# Prototype and reduced nonlinear integrable lattice systems with the
modulated pulson behavior

Oleksiy O. Vakhnenko  Department for Theory of Nonlinear Processes in
Condensed Matter, Bogolyubov Institute for Theoretical Physics, The National
Academy of Sciences of Ukraine, 14-B Metrologichna Street, Kyïv 03143, Ukraïna
<EMAIL_ADDRESS>
###### Abstract

A multi-component semi-discrete nonlinear integrable system associated with
the relevant third-order auxiliary linear problem is claimed to be the
prototype system for several reduced integrable systems formulated in terms of
true dynamical field variables. The main conservation laws related to the
prototype system are found in the framework of generalized recurrent approach.
The two-fold Darboux-Bäcklund dressing technique as applied to the integration
of prototype system is developed in details. The novel reduced complex-valued
nonlinear integrable system embracing three coupled dynamical subsystems on a
quasi-one-dimensional lattice is proposed and its concise Lagrangian and
Hamiltonian representations are written down. The essentially nontrivial
couplings between the two complex-valued Toda-like subsystems are shown to be
mediated by the intermediate subsystem both in their kinetic and potential
parts. The explicit multi-component solution of a modulated pulson type
related to the reduced integrable nonlinear ternary system is presented and
analyzed.

###### keywords:

Semi-discrete nonlinear integrable system , Conservation laws ,
Darboux–Bäcklund transformation , Lagrangian and Hamiltonian representations ,
Modulated pulson

††journal: arXiv.org

##  1 Introduction

The seminal suggestion of one-dimensional dynamical lattice system with the
exponential nonlinearity by Toda [1, 2, 3] and the subsequent discovery of its
complete integrability by Manakov [4] and Flaschka [5] had undoubtedly been
and continue to be the powerful motivating forces for the widespread
development of numerous semi-discrete (i.e. continuous in time and discrete in
spatial coordinate) integrable models [6, 7, 8, 9, 10, 11, 12, 13] including
the multi-component ones [14, 15, 16, 17, 18, 19]. This stream of researches
is accompanied by the recognized applicability of semi-discrete integrable
nonlinear systems to the various branches of physics [20] inasmuch as the vast
number of real physical objects are given on one or another type of quasi-one-
dimensional lattices.

Among other semi-discrete nonlinear integrable systems the integrable systems
generated by the various auxiliary linear problems of third [21, 22, 23, 24,
25, 27, 28] or more higher [29, 30] order deserve a special attention as the
systems comprising several subsystems of distinct physical origins.

Thus, in two our papers [22, 23] we have introduced the general integrable
nonlinear system

$\dot{p}_{11}(n)=F_{12}(n)G_{21}(n-1)-F_{12}(n+1)G_{21}(n)$ (1.1)
$\dot{p}_{13}(n)=F_{12}(n)G_{23}(n-1)-F_{12}(n+1)G_{23}(n)$ (1.2)
$\dot{p}_{31}(n)=F_{32}(n)G_{21}(n-1)-F_{32}(n+1)G_{21}(n)$ (1.3)
$\dot{p}_{33}(n)=F_{32}(n)G_{23}(n-1)-F_{32}(n+1)G_{23}(n)$ (1.4)
$\dot{F}_{12}(n)=p_{11}(n)F_{12}(n)+p_{13}(n)F_{32}(n)$ (1.5)
$\dot{G}_{21}(n)=-G_{21}(n)p_{11}(n)-G_{23}(n)p_{31}(n)$ (1.6)
$\dot{G}_{23}(n)=-G_{21}(n)p_{13}(n)-G_{23}(n)p_{33}(n)$ (1.7)
$\dot{F}_{32}(n)=p_{31}(n)F_{12}(n)+p_{33}(n)F_{32}(n)$ (1.8)

permitting a number of reductions from the so-called prototype $p_{11}(n)$,
$p_{13}(n)$, $p_{31}(n)$, $p_{33}(n)$, $F_{12}(n)$, $G_{21}(n)$, $G_{23}(n)$,
$F_{32}(n)$ to the actual physical field variables. Here the overdot marks the
derivative with respect to time $\tau$, while the discrete spatial coordinate
$n$ is assumed to span all integers from minus to plus infinity. To integrate
this system (1.1)–(1.8) we have developed rather complicated inverse
scattering technique and found some simplest solutions related to the reduced
system encompassing two real-valued Toda-like translational subsystems coupled
with one orientational subsystem. Unfortunately, the orientational component
of the obtained solution suffers to be unexcited [22, 23].

In present paper we try to fill in this gap in the framework of specially
developed two-fold Darboux–Backlund dressing approach which allow to construct
the nonlinear wave packet embracing all involved field components. In
addition, we apply the general recurrent scheme capable to generate the
infinite hierarchy of local conservation laws and write down the most
important of them explicitly. Moreover, we propose early unknown reduction to
the actual field variables appropriate for the practical implementation of
obtained results to the low-dimensional regular physical objects. We formulate
the reduced nonlinear integrable system both in terms of Lagrange and Hamilton
dynamic approaches and underline that two of its three coupled subsystems have
a nontrivial fashion of complex-valued Toda-like type. In so doing, the
respective Hamilton function is proved to emanate from one of the conserved
quantities.

To proceed with the most coherent consideration concerning the general
nonlinear system (1.1)–(1.8) we begin with the confirmation of its complete
integrability.

##  2 Complete integrability of the general nonlinear system

One can readily verify that the general semi-discrete nonlinear system of our
interest (1.1)–(1.8) is presentable in the form of matrix-valued semi-discrete
zero-curvature equation

$\dot{L}(n|z)=A(n+1|z)L(n|z)-L(n|z)A(n|z)$ (2.1)

with the spectral $L(n|z)$ and evolution $A(n|z)$ operators given by the
following $3\times 3$ square matrices

$L(n|z)=\left(\begin{array}[]{ccccc}p_{11}(n)+\lambda(z)&&F_{12}(n)&&p_{13}(n)\\\
G_{21}(n)&&0&&G_{23}(n)\\\ p_{31}(n)&&F_{32}(n)&&p_{33}(n)+\lambda(z)\\\
\end{array}\right)$ (2.2)
$A(n|z)=\left(\begin{array}[]{ccccc}0&&-F_{12}(n)&&0\\\
-G_{21}(n-1)&&\lambda(z)&&-G_{23}(n-1)\\\ 0&&-F_{32}(n)&&0\\\
\end{array}\right)\,.$ (2.3)

Here the symbols $\lambda(z)$ and $z$ denote the authentic and rationalizes
time-independent spectral parameters, respectively. According to the commonly
referred terminology [31, 32, 33] this representation (2.1)–(2.3) determines
the complete integrability of the declared semi-discrete nonlinear system
(1.1)–(1.8) in the Lax sense.

Let us remind that the zero-curvature equation itself (2.1) serves as the
compatibility condition between two linear matrix-valued equations

$X(n+1|z)=L(n|z)X(n|z)$ (2.4) $\dot{X}(n|z)=A(n|z)X(n|z)$ (2.5)

referred to as the auxiliary linear problem. Here the notation $X(n|z)$ stands
for the auxiliary matrix-function of discrete spatial coordinate $n$,
continuous time $\tau$ and time-independent spectral parameter $z$. By and
large, the auxiliary linear problem (2.4) and (2.5) is capable to lay the
foundation for the development of various methods of system’s integration such
as the method of inverse scattering transform [34, 12, 22, 23] and the
Darboux-Bäcklund dressing method [35, 36, 37, 38, 39]. Moreover, it might
provide an indispensable starting tool in recurrent searching for an infinite
hierarchy of local conservation laws [29].

##  3 Natural constraints and the main local densities

It is well known that any nonlinear integrable system on an infinite regular
chain possesses the infinite number of local conservation laws. The most
straightforward way to isolate some of them is based upon the universal local
conservation law

$\frac{d}{d\tau}\ln[\det L(n|z)]=\mathrm{Sp}A(n+1|z)-\mathrm{Sp}A(n|z)$ (3.1)

appearing as a simple contraction of system’s zero-curvature representation
(2.1).

Inasmuch as the determinant $\det L(n|z)$ of spectral matrix $L(n|z)$ depends
on two distinct powers of spectral parameter $\lambda(z)$ and the expression
$\mathrm{Sp}A(n+1|z)-\mathrm{Sp}A(n|z)$ is equal to zero, the universal local
conservation law (3.1) produces two unicellular conservation laws imposing two
natural constraints onto the set of prototype field variables. The explicit
record of natural constraints depends on a particular choice of boundary
conditions for the prototype field variables and on an expected physical sense
of reduced field variables. Assuming the underlying lattice being spatially
uniform and pinned to the immovable frame of reference we take the natural
constraints in the following form

$p_{11}(n)F_{32}(n)G_{23}(n)-p_{31}(n)F_{12}(n)G_{23}(n)+p_{33}(n)F_{12}(n)G_{21}(n)-p_{13}(n)F_{32}(n)G_{21}(n)=0$
(3.2) $F_{12}(n)G_{21}(n)+F_{32}(n)G_{23}(n)=-1\,.$ (3.3)

These constraints (3.2) and (3.3) are consistent with the following boundary
conditions for the prototype fields:
$\lim_{n\rightarrow-\infty}p_{11}(n)\rightarrow 0$,
$\lim_{n\rightarrow-\infty}p_{13}(n)\rightarrow 0$,
$\lim_{n\rightarrow-\infty}p_{33}(n)\rightarrow 0$,
$\lim_{n\rightarrow-\infty}p_{31}(n)\rightarrow 0$ and
$\lim_{n\rightarrow-\infty}F_{12}(n)\rightarrow F_{12}$,
$\lim_{n\rightarrow-\infty}G_{21}(n)\rightarrow G_{21}$,
$\lim_{n\rightarrow-\infty}F_{32}(n)\rightarrow F_{32}$,
$\lim_{n\rightarrow-\infty}G_{23}(n)\rightarrow G_{23}$, where
$F_{12}G_{21}+F_{32}G_{23}=-1$. As a consequence, the limiting eigenvalue
problem

$L(z)X(z)=X(z)\zeta(z)$ (3.4)

(where $L(z)=\lim_{n\rightarrow-\infty}L(n|z)$) prescribes the functional
relationship $\lambda(z)=z+1/z$ in view of very simple resulting expressions

$\zeta_{1}(z)=z$ (3.5) $\zeta_{2}(z)=\lambda(z)\equiv z+1/z$ (3.6)
$\zeta_{3}(z)=1/z$ (3.7)

for the eigenvalues $\zeta_{j}(z)$. It is interesting to note that the
functional dependence $\lambda(z)=z+1/z$ establishes a particular realization
of the Zhukovskiy transformation [40, 41] well known in aerodynamics.

The capability of universal local conservation law (3.1) in generating
system’s local conservation laws is seen to be restricted only by two
specimens.

In contrast, there exists the generalized direct procedure [29, 24, 27, 28]
permitting to develop an infinite set of local conservation laws recursively
without references to auxiliary spectral data. By definition, any local
conservation law associated with some semi-discrete system given on an
infinite quasi-one-dimensional lattice can be written in the form

$\dot{\rho}(n)=J(n|n-1)-J(n+1|n)\,,$ (3.8)

where the quantities $\rho(n)$ and $J(n+1/2|n-1/2)$ are referred to as the
local density and the local current, respectively. Bearing in mind this
general definition (3.8) we must find the recursive presentation (i.e
presentation based upon some proper expansion in spectral parameter $z$ or
inverse spectral parameter $1/z$) for the auxiliary quantities
$\Gamma_{jk}(n|z)$ governed by the following set of spatial Riccati equations

$\Gamma_{jk}(n+1|z)\sum_{i=1}^{3}L_{ki}(n|z)\Gamma_{ik}(n|z)=\sum_{i=1}^{3}L_{ji}(n|z)\Gamma_{ik}(n|z)$
(3.9)

with the restrictions

$\Gamma_{ji}(n|z)\Gamma_{ik}(n|z)=\Gamma_{jk}(n|z)$ (3.10)

being taken into account. The obtained series should be substituted into the
collection of three ($j=1,2,3$) generating equations

$\frac{d}{d\tau}\ln M_{jj}(n|z)=B_{jj}(n+1|z)-B_{jj}(n|z)\,.$ (3.11)

Here the composite functions

$M_{jj}(n|z)=\sum_{i=1}^{3}L_{ji}(n|z)\Gamma_{ij}(n|z)$ (3.12)

and

$B_{jj}(n|z)=\sum_{i=1}^{3}A_{ji}(n|z)\Gamma_{ij}(n|z)$ (3.13)

serve to generate the hierarchy of local densities and the hierarchy of local
currents, respectively. In so doing, the quantities $L_{jk}(n|z)$ and
$A_{jk}(n|z)$ denote the matrix elements of spectral $L(n|z)$ and evolution
$A(n|z)$ operators, respectively. Collecting terms with the same powers of
spectral parameter in each of three ($j=1,2,3$) generating series (3.11) it is
possible to recover any required number of local conservation laws from their
infinite hierarchy.

The most productive is the second ($j=2$) of generating series (3.11). To
develop the second generating series it is sufficient to consider two
auxiliary functions $\Gamma_{12}(n|z)$ and $\Gamma_{32}(n|z)$ since
$\Gamma_{jj}(n|z)\equiv 1$ in view of properties (3.10). Due to the evident
symmetry $\lambda(z)=\lambda(1/z)$ of the functional spectral parameter
$\lambda(z)$ we restrict ourselves only to serial expansions at the center
$|z|\rightarrow 0$ of a rationalized complex spectral plane and seek the
auxiliary functions $\Gamma_{12}(n|z)$ and $\Gamma_{32}(n|z)$ as follows

$\Gamma_{12}(n|z)=z\sum_{j=0}^{\infty}\gamma_{12}(n|m)z^{m}$ (3.14)
$\Gamma_{32}(n|z)=z\sum_{j=0}^{\infty}\gamma_{32}(n|m)z^{m}~{}.$ (3.15)

Then for the generating function $\ln M_{22}(n|z)$ written up to the second
power in spectral parameter $z$ we obtain

$\displaystyle\ln M_{22}(n|z)$ $\displaystyle=$ $\displaystyle\ln
z-\left[p_{11}(n)+p_{33}(n)\right]z+$ (3.16) $\displaystyle+$
$\displaystyle\left[p_{11}^{2}(n)/2+p_{11}^{2}(n)/2+p_{13}(n)p_{31}(n)-G_{21}(n)F_{12}(n+1)-G_{23}(n)F_{32}(n+1)-1\right]z^{2}\,.$

By virtue of second generating equation (i. e. equation (3.11) taken at $j=2$)
the quantities

$p(n)=p_{11}(n)+p_{33}(n)$ (3.17)
$h(n)=p_{11}^{2}(n)/2+p_{11}^{2}(n)/2+p_{13}(n)p_{31}(n)-G_{21}(n)F_{12}(n+1)-G_{23}(n)F_{32}(n+1)-1$
(3.18)

acquire the status of local densities in two the most important local
conservation laws.

The comprehensive analysis shows that the former (3.17) of these local
densities should be identified with the density of system’s momentum, whereas
the latter one (3.18) – with the density of system’s energy.

By invoking the revealed natural constraints (3.2) and (3.3) it is possible to
replace eight original prototype field variables by means of six properly
chosen actual dynamical field variables. A particular realization of such a
reduction is not unique. However, in any case we come to some reduced
nonlinear lattice system comprising three coupled dynamical subsystems. As to
the local density $h(n)$ its expression (3.18) will be useful while writing
the Hamilton function of reduced nonlinear dynamical system in terms actual
fields.

We postpone both of above tasks up to the Section 9 and concentrate now on the
development of Darboux–Bäcklund dressing integration scheme in terms of
prototype field variables applicable to the general nonlinear system of our
interest (1.1)–(1.8).

##  4 Fundamentals of the Darboux–Bäcklund dressing technique

In this Section we itemize the main steps elucidating the essence of the
Darboux–Bäcklund dressing scheme suitable for the integration of the prototype
nonlinear lattice system under study (1.1)–(1.8).

We start with the general definition of Darboux transformation [37]

$^{c}X(n|z)=~{}^{cs}D(n|z)~{}^{s}X(n|z)\,.$ (4.1)

The Darboux transformation (4.1) connects the seed (a priori known)
${}^{s}X(n|z)$ and crop (required) ${}^{c}X(n|z)$ solutions of the auxiliary
linear problem (2.4) and (2.5) via the Darboux matrix ${}^{cs}D(n|z)$, which
should be properly chosen in accordance with the particular realizations (2.2)
and (2.3) of auxiliary spectral $L(n|z)$ and evolution $A(n|z)$ operators. The
Darboux matrix ${}^{cs}D(n|z)$ must obey the set of matrix equations

$^{cs}D(n+1|z)~{}^{s}L(n|z)=~{}^{c}L(n|z)~{}^{cs}D(n|z)$ (4.2)
$^{cs}\dot{D}(n|z)=~{}^{c}A(n|z)~{}^{cs}D(n|z)-~{}^{cs}D(n|z)~{}^{s}A(n|z)$
(4.3)

serving as the spatial (4.2) and temporal (4.3) compatibility conditions
between the seed

$^{s}X(n+1|z)=~{}^{s}L(n|z)~{}^{s}X(n|z)$ (4.4)
$^{s}\dot{X}(n|z)=~{}^{s}A(n|z)~{}^{s}X(n|z)$ (4.5)

and crop

$^{c}X(n+1|z)=~{}^{c}L(n|z)~{}^{c}X(n|z)$ (4.6)
$^{c}\dot{X}(n|z)=~{}^{c}A(n|z)~{}^{c}X(n|z)$ (4.7)

embodiments of auxiliary linear problem (2.4) and (2.5). Here and later on the
upper-front single index $s$ or $c$ indicates a quantity associated with the
seed or crop solution, respectively. The upper-front double index $cs$ marks a
quantity related to or induced by the Darboux matrix. According to the general
rules [20, 38, 39] the spatial compatibility condition (4.2) establishes the
implicit Bäcklund transformation between the seed and crop solutions for the
prototype fields, while the temporal compatibility condition (4.3) allows to
uncover the crucial spectral properties of Darboux matrix sufficient to
restore explicitly its components indispensable for the development of the
whole Darboux-Bäcklund dressing integration scheme.

Seeking for the Darboux matrix ${}^{cs}D(n|z)$ we assume the following two-
fold ($c=s+2$) ansatz

$^{cs}D(n|z)=\left(\begin{array}[]{ccccc}{}^{cs}K_{11}(n)&{}\hfil&{}^{cs}C_{12}(n)\lambda(z)+~{}^{cs}T_{12}(n)&{}\hfil&{}^{cs}K_{13}(n)\\\\[5.69054pt]
{}^{cs}E_{21}(n)\lambda(z)+~{}^{cs}V_{21}(n)&{}\hfil&\lambda^{2}(z)+~{}^{cs}D_{22}(n)\lambda(z)+~{}^{cs}U_{22}(n)&{}\hfil&{}^{cs}E_{23}(n)\lambda(z)+~{}^{cs}V_{23}(n)\\\\[5.69054pt]
{}^{cs}K_{31}(n)&{}\hfil&{}^{cs}C_{32}(n)\lambda(z)+~{}^{cs}T_{32}(n)&{}\hfil&{}^{cs}K_{33}(n)\\\
\end{array}\right),$ (4.8)

which proves to be consistent with the set of governing matrix equations
(4.2)–(4.3). The notion “two-fold ($c=s+2$)” implies that each matrix element
of adopted ansatz (4.8) written as a polynomial of the spectral parameter
$\lambda(z)$ has the same leading term as the respective matrix element
appearing in the product of two evolution matrices. The Darboux matrix taken
in the two-fold form (4.8) are able to generate nontrivial spatially finite
solutions (from the vacuum one) embracing all three subsystems of a reduced
nonlinear dynamical system.

The spectral properties of Darboux matrix ${}^{cs}D(n|z)$ follow directly from
the contracted form

$\frac{d}{d\tau}\ln[\det\,^{cs}D(n|z)]=\mathrm{Sp}\,^{c}A(n|z)-\mathrm{Sp}\,^{s}A(n|z)$
(4.9)

of temporal compatibility condition (4.3). Indeed, in view of the identity

$\mathrm{Sp}\,^{c}A(n|z)-\mathrm{Sp}\,^{s}A(n|z)\equiv 0\,,$ (4.10)

dictated by the specific form (2.2) of evolution matrix $A(n|z)$, the
contracted equation (4.9) with the adopted ansatz (4.8) for the Darboux matrix
$~{}^{cs}D(n|z)$ yields

$\det\,^{cs}D(n|z)=\left[\lambda(z)-\lambda(~{}^{cs}z_{+})\right]\left[\lambda(z)-\lambda(~{}^{cs}z_{-})\right]~{}^{cs}W(n)\,,$
(4.11)

where the quantity ${}^{cs}W(n)$ and the spectral data ${}^{cs}z_{+}$,
${}^{cs}z_{-}$ are proved to be time-independent ${}^{cs}\dot{W}(n)=0$,
${}^{cs}\dot{z}_{+}=0$, ${}^{cs}\dot{z}_{-}=0$.

Evidently, $\det\,^{cs}D(n|~{}^{cs}z_{\pm})=0$ and hence the Darboux
transformation (4.1) yields $\det\,^{c}X(n|~{}^{cs}z_{\pm})=0$. Therefore the
condition

$\sum_{k=1}^{3}~{}^{c}X_{jk}(n|\,^{cs}z_{\pm})~{}^{cs}\varepsilon_{k}(\,^{cs}z_{\pm})=0$
(4.12)

should be satisfied. The detailed form of this condition (4.12) is given by
formula

$\sum_{i=1}^{3}\sum_{k=1}^{3}\,{}^{cs}D_{ji}(n|\,^{cs}z_{\pm})~{}^{s}X_{ik}(n|\,^{cs}z_{\pm})~{}^{cs}\varepsilon_{k}(\,^{cs}z_{\pm})=0\,.$
(4.13)

Here the functions ${}^{cs}D_{ji}(n|z)$ and ${}^{s}X_{ik}(n|z)$ stand for the
elements of respective matrices ${}^{cs}D(n|z)$ and ${}^{s}X(n|z)$, while the
time- and space-independent parameters
${}^{cs}\varepsilon_{k}(\,^{cs}z_{\pm}$) serve as the spectral data. The
invariability of spectral parameters ${}^{cs}\varepsilon_{k}(\,^{cs}z_{\pm}$)
in time and space has a status of rigorously proved theorem.

The obtained condition (4.13) encompasses six linear equations for fourteen
Darboux functions $~{}^{cs}K_{11}(n)$, $~{}^{cs}C_{12}(n)$,
$~{}^{cs}T_{12}(n)$, $~{}^{cs}K_{13}(n)$, $~{}^{cs}E_{21}(n)$,
$~{}^{cs}V_{21}(n)$, $~{}^{cs}D_{22}(n)$, $~{}^{cs}U_{22}(n)$,
$~{}^{cs}E_{23}(n)$, $~{}^{cs}V_{23}(n)$, $~{}^{cs}K_{31}(n)$,
$~{}^{cs}C_{32}(n)$, $~{}^{cs}T_{32}(n)$, $~{}^{cs}K_{33}(n)$. Fortunately
only six $~{}^{cs}C_{12}(n)$, $~{}^{cs}T_{12}(n)$, $~{}^{cs}D_{22}(n)$,
$~{}^{cs}U_{22}(n)$, $~{}^{cs}C_{32}(n)$, $~{}^{cs}T_{32}(n)$ of them turn out
to be truly independent. We prove this statement in Section 5 by invoking the
implicit Bäcklund transformation (4.2) rewritten as twenty two detailed
equations (5.1)–(5.22). In so doing, two Darboux functions $~{}^{cs}E_{21}(n)$
and $~{}^{cs}E_{23}(n)$ are proved to be determined explicitly by the a priori
known seed values $~{}^{s}G_{21}(n)$ and $~{}^{s}G_{23}(n)$ of the prototype
functions $G_{21}(n)$ and $G_{23}(n)$. By dint of all these observations the
above written formula (4.13) produces the set of six nonuniform linear
equations serving to restore the unknown Darboux functions relying upon the
known ones. In so doing, the seed solution ${}^{s}X(n|z)$ to the auxiliary
linear problem (2.4) and (2.5) must be found beforehand.

Once the necessary elements of Darboux matrix have been found, the proper
equations taken among the extended form (5.1)–(5.22) of implicit Bäcklund
transformation (4.2) allow to obtain explicit crop solutions for the prototype
field functions.

##  5 Peculiarities of the Darboux–Bäcklund dressing technique stipulated by
the chosen two-fold Darboux matrix

As we have already mentioned, the first (4.2) of two matrix equations for the
Darboux matrix should be treated as the implicit Bäcklund transformation
between the seed $~{}^{s}p_{11}(n)$, $~{}^{s}p_{13}(n)$, $~{}^{s}p_{31}(n)$,
$~{}^{s}p_{33}(n)$, $~{}^{s}F_{12}(n)$, $~{}^{s}G_{21}(n)$,
$~{}^{s}G_{23}(n)$, $~{}^{s}F_{32}(n)$ and crop $~{}^{c}p_{11}(n)$,
$~{}^{c}p_{13}(n)$, $~{}^{c}p_{31}(n)$, $~{}^{c}p_{33}(n)$,
$~{}^{c}F_{12}(n)$, $~{}^{c}G_{21}(n)$, $~{}^{c}G_{23}(n)$, $~{}^{c}F_{32}(n)$
solutions for the prototype fields. To justify this statement in the case of
adopted two-fold ansatz (4.8) for the Darboux matrix it is sufficient to
observe that the relevant matrix formula (4.2) comprises the following twenty
two equations

$^{cs}K_{11}(n+1)+~{}^{cs}C_{12}(n+1)~{}^{s}G_{21}(n)=~{}^{cs}K_{11}(n)+~{}^{c}F_{12}(n)~{}^{cs}E_{21}(n)$
(5.1) ${}^{cs}K_{11}(n+1)~{}^{s}p_{11}(n)$ $\displaystyle+$
$\displaystyle~{}^{cs}T_{12}(n+1)~{}^{s}G_{21}(n)+~{}^{cs}K_{13}(n+1)~{}^{s}p_{31}(n)=$
(5.2) $\displaystyle=$
${}^{c}p_{11}(n)~{}^{cs}K_{11}(n)+~{}^{c}F_{12}(n)~{}^{cs}V_{21}(n)+~{}^{c}p_{13}(n)^{cs}K_{31}(n)$
$^{cs}C_{12}(n)+~{}^{c}F_{12}(n)=0$ (5.3)
$^{cs}T_{12}(n)+~{}^{c}p_{11}(n)~{}^{cs}C_{12}(n)+~{}^{c}F_{12}(n)~{}^{cs}D_{22}(n)+~{}^{c}p_{13}(n)~{}^{cs}C_{32}(n)=0$
(5.4) ${}^{cs}K_{11}(n+1)~{}^{s}F_{12}(n)$ $\displaystyle+$
$\displaystyle~{}^{cs}K_{13}(n+1)~{}^{s}F_{32}(n)=$ (5.5) $\displaystyle=$
${}^{c}p_{11}(n)~{}^{cs}T_{12}(n)+~{}^{c}F_{12}(n)~{}^{cs}U_{22}(n)+~{}^{c}p_{13}(n)^{cs}T_{32}(n)$
$^{cs}C_{12}(n+1)~{}^{s}G_{23}(n)+~{}^{cs}K_{13}(n+1)=~{}^{cs}K_{13}(n)+~{}^{c}F_{12}(n)~{}^{cs}E_{23}(n)$
(5.6) ${}^{cs}K_{11}(n+1)~{}^{s}p_{13}(n)$ $\displaystyle+$
$\displaystyle~{}^{cs}T_{12}(n+1)~{}^{s}G_{23}(n)+~{}^{cs}K_{13}(n+1)~{}^{s}p_{33}(n)=$
(5.7) $\displaystyle=$
${}^{c}p_{11}(n)~{}^{cs}K_{13}(n)+~{}^{c}F_{12}(n)~{}^{cs}V_{23}(n)+~{}^{c}p_{13}(n)^{cs}K_{33}(n)$
$^{cs}E_{21}(n+1)+~{}^{s}G_{21}(n)=0$ (5.8)
$^{cs}E_{21}(n+1)~{}^{s}p_{11}(n)+~{}^{cs}V_{21}(n+1)+~{}^{cs}D_{22}(n+1)~{}^{s}G_{21}(n)+~{}^{cs}E_{21}(n+1)~{}^{s}p_{31}(n)=0$
(5.9) ${}^{cs}V_{21}(n+1)~{}^{s}p_{11}(n)$ $\displaystyle+$
$\displaystyle~{}^{cs}U_{22}(n+1)~{}^{s}G_{21}(n)+^{cs}V_{23}(n+1)~{}^{s}p_{31}(n)=$
(5.10) $\displaystyle=$
${}^{c}G_{21}(n)~{}^{cs}K_{11}(n)+~{}^{c}G_{23}(n)~{}^{cs}K_{31}(n)$
$^{cs}E_{21}(n+1)~{}^{s}F_{12}(n)+~{}^{cs}E_{23}(n+1)~{}^{s}F_{32}(n)=~{}^{c}G_{21}(n)~{}^{cs}C_{12}(n)+~{}^{c}G_{23}(n)~{}^{cs}C_{32}(n)$
(5.11)
$^{cs}V_{21}(n+1)~{}^{s}F_{12}(n)+~{}^{cs}V_{23}(n+1)~{}^{s}F_{32}(n)=~{}^{c}G_{21}(n)~{}^{cs}T_{12}(n)+~{}^{c}G_{23}(n)~{}^{cs}T_{32}(n)$
(5.12) $^{cs}E_{23}(n+1)+~{}^{s}G_{23}(n)=0$ (5.13)
$^{cs}E_{21}(n+1)~{}^{s}p_{13}(n)+~{}^{cs}D_{22}(n+1)~{}^{s}G_{23}(n)+~{}^{cs}E_{23}(n+1)~{}^{s}p_{33}(n)+~{}^{cs}V_{23}(n+1)=0$
(5.14) ${}^{cs}V_{21}(n+1)~{}^{s}p_{13}(n)$ $\displaystyle+$
$\displaystyle~{}^{cs}U_{22}(n+1)~{}^{s}G_{23}(n)+^{cs}V_{23}(n+1)~{}^{s}p_{33}(n)=$
(5.15) $\displaystyle=$
${}^{c}G_{21}(n)~{}^{cs}K_{13}(n)+~{}^{c}G_{23}(n)~{}^{cs}K_{33}(n)$
$^{cs}K_{31}(n+1)+~{}^{cs}C_{32}(n+1)~{}^{s}G_{21}(n)=~{}^{c}F_{32}(n)~{}^{cs}E_{21}(n)+~{}^{cs}K_{31}(n)$
(5.16) ${}^{cs}K_{31}(n+1)~{}^{s}p_{11}(n)$ $\displaystyle+$
$\displaystyle~{}^{cs}T_{32}(n+1)~{}^{s}G_{21}(n)+~{}^{cs}K_{33}(n+1)~{}^{s}p_{31}(n)=$
(5.17) $\displaystyle=$
${}^{c}p_{31}(n)~{}^{cs}K_{11}(n)+~{}^{c}F_{32}(n)~{}^{cs}V_{21}(n)+~{}^{c}p_{33}(n)^{cs}K_{31}(n)$
$^{cs}C_{32}(n)+~{}^{c}F_{32}(n)=0$ (5.18)
$^{c}p_{31}(n)~{}^{cs}C_{12}(n)+~{}^{c}F_{32}(n)~{}^{cs}D_{22}(n)=~{}^{cs}T_{32}(n)+~{}^{c}p_{33}(n)~{}^{cs}C_{32}(n)$
(5.19) ${}^{cs}K_{31}(n+1)~{}^{s}F_{12}(n)$ $\displaystyle+$
$\displaystyle~{}^{cs}K_{33}(n+1)~{}^{s}F_{32}(n)=$ (5.20) $\displaystyle=$
${}^{c}p_{31}(n)~{}^{cs}T_{12}(n)+~{}^{c}F_{32}(n)~{}^{cs}U_{22}(n)+~{}^{c}p_{33}(n)^{cs}T_{32}(n)$
$^{cs}C_{32}(n+1)~{}^{s}G_{23}(n)+~{}^{cs}K_{33}(n+1)=~{}^{c}F_{32}(n)~{}^{cs}E_{23}(n)+~{}^{cs}K_{33}(n)$
(5.21) ${}^{cs}K_{31}(n+1)~{}^{s}p_{13}(n)$ $\displaystyle+$
$\displaystyle~{}^{cs}T_{32}(n+1)~{}^{s}G_{23}(n)+~{}^{cs}K_{33}(n+1)~{}^{s}p_{33}(n)=$
(5.22) $\displaystyle=$
${}^{c}p_{31}(n)~{}^{cs}K_{13}(n)+~{}^{c}F_{32}(n)~{}^{cs}V_{23}(n)+~{}^{c}p_{33}(n)^{cs}K_{33}(n)\,.$

Provided all fourteen Darboux functions $~{}^{cs}K_{11}(n)$,
$~{}^{cs}C_{12}(n)$, $~{}^{cs}T_{12}(n)$, $~{}^{cs}K_{13}(n)$,
$~{}^{cs}E_{21}(n)$, $~{}^{cs}V_{21}(n)$, $~{}^{cs}D_{22}(n)$,
$~{}^{cs}U_{22}(n)$, $~{}^{cs}E_{23}(n)$, $~{}^{cs}V_{23}(n)$,
$~{}^{cs}K_{31}(n)$, $~{}^{cs}C_{32}(n)$, $~{}^{cs}T_{32}(n)$,
$~{}^{cs}K_{33}(n)$ in these detailed relationships (5.1)–(5.22) have been
excluded, we come to just eight equations establishing explicit Bäcklund
transformation between eight seed and eight crop prototype field functions.

Though having been very impractical for the actual calculations of solitonic
or any other solutions the explicit Bäcklund transformation as such provides a
strong argument for the original implicit Bäcklund transformation (5.1)–(5.22)
to be naturally incorporated into the Darboux–Bäcklund integration procedure
as its crucial and intrinsically noncontradictory element.

First of all, two Bäcklund relations (5.8) and (5.13) clearly indicate that
two Darboux functions $~{}^{cs}E_{21}(n)$ and $~{}^{cs}E_{23}(n)$ are actually
known $~{}^{cs}E_{21}(n)=-~{}^{s}G_{21}(n-1)$,
$~{}^{cs}E_{23}(n)=-~{}^{s}G_{23}(n-1)$. As a consequence, the formulas (5.1),
(5.6), (5.16), (5.21) accompanied by another two Bäcklund relations (5.3) and
(5.18) yield

$^{cs}K_{11}(n)=~{}^{cs}K_{11}-~{}^{cs}C_{12}(n)~{}^{s}G_{21}(n-1)$ (5.23)
$^{cs}K_{13}(n)=~{}^{cs}K_{13}-~{}^{cs}C_{12}(n)~{}^{s}G_{23}(n-1)$ (5.24)
$^{cs}K_{31}(n)=~{}^{cs}K_{31}-~{}^{cs}C_{32}(n)~{}^{s}G_{21}(n-1)$ (5.25)
$^{cs}K_{33}(n)=~{}^{cs}K_{33}-~{}^{cs}C_{32}(n)~{}^{s}G_{23}(n-1)\,.$ (5.26)

Here the coordinate-independent parameters $~{}^{cs}K_{jk}$ are proved to be
time-independent too. To confirm their time-independence it is sufficient to
combine the expressions (5.23)–(5.26) for $~{}^{cs}K_{jk}(n)$ with the proper
evolution equations selected from the matrix-valued evolution equation for the
Darboux matrix (4.3) and to use the equalities (5.3), (5.8), (5.13), (5.18).
On the other hand, the formulas (5.9), (5.14) yield

$^{cs}V_{21}(n)=~{}^{s}G_{21}(n-1)~{}^{s}p_{11}(n-1)+~{}^{s}G_{23}(n-1)~{}^{s}p_{31}(n-1)-~{}^{cs}D_{22}(n)~{}^{s}G_{21}(n-1)$
(5.27)
$^{cs}V_{23}(n)=~{}^{s}G_{21}(n-1)~{}^{s}p_{13}(n-1)+~{}^{s}G_{23}(n-1)~{}^{s}p_{33}(n-1)-~{}^{cs}D_{22}(n)~{}^{s}G_{23}(n-1)\,.$
(5.28)

These six relationships (5.23)–(5.28) accompanied by equalities (5.8) and
(5.13) say that only six independent Darboux functions $~{}^{cs}C_{12}(n)$,
$~{}^{cs}T_{12}(n)$, $~{}^{cs}D_{22}(n)$, $~{}^{cs}U_{22}(n)$,
$~{}^{cs}C_{32}(n)$, $~{}^{cs}T_{32}(n)$ have to be found relying upon the
spectral properties of Darboux matrix.

By another words, the number of nonuniform linear equations encoded in
expression (4.13) indeed coincides with the number of unknown independent
Darboux functions. Hence, all six independent Darboux functions
$~{}^{cs}C_{12}(n)$, $~{}^{cs}T_{12}(n)$, $~{}^{cs}D_{22}(n)$,
$~{}^{cs}U_{22}(n)$, $~{}^{cs}C_{32}(n)$, $~{}^{cs}T_{32}(n)$ can be
calculated in terms of seed solution $~{}^{s}X(n|z)$ to the auxiliary linear
problem (2.4) and (2.5) supplemented by the spectral characteristics
$~{}^{cs}z_{\pm}$ and $~{}^{cs}\varepsilon(\,^{cs}z_{\pm})$ of requested
Darboux matrix (4.8). In so doing, the relevant set of six nonuniform linear
equations is split into three separate subsets given by formulas

${}^{cs}C_{12}(n)\left[~{}^{s}G_{21}(n-1)~{}^{s}Y_{1}(n|\,^{cs}z_{\pm})-\lambda(\,^{cs}z_{\pm})~{}^{s}Y_{2}(n|\,^{cs}z_{\pm})+~{}^{s}G_{23}(n-1)~{}^{s}Y_{3}(n|\,^{cs}z_{\pm})\,\right]-$
(5.29) $\displaystyle-$
$\displaystyle~{}^{cs}T_{12}(n)~{}^{s}Y_{2}(n|\,^{cs}z_{\pm})=~{}^{cs}K_{11}~{}^{s}Y_{1}(n|\,^{cs}z_{\pm})+~{}^{cs}K_{13}~{}^{s}Y_{3}(n|\,^{cs}z_{\pm})$
${}^{cs}D_{22}(n)\left[~{}^{s}G_{21}(n-1)~{}^{s}Y_{1}(n|\,^{cs}z_{\pm})-\lambda(\,^{cs}z_{\pm})~{}^{s}Y_{2}(n|\,^{cs}z_{\pm})+~{}^{s}G_{23}(n-1)~{}^{s}Y_{3}(n|\,^{cs}z_{\pm})\,\right]-$
(5.30) $\displaystyle-$
$\displaystyle~{}^{cs}U_{22}(n)~{}^{s}Y_{2}(n|\,^{cs}z_{\pm})=$
$\displaystyle=$
$\displaystyle\left[~{}^{s}G_{21}(n-1)~{}^{s}p_{11}(n-1)+~{}^{s}G_{23}(n-1)~{}^{s}p_{31}(n-1)-~{}^{s}G_{21}(n-1)\,\lambda(\,^{cs}z_{\pm})\,\right]~{}^{s}Y_{1}(n|\,^{cs}z_{\pm})+$
$\displaystyle+$
$\displaystyle\lambda^{2}(\,^{cs}z_{\pm})~{}^{s}Y_{2}(n|\,^{cs}z_{\pm})+$
$\displaystyle+$
$\displaystyle\left[~{}^{s}G_{21}(n-1)~{}^{s}p_{13}(n-1)+~{}^{s}G_{23}(n-1)~{}^{s}p_{33}(n-1)-~{}^{s}G_{23}(n-1)\,\lambda(\,^{cs}z_{\pm})\,\right]~{}^{s}Y_{3}(n|\,^{cs}z_{\pm})$
${}^{cs}C_{32}(n)\left[~{}^{s}G_{21}(n-1)~{}^{s}Y_{1}(n|\,^{cs}z_{\pm})-\lambda(\,^{cs}z_{\pm})~{}^{s}Y_{2}(n|\,^{cs}z_{\pm})+~{}^{s}G_{23}(n-1)~{}^{s}Y_{3}(n|\,^{cs}z_{\pm})\,\right]-$
(5.31) $\displaystyle-$
$\displaystyle~{}^{cs}T_{32}(n)~{}^{s}Y_{2}(n|\,^{cs}z_{\pm})=~{}^{cs}K_{31}~{}^{s}Y_{1}(n|\,^{cs}z_{\pm})+~{}^{cs}K_{33}~{}^{s}Y_{3}(n|\,^{cs}z_{\pm})\,.$

Here the shorthand notation

$^{s}Y_{j}(n|\,^{cs}z_{\pm})=\sum_{k=1}^{3}~{}^{s}X_{jk}(n|\,^{cs}z_{\pm})~{}^{cs}\varepsilon(\,^{cs}z_{\pm})$
(5.32)

has been introduced. The first set (5.29) is seen to produce the expressions
for $~{}^{cs}C_{12}(n)$ and $~{}^{cs}T_{12}(n)$. The second set (5.30) gives
rise to the expressions for $~{}^{cs}D_{22}(n)$ and $~{}^{cs}U_{22}(n)$. At
last, the third set (5.31) allows to obtain the expressions for
$~{}^{cs}C_{32}(n)$ and $~{}^{cs}T_{32}(n)$. By dint of relationships
(5.23)–(5.28) between the dependent and independent Darboux functions the
dependent Darboux functions $~{}^{cs}K_{11}(n)$, $~{}^{cs}K_{13}(n)$,
$~{}^{cs}K_{31}(n)$, $~{}^{cs}K_{33}(n)$ and $~{}^{cs}V_{21}(n)$,
$~{}^{cs}V_{23}(n)$ are also readily restorable.

Thus, the procedure of obtaining the expressions for Darboux functions relying
upon the spectral properties of Darboux matrix is seen to be complete .

The next step in our task is to select eight formulas among the implicit
Bäcklund transformation (5.1)–(5.22) suitable to convert the expressions for
Darboux functions into the expressions for crop field functions. Below we
summarize the recipes for relevant calculations.

(1) The expressions for $~{}^{c}F_{12}(n)$ and $~{}^{c}F_{32}(n)$ are given
directly by formulas (5.3) and (5.18).

(2) The expressions for $~{}^{c}G_{21}(n)$ and $~{}^{c}G_{23}(n)$ follow from
the set of two linear equations (5.10) and (5.15).

(3) The expressions for $~{}^{c}p_{11}(n)$ and $~{}^{c}p_{13}(n)$ are
obtainable from the set of two linear equations (5.2) and (5.7) with the
relationship (5.3) for $~{}^{c}F_{12}(n)$ taken into account.

(4) At last, the expressions for $~{}^{c}p_{31}(n)$ and $~{}^{c}p_{33}(n)$ are
obtainable from the set of two linear equations (5.17) and (5.22) with the
relationship (5.18) for $~{}^{c}F_{32}(n)$ taken into account.

##  6 Solution to the auxiliary linear problem versus the vacuum seed solution
to the prototype nonlinear system

In order to complete the development of Darboux–Bäcklund dressing scheme we
must rely upon some known solution to the prototype nonlinear system
(1.1)–(1.8) as the seed one. In practice the most preferable seed solution is
chosen to be a trivial (presumably vacuum) solution ensuring the respective
seed solution to the auxiliary linear problem (2.4) and (2.5) being
presentable in an explicit closed form.

The most suitable seed solution to the prototype nonlinear system of our
interest (1.1)–(1.8) looks as follows $~{}^{s}p_{11}(n)=0$,
$~{}^{s}p_{13}(n)=0$, $~{}^{s}p_{31}(n)=0$, $~{}^{s}p_{33}(n)=0$,
$~{}^{s}F_{12}(n)=~{}^{0}F_{12}$ , $~{}^{s}G_{21}(n)=~{}^{0}G_{21}$ ,
$~{}^{s}G_{23}(n)=~{}^{0}G_{23}$ , $~{}^{s}F_{32}(n)=~{}^{0}F_{32}$ , where
$~{}^{0}F_{12}$ , $~{}^{0}G_{21}$ , $~{}^{0}G_{23}$ , $~{}^{0}F_{12}$ are
coordinate- and time-independent quantities subjected to the natural condition

$~{}^{0}F_{12}~{}^{0}G_{21}+~{}^{0}F_{32}~{}^{0}G_{23}=-1\,.$ (6.1)

Consequently, the seed spectral matrix $~{}^{s}L(n|z)$ and the seed evolution
matrix $~{}^{s}A(n|z)$ become coordinate- and time-independent
$~{}^{s}L(n|z)=~{}^{0}L(z)$ and $~{}^{s}A(n|z)=~{}^{0}A(z)$. Moreover, the
zero-curvature equation (2.1) yields the strict commutativity relation

$~{}^{0}A(z)~{}^{0}L(z)-~{}^{0}L(z)~{}^{0}A(z)=0$ (6.2)

and hence the matrices $~{}^{0}L(z)$ and $~{}^{0}A(z)$ must possess the same
set of eigenfunctions. However, the eigenvalues $~{}^{0}\zeta_{k}(z)$ of
matrix $~{}^{0}L(z)$ are proved to be distinct from the eigenvalues
$~{}^{0}\eta_{k}(z)$ of matrix $~{}^{0}A(z)$. Precisely we have

$^{0}\zeta_{1}(z)=z$ (6.3) $^{0}\zeta_{2}(z)=\lambda(z)\equiv z+1/z$ (6.4)
$^{0}\zeta_{3}(z)=1/z$ (6.5)

and

$^{0}\eta_{1}(z)=1/z$ (6.6) $^{0}\eta_{2}(z)=0$ (6.7) $^{0}\eta_{3}(z)=z\,.$
(6.8)

As for the seed solution $~{}^{s}X(n|z)=~{}^{0}X(n|z)$ to the auxiliary linear
problem (2.4) and (2.5) we come to the expression given by matrix

$^{0}X(n|z)=\left(\begin{array}[]{ccccc}-~{}^{0}F_{12}~{}z^{n}\exp(\tau/z)&{}{}{}\hfil&+\mathrm{i}~{}^{0}G_{23}~{}[z+1/z]^{n}&{}{}{}\hfil&-~{}^{0}F_{12}~{}z^{-n}\exp(\tau
z)\\\\[5.69054pt]
~{}z^{n-1}\exp(\tau/z)&{}{}{}\hfil&0&{}{}{}\hfil&~{}z^{1-n}\exp(\tau
z)\\\\[5.69054pt]
-~{}^{0}F_{32}~{}z^{n}\exp(\tau/z)&{}{}{}\hfil&-\mathrm{i}~{}^{0}G_{21}~{}[z+1/z]^{n}&{}{}{}\hfil&-~{}^{0}F_{32}~{}z^{-n}\exp(\tau
z)\\\ \end{array}\right).$ (6.9)

##  7 Symmetries between the prototype field functions and their implications
for the Darboux–Bäcklund scheme

The symmetries between the dynamical fields of an integrable nonlinear
dynamical system are known to stipulate the symmetries between the scattering
data of relevant auxiliary spectral problem [12, 38, 39]. The practical role
of these symmetries for the explicit analytical integration of nonlinear
systems is also indisputable [12, 38, 39].

Analyzing the prototype nonlinear system of our interest (1.1)–(1.8) one can
readily observe that its equations are consistent with the symmetries of
complex conjugation

$p_{11}(n)=p_{33}^{\ast}(n)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}p_{33}(n)=p_{11}^{\ast}(n)$
(7.1)
$p_{13}(n)=p_{31}^{\ast}(n)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}p_{31}(n)=p_{13}^{\ast}(n)$
(7.2)
$F_{12}(n)=F_{32}^{\ast}(n)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}F_{32}(n)=F_{12}^{\ast}(n)$
(7.3)
$G_{21}(n)=G_{23}^{\ast}(n)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}G_{23}(n)=G_{32}^{\ast}(n)$
(7.4)

between the prototype field functions. In view of these symmetries (7.1)–(7.4)
we inevitably come to the following symmetry relations

$\Omega~{}\left[L(n|1/z^{\ast})\right]^{\ast}\,\Omega=L(n|z)$ (7.5)
$\Omega~{}\left[A(n|1/z^{\ast})\right]^{\ast}\,\Omega=A(n|z)$ (7.6)

for the spectral $L(n|z)$ and evolution $A(n|z)$ matrices written in terms of
prototype fields. Here the involutory matrix $\Omega$ is given by formula

$\Omega=\left(\begin{array}[]{ccccc}0&{}{}\hfil&0&{}{}\hfil&1\\\
0&{}{}\hfil&1&{}{}\hfil&0\\\ 1&{}{}\hfil&0&{}{}\hfil&0\\\
\end{array}\right)\,.$ (7.7)

We impose the similar symmetries

$\Omega~{}\left[X(n|1/z^{\ast})\right]^{\ast}\,\Omega=X(n|z)$ (7.8)
$\Omega~{}\left[\,{}^{cs}D(n|1/z^{\ast})\right]^{\ast}\,\Omega=~{}^{cs}D(n|z)$
(7.9)

to be valid also for the matrix-solution $X(n|z)$ to the auxiliary linear
problem (2.4) and (2.5) as well as for the Darboux matrix $~{}^{cs}D(n|z)$
(4.8).

By the way, the expression (6.9) for the zero-labeled seed matrix
$~{}^{0}X(n|z)$ has been purposely arranged to be consistent with the just
adopted symmetries.

By means of the last two symmetry relations (7.9) and (7.8) one can establish
the symmetry relations

$^{cs}z^{\ast}_{+}~{}^{cs}z_{-}=1=~{}^{cs}z^{\ast}_{-}~{}^{cs}z_{+}$ (7.10)

and

${\left[\,{}^{cs}\varepsilon_{1}(\,^{cs}z_{+})\right]^{\ast}}/{~{}^{cs}\varepsilon_{3}(\,^{cs}z_{-})}=1={\left[\,{}^{cs}\varepsilon_{3}(\,^{cs}z_{-})\right]^{\ast}}/{~{}^{cs}\varepsilon_{1}(\,^{cs}z_{+})}$
(7.11)
${\left[\,{}^{cs}\varepsilon_{2}(\,^{cs}z_{+})\right]^{\ast}}/{~{}^{cs}\varepsilon_{2}(\,^{cs}z_{-})}=1={\left[\,{}^{cs}\varepsilon_{2}(\,^{cs}z_{-})\right]^{\ast}}/{~{}^{cs}\varepsilon_{2}(\,^{cs}z_{+})}$
(7.12)
${\left[\,{}^{cs}\varepsilon_{3}(\,^{cs}z_{+})\right]^{\ast}}/{~{}^{cs}\varepsilon_{1}(\,^{cs}z_{-})}=1={\left[\,{}^{cs}\varepsilon_{1}(\,^{cs}z_{-})\right]^{\ast}}/{~{}^{cs}\varepsilon_{3}(\,^{cs}z_{+})}$
(7.13)

for the spectral data $~{}^{cs}z_{\pm}$ and
$~{}^{cs}\varepsilon_{k}(\,^{cs}z_{\pm})$. When tackling these symmetries
(7.10)–(7.13) the expression (4.11) for $\det\,^{cs}D(n|z)$ and the condition
(4.12) supporting the property $\det\,^{c}X(n|~{}^{cs}z_{\pm})=0$ have also to
be invoked.

##  8 Prototype crop solution generated by the two-fold Darboux–Bäcklund
transformation

In this section we consider the practical implementation of two-fold
Darboux–Bäcklund integration scheme with regard to the prototype integrable
nonlinear system (1.1)–(1.8). The last four equations (1.5)–(1.8) of this
system clearly indicate that the basic calculations can be safely concentrated
only on four prototype functions $F_{12}(n)$, $G_{21}(n)$, $G_{23}(n)$,
$F_{32}(n)$.

To implement our task let us summarize the relevant steps to be made in the
framework of two-fold Darboux–Bäcklund integration scheme already reported in
the previous Sections. Thus, three sets (5.29)–(5.31) of nonuniform linear
equations provide us with the solutions for the Darboux functions
$~{}^{cs}C_{12}(n)$, $~{}^{cs}T_{12}(n)$, $~{}^{cs}D_{22}(n)$,
$~{}^{cs}U_{22}(n)$, $~{}^{cs}C_{32}(n)$, $~{}^{cs}T_{32}(n)$. By virtue of
relationships (5.23)–(5.26) the obtained expressions for $~{}^{cs}C_{12}(n)$
and $~{}^{cs}C_{32}(n)$ give rise to the solutions for the Darboux functions
$~{}^{cs}K_{11}(n)$, $~{}^{cs}K_{13}(n)$, $~{}^{cs}K_{31}(n)$,
$~{}^{cs}K_{33}(n)$. The expressions for another two Darboux functions
$~{}^{cs}V_{21}(n)$ and $~{}^{cs}V_{23}(n)$ follow from the relationships
(5.27) and (5.28) by the use of already known expression for
$~{}^{cs}D_{22}(n)$. Once the explicit expressions for the Darboux functions
$~{}^{cs}C_{12}(n)$ and $~{}^{cs}C_{32}(n)$ have been found, the prototype
crop functions $~{}^{c}F_{12}(n)$, $~{}^{c}F_{32}(n)$ are given by two proper
formulas (5.3) and (5.18) of implicit Bäcklund transformation. Another two
prototype crop functions $~{}^{c}G_{21}(n)$ and $~{}^{c}G_{23}(n)$ are
obtainable as the solutions to the set of two nonuniform linear equations
(5.10) and (5.15) of implicit Bäcklund transformation inasmuch as all involved
Darboux functions $~{}^{cs}K_{11}(n)$, $~{}^{cs}K_{13}(n)$,
$~{}^{cs}K_{31}(n)$, $~{}^{cs}K_{33}(n)$ and $~{}^{cs}V_{21}(n)$,
$~{}^{cs}U_{22}(n)$, $~{}^{cs}V_{23}(n)$ have been restored beforehand.

Though just outlined dressing procedure is valid to start with the prototype
seed solution $~{}^{s}p_{11}(n)$, $~{}^{s}p_{13}(n)$, $~{}^{s}p_{31}(n)$,
$~{}^{s}p_{33}(n)$, $~{}^{s}F_{12}(n)$ , $~{}^{s}G_{21}(n)$ ,
$~{}^{s}G_{23}(n)$ , $~{}^{s}F_{32}(n)$ of an arbitrary order
($s=0,1,2,3,...$), we apply it to a priori known trivial solution
$~{}^{0}p_{11}(n)=0$, $~{}^{0}p_{13}(n)=0$, $~{}^{0}p_{31}(n)=0$,
$~{}^{0}p_{33}(n)=0$, $~{}^{0}F_{12}(n)=~{}^{0}F_{12}$ ,
$~{}^{0}G_{21}(n)=~{}^{0}G_{21}$ , $~{}^{0}G_{23}(n)=~{}^{0}G_{23}$ ,
$~{}^{0}F_{32}(n)=~{}^{0}F_{32}$ labeled by index $s=0$. Then, taking into
account the explicit formula (6.9) for the zeroth seed solution
$~{}^{0}X(n|z)$ to the auxiliary linear problem (2.4) and (2.5) we find the
prototype crop functions $~{}^{2}F_{12}(n)$, $~{}^{2}F_{32}(n)$,
$~{}^{2}G_{21}(n)$, $~{}^{2}G_{23}(n)$ in the form

${}^{2}F_{12}(n)$ $\displaystyle=$
$\displaystyle(\,^{20}K_{11}~{}^{0}F_{12}+~{}^{20}K_{13}~{}^{0}F_{32})\times$
(8.1) $\displaystyle\times$
$\displaystyle\frac{{}^{0}R(n|\,^{20}z_{+})~{}^{0}R(n-1|\,^{20}z_{-})-~{}^{0}R(n|\,^{20}z_{-})~{}^{0}R(n-1|\,^{20}z_{+})}{{}^{0}R(n-1|\,^{20}z_{+})~{}^{0}R(n-2|\,^{20}z_{-})-~{}^{0}R(n-1|\,^{20}z_{-})~{}^{0}R(n-2|\,^{20}z_{+})}-$
$\displaystyle-$
$\displaystyle\mathrm{i}\,(\,^{0}G_{23}~{}^{20}K_{11}-~{}^{0}G_{21}~{}^{20}K_{13})\times$
$\displaystyle\times$
$\displaystyle\frac{{}^{0}S(n|\,^{20}z_{+})~{}^{0}R(n-1|\,^{20}z_{-})-~{}^{0}S(n|\,^{20}z_{-})~{}^{0}R(n-1|\,^{20}z_{+})}{{}^{0}R(n-1|\,^{20}z_{+})~{}^{0}R(n-2|\,^{20}z_{-})-~{}^{0}R(n-1|\,^{20}z_{-})~{}^{0}R(n-2|\,^{20}z_{+})}$
${}^{2}F_{32}(n)$ $\displaystyle=$
$\displaystyle(\,^{20}K_{33}~{}^{0}F_{32}+~{}^{20}K_{31}~{}^{0}F_{12})\times$
(8.2) $\displaystyle\times$
$\displaystyle\frac{{}^{0}R(n|\,^{20}z_{+})~{}^{0}R(n-1|\,^{20}z_{-})-~{}^{0}R(n|\,^{20}z_{-})~{}^{0}R(n-1|\,^{20}z_{+})}{{}^{0}R(n-1|\,^{20}z_{+})~{}^{0}R(n-2|\,^{20}z_{-})-~{}^{0}R(n-1|\,^{20}z_{-})~{}^{0}R(n-2|\,^{20}z_{+})}+$
$\displaystyle+$
$\displaystyle\mathrm{i}\,(\,^{0}G_{21}~{}^{20}K_{33}-~{}^{0}G_{23}~{}^{20}K_{31})\times$
$\displaystyle\times$
$\displaystyle\frac{{}^{0}S(n|\,^{20}z_{+})~{}^{0}R(n-1|\,^{20}z_{-})-~{}^{0}S(n|\,^{20}z_{-})~{}^{0}R(n-1|\,^{20}z_{+})}{{}^{0}R(n-1|\,^{20}z_{+})~{}^{0}R(n-2|\,^{20}z_{-})-~{}^{0}R(n-1|\,^{20}z_{-})~{}^{0}R(n-2|\,^{20}z_{+})}$
${}^{2}G_{21}(n)$ $\displaystyle=$
$\displaystyle\frac{{}^{0}G_{21}~{}^{20}K_{33}-~{}^{0}G_{23}~{}^{20}K_{31}}{{}^{20}K_{11}~{}^{20}K_{33}-~{}^{20}K_{13}~{}^{20}K_{31}}\times$
(8.3) $\displaystyle\times$
$\displaystyle\frac{{}^{0}R(n-1|\,^{20}z_{+})~{}^{0}R(n-2|\,^{20}z_{-})-~{}^{0}R(n-1|\,^{20}z_{-})~{}^{0}R(n-2|\,^{20}z_{+})}{{}^{0}R(n|\,^{20}z_{+})~{}^{0}R(n-1|\,^{20}z_{-})-~{}^{0}R(n|\,^{20}z_{-})~{}^{0}R(n-1|\,^{20}z_{+})}$
${}^{2}G_{23}(n)$ $\displaystyle=$
$\displaystyle\frac{{}^{0}G_{23}~{}^{20}K_{11}-~{}^{0}G_{21}~{}^{20}K_{13}}{{}^{20}K_{33}~{}^{20}K_{11}-~{}^{20}K_{31}~{}^{20}K_{13}}\times$
(8.4) $\displaystyle\times$
$\displaystyle\frac{{}^{0}R(n-1|\,^{20}z_{+})~{}^{0}R(n-2|\,^{20}z_{-})-~{}^{0}R(n-1|\,^{20}z_{-})~{}^{0}R(n-2|\,^{20}z_{+})}{{}^{0}R(n|\,^{20}z_{+})~{}^{0}R(n-1|\,^{20}z_{-})-~{}^{0}R(n|\,^{20}z_{-})~{}^{0}R(n-1|\,^{20}z_{+})}~{}.$

Here

$^{0}R(n|\,^{20}z_{\pm})=(\,^{20}z_{\pm})^{n}~{}\exp(\tau/\,^{20}z_{\pm})~{}^{20}\varepsilon_{1}(\,^{20}z_{\pm})+(1/\,^{20}z_{\pm})^{n}~{}\exp(\tau~{}^{20}z_{\pm})~{}^{20}\varepsilon_{3}(\,^{20}z_{\pm})$
(8.5)
$^{0}S(n|\,^{20}z_{\pm})=~{}^{20}\varepsilon_{2}(\,^{20}z_{\pm})\,(\,^{20}z_{\pm}+1/\,^{20}z_{\pm})^{n}.$
(8.6)

In what follows we take

$^{20}K_{11}=1=~{}^{20}K_{33}$ (8.7)

and

$^{20}K_{13}=0=~{}^{20}K_{31}$ (8.8)

without the loss of generality.

By virtue of the symmetry relations (7.10)–(7.13) it is reasonable to
parameterize the spectral data by formulas

$^{20}z_{\pm}=\exp(\pm\mu+\mathrm{i}k)$ (8.9)
$^{20}\varepsilon_{1}(\,^{20}z_{\pm})=\exp(\pm\alpha+\mathrm{i}\beta)$ (8.10)
$^{20}\varepsilon_{2}(\,^{20}z_{\pm})=g\exp(\pm\mathrm{i}\delta)$ (8.11)
$^{20}\varepsilon_{3}(\,^{20}z_{\pm})=\exp(\mp\alpha-\mathrm{i}\beta)\,,$
(8.12)

where $k$, $\alpha$, $\beta$, $g$, $\delta$ are arbitrary real parameters, and
$\mu$ is the real positive parameter. To suppress systematic divergences of
crop field functions at spatial and temporal infinities we are obliged to
impose two restrictions

$\mu>\ln(\sqrt{2})$ (8.13) $k=\sigma\pi/2$ (8.14)

onto the parameters $\mu$ and $k$ with the coefficient $\sigma$ being defined
by the equality

$\sigma^{2}=1\,.$ (8.15)

In order to apply the general results of this Section to an actual nonlinear
integrable dynamical system we must adopt some proper parametrization of
prototype field variables compatible both with the natural constraints (3.17),
(3.18) and the earlier adopted symmetries (7.1)–(7.4).

##  9 Reduced ternary system in the framework of Lagrangian and Hamiltonian
formulations

As we have already mentioned in Section 3, the natural constraints (3.17) and
(3.18) are empowered to reduce eight prototype field variables to six actual
dynamical ones. In so doing, the dynamics of the reduced system will be
governed by three nonlinear Lagrange equations or alternatively by six
nonlinear Hamilton equations. The key idea to introduce appropriate dynamical
variables is to invent parametrization formulas converting both of natural
constraints (3.2) and (3.3) into identities with the symmetry relations
(7.1)–(7.4) taken into account. A particular realization of such
parameterizations is not unique [22, 23] and here we consider the following
one

$F_{12}(n)=+\exp[+q_{-}(n)]\sqrt{[1+\mathrm{i}s(n)]/2}$ (9.1)
$G_{21}(n)=-\exp[-q_{-}(n)]\sqrt{[1+\mathrm{i}s(n)]/2}$ (9.2)
$G_{23}(n)=-\exp[-q_{+}(n)]\sqrt{[1-\mathrm{i}s(n)]/2}$ (9.3)
$F_{32}(n)=+\exp[+q_{+}(n)]\sqrt{[1-\mathrm{i}s(n)]/2}$ (9.4)
$p_{11}(n)=\frac{1}{4}[\,\dot{q}_{-}(n)-\dot{q}_{+}(n)][1+s^{2}(n)]+\frac{1}{2}\dot{q}_{-}(n)[1+\mathrm{i}s(n)]$
(9.5)
$p_{13}(n)=\frac{\exp[\,q_{-}(n)-q_{+}(n)]}{4\sqrt{1+s^{2}(n)}}\Bigl{\\{}\dot{q}_{+}(n)[1+s^{2}(n)][1+\mathrm{i}s(n)]+\dot{q}_{-}(n)[1+s^{2}(n)][1-\mathrm{i}s(n)]+2\mathrm{i}s(n)\Bigr{\\}}$
(9.6)
$p_{31}(n)=\frac{\exp[\,q_{+}(n)-q_{-}(n)]}{4\sqrt{1+s^{2}(n)}}\Bigl{\\{}\dot{q}_{-}(n)[1+s^{2}(n)][1-\mathrm{i}s(n)]+\dot{q}_{+}(n)[1+s^{2}(n)][1+\mathrm{i}s(n)]-2\mathrm{i}s(n)\Bigr{\\}}$
(9.7)
$p_{33}(n)=\frac{1}{4}[\,\dot{q}_{+}(n)-\dot{q}_{-}(n)][1+s^{2}(n)]+\frac{1}{2}\dot{q}_{+}(n)[1-\mathrm{i}s(n)]\,.$
(9.8)

These parametrization formulas (9.1)–(9.8) are seen to support the complex
conjugate symmetries $q_{-}(n)=q_{+}^{\ast}(n)$ and $q_{+}(n)=q_{-}^{\ast}(n)$
between the field variables $q_{-}(n)$ and $q_{+}(n)$ supplemented by the
reality $s(n)=s^{\ast}(n)$ of field variable $s(n)$. Evidently, the same
symmetries take place for the time derivatives of field variables.

By dint of the adopted parameterizations (9.1)–(9.8), the prototype nonlinear
system (1.1)–(1.8) is reduced to an actual nonlinear lattice system governed
by the set of three Lagrange equations

$\frac{d}{d\tau}[\partial{\mathcal{L}}/\partial\dot{q}_{+}(n)]=\partial{\mathcal{L}}/\partial
q_{+}(n)$ (9.9)
$\frac{d}{d\tau}[\partial{\mathcal{L}}/\partial\dot{s}(n)]=\partial{\mathcal{L}}/\partial
s(n)$ (9.10)
$\frac{d}{d\tau}[\partial{\mathcal{L}}/\partial\dot{q}_{-}(n)]=\partial{\mathcal{L}}/\partial
q_{-}(n)$ (9.11)

with the Lagrange function given by expression

$\displaystyle{\mathcal{L}}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\,\sum_{m=-\infty}^{\infty}[1-\mathrm{i}s(m)]\dot{q}^{2}_{+}(m)+\frac{1}{4}\,\sum_{m=-\infty}^{\infty}[1+\mathrm{i}s(m)]\dot{q}^{2}_{-}(m)+\frac{1}{8}\,\sum_{m=-\infty}^{\infty}\,[1+s^{2}(m)][\dot{q}_{+}(m)-\dot{q}_{-}(m)]^{2}$
(9.12) $\displaystyle+$
$\displaystyle\frac{1}{4}\,\sum_{m=-\infty}^{\infty}\frac{\dot{s}^{2}(m)}{1+s^{2}(m)}-\frac{1}{2}\,\sum_{m=-\infty}^{\infty}\exp\bigl{[}+q_{+}(m+1)-q_{+}(m)\bigr{]}\sqrt{[1-\mathrm{i}s(m+1)][1-\mathrm{i}s(m)]}$
$\displaystyle-$
$\displaystyle\frac{1}{2}\,\sum_{m=-\infty}^{\infty}\exp\bigl{[}+q_{-}(m+1)-q_{-}(m)\bigr{]}\sqrt{[1+\mathrm{i}s(m+1)][1+\mathrm{i}s(m)]}+\sum_{m=-\infty}^{\infty}1\,.$

The relevant Hamiltonian formulation of reduced nonlinear system in question
is based upon the set of Hamilton equations

$\dot{q}_{+}(n)=\partial{\mathcal{H}}/\partial
p_{+}(n)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\dot{p}_{+}(n)=-\partial{\mathcal{H}}/\partial
q_{+}(n)$ (9.13) $\dot{s}(n)=\partial{\mathcal{H}}/\partial
r(n)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\dot{r}(n)=-\partial{\mathcal{H}}/\partial
s(n)$ (9.14) $\dot{q}_{-}(n)=\partial{\mathcal{H}}/\partial
p_{-}(n)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\dot{p}_{-}(n)=-\partial{\mathcal{H}}/\partial
q_{-}(n)$ (9.15)

with the Hamilton function given by expression

$\displaystyle{\mathcal{H}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\,\sum_{m=-\infty}^{\infty}\frac{p^{2}_{+}(m)}{1-\mathrm{i}s(m)}+\frac{1}{2}\,\sum_{m=-\infty}^{\infty}\frac{p^{2}_{-}(m)}{1+\mathrm{i}s(m)}+\frac{1}{4}\,\sum_{m=-\infty}^{\infty}\,[\,p_{+}(m)+p_{-}(m)]^{2}$
(9.16) $\displaystyle+$
$\displaystyle\sum_{m=-\infty}^{\infty}\,[1+s^{2}(m)]r^{2}(m)+\frac{1}{2}\,\sum_{m=-\infty}^{\infty}\exp\bigl{[}\,q_{+}(m+1)-q_{+}(m)\bigr{]}\sqrt{[1-\mathrm{i}s(m+1)][1-\mathrm{i}s(m)]}$
$\displaystyle+$
$\displaystyle\frac{1}{2}\,\sum_{m=-\infty}^{\infty}\exp\bigl{[}\,q_{-}(m+1)-q_{-}(m)\bigr{]}\sqrt{[1+\mathrm{i}s(m+1)][1+\mathrm{i}s(m)]}-\sum_{m=-\infty}^{\infty}1~{}.$

The standard Hamiltonian form (9.13)–(9.15) of above dynamical equations
points out on the standard form of related Poisson structure.

In either of its incarnations (9.9)–(9.12) or (9.13)–(9.16) the reduced
nonlinear dynamical system comprises three dynamical subsystems coupled both
in their kinetic and potential parts and it can be referred to as the
nonlinear integrable ternary system with the combined inter-mode couplings.
Two of its subsystems are seen to be qualified as the subsystems of complex-
valued Toda-like type.

One can readily verify that the Lagrange ${\mathcal{L}}$ (9.12) and Hamilton
${\mathcal{H}}$ (9.16) functions are related via the Legendre transformation

${\mathcal{H}}+{\mathcal{L}}=\sum_{m=-\infty}^{\infty}[p_{+}(m)\dot{q}_{+}(m)+r(m)\dot{s}(m)+p_{-}(m)\dot{q}_{-}(n)]\\\
$

with the quantities $p_{+}(n)$, $r(n)$, $p_{-}(n)$ being defined by the
conventional relationships

$p_{+}(n)=\partial{\mathcal{L}}/\partial\dot{q}_{+}(n)$ (9.17)
$r(n)=\partial{\mathcal{L}}/\partial\dot{s}(n)$ (9.18)
$p_{-}(n)=\partial{\mathcal{L}}/\partial\dot{q}_{-}(n)\,.$ (9.19)

##  10 Modulated pulson solution to the nonlinear integrable ternary system

According to the Lagrange representation (9.9)–(9.12) the integration of
reduced nonlinear system can be safely restricted only by calculations of
coordinate field functions $q_{+}(n)$, $s(n)$, $q_{-}(n)$. To accomplish this
purpose we must invert earlier obtained results (8.1)–(8.4) for
$~{}^{2}F_{12}(n)$, $~{}^{2}G_{21}(n)$, $~{}^{2}G_{23}(n)$, $~{}^{2}F_{32}(n)$
into formulas for $~{}^{2}q_{+}(n)$, $~{}^{2}s(n)$, $~{}^{2}q_{-}(n)$ by means
of respective parametrization formulas (9.1)–(9.4). In so doing, the
specification formulas (8.5)–(8.15) supplemented by the parametrization
formulas

$^{0}F_{12}=+\exp(+q_{-})\sqrt{(1+\mathrm{i}s)/2}$ (10.1)
$^{0}G_{21}=-\exp(-q_{-})\sqrt{(1+\mathrm{i}s)/2}$ (10.2)
$^{0}G_{23}=-\exp(-q_{+})\sqrt{(1-\mathrm{i}s)/2}$ (10.3)
$^{0}F_{32}=+\exp(+q_{+})\sqrt{(1-\mathrm{i}s)/2}$ (10.4)

for the prototype seed quantities $~{}^{0}F_{12}$ , $~{}^{0}G_{21}$ ,
$~{}^{0}G_{23}$ , $~{}^{0}F_{32}$ turn out to be useful. The final result for
the crop solution to the reduced integrable nonlinear lattice system of our
interest (9.9)–(9.12) obtainable in the framework of two-fold Darboux–Bäcklund
dressing technique reads as follows

$\displaystyle q_{+}(n)$ $\displaystyle=$ $\displaystyle
q_{+}+\frac{1}{2}\ln\left[\frac{\Phi^{2}(n)}{\Phi^{2}(n-1)}\right]+$ (10.5)
$\displaystyle+$
$\displaystyle\frac{1}{4}\ln\left\\{\left[1+s~{}\frac{\exp(-q_{+}-q_{-})}{\sqrt{1+s^{2}}}~{}\frac{\Theta(n)}{\Phi(n)}\right]^{2}+\frac{\exp(-2q_{+}-2q_{-})}{1+s^{2}}~{}\frac{\Theta^{2}(n)}{\Phi^{2}(n)}\right\\}-$
$\displaystyle-$
$\displaystyle\frac{\mathrm{i}}{2}\arctan\left[\frac{\exp(-q_{+}-q_{-})~{}\Theta(n)}{\sqrt{1+s^{2}}~{}\Phi(n)+s~{}\exp(-q_{-}-q_{+})~{}\Theta(n)}\right]$
$s(n)=s+\exp(-q_{+}-q_{-})\,\sqrt{1+s^{2}}\,~{}\frac{\Theta(n)}{\Phi(n)}$
(10.6) $\displaystyle q_{-}(n)$ $\displaystyle=$ $\displaystyle
q_{-}+\frac{1}{2}\ln\left[\frac{\Phi^{2}(n)}{\Phi^{2}(n-1)}\right]+$ (10.7)
$\displaystyle+$
$\displaystyle\frac{1}{4}\ln\left\\{\left[1+s~{}\frac{\exp(-q_{+}-q_{-})}{\sqrt{1+s^{2}}}~{}\frac{\Theta(n)}{\Phi(n)}\right]^{2}+\frac{\exp(-2q_{+}-2q_{-})}{1+s^{2}}~{}\frac{\Theta^{2}(n)}{\Phi^{2}(n)}\right\\}+$
$\displaystyle+$
$\displaystyle\frac{\mathrm{i}}{2}\arctan\left[\frac{\exp(-q_{+}-q_{-})~{}\Theta(n)}{\sqrt{1+s^{2}}~{}\Phi(n)+s~{}\exp(-q_{-}-q_{+})~{}\Theta(n)}\right]\,.$

Here the upper front indices in field functions $~{}^{2}q_{+}(n)$,
$~{}^{2}s(n)$, $~{}^{2}q_{-}(n)$ have been omitted for the stylistic purposes
and the shorthand notations

$\displaystyle\Phi(n)=2\sigma\sinh(2\mu
n-\mu+2\alpha)+2\sinh(\mu)\,\sin[\sigma\pi
n-\sigma\pi/2+2\beta-2\tau\sigma\cosh(\mu)]$ (10.8) $\displaystyle\Theta(n)$
$\displaystyle=$ $\displaystyle g\exp(+\mu
n-\mu+\alpha)[2\sinh(\mu)]^{n}\,\sin[\sigma\pi/2+\delta-\beta+\tau\sigma\exp(-\mu)]+$
(10.9) $\displaystyle+$ $\displaystyle g\exp(-\mu
n+\mu-\alpha)[2\sinh(\mu)]^{n}~{}\sin[\sigma\pi
n-\sigma\pi/2+\delta+\beta-\tau\sigma\exp(+\mu)]$

for the typical functional combinations have been introduced. The obtained
solution (10.5)–(10.7) embraces all three coupled subsystems to be excited.
Considering the expression (10.8) for the function $\Phi(n)$ appearing in
denominators of all three components $q_{+}(n)$, $s(n)$, $q_{-}(n)$ of our
solution (10.5)–(10.7), we clearly see that it has distinct signs at opposite
spatial infinities. Consequently, at some unfortunate values of parameter
$\alpha$ the quantity $\Phi(n)$ as the function of time $\tau$ can change its
sign even at some admissible (i.e. integer) values of spatial coordinate $n$.
At such spatial coordinates all three components $q_{+}(n)$, $s(n)$,
$q_{-}(n)$ as the functions of time $\tau$ are inflicted by the infinite
discontinuities emerging periodically in time with the cyclic frequency equal
to $2\cosh(\mu)$.

In order to eliminate such nonphysical solutions we must impose the additional
condition $\alpha=-\mu l$ onto the parameter $\alpha$, where $l$ is an
arbitrary integer. In this adjusted case we have $\sinh(2\mu n-\mu-2\mu
l)\leq-\sinh(\mu)$ at $n\leq l$ and $\sin(2\mu n-\mu-2\mu l)\geq+\sinh(\mu)$
at $n\geq l+1$. Therefore, the modified quantity

$\displaystyle\Phi(n)=2\sigma\sinh(2\mu n-\mu-2\mu
l)+2\sinh(\mu)\,\sin[\sigma\pi n-\sigma\pi/2+2\beta-2\tau\sigma\cosh(\mu)]$
(10.10)

taken as the function of time $\tau$ can acquire zero values only in two
neighboring spatial points $n=l$ and $n=l+1$ never changing its sign in either
of them.

As a result, the components $q_{+}(n)$ and $q_{-}(n)$ demonstrate splashes of
infinite amplitudes appearing periodically in time with the cyclic frequency
equal to $2\cosh(\mu)$ in three neighboring spatial points $n=l$, $n=l+1$,
$n=l+2$, while the component $s(n)$ demonstrates splashes only in two points
$n=l$ and $n=l+1$. In so doing, all three components remain being continuous
functions of time $\tau$. Evidently, the splashes taking place in three
distinct spatial points $n=l$, $n=l+1$, $n=l+2$ occur in different instances
of time. Moreover, in each of three analyzed formulas (10.5)–(10.7) we observe
contributions modulated by the function $\Theta(n)$ characterized by two low
cyclic frequencies $\exp(+\mu)$ and $\exp(-\mu)$. For this reason the
nonlinear wave packet described by the obtained solution (10.5)–(10.7) could
be referred to as the modulated pulson. In general, all three cyclic
frequencies $2\cosh(\mu)$ and $\exp(+\mu)$, $\exp(-\mu)$ are seen to be
incommensurable. Hence, the spatial pattern of modulated pulson observed at
one instant of time can never be repeated at any other instant.

##  11 Conclusion

By writing this paper we have tried to attract attention to the broad
investigation of multi-component semi-discrete nonlinear integrable systems
associated with the auxiliary linear problems of third and more higher orders.
In view of very sophisticated dynamics prospective for the description of
various physical phenomena such systems require the development of adequate
and understandable methods of their integration. The first fundamental steps
in this direction had been made by Caudrey in his famous work [34] suggested
the most general inverse scattering approach to the integration of semi-
discrete nonlinear integrable systems generated by the auxiliary linear
problems of arbitrary orders. Yet, up to now his pioneer attempts appear to be
underestimated. Maybe such an attitude is caused by the tedious analytical
calculations inevitably arising at practical implementation of Caudrey’s
integration scheme [22, 23].

In present paper we have considered the alternative approach of integration as
applied to the particular prototype semi-discrete nonlinear integrable system
associated with the relevant auxiliary linear problem of third order. The
approach is based upon the two-fold Darboux transformation for the seed
auxiliary matrix function accompanied by the implicit Bäcklund transformation
for a supposedly known seed solution to the prototype semi-discrete nonlinear
integrable system itself. The majority of calculations have been made in terms
of prototype field functions in order that their results to be suitable for
any reduced semi-discrete nonlinear integrable system compatible with the so
called natural constraints and the adopted symmetries for the prototype field
functions.

We have also used the generalized version of direct recurrent approach to find
the main conservation laws applicable to any integrable system associated with
the chosen form of third order auxiliary linear problem. One of the obtained
conservation laws originates the Hamilton function of any reduced semi-
discrete nonlinear integrable system comprising three coupled dynamical
subsystems. This fact is readily verifiable on the reduced ternary system
explicitly written down in terms of two complex-valued Toda-like subsystems
and one intermediate subsystem. The coupling between the Toda-like subsystems
is mediated by the intermediate subsystem both in the kinetic and potential
parts of Hamilton function.

Having taken into account the general findings of two-fold Darboux–Bäcklund
dressing scheme we have obtained the explicit solution to the reduced ternary
system in the form of modulated pulson wave packet.

As the final remark, it is necessary to stress that the complex-valued
dynamical fields in the reduced ternary system can be rearranged into the
purely real fields responsible for the translational and orientational modes
typical of long macromolecules both synthesized and natural origins.

## Acknowledgement

The work has been supported by The National Academy of Sciences of Ukraine
within the Project No 0120U100858 (Functional properties of materials
prospective for nanotechnologies).

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