mu}}\chi^{(n)}-A_{\mu}\chi^{(n)};\,\,\,\,\,\chi^{(0)}=1.$ (4.64)

Next, an inductive procedure is used to show that the (non-local) currents
$J_{\mu}^{(n)}$ are conserved

$\displaystyle\partial_{t}J^{(n)}_{t}-\partial_{x}J^{(n)}_{x}=0,\,\,\,\,n=1,2,3,...$
(4.65)

In fact, the first non-trivial current becomes $J_{\mu}^{(1)}=(A_{x},A_{t})$
whose conservation law $\partial_{t}A-\partial_{x}A_{t}=0$ reproduces the eq.
(4.18). The second order current becomes
$J_{\mu}^{(2)}=(A_{x}-A_{x}\chi^{(1)},A_{t}-A_{t}\chi^{(1)})$, and from the
conservation law (4.65), using the first order conservation law (4.18), one
gets

$\displaystyle\partial_{t}[A_{x}\chi^{(1)}]-\partial_{x}[A_{t}\chi^{(1)}]=0.$
(4.66)

The third order current becomes $J_{\mu}^{(3)}=(\frac{\partial}{\partial
x}\chi^{(2)}-A_{x}\chi^{(2)},\frac{\partial}{\partial
t}\chi^{(2)}-A_{t}\chi^{(2)})$. So, at this order, the conservation law
(4.65), upon using the first (4.18) and second (4.66) order conservation laws,
can be written as

$\displaystyle\partial_{t}[A_{x}\chi^{(2)}]-\partial_{x}[A_{t}\chi^{(2)}]=0.$
(4.67)

where

$\displaystyle\partial_{x}\chi^{(2)}=A_{x}-A_{x}\chi^{(1)},\,\,\,\,\partial_{t}\chi^{(2)}=A_{t}-A_{t}\chi^{(1)}.$
(4.68)

Then, one can write the infinite tower of non-local conservation laws as

$\displaystyle\partial_{t}[A_{x}\chi^{(1)}]-\partial_{x}[A_{t}\chi^{(1)}]$
$\displaystyle=$ $\displaystyle 0,$ (4.69)
$\displaystyle\partial_{t}[A_{x}\chi^{(n)}]-\partial_{x}[A_{t}\chi^{(n)}]$
$\displaystyle=$ $\displaystyle 0,\,\,\,\,\,n=2,3,4,...$ (4.70)
$\displaystyle\partial_{x}\chi^{(n)}$ $\displaystyle=$ $\displaystyle
A_{x}-A_{x}\chi^{(n-1)},\,\,\,\,\,\,\partial_{t}\chi^{(n)}=A_{t}-A_{t}\chi^{(n-1)}.$
(4.71)

A similar procedure can be performed for the sector with the gauge connection
$\left(\widetilde{A}_{x},\,\widetilde{A}_{t}\right)$ in (4.26)-(4.27), giving
rise to another tower of infinite number of non-local conservation laws

$\displaystyle\partial_{t}[\widetilde{A}_{x}\widetilde{\chi}^{(1)}]-\partial_{x}[\widetilde{A}_{t}\widetilde{\chi}^{(1)}]$
$\displaystyle=$ $\displaystyle 0,$ (4.72)
$\displaystyle\partial_{t}[\widetilde{A}_{x}\widetilde{\chi}^{(n)}]-\partial_{x}[\widetilde{A}_{t}\widetilde{\chi}^{(n)}]$
$\displaystyle=$ $\displaystyle 0,\,\,\,\,\,n=2,3,4,...$ (4.73)
$\displaystyle\partial_{x}\widetilde{\chi}^{(n)}$ $\displaystyle=$
$\displaystyle\widetilde{A}_{x}-\widetilde{A}_{x}\widetilde{\chi}^{(n-1)},\,\,\,\,\,\,\partial_{t}\widetilde{\chi}^{(n)}=\widetilde{A}_{t}-\widetilde{A}_{t}\widetilde{\chi}^{(n-1)}.$
(4.74)

Due to the reduction process $MAKNS\rightarrow MNLS$, through the
identification (2.12), the towers of non-local charges constructed above can
directly be reproduced for the modified NLS model (2.13). Moreover, additional
reductions of the standard AKNS system have been reported which define some
integrable non-local NLS, SG and KdV models [29, 30]. So, in the context of
the modified AKNS the relevant NLS-type, SG-type and KdV-type equations will
appear for a convenient choice of the parameters $A,B$ and $C$, as well as the
auxiliary fields like $r$ and $s$, in the MAKNS system (2.1)-(2.2). The
suitable choices have been done in [11] for the modified SG-like and in [12]
for the modified KdV-like systems, respectively. So, our calculations and
results above can be reproduced for the following reductions of the MAKNS
system (2.1)-(2.2)

$\displaystyle\bar{q}(x,t)$ $\displaystyle\equiv$
$\displaystyle\sigma\,q^{\star}(x,t),\,\,\,\,\,\,\,\,\,\sigma=\pm
1,\,\,\,\,\,\star\equiv\mbox{complex conjugation}$ (4.75)
$\displaystyle\bar{q}(x,t)$ $\displaystyle\equiv$
$\displaystyle\sigma\,q(x,-t),$ (4.76) $\displaystyle\bar{q}(x,t)$
$\displaystyle\equiv$ $\displaystyle\sigma\,q^{\star}(-x,t),$ (4.77)
$\displaystyle\bar{q}(x,t)$ $\displaystyle\equiv$
$\displaystyle\sigma\,q(-x,-t),\,\,\,\,q\in\leavevmode\hbox{\,$\vrule
height=6.45831pt,width=0.4pt,depth=0.0pt\kern-3.00003pt{\rm C}$}$ (4.78)
$\displaystyle\bar{q}(x,t)$ $\displaystyle\equiv$
$\displaystyle\sigma\,q^{\star}(-x,-t),$ (4.79) $\displaystyle\bar{q}(x,t)$
$\displaystyle\equiv$
$\displaystyle\sigma\,q(-x,-t),\,\,\,\,q\in\leavevmode\hbox{\rm
I\kern-1.79993ptR}.$ (4.80)

The first reduction (4.75) is just the reduction $MAKNS\rightarrow MNLS$
(2.12) we have discussed in this paper. We expect that the second, third and
fourth reductions above will give rise to non-local MNLS-type equations and
the last two of them to non-local modified KdV-type evolution equations. The
construction of the anomalous charges and the symmetries satisfied by the
relevant anomalies associated to the NLS-type (4.76)-(4.78) and (real or
complex) KdV-type (4.79)-(4.80) reductions deserve careful analysis and we
will postpone those important issues for future research.

So far, the relevant deformations of the SG and KdV models have been
considered in the literature [11, 12], and they share a similar structure
regarding their non-local conservation laws with the one of the AKNS-type
models in (4.69)-(4.71), since their linear formulations possess the same form
as in (4.14)-(4.15) or (4.24)-(4.25). Since the algebra of conserved charges
in certain two-dimensional integrable quantum field theories is also present
in the classical theory as a Poisson-Hopf algebra [31, 32], it would be
interesting to search for those type of classical Yangian algebras related to
the set of non-local currents and charges for the deformations of the
integrable models. The non-local conserved charges, as in the non-linear
$\sigma-$model, would be relevant at the quantum level and they would imply
absence of particle production (see e.g. [33, 35]).

The same technique as above can be used to construct infinite sets of non-
local conserved charges for deformations of other integrable models. The
issues such as the independence and involution of the charges deserve further
analysis and remain as open questions. In order to examine those properties
one must search for a Hamiltonian formulation of the model and compute the
relevant Poisson-like brackets among the charges. An interesting question
arises about the algebra of these infinite number of nonlocal charges. One
expects the algebra to be nonlinear very much like the ones of the nonlinear
sigma model and topological field theory (see e.g. [34] and references
therein).

The AKNS-type models are quite ubiquitous in the nonlinear science and it
would be interesting to investigate the relevance and physical consequences of
the infinite towers of infinitely many anomalous and non-local charges
discussed in this paper. Some remarkable and profound relationships between
integrable models and gauge theories have been uncovered in recent years. For
example, it has been proposed a kind of triality among gauge theories,
integrable models and gravity theories in some UV regime. In particular, the
$(1+1)D$ nonlinear Schrödinger equation corresponds to the $2D\,{\cal
N}=(2,2)^{\star}U(N)$super Yang-Mills theory (see [22] and references
therein).

We have used the Mathematica software for the various symbolic computations.
The symbolic computations turn out to be usefull in non-linear physics, for
example to get soliton solutions (see e.g. [36]).

##  5 Conclusions and discussion

We have performed the Riccati-type pseudo-potential approach to deformations
of the AKNS model in sec. 2, such that the modified NLS is obtained through a
certain reduction. In this framework it has been constructed infinite towers
of quasi-conservation laws and discussed their properties and relationships
with the MNLS model. This construction reproduced the tower of NLS-type quasi-
conserved charges obtained in the anomalous zero-curvature approach of [4].
Moreover, in sec. 3 we have introduced a dual Riccati-type pseudo-potential
approach and uncovered, in that framework, a novel set of infinite number of
quasi-conservation laws, such that it encompasses the quasi-conservation laws
obtained by a direct method starting from the eqs. of motion in sec. 3 of the
companion paper [13].

In the framework of the Riccati-type pseudo-potential approach we have
constructed a couple of linear systems of equations, (4.14)-(4.15) and
(4.24)-(4.25), whose relevant compatibility conditions give rise to the
modified AKNS system of equations (2.10)-(2.11). The second system of linear
eqs. (4.24)-(4.25) is related to the first one (4.14)-(4.15) through the
transformation (2.16) and (2.17). In subsection 4.1 we have constructed two
towers of infinite sets of non-local conservation laws associated to the
linear formulations, respectively. These linear systems and their associated
non-local charges deserve more careful considerations; in particular,
regarding their relationships of their associated non-local currents with the
so-called classical Yangians [31, 32].

In view of the current results, on deformations of SG, KdV and in this paper
on deformations of the AKNS and related NLS models, one can inquire about the
non-local properties of the quasi-integrable systems studied in the
literature, such as the deformations of the Bullough-Dodd, Toda and SUSY sine-
Gordon systems [3, 5, 10], and more specific structures, such as the complete
list of the towers of infinite number of anomalous charges and the (non-local)
exact conservation laws, as discussed in this paper. Moreover, it would be
interesting to consider the general AKNS model and study their (non-local)
reductions (4.76)-(4.80) as proposed in [30, 29], as well as their relevant
deformations in the lines discussed above.

##  6 Acknowledgments

HB thanks FC-UNI (Lima-Perú) and FC-UNASAM (Huaraz-Perú) for hospitality
during the initial stage of the work. MC thanks the Peruvian agency Concytec
for partial financial support. LFdS thanks CEFET Celso Sukow da Fonseca-Rio de
Janeiro-Brazil for kind support. The authors thank A. C. R. do Bonfim, H. F.
Callisaya, C. A. Aguirre, J. P. R. Campos, R. Q. Bellido, J.M.J. Monsalve and
A. Vilela for useful discussions.

##  Appendix A The $u_{n}^{\prime}$s of the first set of charges

The $u_{n}^{\prime}$s can be determined recursively by substituting the
expansion (2.22) into (2.1). Then the first quantities become

$\displaystyle u_{1}$ $\displaystyle=$ $\displaystyle-\frac{1}{2}iq$ (A.1)
$\displaystyle u_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}i\partial_{x}u_{1}$ (A.2) $\displaystyle u_{3}$
$\displaystyle=$
$\displaystyle\frac{1}{2}i\left(-\bar{q}u_{1}^{2}+\partial_{x}u_{2}\right)$
(A.3) $\displaystyle u_{4}$ $\displaystyle=$
$\displaystyle\frac{1}{2}i\left(-2\bar{q}u_{1}u_{2}+\partial_{x}u_{3}\right)$
(A.4) $\displaystyle u_{5}$ $\displaystyle=$
$\displaystyle\frac{1}{2}i\left(-\bar{q}u_{2}^{2}-2\bar{q}u_{1}u_{3}+\partial_{x}u_{4}\right)$
(A.5) $\displaystyle u_{6}$ $\displaystyle=$
$\displaystyle\frac{1}{2}i\left(-2\bar{q}(u_{3}u_{2}+u_{1}u_{4})+\partial_{x}u_{5}\right)$
$\displaystyle............................$

The above sequence can be written for any $n$ (even or odd) as follows

$\displaystyle u_{n}$ $\displaystyle=$
$\displaystyle\frac{1}{2}i\Big{[}-2\bar{q}\sum_{\begin{array}[]{c}i_{1}+i_{2}=n-1\\\
i_{1}\neq
i_{2}\end{array}}u_{i_{1}}u_{i_{2}}+\partial_{x}u_{n-1}\Big{]},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,n=even$
(A.9) $\displaystyle u_{n}$ $\displaystyle=$
$\displaystyle\frac{1}{2}i\Big{[}-\bar{q}\,u_{\frac{n-1}{2}}^{2}-2\bar{q}\sum_{\begin{array}[]{c}i_{1}+i_{2}=n-1\\\
i_{1}\neq
i_{2}\end{array}}\,u_{i_{1}}u_{i_{2}}+\partial_{x}u_{n-1}\Big{]},\,\,\,\,\,\,n=odd.$
(A.12)

In terms of the field components the first six $u_{i}\,(i=1,2,...6)$ become

$\displaystyle u_{1}$ $\displaystyle=$ $\displaystyle-\frac{1}{2}iq$ (A.13)
$\displaystyle u_{2}$ $\displaystyle=$ $\displaystyle\frac{1}{4}\partial_{x}q$
(A.14) $\displaystyle u_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{8}i\left(\bar{q}qq+\partial^{2}_{x}q\right)$ (A.15)
$\displaystyle u_{4}$ $\displaystyle=$
$\displaystyle-\frac{1}{16}[4\bar{q}q\partial_{x}q+qq\partial_{x}\bar{q}+\partial^{3}_{x}q]$
(A.16) $\displaystyle u_{5}$ $\displaystyle=$
$\displaystyle-\frac{1}{32}i[2(\bar{q}q)^{2}q+5\bar{q}(\partial_{x}q)^{2}+6q\left(\partial_{x}q\partial_{x}\bar{q}+\bar{q}\partial^{2}_{x}q\right)+q^{2}\partial^{2}_{x}\bar{q}+\partial^{4}_{x}q]$
(A.17) $\displaystyle u_{6}$ $\displaystyle=$
$\displaystyle\frac{1}{64}\Big{[}6q^{3}\bar{q}\partial_{x}\bar{q}+11\partial_{x}\bar{q}(\partial_{x}q)^{2}+18\bar{q}\partial_{x}q\partial^{2}_{x}q+4q(3\partial^{2}_{x}q\partial_{x}\bar{q}+2\partial_{x}q\partial^{2}_{x}\bar{q}+2\bar{q}\partial^{3}_{x}q)+$
(A.18) $\displaystyle
q^{2}(16\bar{q}^{2}\partial_{x}q+\partial^{3}_{x}\bar{q})+\partial^{5}_{x}q\Big{]}.$

##  Appendix B The $\chi$ components

The components of the expansion of $\chi$ in a recursive form become

$\displaystyle\chi_{1}$ $\displaystyle=$ $\displaystyle-i\bar{q}u_{1}X$ (B.1)
$\displaystyle\chi_{2}$ $\displaystyle=$
$\displaystyle-i\bar{q}u_{2}X+\frac{1}{2}i\partial_{x}\chi_{1}-\frac{1}{2}i\chi_{1}\frac{\partial_{x}\bar{q}}{\bar{q}}$
(B.2) $\displaystyle\chi_{3}$ $\displaystyle=$
$\displaystyle-i\bar{q}u_{3}X-i\bar{q}u_{1}\chi_{1}+\frac{1}{2}i\partial_{x}\chi_{2}-\frac{1}{2}i\chi_{2}\frac{\partial_{x}\bar{q}}{\bar{q}}$
(B.3) $\displaystyle\chi_{4}$ $\displaystyle=$
$\displaystyle-i\bar{q}u_{4}X-i\bar{q}u_{2}\chi_{1}-i\bar{q}u_{1}\chi_{2}+\frac{1}{2}i\partial_{x}\chi_{3}-\frac{1}{2}i\chi_{3}\frac{\partial_{x}\bar{q}}{\bar{q}}$
(B.4) $\displaystyle\chi_{5}$ $\displaystyle=$
$\displaystyle-i\bar{q}u_{5}X-i\bar{q}u_{3}\chi_{1}-i\bar{q}u_{2}\chi_{2}-i\bar{q}u_{1}\chi_{3}+\frac{1}{2}i\partial_{x}\chi_{4}-\frac{1}{2}i\chi_{4}\frac{\partial_{x}\bar{q}}{\bar{q}}$
$\displaystyle.............$

The above sequence can be written for any $n$ as

$\displaystyle\chi_{n}$ $\displaystyle=$
$\displaystyle-i\,\bar{q}u_{n}X-i\bar{q}\sum_{i_{1}+i_{2}=n-1}u_{i_{1}}\chi_{i_{2}}+\frac{1}{2}i\,\partial_{x}\chi_{n-1}-\frac{1}{2}i\chi_{n-1}\frac{\partial_{x}\bar{q}}{\bar{q}};\,\,\,\,\,\,\,\,\,\chi_{0}\equiv
0,\,\,\,\,\,\,\,n=1,2,....$ (B.6)

The first five components become

$\displaystyle\chi_{1}$ $\displaystyle=$ $\displaystyle-\frac{1}{2}\bar{q}qX.$
$\displaystyle\chi_{2}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}i[2\bar{q}\partial_{x}qX+\bar{q}q\partial_{x}X].$
$\displaystyle\chi_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{8}[3\bar{q}\partial_{x}q\partial_{x}X+3\bar{q}X(\bar{q}q^{2}+\partial^{2}_{x}q)+\bar{q}q\partial^{2}_{x}X].$
(B.7) $\displaystyle\chi_{4}$ $\displaystyle=$
$\displaystyle\frac{i}{16}[\bar{q}q^{2}\left(5\bar{q}\partial_{x}X+4X\partial_{x}\bar{q}\right)+6\bar{q}\partial^{2}_{x}q\partial_{x}X+4\bar{q}\partial_{x}q\partial^{2}_{x}X+4\bar{q}X\partial^{3}_{x}q+\bar{q}q\left(16X\bar{q}\partial_{x}q+\partial^{3}_{x}X\right)].$
$\displaystyle\chi_{5}$ $\displaystyle=$
$\displaystyle-\frac{1}{32}\bar{q}\Big{\\{}10q^{3}\bar{q}^{2}X+q^{2}\left(9\partial_{x}X\partial_{x}\bar{q}+7\bar{q}\partial^{2}_{x}X+5X\partial_{x}^{2}\bar{q}\right)+$
$\displaystyle
5[2\partial^{2}_{x}q\partial^{2}_{x}X+2\partial^{3}_{x}q\partial_{x}X+\partial_{x}^{3}X\partial_{x}q+X(5\bar{q}(\partial_{x}q)^{2}+\partial^{4}_{x}q)]+$
$\displaystyle
q[2\partial_{x}q(17\bar{q}\partial_{x}X+15X\partial_{x}\bar{q})+30X\bar{q}\partial^{2}_{x}q+\partial^{4}_{x}X]\Big{\\}}.$

##  Appendix C The first $\bar{u}_{n}$ and $\bar{\chi}_{n}$

Next we provide the lowest order components of the dual potentials $\bar{u}$
and $\bar{\chi}$

$\displaystyle\bar{u}_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}i\bar{q},$ (C.1) $\displaystyle\bar{u}_{2}$
$\displaystyle=$ $\displaystyle\frac{1}{4}\partial_{x}\bar{q},$ (C.2)
$\displaystyle\bar{u}_{3}$ $\displaystyle=$
$\displaystyle-\frac{1}{8}i\left(q\bar{q}\bar{q}+\partial^{2}_{x}\bar{q}\right),$
(C.3) $\displaystyle\bar{u}_{4}$ $\displaystyle=$
$\displaystyle-\frac{1}{16}[4q\bar{q}\partial_{x}\bar{q}+\bar{q}\bar{q}\partial_{x}q+\partial^{3}_{x}\bar{q}],$
(C.4) $\displaystyle\bar{u}_{5}$ $\displaystyle=$
$\displaystyle\frac{1}{32}i[2(\bar{q}q)^{2}\bar{q}+5q(\partial_{x}\bar{q})^{2}+6\bar{q}\left(\partial_{x}\bar{q}\partial_{x}q+q\partial^{2}_{x}\bar{q}\right)+\bar{q}^{2}\partial^{2}_{x}q+\partial^{4}_{x}\bar{q}],$
(C.5) $\displaystyle\bar{u}_{6}$ $\displaystyle=$
$\displaystyle\frac{1}{64}\Big{[}6\bar{q}^{3}q\partial_{x}q+11\partial_{x}q(\partial_{x}\bar{q})^{2}+18q\partial_{x}\bar{q}\partial^{2}_{x}\bar{q}+4\bar{q}(3\partial^{2}_{x}\bar{q}\partial_{x}q+2\partial_{x}\bar{q}\partial^{2}_{x}q+2q\partial^{3}_{x}\bar{q})+$
(C.6)
$\displaystyle\bar{q}^{2}(16q^{2}\partial_{x}\bar{q}+\partial^{3}_{x}q)+\partial^{5}_{x}\bar{q}\Big{]}$

and

$\displaystyle\bar{\chi}_{1}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\bar{q}qX,$ $\displaystyle\bar{\chi}_{2}$
$\displaystyle=$
$\displaystyle\frac{1}{4}i[2q\partial_{x}\bar{q}X+\bar{q}q\partial_{x}X],$
$\displaystyle\bar{\chi}_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{8}[3q\partial_{x}\bar{q}\partial_{x}X+3qX(q\bar{q}^{2}+\partial^{2}_{x}\bar{q})+\bar{q}q\partial^{2}_{x}X],$
(C.7) $\displaystyle\bar{\chi}_{4}$ $\displaystyle=$
$\displaystyle-\frac{i}{16}[q\bar{q}^{2}\left(5q\partial_{x}X+4X\partial_{x}q\right)+6q\partial^{2}_{x}\bar{q}\partial_{x}X+4q\partial_{x}\bar{q}\partial^{2}_{x}X+4qX\partial^{3}_{x}\bar{q}+\bar{q}q\left(16Xq\partial_{x}\bar{q}+\partial^{3}_{x}X\right)],$
$\displaystyle\bar{\chi}_{5}$ $\displaystyle=$
$\displaystyle-\frac{1}{32}q\Big{\\{}10\bar{q}^{3}q^{2}X+\bar{q}^{2}\left(9\partial_{x}X\partial_{x}q+7q\partial^{2}_{x}X+5X\partial_{x}^{2}q\right)+$
$\displaystyle
5[2\partial^{2}_{x}\bar{q}\partial^{2}_{x}X+2\partial^{3}_{x}\bar{q}\partial_{x}X+\partial_{x}^{3}X\partial_{x}\bar{q}+X(5q(\partial_{x}\bar{q})^{2}+\partial^{4}_{x}\bar{q})]+$
$\displaystyle\bar{q}[2\partial_{x}\bar{q}(17q\partial_{x}X+15X\partial_{x}q)+30Xq\partial^{2}_{x}\bar{q}+\partial^{4}_{x}X]\Big{\\}}.$

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