# The vector form of Kundu-Eckhaus equation and its simplest solutions

Aleksandr O. Smirnov, Aleksandra A. Caplieva
St.-Petersburg State University of Aerospace Instrumentation
B. Morskaya str., 67A, St.-Petersburg, 1900000, Russia

###### Abstract

In our work a hierarchy of integrable vector nonlinear differential equations
depending on the functional parameter $r$ is constructed using a monodromy
matrix. The first equation of this hierarchy for
$r=\alpha(\mathbf{p}^{t}\mathbf{q})$ is vector analogue of the Kundu-Eckhaus
equation. When $\alpha=0$, the equations of this hierarchy turn into equations
of the Manakov system hierarchy. New elliptic solutions to vector analogue of
the Kundu-Eckhaus and Manakov system are presented. In conclusion, it is shown
that there exist linear transformations of solutions to vector integrable
nonlinear equations into other solutions to the same equations.

Keywords: Monodromy matrix, spectral curve, derivative nonlinear Schrodinger
equation, vector integrable nonlinear equation

## Introduction

It is well known that the derived nonlinear Schrodinger equations [23, 8, 16,
17, 33, 31] have numerous applications in various fields of physics and
mathematics. In this regard, studies of various types of solutions to these
equations are constantly being carried out (see, for example, [5, 21, 38, 40,
30, 2]). At the same time, it should be noted that along with the Kaup-Newell
[23] equation

$ip_{z}+p_{tt}+i(\left\lvert p\right\rvert^{2}p)_{t}=0,$ (1)

Chen-Lee-Liu equation [8]

$ip_{z}+p_{tt}+i\left\lvert p\right\rvert^{2}p_{t}=0,$ (2)

and Gerdjikov-Ivanov equation [16, 17]

$ip_{z}+p_{tt}-ip^{2}p_{t}^{\ast}+\dfrac{1}{2}\left\lvert
p\right\rvert^{4}p=0,$ (3)

there exists the Kundu-Eckhaus equation [24, 6, 25, 39, 41]

$ip_{z}+p_{tt}-2\sigma\left\lvert p\right\rvert^{2}p+\alpha^{2}\left\lvert
p\right\rvert^{4}p+2i\alpha\partial_{t}\left(\left\lvert
p\right\rvert^{2}\right)p=0,\quad\sigma=\pm 1.$ (4)

Equation (4), as well as equations (1)-(3), contains first derivatives and
also has numerous applications.

However, there is a significant difference between equations (1)-(3) and (4).
The first three equations are consequences of compatibility conditions of Lax
pairs with quadratic in spectral parameter Lax operators. Equation (4), in
contrast to equations (1)-(3), is a result of the gauge transformation

$p=\widehat{p}e^{-i\theta},\quad\theta=\alpha\int\left\lvert
p\right\rvert^{2}dt$

of a solution $\widehat{p}$ to the nonlinear Schrödinger equation

$i\widehat{p}_{z}+\widehat{p}_{tt}-2\sigma\left\lvert\widehat{p}\right\rvert^{2}\widehat{p}=0.$

The ever-growing traffic in networks requires finding ways to increase
bandwidth of optical fibers. Therefore, researchers are actively working on
vector models of nonlinear optical waves propagation [19, 14, 15]. Many of
these models have been known for a long time. One such model is the Manakov
system [27, 12, 13, 34, 18, 32]

$\displaystyle\partial_{z}p_{1}=i\partial^{2}_{t}p_{1}-2i\sigma(\left\lvert
p_{1}\right\rvert^{2}+\left\lvert p_{2}\right\rvert^{2})p_{1},$ (5)
$\displaystyle\partial_{z}p_{2}=i\partial^{2}_{t}p_{2}-2i\sigma(\left\lvert
p_{1}\right\rvert^{2}+\left\lvert p_{2}\right\rvert^{2})p_{2}.$

Let us note that vector nonlinear Schrödinger equations also have derivative
forms, one of which is the equations [28, 37, 26, 7, 20]

$\displaystyle
i\partial_{z}p_{1}=-\partial^{2}_{t}p_{1}-\dfrac{2i}{3}\partial_{t}\left[\left(\left\lvert
p_{1}\right\rvert^{2}+\left\lvert p_{2}\right\rvert^{2}\right)p_{1}\right],$
(6) $\displaystyle
i\partial_{z}p_{2}=-\partial^{2}_{t}p_{2}-\dfrac{2i}{3}\partial_{t}\left[\left(\left\lvert
p_{1}\right\rvert^{2}+\left\lvert p_{2}\right\rvert^{2}\right)p_{2}\right].$

In contrast to the above works, we decided to investigate a vector analogue of
the Kundu-Eckhaus equation. In this paper, we use a monodromy matrix to derive
equations from vector analogue of the Kundu-Eckhaus equation hierarchy and
construct the simplest nontrivial solutions to first equation. We hope that
the vector equation we have obtained, as well as the scalar one, will have
numerous applications in physics and mathematics.

The work consists of an introduction, four sections, and concluding remarks.
In the first section, we define the Lax operator

$i\Psi_{t}+U\Psi=\boldsymbol{0},$

which depends on a functional parameter $r\in\mathbb{R}$, and investigate
properties of corresponding monodromy matrix. Since the spectral curve
equation is a characteristic equation of the monodromy matrix [10], it is not
difficult to obtain properties of the spectral curves equations from
properties of the monodromy matrix. As in the case of the Manakov system [32],
the spectral curves equations are quite cumbersome, so we will not give them
in this paper. Let us note only that, as is shown in [32], linear dependence
of the functions $p_{j}$ leads to factorization of the spectral curve equation
into separate components. Therefore, from our point of view, solutions with
linearly independent $p_{j}$ are more interesting to study and use in
applications.

In section 2 we derive stationary equations for multiphase solutions. These
equations are analogs of the Novikov equations for the Korteweg-de Vries
hierarchy. Also in this section, we define a hierarchy of the second operators
of the Lax pair

$i\Psi_{z_{k}}+W_{k}\Psi=\boldsymbol{0},$

which depend on the functional parameter $r_{k}$:
$\partial_{t}r_{k}=\partial_{z_{k}}r$. The Lax pair compatibility conditions
give a hierarchy of vector derivative nonlinear Schrödinger equations with an
additional functional parameter $\phi$: $r=\partial_{t}\phi$ and
$r_{k}=\partial_{z_{k}}\phi$. When $r=\alpha(\mathbf{p}^{t}\mathbf{q})$ these
equations are vector analogue of the Kundu-Eckhaus equation and its higher
forms. Another choice of the functional parameter leads to other vector
nonlinear equations. For $\phi\equiv 0$ these equations will turn into
equations from the Manakov hierarchy [32]. Let us note that an existence of a
Lax pair makes it possible to use the Darboux transformation to construct new
solutions to vector analogue of the Kundu-Eckhaus equation.

In section 3, we construct one-phase solutions to vector analogue of the
Kundu-Eckhaus equation. The first three solutions are expressed in terms of
elliptic Jacobi functions, and for $\phi\equiv 0$ they will be new elliptic
solutions to the Manakov system. Let us recall, that obtained in [32] elliptic
solutions to the Manakov system were expressed in terms of Weierstrass
functions. Then we construct solutions expressed in terms of hyperbolic
functions. At the end of the section, we consider one-phase two-gap solutions.
Despite the fact that in the last case the spectral curve has a genus equal to
2, the corresponding solution is a traveling wave.

In the section 4, we show that there exist linear transformations of solutions
to vector integrable nonlinear equations into other solutions to the same
equations. Original and transformed solutions are associated with the same
spectral curve, but they correspond to different Baker-Akhiezer functions. One
of these Baker-Achiezer functions differs from the other by an orthogonal
matrix factor. I.e., the Baker-Achiezer function for considered vector
nonlinear Schrödinger equation is determined up to orthogonal transformation.

##  1 The monodromy matrix and its properties

Let first equation of a Lax pair have the form

$i\Psi_{t}+U\Psi=\boldsymbol{0},$ (7)

where

$U=U_{0}+rJ,\quad U_{0}=-\lambda J+Q,$
$J=\dfrac{1}{3}\begin{pmatrix}2&\boldsymbol{0}^{t}\\\
\boldsymbol{0}&-I\end{pmatrix},\quad Q=\begin{pmatrix}0&\mathbf{p}^{t}\\\
-\mathbf{q}&\boldsymbol{0}\end{pmatrix},$

$\mathbf{p}^{t}=(p_{1},p_{2})$, $\mathbf{q}^{t}=(q_{1},q_{2})$, $I$ is
identity matrix, $r\in\mathbb{R}$ is a some function, and $\lambda$ is a
spectral parameter.

Following [10, 32], assume that there exists a monodromy matrix $M$ such that
the matrix function $\widehat{\Psi}=M\Psi$ is also an eigenfunction of Lax
operator (7). Then the matrix $M$ satisfies the equation

$iM_{t}+UM-MU=\boldsymbol{0}.$ (8)

In the case of a finite-gap matrix potential $Q$, the monodromy matrix $M$ is
a polynomial in the spectral parameter $\lambda$ [10, 32]

$M=\sum_{j=0}^{n}m_{j}(t)\lambda^{j}.$ (9)

Substituting (9) in (8) and simplifying, we get that the matrix $M$ has the
following structure

$M=V_{n}+\sum_{k=1}^{n-1}c_{k}V_{n-k}+c_{n}U_{0}+J_{n},$

where $V_{1}=\lambda U_{0}+V_{1}^{0}$,

$\displaystyle V_{k+1}=\lambda V_{k}+V_{k+1}^{0},\quad
V_{k}^{0}=\begin{pmatrix}-\mathcal{F}_{k}&\mathbf{H}_{k}^{t}\\\
\mathbf{G}_{k}&F_{k}\end{pmatrix},\quad k\geq 1,$ $\displaystyle
J_{n}=\begin{pmatrix}-c_{n+1}-c_{n+2}&0&0\\\ 0&c_{n+1}&c_{n+3}\\\
0&c_{n+4}&c_{n+2}\end{pmatrix},$

$c_{j}$ are some constants, $\mathcal{F}_{k}=\mathrm{Tr}F_{k}$.

From equation (8) it follows that elements of the matrices $V_{k}^{0}$ satisfy
recurrence relations

$\displaystyle\mathbf{H}_{1}=i\partial_{t}\mathbf{p}+r\mathbf{p},$ (10)
$\displaystyle\mathbf{G}_{1}=i\partial_{t}\mathbf{q}-r\mathbf{q},$
$\displaystyle
F_{k}=-i\partial_{t}^{-1}\left(\mathbf{G}_{k}\mathbf{p}^{t}+\mathbf{q}\mathbf{H}_{k}^{t}\right),$
$\displaystyle\mathbf{H}_{k+1}=i\partial_{t}\mathbf{H}_{k}+r\mathbf{H}_{k}+\left(F_{k}^{t}+\mathcal{F}_{k}I\right)\mathbf{p},$
$\displaystyle\mathbf{G}_{k+1}=-i\partial_{t}\mathbf{G}_{k}+r\mathbf{G}_{k}-\left(F_{k}+\mathcal{F}_{k}I\right)\mathbf{q}.$

In particular, $F_{1}=\mathbf{q}\mathbf{p}^{t}$,
$\mathcal{F}_{1}=\mathbf{p}^{t}\mathbf{q}=\mathbf{q}^{t}\mathbf{p}$,

$\displaystyle\mathbf{H}_{2}=-\partial_{t}^{2}\mathbf{p}+2ir\partial_{t}\mathbf{p}+\left(2\mathbf{p}^{t}\mathbf{q}+r^{2}+i\partial_{t}r\right)\mathbf{p},$
$\displaystyle\mathbf{G}_{2}=\partial_{t}^{2}\mathbf{q}+2ir\partial_{t}\mathbf{q}-\left(2\mathbf{p}^{t}\mathbf{q}+r^{2}-i\partial_{t}r\right)\mathbf{q},$
$\displaystyle
F_{2}=2\left(\mathbf{q}\mathbf{p}^{t}\right)r+i\left(\mathbf{q}\partial_{t}\mathbf{p}^{t}-\partial_{t}\mathbf{q}\mathbf{p}^{t}\right)=\mathbf{q}\mathbf{H}_{1}^{t}-\mathbf{G}_{1}\mathbf{p}^{t},$
$\displaystyle\mathcal{F}_{2}=2(\mathbf{p}^{t}\mathbf{q})r+i(\mathbf{q}^{t}\partial_{t}\mathbf{p}-\mathbf{p}^{t}\partial_{t}\mathbf{q})=\mathbf{H}_{1}^{t}\mathbf{q}-\mathbf{p}^{t}\mathbf{G}_{1},$
$\displaystyle\mathbf{H}_{3}=-i\partial_{t}^{3}\mathbf{p}-3r\partial_{t}^{2}\mathbf{p}+3i\left(\mathbf{p}^{t}\mathbf{q}+r^{2}+i\partial_{t}r\right)\partial_{t}\mathbf{p}$
$\displaystyle\qquad+\left(3i\partial_{t}\mathbf{p}^{t}\mathbf{q}+r^{3}+6r\mathbf{p}^{t}\mathbf{q}+3ir\partial_{t}r-\partial_{t}^{2}r\right)\mathbf{p},$
$\displaystyle\mathbf{G}_{3}=-i\partial_{t}^{3}\mathbf{q}+3r\partial_{t}^{2}\mathbf{q}+3i\left(\mathbf{p}^{t}\mathbf{q}+r^{2}-i\partial_{t}r\right)\partial_{t}\mathbf{q}$
$\displaystyle\qquad+\left(3i\mathbf{p}^{t}\partial_{t}\mathbf{q}-r^{3}-6r\mathbf{p}^{t}\mathbf{q}+3ir\partial_{r}r+\partial_{t}^{2}r\right)\mathbf{q},$
$\displaystyle
F_{3}=3(\mathbf{q}\mathbf{p}^{t})r^{2}+3i(\mathbf{q}\partial_{t}\mathbf{p}^{t}-\partial_{t}\mathbf{q}\mathbf{p}^{t})r+\partial_{t}\mathbf{q}\partial_{t}\mathbf{p}^{t}$
$\displaystyle\qquad-\mathbf{q}\partial_{t}^{2}\mathbf{p}^{t}-\partial_{t}^{2}\mathbf{q}\mathbf{p}^{t}+3(\mathbf{p}^{t}\mathbf{q})\mathbf{q}\mathbf{p}^{t}$
$\displaystyle\qquad=\mathbf{q}\mathbf{H}_{2}^{t}-\mathbf{G}_{2}\mathbf{p}^{t}-\mathbf{G}_{1}\mathbf{H}_{1}^{t}-(\mathbf{p}^{t}\mathbf{q})\mathbf{q}\mathbf{p}^{t},$
$\displaystyle\mathcal{F}_{3}=3(\mathbf{p}^{t}\mathbf{q})r^{2}+3i(\mathbf{q}^{t}\partial_{t}\mathbf{p}-\mathbf{p}^{t}\partial_{t}\mathbf{q})r+3(\mathbf{p}^{t}\mathbf{q})^{2}$
$\displaystyle\qquad+\partial_{t}\mathbf{p}^{t}\partial_{t}\mathbf{q}-\mathbf{p}^{t}\partial_{t}^{2}\mathbf{q}-\mathbf{q}^{t}\partial_{t}^{2}\mathbf{p}.$

For $r\equiv 0$, all the above equations turn into corresponding equations for
the Manakov system [32].

##  2 Stationary and evolutionary equations

From equation (8) the following stationary equations follow:

$\displaystyle\mathbf{H}_{n+1}+\sum_{k=1}^{n}c_{k}\mathbf{H}_{n+1-k}+C_{n}^{t}\mathbf{p}=\boldsymbol{0},$
(11)
$\displaystyle\mathbf{G}_{n+1}+\sum_{k=1}^{n}c_{k}\mathbf{G}_{n+1-k}-C_{n}\mathbf{q}=\boldsymbol{0},$

where

$C_{n}=\begin{pmatrix}2c_{n+1}+c_{n+2}&c_{n+3}\\\
c_{n+4}&c_{n+1}+2c_{n+2}\end{pmatrix}.$

All multiphase solutions to evolutionary integrable nonlinear equations are
simultaneously solutions to some stationary equations.

In the case of reduction $\mathbf{q}=\sigma\mathbf{p}^{\ast}$ ($\sigma=\pm
1$), the following identities

$\displaystyle\mathbf{G}_{k}^{\ast}=-\sigma\mathbf{H}_{k},\quad\mathbf{H}_{k}^{\ast}=-\sigma\mathbf{G}_{k},\quad
F_{k}^{\ast}=F_{k}^{t},\quad\mathcal{F}_{k}\in\mathbb{R}$

follow from the recurrence relations (10). Therefore, the constants $c_{j}$ in
stationary equations (11) must satisfy the following conditions:
$c_{k}\in\mathbb{R}$ ($1\leq k\leq n+2$), $c_{n+4}=c_{n+3}^{\ast}$.

Let a second operator of a Lax pair have the form

$i\Psi_{z_{k}}+W_{k}\Psi=\boldsymbol{0}$ (12)

where $W_{k}=V_{k}+r_{k}J$. Then from compatibility condition of equations (7)
and (12), or from equation

$i\partial_{t}W_{k}-i\partial_{z_{k}}U+UW_{k}-W_{k}U=\boldsymbol{0}$

the evolutionary integrable nonlinear equations

$i\partial_{z_{k}}\mathbf{p}=\mathbf{H}_{k+1}-r_{k}\mathbf{p},\quad
i\partial_{z_{k}}\mathbf{q}=\mathbf{G}_{k+1}+r_{k}\mathbf{q},$ (13)

and the additional relation

$\partial_{z_{k}}r=\partial_{t}r_{k}$ (14)

follow. It follows from (14) that there exists a function $\phi$ such that

$r=\partial_{t}\phi,\quad r_{k}=\partial_{z_{k}}\phi.$

The first systems of integrable nonlinear equations from hierarchy (13) have
the form

$\displaystyle
i\partial_{z_{1}}\mathbf{p}=-\partial_{t}^{2}\mathbf{p}+2ir\partial_{t}\mathbf{p}+\left(2\mathbf{p}^{t}\mathbf{q}+r^{2}+i\partial_{t}r-r_{1}\right)\mathbf{p},$
(15) $\displaystyle
i\partial_{z_{1}}\mathbf{q}=\partial_{t}^{2}\mathbf{q}+2ir\partial_{t}\mathbf{q}-\left(2\mathbf{p}^{t}\mathbf{q}+r^{2}-i\partial_{t}r-r_{1}\right)\mathbf{q},$

and

$\displaystyle\partial_{z_{2}}\mathbf{p}=-\partial_{t}^{3}\mathbf{p}+3ir\partial_{t}^{2}\mathbf{p}+3\left(\mathbf{p}^{t}\mathbf{q}+r^{2}+i\partial_{t}r\right)\partial_{t}\mathbf{p}$
(16)
$\displaystyle\qquad+\left(3\partial_{t}\mathbf{p}^{t}\mathbf{q}-ir^{3}-6ir\mathbf{p}^{t}\mathbf{q}+3r\partial_{t}r+i\partial_{t}^{2}r+ir_{2}\right)\mathbf{p},$
$\displaystyle\partial_{z_{2}}\mathbf{q}=-\partial_{t}^{3}\mathbf{q}-3ir\partial_{t}^{2}\mathbf{q}+3\left(\mathbf{p}^{t}\mathbf{q}+r^{2}-i\partial_{t}r\right)\partial_{t}\mathbf{q}$
$\displaystyle\qquad+\left(3\mathbf{p}^{t}\partial_{t}\mathbf{q}+ir^{3}+6ir\mathbf{p}^{t}\mathbf{q}+3r\partial_{r}r-i\partial_{t}^{2}r-ir_{2}\right)\mathbf{q}.$

Equation (15) is one of a vector derivative nonlinear Schrödinger equations,
and (16) is a vector modified Korteweg-de Vries equation. Both equations are
parametrized by an arbitrary real function $\phi$.

From (10) and (13) the following equalities

$\partial_{z_{k}}F_{1}=\partial_{t}F_{k+1}\quad\text{and}\quad\partial_{z_{k}}\mathcal{F}_{1}=\partial_{t}\mathcal{F}_{k+1}$

follow. Therefore, there exist functions $\Phi$ and $\widetilde{\phi}$ such
that

$F_{1}=\partial_{t}\Phi,\quad
F_{k+1}=\partial_{z_{k}}\Phi\quad\text{and}\quad\mathcal{F}_{1}=\partial_{t}\widetilde{\phi},\quad\mathcal{F}_{k+1}=\partial_{z_{k}}\widetilde{\phi}.$

Therefore, if we put $\phi=\alpha\widetilde{\phi}$ or

$r=\alpha\mathcal{F}_{1}\quad\text{and}\quad r_{k}=\alpha\mathcal{F}_{k+1},$
(17)

then equations (13), (17) will determine evolutionary integrable nonlinear
equations from the hierarchy of a vector analogue of the Kundu-Eckhaus
equation.

In particular, for $k=1$, or for

$r=\alpha\mathcal{F}_{1}=\alpha(\mathbf{p}^{t}\mathbf{q})\quad\text{and}\quad
r_{1}=\alpha\mathcal{F}_{2}=2\alpha^{2}(\mathbf{p}^{t}\mathbf{q})^{2}+i\alpha(\mathbf{q}^{t}\partial_{t}\mathbf{p}-\mathbf{p}^{t}\partial_{t}\mathbf{q}),$

equation (15) will have the form

$\displaystyle
i\partial_{z_{1}}\mathbf{p}=-\partial_{t}^{2}\mathbf{p}+2i\alpha(\mathbf{p}^{t}\mathbf{q})\partial_{t}\mathbf{p}+\left(2\mathbf{p}^{t}\mathbf{q}-\alpha^{2}(\mathbf{p}^{t}\mathbf{q})^{2}+2i\alpha\mathbf{p}^{t}\partial_{t}\mathbf{q}\right)\mathbf{p},$
(18) $\displaystyle
i\partial_{z_{1}}\mathbf{q}=\partial_{t}^{2}\mathbf{q}+2i\alpha(\mathbf{p}^{t}\mathbf{q})\partial_{t}\mathbf{q}-\left(2\mathbf{p}^{t}\mathbf{q}-\alpha^{2}(\mathbf{p}^{t}\mathbf{q})^{2}-2i\alpha\partial_{t}\mathbf{p}^{t}\mathbf{q}\right)\mathbf{q}.$

When $\mathbf{q}=S\mathbf{p}^{\ast}$,
$S=\mathrm{diag}(\sigma_{1},\sigma_{2})$, $\sigma_{j}=\pm 1$, equations (18)
transform into a vector analogue of the Kundu-Eckhaus equation. It is not
difficult to see that for $\alpha=0$ equations (18) transform into the Manakov
system [32].

##  3 One-phase solutions

###  3.1 Solutions in elliptic Jacobi functions

For $n=1$ stationary equations have the form

$\displaystyle\mathbf{H}_{2}+c_{1}\mathbf{H}_{1}+C_{1}^{t}\mathbf{p}=\boldsymbol{0},$
(19)
$\displaystyle\mathbf{G}_{2}+c_{1}\mathbf{G}_{1}-C_{1}\mathbf{q}=\boldsymbol{0}$

or (for $c_{4}=c_{5}=0$ and $r=\alpha\mathcal{F}_{1}$,
$r_{1}=\alpha\mathcal{F}_{2}$)

$\displaystyle\partial_{t}^{2}p_{1}=i(c_{1}+2\alpha(\mathbf{p}^{t}\mathbf{q}))\partial_{t}p_{1}$
(20)
$\displaystyle\qquad+(2c_{2}+c_{3}+(2+c_{1}\alpha)\mathbf{p}^{t}\mathbf{q}+\alpha^{2}(\mathbf{p}^{t}\mathbf{q})^{2}+i\alpha\partial_{t}(\mathbf{p}^{t}\mathbf{q}))p_{1},$
$\displaystyle\partial_{t}^{2}p_{2}=i(c_{1}+2\alpha(\mathbf{p}^{t}\mathbf{q}))\partial_{t}p_{2}$
$\displaystyle\qquad+(c_{2}+2c_{3}+(2+c_{1}\alpha)\mathbf{p}^{t}\mathbf{q}+\alpha^{2}(\mathbf{p}^{t}\mathbf{q})^{2}+i\alpha\partial_{t}(\mathbf{p}^{t}\mathbf{q}))p_{2},$
$\displaystyle\partial_{t}^{2}q_{1}=-i(c_{1}+2\alpha(\mathbf{p}^{t}\mathbf{q}))\partial_{t}q_{1}$
$\displaystyle\qquad+(2c_{2}+c_{3}+(2+c_{1}\alpha)\mathbf{p}^{t}\mathbf{q}+\alpha^{2}(\mathbf{p}^{t}\mathbf{q})^{2}-i\alpha\partial_{t}(\mathbf{p}^{t}\mathbf{q}))q_{1},$
$\displaystyle\partial_{t}^{2}q_{2}=-i(c_{1}+2\alpha(\mathbf{p}^{t}\mathbf{q}))\partial_{t}q_{2}$
$\displaystyle\qquad+(c_{2}+2c_{3}+(2+c_{1}\alpha)\mathbf{p}^{t}\mathbf{q}+\alpha^{2}(\mathbf{p}^{t}\mathbf{q})^{2}-i\alpha\partial_{t}(\mathbf{p}^{t}\mathbf{q}))q_{2}.$

Replacing functions $p_{j}$ and $q_{j}$ by formulas

$p_{j}=\widehat{p}_{j}e^{i\theta},\quad
q_{j}=\widehat{q}_{j}e^{-i\theta},\quad\partial_{t}\theta=\dfrac{1}{2}c_{1}+\alpha\mathbf{p}^{t}\mathbf{q},$

we obtain the following equalities

$\displaystyle\partial_{t}^{2}\widehat{p}_{1}=\left(2\widehat{\mathbf{p}}^{t}\widehat{\mathbf{q}}+2c_{2}+c_{3}-\dfrac{1}{4}c_{1}^{2}\right)\widehat{\mathbf{p}}_{1},$
(21)
$\displaystyle\partial_{t}^{2}\widehat{p}_{2}=\left(2\widehat{\mathbf{p}}^{t}\widehat{\mathbf{q}}+c_{2}+2c_{3}-\dfrac{1}{4}c_{1}^{2}\right)\widehat{\mathbf{p}}_{2},$
$\displaystyle\partial_{t}^{2}\widehat{q}_{1}=\left(2\widehat{\mathbf{p}}^{t}\widehat{\mathbf{q}}+2c_{2}+c_{3}-\dfrac{1}{4}c_{1}^{2}\right)\widehat{\mathbf{q}}_{1},$
$\displaystyle\partial_{t}^{2}\widehat{q}_{2}=\left(2\widehat{\mathbf{p}}^{t}\widehat{\mathbf{q}}+c_{2}+2c_{3}-\dfrac{1}{4}c_{1}^{2}\right)\widehat{\mathbf{q}}_{2}.$

It is not difficult to see that functions $\widehat{p}_{j}$ and
$\widehat{q}_{j}$ are solutions of the same second-order linear differential
equations. Hence, their products $u_{j}=\widehat{p}_{j}\widehat{q}_{j}$
satisfy the corresponding Appel’s equations ([35], Part II, Chapter 14,
Example 10; [4])

$\displaystyle\partial_{t}^{3}u_{1}-(8u_{1}+8u_{2}+8c_{2}+4c_{3}-c_{1}^{2})\partial_{t}u_{1}-4\partial_{t}(u_{1}+u_{2})u_{1}=0,$
(22)
$\displaystyle\partial_{t}^{3}u_{2}-(8u_{1}+8u_{2}+4c_{2}+8c_{3}-c_{1}^{2})\partial_{t}u_{2}-4\partial_{t}(u_{1}+u_{2})u_{2}=0.$

Let us introduce the notations: $u_{1}+u_{2}=u$, $u_{1}-u_{2}=v$. In these
notations, equations (22) take the form

$\displaystyle\partial_{t}^{3}u+(c_{1}^{2}-6c_{2}-6c_{3}-12u)\partial_{t}u=2(c_{2}-c_{3})\partial_{t}v,$
(23)
$\displaystyle\partial_{t}^{3}v+(c_{1}^{2}-6c_{2}-6c_{3}-8u)\partial_{t}v=2(c_{2}-c_{3}+2v)\partial_{t}u.$

The simplest solutions of equations (23) can be obtained when
$v=(c_{3}-c_{2})/2$. In this case, the function $u$ satisfies the equation

$\partial_{t}^{3}u+(c_{1}^{2}-6c_{2}-6c_{3}-12u)\partial_{t}u=0$

or

$\partial_{t}^{2}u+(c_{1}^{2}-6c_{2}-6c_{3})u-6u^{2}=\widehat{c}_{1},$ (24)

where $\widehat{c}_{1}$ is an integration constant. Simplifying relation (24),
we obtain the equation

$(\partial_{t}u)^{2}=4u^{3}-(c_{1}^{2}-6c_{2}-6c_{3})u^{2}+2\widehat{c}_{1}u+\widehat{c}_{2},$
(25)

where $\widehat{c}_{2}$ is a second integration constant. It is well known
that solutions to equation (25) are elliptic functions or their degenerations.

It is easy to verify that one of the non-degenerate solutions to the equation
(25) has the form

$u=k^{2}\operatorname{\mathrm{sn}}^{2}(t;k)+\dfrac{1}{12}c_{1}^{2}-\dfrac{1}{2}(c_{2}+c_{3})-\dfrac{1}{3}(1+k^{2}),$
(26)

where $\operatorname{\mathrm{sn}}(t;k)$ is an elliptic Jacobi function [3, 1],
that satisies the equation

$[\operatorname{\mathrm{sn}}^{\prime}(t)]^{2}=(1-\operatorname{\mathrm{sn}}^{2}(t))(1-k^{2}\operatorname{\mathrm{sn}}^{2}(t)).$

The integration constant for solution (26) is equal to

$\displaystyle\widehat{c}_{1}=\dfrac{1}{24}c_{1}^{4}-\dfrac{1}{2}(c_{2}+c_{3})c_{1}^{2}+\dfrac{3}{2}(c_{2}+c_{3})^{2}-\dfrac{2}{3}(1-k^{2}+k^{4}),$
$\displaystyle\widehat{c}_{2}=-\dfrac{1}{432}c_{1}^{6}+\dfrac{1}{24}(c_{2}+c_{3})c_{1}^{4}-\dfrac{1}{4}(c_{2}+c_{3})^{2}c_{1}^{2}+\dfrac{1}{9}(1-k^{2}+k^{4})c_{1}^{2}$
$\displaystyle\qquad+\dfrac{1}{2}(c_{2}+c_{3})^{3}-\dfrac{2}{3}(c_{2}+c_{3})(1-k^{2}+k^{4})-\dfrac{4}{27}(2-3k^{2}-3k^{4}+2k^{6}).$

Knowing functions $u$ and $v$, we obtain functions $u_{j}$

$\displaystyle
u_{1}=\dfrac{1}{2}(u+v)=\dfrac{k^{2}}{2}\operatorname{\mathrm{sn}}^{2}(t;k)+\dfrac{c_{1}^{2}}{24}-\dfrac{c_{2}}{2}-\dfrac{1+k^{2}}{6},$
(27) $\displaystyle
u_{2}=\dfrac{1}{2}(u-v)=\dfrac{k^{2}}{2}\operatorname{\mathrm{sn}}^{2}(t;k)+\dfrac{c_{1}^{2}}{24}-\dfrac{c_{3}}{2}-\dfrac{1+k^{2}}{6}.$

Thus, functions $\widehat{p}_{j}$ and $\widehat{q}_{j}$ are solutions to the
equations

$\displaystyle\partial_{t}^{2}\widehat{p}_{1}=\left(2k^{2}\operatorname{\mathrm{sn}}^{2}(t;k)-\dfrac{2}{3}(1+k^{2})-\dfrac{1}{12}c_{1}^{2}+c_{2}\right)\widehat{p}_{1},$
(28)
$\displaystyle\partial_{t}^{2}\widehat{p}_{2}=\left(2k^{2}\operatorname{\mathrm{sn}}^{2}(t;k)-\dfrac{2}{3}(1+k^{2})-\dfrac{1}{12}c_{1}^{2}+c_{3}\right)\widehat{p}_{2}.$

Since functions $\widehat{q}_{j}$ satisfy the same equations as
$\widehat{p}_{j}$, and the Wronskian of these solutions are constant:

$W[\widehat{p}_{j},\widehat{q}_{j}]=2iW_{j},$

functions $\widehat{p}_{j}$ and $\widehat{q}_{j}$ are equal to

$\widehat{p}_{j}=\sqrt{u_{j}}\exp\left\\{-iW_{j}\int\dfrac{dt}{u_{j}}\right\\},\quad\widehat{q}_{j}=\sqrt{u_{j}}\exp\left\\{iW_{j}\int\dfrac{dt}{u_{j}}\right\\},$
(29)

where $u_{j}$ are defined by formulas (27). Substituting expressions (29) into
equation (28) and simplifying, we get

$W_{j}^{2}=\dfrac{1}{6912}(c_{1}^{2}-12c_{j+1}-4-4k^{2})(c_{1}^{2}-12c_{j+1}+8-4k^{2})(c_{1}^{2}-12c_{j+1}-4+8k^{2}).$
(30)

It follows from equation (30) that there are three cases when
$\widehat{p}_{j}=\widehat{q}_{j}$ and $\widehat{p}_{1}\neq\widehat{p}_{2}$. If

$c_{2}=\dfrac{1}{12}(c_{1}^{2}+8-4k^{2}),\quad
c_{3}=\dfrac{1}{12}(c_{1}^{2}+8k^{2}-4),$

then

$\widehat{p}_{1}=\widehat{q}_{1}=\dfrac{i}{\sqrt{2}}\operatorname{\mathrm{dn}}(t;k),\quad\widehat{p}_{2}=\widehat{q}_{2}=\dfrac{ik}{\sqrt{2}}\operatorname{\mathrm{cn}}(t;k).$

In this case, the solution to equations (18) has the form

$\displaystyle p_{1}=i\mathfrak{p}_{1}(t-c_{1}z_{1})e^{i\theta},\quad
q_{1}=-p_{1}^{\ast},$ (31) $\displaystyle
p_{2}=i\mathfrak{p}_{2}(t-c_{1}z_{1})e^{i\theta},\quad q_{2}=-p_{2}^{\ast},$

where

$\displaystyle\mathfrak{p}_{1}(T)=\dfrac{1}{\sqrt{2}}\operatorname{\mathrm{dn}}(T;k),\quad\mathfrak{p}_{2}(T)=\dfrac{k}{\sqrt{2}}\operatorname{\mathrm{cn}}(T;k),$
$\displaystyle\theta=\dfrac{c_{1}}{2}t+\left(1-\dfrac{c_{1}^{2}}{4}\right)z_{1}+\alpha\int\left(k^{2}\operatorname{\mathrm{sn}}^{2}(t-c_{1}z_{1};k)-\dfrac{1+k^{2}}{2}\right)dt.$

Magnitudes of solutions (31) are shown in Fig. 1.

|
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Magnitude $\mathfrak{p}_{1}(t-c_{1}z_{1})$ |  Magnitude $\mathfrak{p}_{2}(t-c_{1}z_{1})$
Figure 1: Magnitudes of solutions (31) for $k=0.7$, $c_{1}=1$

For

$c_{2}=\dfrac{1}{12}(c_{1}^{2}+8-4k^{2}),\quad
c_{3}=\dfrac{1}{12}(c_{1}^{2}-4k^{2}-4),$

we have

$\widehat{p}_{1}=\widehat{q}_{1}=\dfrac{i}{\sqrt{2}}\operatorname{\mathrm{dn}}(t;k),\quad\widehat{p}_{2}=\widehat{q}_{2}=\dfrac{k}{\sqrt{2}}\operatorname{\mathrm{sn}}(t;k).$

The corresponding solution to equations (18) has the form

$\displaystyle p_{1}=i\mathfrak{p}_{1}(t-c_{1}z_{1})e^{i\theta},\quad
q_{1}=-p_{1}^{\ast},$ (32) $\displaystyle
p_{2}=\mathfrak{p}_{2}(t-c_{1}z_{1})e^{i\theta},\quad q_{2}=p_{2}^{\ast},$

where

$\displaystyle\mathfrak{p}_{1}(T)=\dfrac{1}{\sqrt{2}}\operatorname{\mathrm{dn}}(T;k),\quad\mathfrak{p}_{2}(T)=\dfrac{k}{\sqrt{2}}\operatorname{\mathrm{sn}}(T;k),$
$\displaystyle\theta=\dfrac{c_{1}}{2}t+\left(1-k^{2}-\dfrac{c_{1}^{2}}{4}\right)z_{1}+\alpha\int\left(k^{2}\operatorname{\mathrm{sn}}^{2}(t-c_{1}z_{1};k)-\dfrac{1}{2}\right)dt.$

If

$c_{2}=\dfrac{1}{12}(c_{1}^{2}+8k^{2}-4),\quad
c_{3}=\dfrac{1}{12}(c_{1}^{2}-4k^{2}-4),$

we have

$\widehat{p}_{1}=\widehat{q}_{1}=\dfrac{ik}{\sqrt{2}}\operatorname{\mathrm{cn}}(t;k),\quad\widehat{p}_{2}=\widehat{q}_{2}=\dfrac{k}{\sqrt{2}}\operatorname{\mathrm{sn}}(t;k).$

In this case, the solution to equations (18) has the form

$\displaystyle p_{1}=i\mathfrak{p}_{1}(t-c_{1}z_{1})e^{i\theta},\quad
q_{1}=-p_{1}^{\ast},$ (33) $\displaystyle
p_{2}=\mathfrak{p}_{2}(t-c_{1}z_{1})e^{i\theta},\quad q_{2}=p_{2}^{\ast},$

where

$\displaystyle\mathfrak{p}_{1}(T)=\dfrac{k}{\sqrt{2}}\operatorname{\mathrm{cn}}(T;k),\quad\mathfrak{p}_{2}(T)=\dfrac{k}{\sqrt{2}}\operatorname{\mathrm{sn}}(T;k),$
$\displaystyle\theta=\dfrac{c_{1}}{2}t+\left(k^{2}-1-\dfrac{c_{1}^{2}}{4}\right)z_{1}+\alpha\int\left(k^{2}\operatorname{\mathrm{sn}}^{2}(t-c_{1}z_{1};k)-\dfrac{k^{2}}{2}\right)dt.$

Dependency of solutions (31)-(33) from the variable $z_{1}$ was found from
equations (18).

###  3.2 Solutions in hyperbolic functions

The equation (25) is well studied. In particular, it has the following
solution

$u=a^{2}\tanh^{2}(at)+\dfrac{1}{12}\left((c_{1}^{2}-6c_{2}-6c_{3})-8a^{2}\right).$

Integration constants for a given function $u$ are equal to

$\displaystyle\widehat{c}_{1}=\dfrac{1}{24}\left((c_{1}^{2}-6c_{2}-6c_{3})^{2}-16a^{4}\right),$
$\displaystyle\widehat{c}_{2}=\dfrac{1}{432}\left(4a^{2}+(c_{1}^{2}-6c_{2}-6c_{3})\right)^{2}\left(8a^{2}-(c_{1}^{2}-6c_{2}-6c_{3})\right).$

In this case, functions $\widehat{p}_{j}$ and $\widehat{q}_{j}$ satisfy the
following equations:

$\partial_{t}^{2}\widehat{p}_{j}=\left(2a^{2}\tanh^{2}(at)-\dfrac{4a^{2}}{3}-\dfrac{c_{1}^{2}}{12}+c_{j+1}\right)\widehat{p}_{j}$
(34)

and

$u_{j}=\widehat{p}_{j}\widehat{q}_{j}=\dfrac{a^{2}}{2}\tanh^{2}(at)+\dfrac{1}{24}\left(c_{1}^{2}-8a^{2}-12c_{j+1}\right).$
(35)

Let us recall that functions $\widehat{q}_{j}$ also satisfy equations (34).

Solving equations (34) with conditions (35), we get

$\widehat{p}_{j}=\dfrac{1}{\sqrt{2}}(k_{j}+ia\tanh(at))e^{ik_{j}t},\quad\widehat{q}_{j}=\dfrac{1}{\sqrt{2}}(k_{j}-ia\tanh(at))e^{-ik_{j}t},$
(36)

where

$k_{j}^{2}=\dfrac{1}{12}\left(c_{1}^{2}-8a^{2}-12c_{j+1}\right)\quad\text{or}\quad
c_{j+1}=\dfrac{1}{12}\left(c_{1}^{2}-8a^{2}-12k_{j}^{2}\right).$

The corresponding solution to equations (18) has the form

$\displaystyle
p_{j}=\dfrac{1}{\sqrt{2}}\left(k_{j}+ia\tanh[a(t-c_{1}z_{1})]\right)e^{i\theta_{j}},\quad
q_{j}=p_{j}^{\ast},$ (37)
$\displaystyle\theta_{j}=\left(\dfrac{c_{1}}{2}+k_{j}\right)t+m_{j}z_{1}+\alpha\int\left(a\tanh^{2}(at-
ac_{1}z_{1})+\dfrac{k_{1}^{2}+k_{2}^{2}}{2a}\right)dt,$

where

$m_{j}=-2a^{2}-\dfrac{1}{4}(c_{1}+2k_{j})^{2}-k_{1}^{2}-k_{2}^{2}-\alpha(k_{1}+k_{2})(a^{2}+k_{1}^{2}-k_{1}k_{2}+k_{2}^{2}).$

Magnitudes of solutions (37) are shown in Fig. 2.

|
---|---
Magnitude $\left\lvert p_{1}(t,z_{1})\right\rvert$ |  Magnitude $\left\lvert p_{2}(t,z_{1})\right\rvert$
Figure 2: Magnitudes of solutions (37) for $a=1$, $k_{1}=2$, $k_{2}=3$,
$c_{1}=1$.

###  3.3 Two-gap one-phase solutions

If $n=1$, $v\neq const,$ and $c_{3}\neq c_{2}$, then from (23) we have

$v=\dfrac{1}{2(c_{2}-c_{3})}\left(u_{tt}-6u^{2}+(c_{1}^{2}-6c_{2}-6c_{3})u\right)+C_{1},$
(38)

and

$u_{tttt}+2(c_{1}^{2}-6c_{2}-6c_{3}-10u)u_{tt}-10\left(u_{t}\right)^{2}+40u^{3}-12(c_{1}^{2}-6c_{2}-6c_{3})u^{2}\\\
+\left(c_{1}^{4}-12c_{1}^{2}(c_{2}+c_{3})+8\left(4c_{2}^{2}+10c_{2}c_{3}+4c_{3}^{3}-C_{1}c_{2}+C_{1}c_{3}\right)\right)u+C_{2}=0,$
(39)

where $C_{1}$ and $C_{2}$ are integration constants.

We can rewrite equation (39) in the form

$I_{2}+2AI_{1}+(A^{2}-4B^{2}+8C_{1}B)u+C_{2}=0,$ (40)

where $A=c_{1}^{2}-6(c_{3}+c_{2})$, $B=c_{3}-c_{2}$,

$I_{2}=u_{tttt}-20uu_{tt}-10\left(u_{t}\right)^{2}+40u^{3},\quad
I_{1}=u_{tt}-6u^{2}.$ (41)

It follows from equations (40) and (41) that the function $2u(t)$ is a two-gap
potential of the Schrödinger operator [29, 11]

$\psi_{tt}-2u\psi=E\psi.$ (42)

It is well known that linear independent solutions to equation (42) with two-
gap potential $2u(t)$ can be written as

$\psi_{1,2}=\varepsilon_{1,2}\sqrt{\Psi}\exp\left\\{\pm\nu(E)\int\dfrac{dt}{\Psi}\right\\},$
(43)

where

$\Psi=E^{2}+(\gamma_{1}-u)E+\gamma_{2}-\gamma_{1}u-\dfrac{1}{4}I_{1},$

and

$\gamma_{1}=\dfrac{A}{2},\quad\gamma_{2}=\dfrac{1}{16}\left(A^{2}-4B^{2}+8C_{1}B\right).$

Substituing (43) in (42) and simplifying, we obtain an equation for spectral
curve of two-gap potential $u(t)$:

$\nu^{2}=E^{5}+2\gamma_{1}E^{4}+(\gamma_{1}^{2}+2\gamma_{2})E^{3}+\left(2\gamma_{1}\gamma_{2}-\dfrac{1}{8}C_{2}\right)E^{2}+C_{3}E+C_{4},$
(44)

where

$\displaystyle
C_{3}=\dfrac{1}{8}u_{t}u_{ttt}-\dfrac{1}{16}u_{tt}^{2}+\dfrac{1}{4}(\gamma_{1}-5u)u_{t}^{2}+\dfrac{5}{4}u^{4}-\gamma_{1}u^{3}+\gamma_{2}u^{2}+\dfrac{1}{8}C_{2}u+\gamma_{2}^{2}-\dfrac{1}{8}\gamma_{1}C_{2},$
$\displaystyle
C_{4}=\dfrac{1}{64}u_{ttt}^{2}+\dfrac{1}{8}(\gamma_{1}-3u)u_{t}u_{ttt}-\dfrac{1}{8}uu_{tt}^{2}+\dfrac{1}{32}\left(C_{2}+16\gamma_{2}u-24\gamma_{1}u^{2}+40u^{3}+2u_{t}^{2}\right)u_{tt}$
$\displaystyle\qquad+\dfrac{1}{8}\left(2\gamma_{1}^{2}-2\gamma_{2}-10\gamma_{1}u+15u^{2}\right)u_{t}^{2}-3u^{5}+\dfrac{7}{2}\gamma_{1}u^{4}-(\gamma_{1}^{2}+2\gamma_{2})u^{3}$
$\displaystyle\qquad+\dfrac{1}{16}(16\gamma_{1}\gamma_{2}-3C_{2})u^{2}+\dfrac{1}{8}\gamma_{1}C_{2}u-\dfrac{1}{8}\gamma_{2}C_{2}.$

From (38) we have

$\displaystyle
u_{1}=\dfrac{1}{2}(u+v)=-\dfrac{1}{4B}I_{1}+\left(\dfrac{1}{2}-\dfrac{A}{4B}\right)u+\dfrac{1}{2}C_{1},$
(45) $\displaystyle
u_{2}=\dfrac{1}{2}(u-v)=\dfrac{1}{4B}I_{1}+\left(\dfrac{1}{2}+\dfrac{A}{4B}\right)u-\dfrac{1}{2}C_{1}.$

Therefore, functions $\widehat{p}_{j}$ and $\widehat{q}_{j}$ are solutions to
equations

$\displaystyle\partial_{t}^{2}\widehat{p}_{1}-2u\widehat{p}_{1}=-\dfrac{1}{4}\left(A+2B\right)\widehat{p}_{1},$
(46)
$\displaystyle\partial_{t}^{2}\widehat{p}_{2}-2u\widehat{p}_{2}=-\dfrac{1}{4}\left(A-2B\right)\widehat{p}_{2}.$

It is following from (42), (43), (45), and (46) that

$\displaystyle\widehat{p}_{1}=\left.\varepsilon_{11}\sqrt{\Psi}\exp\left\\{\nu(E)\int\dfrac{dt}{\Psi}\right\\}\right|_{E=E_{1}},$
$\displaystyle\widehat{q}_{1}=\left.\varepsilon_{12}\sqrt{\Psi}\exp\left\\{-\nu(E)\int\dfrac{dt}{\Psi}\right\\}\right|_{E=E_{1}},$
$\displaystyle\widehat{p}_{2}=\left.\varepsilon_{21}\sqrt{\Psi}\exp\left\\{\nu(E)\int\dfrac{dt}{\Psi}\right\\}\right|_{E=E_{2}},$
$\displaystyle\widehat{q}_{2}=\left.\varepsilon_{22}\sqrt{\Psi}\exp\left\\{-\nu(E)\int\dfrac{dt}{\Psi}\right\\}\right|_{E=E_{2}},$

where

$\varepsilon_{11}\varepsilon_{12}=1/B,\quad\varepsilon_{21}\varepsilon_{22}=-1/B,\quad
E_{1}=-(A+2B)/4,\quad E_{2}=-(A-2B)/4.$

It is not difficult to see that functions $\widehat{p}_{j}$ and
$\widehat{q}_{j}$ will be bounded when $E_{j}$ satisfies to conditions
$\mathrm{Re}(\nu(E_{j}))=0$.

Two-soliton potential of operator (42) has the form ($b>a>0$)

$u(t)=\dfrac{(a^{2}-b^{2})(b^{2}-a^{2}+a^{2}\cosh(2bt)+b^{2}\cosh(2at)}{2(b\cosh(bt)\cosh(at)-a\sinh(bt)\sinh(at))^{2}}.$
(47)

Substituting (47) into (40), we get

$A=-2(a^{2}+b^{2}),\quad C_{1}=-\dfrac{(a^{2}-b^{2})^{2}-B^{2}}{2B},\quad
C_{2}=0.$

The spectral curve (44) of potential (47) is determined by the equation

$\nu^{2}=E(E-a^{2})^{2}(E-b^{2})^{2}.$

Calculating $\nu^{2}(E_{j})$, we have

$\displaystyle\nu^{2}(E_{1})=\dfrac{1}{32}(a^{2}+b^{2}-B)\left((a^{2}-b^{2})^{2}-B^{2}\right),$
$\displaystyle\nu^{2}(E_{2})=\dfrac{1}{32}(a^{2}+b^{2}+B)\left((a^{2}-b^{2})^{2}-B^{2}\right).$

From conditions $\nu^{2}(E_{j})\leq 0$ ($j=1,2$) we have $B=\pm(b^{2}-a^{2})$.

If $B=b^{2}-a^{2}$ then $E_{1}=b^{2}$, $E_{2}=a^{2}$,

$c_{2}=\dfrac{1}{12}(c_{1}^{2}+8b^{2}-4a^{2}),\quad
c_{3}=\dfrac{1}{12}(c_{1}^{2}+8a^{2}-4b^{2}),$

and

$\displaystyle\widehat{p}_{1}=\widehat{q}_{1}=\dfrac{ib\sqrt{b^{2}-a^{2}}\cosh(at)}{(b\cosh(bt)\cosh(at)-a\sinh(bt)\sinh(at))},\quad\widehat{q}_{1}=-\widehat{p}_{1}^{\ast},$
$\displaystyle\widehat{p}_{2}=\widehat{q}_{2}=\dfrac{ia\sqrt{b^{2}-a^{2}}\sinh(bt)}{(b\cosh(bt)\cosh(at)-a\sinh(bt)\sinh(at))},\quad\widehat{q}_{2}=-\widehat{p}_{2}^{\ast}.$

The corresponding solution to equations (18) has the form

$\displaystyle
p_{1}(t,z_{1})=i\mathfrak{p}_{1}(t-c_{1}z_{1})e^{i\theta_{1}(t,z_{1})},\quad
q_{1}(t,z_{1})=-p_{1}^{\ast}(t,z_{1}),$ (48) $\displaystyle
p_{2}(t,z_{1})=i\mathfrak{p}_{2}(t-c_{1}z_{1})e^{i\theta_{2}(t,z_{1})},\quad
q_{2}(t,z_{1})=-p_{2}^{\ast}(t,z_{1}),$

where

$\displaystyle\mathfrak{p}_{1}(T)=\dfrac{b\sqrt{b^{2}-a^{2}}\cosh(aT+t_{a})}{(b\cosh(bT+t_{b})\cosh(aT+t_{a})-a\sinh(bT+t_{b})\sinh(aT+t_{a}))},$
$\displaystyle\mathfrak{p}_{2}(T)=\dfrac{a\sqrt{b^{2}-a^{2}}\sinh(bT+t_{b})}{(b\cosh(bT+t_{b})\cosh(aT+t_{a})-a\sinh(bT+t_{b})\sinh(aT+t_{a}))},$

and

$\displaystyle\theta_{1}(t,z_{1})=\dfrac{c_{1}}{2}t+\left(b^{2}-\dfrac{1}{4}c_{1}^{2}\right)z_{1}-\alpha\int\left(\mathfrak{p}_{1}^{2}(T)+\mathfrak{p}_{2}^{2}(T)\right)dt,$
$\displaystyle\theta_{2}(t,z_{1})=\dfrac{c_{1}}{2}t+\left(a^{2}-\dfrac{1}{4}c_{1}^{2}\right)z_{1}-\alpha\int\left(\mathfrak{p}_{1}^{2}(T)+\mathfrak{p}_{2}^{2}(T)\right)dt.$

Here $(t_{a},t_{b})$ is an initial two-dimensional phase. Magnitudes of
solutions (31) are shown in Fig. 3.

|
---|---
Magnitude $\mathfrak{p}_{1}(t-c_{1}z_{1})$ |  Magnitude $\mathfrak{p}_{2}(t-c_{1}z_{1})$
Figure 3: Magnitudes of solutions (48) for $a=3$, $b=5$, $c_{1}=1$, $t_{a}=2$,
$t_{b}=3$.

##  4 Orthogonal transformation

Since the matrix $J$ has two equal diagonal elements, an orthogonal
transformation of the vectors of solutions to equations (13) again gives
solutions to these equations. To prove this statement, we consider the
equation

$i\widetilde{\Psi}_{t}+\widetilde{U}\widetilde{\Psi}=\boldsymbol{0},$ (49)

where

$\displaystyle\widetilde{U}=\widetilde{U}_{0}+rJ,\quad\widetilde{U}_{0}=-\lambda
J+\widetilde{Q},$
$\displaystyle\widetilde{\Psi}=\widetilde{T}\Psi,\quad\widetilde{T}=\begin{pmatrix}1&\boldsymbol{0}^{t}\\\
\boldsymbol{0}&T\end{pmatrix}.$

From equations (7) and (49) it follows that
$\widetilde{Q}\widetilde{T}=\widetilde{T}Q$. Therefore, the following
equalities hold:

$\widetilde{\mathbf{q}}=T\mathbf{q},\quad\widetilde{\mathbf{p}}=\left(T^{t}\right)^{-1}\mathbf{p}.$
(50)

Thus, if the matrix $T$ satisfies to the condition

$TST^{\dagger}=S,$ (51)

then vectors $\mathbf{p}$ and $\mathbf{q}$ ($\widetilde{\mathbf{p}}$ and
$\widetilde{\mathbf{q}}$) are connected by the relation

$\mathbf{q}=S\mathbf{p}^{\ast},\quad\widetilde{\mathbf{q}}=S\widetilde{\mathbf{p}}^{\ast}.$

The following equalities

$\widetilde{\mathbf{G}}_{k}=T\mathbf{G}_{k},\quad\widetilde{\mathbf{H}}_{k}=\left(T^{t}\right)^{-1}\mathbf{H}_{k},\quad\widetilde{F}_{k}=TF_{k}T^{-1},\quad\widetilde{\mathcal{F}}_{k}=\mathcal{F}_{k}$

follow from the recurrence relations (10). Therefore, if vectors $\mathbf{p}$
and $\mathbf{q}$ are solutions to evolutionary equations (13), then vectors
$\widetilde{\mathbf{p}}$ and $\widetilde{\mathbf{q}}$ are also solutions to
the same equations. Thus, using formula (50) with the matrix

$T_{1}=\begin{pmatrix}\cos\varphi&\sin\varphi\\\
-\sin\varphi&\cos\varphi\end{pmatrix}$

and solution (31), it is possible to construct new elliptic solutions to
equation (18):

$\displaystyle\widetilde{p}_{1}=\dfrac{i}{\sqrt{2}}\left(\cos\varphi\operatorname{\mathrm{dn}}(t-c_{1}z_{1};k)+k\sin\varphi\operatorname{\mathrm{cn}}(t-c_{1}z_{1};k)\right)e^{i\theta},\quad\widetilde{q}_{1}=-\widetilde{p}_{1}^{\ast},$
(52)
$\displaystyle\widetilde{p}_{2}=-\dfrac{i}{\sqrt{2}}\left(\sin\varphi\operatorname{\mathrm{dn}}(t-c_{1}z_{1};k)-k\cos\varphi\operatorname{\mathrm{cn}}(t-c_{1}z_{1};k)\right)e^{i\theta},\quad\widetilde{q}_{2}=-\widetilde{p}_{2}^{\ast}.$

Let us note that the vectors $\widetilde{p}$ and $\widetilde{q}$ satisfy the
stationary equation (11) with a non-diagonal matrix

$\widetilde{C}_{n}=TC_{n}T^{-1}.$

For solution (52) the matrix $\widetilde{C}_{1}$ has the form

$\widetilde{C}_{1}=\dfrac{1}{2}\begin{pmatrix}3(c_{3}+c_{2})-(c_{3}-c_{2})\cos(2\varphi)&(c_{3}-c_{2})\sin(2\varphi)\\\
(c_{3}-c_{2})\sin(2\varphi)&3(c_{3}+c_{2})+(c_{3}-c_{2})\cos(2\varphi)\end{pmatrix}.$

Therefore, for transformed solutions the constants $\widetilde{c}_{j}$ are
equal:

$\displaystyle\widetilde{c}_{2}=\dfrac{1}{2}(c_{3}+c_{2})-\dfrac{1}{2}(c_{3}-c_{2})\cos(2\varphi),$
$\displaystyle\widetilde{c}_{3}=\dfrac{1}{2}(c_{3}+c_{2})+\dfrac{1}{2}(c_{3}-c_{2})\cos(2\varphi),$
$\displaystyle\widetilde{c}_{4}=\widetilde{c}_{5}=\dfrac{1}{2}(c_{3}-c_{2})\sin(2\varphi).$

At the same time, since the monodromy matrices of the functions
$\widetilde{\Psi}$ and $\Psi$ are similar

$\widetilde{M}=\widetilde{T}M\widetilde{T}^{-1},$

original and transformed solutions are associated with the same spectral
curve. Therefore, the Baker-Akhiezer function $\Psi$ can be constructed from a
spectral curve only up to a linear transformation $\widetilde{T}$.

## Discussions and conclusions

In many works that previously investigated finite-gap solutions of the Manakov
system (see, for example [12, 36, 34, 9, 13, 22]), following factors that we
found recently were not taken into account. First, as we showed in [32], if
functions $p_{j}$ are linearly dependent, then eigenfunctions of Lax operator
(7) are determined on two separated spectral curves. Secondly, as the authors
know, other researchers have not previously taken into account preserving
spectral curves orthogonal transformations of solutions.

And finally, as we have seen from the considered in the section 3 examples,
the number of phases of the solution is less than the genus of the
corresponding spectral curve. Indeed, it follows from equations (11) and (13)
that the following relations hold:

$\displaystyle\partial_{z_{n}}\mathbf{p}=-\sum_{k=1}^{n-1}c_{k}\partial_{z_{n-k}}\mathbf{p}-c_{n}\partial_{t}\mathbf{p}+i\left(r_{n}+\sum_{k=1}^{n-1}c_{k}r_{n-k}+c_{n}r_{n}\right)\mathbf{p}+iC_{n}^{t}\mathbf{p},$
$\displaystyle\partial_{z_{n}}\mathbf{q}=-\sum_{k=1}^{n-1}c_{k}\partial_{z_{n-k}}\mathbf{q}-c_{n}\partial_{t}\mathbf{p}-i\left(r_{n}+\sum_{k=1}^{n-1}c_{k}r_{n-k}+c_{n}r_{n}\right)\mathbf{q}-iC_{n}\mathbf{q}.$

Therefore, solutions $p_{j}$ and $q_{j}$ up to exponential multipliers are
$n$-phase functions (functions with $n$ arguments):

$\displaystyle
p_{j}(t,z_{1},\dots,z_{n})=\mathfrak{p}_{j}(t-c_{n}z_{n},z_{1}-c_{n-1}z_{n},\dots,z_{n-1}-c_{1}z_{n})e^{i\theta_{j}(t,z_{1},\dots,z_{n})},$
$\displaystyle
q_{j}(t,z_{1},\dots,z_{n})=\mathfrak{q}_{j}(t-c_{n}z_{n},z_{1}-c_{n-1}z_{n},\dots,z_{n-1}-c_{1}z_{n})e^{-i\theta_{j}(t,z_{1},\dots,z_{n})}.$

An equation for a spectral curve $\Gamma=\\{(\mu,\lambda)\\}$ has the form

$\mathcal{R}(\mu,\lambda)=\det(\mu
I-M)=\mu^{3}+\mathcal{A}(\lambda)\mu+\mathcal{B}(\lambda)=0,$ (53)

where

$\displaystyle\mathcal{A}(\lambda)=-\dfrac{1}{3}\lambda^{2n+2}-\dfrac{2c_{1}}{3}\lambda^{2n+1}+\sum_{j=2}^{2n+2}A_{j}\lambda^{2n+2-j},$
$\displaystyle\mathcal{B}(\lambda)=\dfrac{2}{27}\lambda^{3n+3}+\dfrac{2c_{1}}{9}\lambda^{3n+2}+\sum_{j=2}^{3n+3}B_{j}\lambda^{3n+3-j}.$

If $n\leq 3$, then a discriminant of (53), as a polynomial of $\mu$, is equal
to:

$\Delta(\lambda)=(c_{n+1}-c_{n+2})^{2}\lambda^{4n+4}+\sum_{j=1}^{4n+4}D_{j}\lambda^{4n+4-j}.$
(54)

Perhaps equality (54) is also true for other values of $n$. It follows from
equation (54) that when condition $c_{n+1}\neq c_{n+2}$ is fulfilled, the
curve $\Gamma$ has $(4n+4)$ branching points. Using the Riemann-Hurwitz
formula

$g=\dfrac{M}{2}-N+1,$

where $M$ is a number of branching points, $N$ is a number of sheets of a
covering, we get that the genus of the spectral curve $\Gamma$ is equal

$g=\dfrac{4n+4}{2}-3+1=2n.$

Thus, to construct finite-gap solutions to the Manakov system or to the vector
Kundu-Eckhaus equation, it is necessary to use trigonal curves, the genus of
which is twice the number of phases of solutions.

## Aknowledgements

The research was supported by the Russian Science Foundation (grant agreement
No 22-11-00196), https://rscf.ru/project/22-11-00196/

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|  Quantifying Intuition: Bayesian Approach to Figures of Merit in EXAFS
Analysis of Magic Size Clusters
---|---
|  Lucy Haddad,a,b Diego Gianolio,b David J. Dunstan, a and Andrei Sapelkina
|  Analysis of the extended X-ray absorption fine structure (EXAFS) can yield
local structural information in magic size clusters even when other structural
methods (such as X-ray diffraction) fail, but typically requires an initial
guess – an atomistic model. Model comparison is thus one of the most crucial
steps in establishing atomic structure of nanoscale systems and relies
critically on the corresponding figures of merit (delivered by the data
analysis) to make a decision on the most suitable model of atomic
arrangements. However, none of the currently used statistical figures of merit
take into account such significant factor as parameter correlations. Here we
show that ignoring such correlations may result in a selection of an incorrect
structural model. We then report on a new metric based on Bayes theorem that
addresses this problem. We show that our new metric is superior to the
currently used in EXAFS analysis as it reliably yields correct structural
models even in cases when other statistical criteria may fail. We then
demonstrate the utility of the new figure of merit in comparison of structural
models for CdS magic-size clusters using EXAFS data.

##

††footnotetext: a QMUL, Mile End Road, London E1 4NS UK; Tel: ; E-mail:
<EMAIL_ADDRESS>b Diamond Light Source, Diamond House Harwell
Science & Innovation Campus, Didcot OX11 0DE, UK. ††footnotetext: † Electronic
Supplementary Information (ESI) available: See DOI: 00.0000/00000000.

##  1 Introduction

Establishing the atomic structures of materials is a fundamental step in
understanding their mechanical, electronic and optical properties and is
essential for material applications. However, recovering the atomic structures
of nanomaterials is particularly challenging using standard structural
analysis techniques (e.g. X-ray and electron diffraction, Raman scattering,
etc.) due to loss of periodicity at atomic level and potential presence of
novel metastable atomic arrangements. This is especially true of the recently
discovered ultra-small truly mono-disperse nanoparticles — magic-size clusters
(MSCs) 1, 2, 3. As a consequence, several advanced structural methods such as
X-ray absorption spectroscopy (XAS) and pair distribution function (PDF)
analysis have been utilised to investigate atomic structure of MSCs 4,5,6.
XAS, in particular, has been shown to be sensitive to the atomic arrangements
and structural changes in MSCs delivering information about sample
stoichiometry and cluster symmetry discriminating between variety of
structural models 6.

This new class of nanoscale systems pushes XAS capabilities to the limit both
in terms of the quality of the data required and of analysis methods for the
two key parts of the X-ray absorption spectrum: X-ray absorption near edge
structure (XANES) and extended X-ray absorption fine structure (EXAFS). The
former is sensitive to the symmetry around the absorbing atom of interest
(e.g. Cd in CdS MSCs 6) and its oxidation state, while the latter provides
information about local coordination numbers, interatomic distances and local
atomic dynamics (see Fig. 1).

Analysis of EXAFS data typically involves background subtraction and
normalisation followed by theoretical EXAFS calculations for a selected
structural model (or a selection of model structures), comparisons of the
calculated spectrum with the data and parameter refinement to obtain the best
fit and the corresponding structural information 7 8. Theoretical calculations
and subsequent refinement are some of the most crucial steps and require a
suitable atomistic model, thus implying some prior knowledge of the atomic
structure or having an informed guess (e.g. based on molecular dynamics, DFT
calculations or similar material, etc.). Recovering atomic structure of MSCs
puts particularly stringent demands on model comparison in EXAFS because local
atomic arrangements can be quite similar 9. When theoretical and experimental
spectra are compared, in most EXAFS analysis programs (such as Artemis 10 and
Larch 11) there are a number of figures of merit (FoMs) available to provide
quantitative model evaluation to answer the question of whether the model is a
suitable match for the experimental data. However, none of the commonly used
FoMs take into account parameter correlation. At the same time, it is well-
known 12,13,14 that correlation can have significant negative consequence on
data refinement (i.e. larger errors) and, most importantly, on model
verification and selection.

Fig. 1: EXAFS analysis is the study and interpretation of the fluctuations in
the post-edge X-ray absorption spectrum. The fluctuations in the signal (in
the purple highlight) are the result of interference of the outgoing
photoelectron wave with the portion scattered by the neighboring atoms. The
EXAFS equation used for modelling these oscillations. $\chi(k)$ is related to
the plotted absorption $\mu(E)$ by the transform:
$k=\sqrt{\frac{2m_{e}}{\hbar^{2}}(\hbar\omega-E_{0})}$, $E_{0}$ being
absorption edge energy.

This shortcoming of the EXAFS FoMs has long been recognised as a problem and
in the latest development of the Artemis (one of the most commonly used EAXFS
analysis package) a heuristic "happiness parameter" is offered to provide in-
code indication of the fit quality. This parameter is based on decades of
EAXFS analysis experience and includes, with varying weighting, an R-factor (a
numerical measure of how well the fit over-plots the data), penalties for
parameter correlations, restraints, the number of independent parameters, etc.
15. While recognised as an important guide during data analysis, being a
heuristic parameter, it has no firm basis in statistics and therefore cannot
be quoted in publications.

In this article we introduce for the first time in EXAFS analysis an FoM that
explicitly includes parameter correlations - the Bayes Factor Integral (BFI)16
We use EXAFS data for crystalline Ge at low temperature to demonstrate that
the BFI is more sensitive than the typical FoMs used for EXAFS analysis to
model choice. We then demonstrate that the BFI consistently points towards the
correct structure as preferred model. We then use the BFI to compare a
selection of models for a material with unknown structure: CdS Magic Sized
Clusters (MSCs 311 and 322) 6 5 17. With these examples, we introduce the BFI
as a numerical metric for quantifying intuition in EXAFS model comparison.

##  2 Methods

###  2.1 Figures of Merit in EXAFS analysis

Least-squares fitting (LS, the minimisation of the sums of squares of
residuals to optimise a model) is a commonly used method to fit data, to
estimate parameters and to make decisions about model selection. In EXAFS LS
fitting, reported FoMs in Larch and Artemis are $\chi^{2},\chi_{\nu}^{2}$,
R-factor, AIC and BIC.

The first is the well-known statistical value characterising the residuals
between the model and experimental data. It is a simple statistical measure of
how small the fit residuals are: i.e. how closely does the model fit the data,
However, it is has been well-established that the number of independent
variables (fitting parameters) can significantly influence the value of
$\chi^{2}$. The total number of parameters available in EXAFS analysis is
limited by the sampling theorem of the Fourier analysis 18 (this is also known
as the Nyquist criterion/theorem in EXAFS community). Therefore, the most
commonly reported fitting statistic in EXAFS is the so-called reduced chi-
squared, $\chi_{\nu}^{2}$, based on $\chi^{2}$ but with a modification to
include normalisation of $\chi^{2}$ by degrees of freedom such that once the
maximum number of free parameters allowed for the data ($N_{ind}$) is reached
18, it will become negative and provides a clear indication of over-fitting.
The R-factor is another variation of the $\chi^{2}$ criterion with a different
normalisation factor.

The AIC and BIC are not found in Artemis, but are used as FoMs in Larch 11 to
aid model comparison: both are based on the Likelihood function (rather than
$\chi^{2}$ ) while also including a penalty term for adding parameters to the
model (adding fitting parameters to a model — physically meaningful or not —
normally increases the likelihood of the model while reducing the probability
that the model is correct).

There are a number of problems with the figures of merit described above. They
treat all parameters alike, whether physically-meaningful or not. Apart from
the number of parameters approaching $N_{ind}$, there is not much help from
these FoMs to tell whether one has a physically meaningful fit. Crucially,
none of them include parameter correlations, while it is well documented
12,13,14 that parameter correlations indicate over-fitting and have
significant consequences on the refinement errors and model selection. For
example, in EXAFS analysis it is well-established that there exist
correlations between fitting parameters even when $N_{ind}$ 18 is not exceeded
19,7,11 ,20,21. Both Artemis and Larch do provide functionality to calculate
parameter correlations, but these are almost never used to assess the quality
of the fit nor to aid model justification or selection. To compensate for that
and to help guide users during the refinement in Artemis there is an inbuilt
FoM that does include correlations: the Happiness parameter 15. However, it
cannot be reported in publications since it has no mathematical basis: it is
an empirical FoM that can be adjusted between fits to accord with the user’s
preferences. Hence, there is a need for a FoM rooted in statistics that does
include parameter correlations.

###  2.2 Bayes Theorem, Bayes Factor and Bayes Factor Integral

The goal of EXAFS analysis can be described as “to find the best model
parameters that fit the data” or, more generally, “to select the best model
that fits the data”. This lends itself naturally to the Bayesian statistical
analysis and the use of Bayes theorem:

$P(M|D)=\frac{P(M)P(D|M)}{P(D)}$ (1)

where $P(M|D)$ represents the conditional probability of the model $M$, given
the data $D$. In the case of multi-parameter fitting (including EXAFS) this
can be rewritten as (see, for example 22):

$P(\mathbf{w}|D,M)=\frac{P(D|\mathbf{w},M)P(\mathbf{w}|M)}{P(D|M)}.$ (2)

where $\mathbf{w}$ is the vector of parameter values. Models can then be
compared by, for example, taking a ratio of their conditional probabilities
$P(\mathbf{w}|D,M)$. This ratio of probabilities of two models (e.g. $i$ and
$j$) represents the odds ratio in favour of one model over the other 23:

$O_{ij}=\frac{P(M_{i}|D,I)}{P(M_{j}|D,I)}$ (3)

where $I$ is prior information we have about the models. Using Eq. 2 it is
straightforward to show that:

$O_{ij}=\frac{P(M_{i}|I)P(D|M_{i},I)}{P(M_{j}|I)P(D|M_{j},I)}=\frac{P(M_{i}|I)}{P(M_{j}|I)}BF_{ij}$
(4)

where $BF_{ij}$ is the Bayes factor23. The ratio on the right hand side is the
prior odds ratio of the two models and throughout this work we consider this
to be unity (i.e. no preference of one model over another). Thus, to compare
the models we need to compute $P(D|M_{i},I)$ — probability of the data given
the model and prior information. However, expressions for $P(D|M_{i},I)$ can
be rather complicated and for analysis involving many (in general correlated)
parameters they include the evaluation of a multi-dimensional integral over
the parameter space (the MLI – marginal likelihood integral). Assuming uniform
top-hat priors and a Gaussian error distribution for independent identically
distributed experimental data points gives (see for example 23, p. 276):

$P(D|M_{i},I)=\frac{1}{\prod_{i=1}^{m}\Delta
p_{i}}\frac{1}{\sigma^{n}(2\pi)^{n/2}}\int_{\Delta
p}d^{m}\mathbf{p}e^{-\frac{1}{2}{\mathbf{p^{T}}\mathbf{Cov_{p}}^{-1}\mathbf{p}}}.$
(5)

where $\Delta p_{i}$ are prior parameter ranges, $n$ is the number of the data
points, $m$ is the number of the model parameters
$\mathbf{p}=[p_{1}p_{2}...p_{m}]^{{\text{T}}}$ and $\mathbf{Cov_{p}}$ is the
parameter covariance matrix. Thus, although the Bayesian approach has already
been demonstrated in application to EXAFS analysis 2425, it has not been used
to any significant extent, as far as we can tell, on account of its
complexity. Indeed, parameter correlation is almost always the case in EXAFS
and would normally require evaluation of the multidimensional integral in Eq.
5. That can be addressed by constructing an orthonormal set of the model basis
functions 23 (model parameters) so that the new parameters will have no
correlation and hence the multidimensional integral can be replaced by the
product of multiple single integrals. However, this would require redefining
the problem in terms of the new (orthogonal) parameter set, repeating the fit
and then back-transforming the new parameters to recover the original ones.

Here we propose a simple alternative FoM that requires only trivial
modifications to the statistical procedures already existing in EXAFS
analysis. We note that the multidimensional integral on the far right of the
Eq. 5 constitutes the volume in the parameter distribution space. We also note
that $\mathbf{Cov_{p}}$ is symmetric positive definite, hence its
diagonalization involves basis rotation. However, the volume (and also
$\text{det}(\bf{Cov_{p}})$) does not change under rotation of the parameter
space required for the transformation to the orthonormal basis. Hence, the
following expression normally corresponding to the orthonormal parameter set
can be used for calculation of $P(D|M_{i},I)$ — Bayes Factor Integral (BFI) —
for a model with parameter correlations present 16:

$BFI=(2\pi)^{m/2}L_{\text{max}}\frac{\sqrt{\text{det}(\bf{Cov_{p}})}}{\prod_{i=1}^{m}\Delta
p_{i}}$ (6)

where $\Delta p_{i}$ are the initial parameter ranges and $L_{\text{max}}$ is
the likelihood for the model. Thus defined $BFI$ can then be used for model
comparison (giving preference for a model with the larger value of $BFI$)
following EXAFS data fitting without the need to redefine the problem in the
new orthonormal parameter set. We call this $BFI$ (rather than, for example,
MLI) to distinguish from a more common case when the orthogonal parameter set
is used to obtain Eq. 6 (and therefore the $\bf{Cov_{p}}$ is a diagonal
matrix). Crucially, the FoM in Eq. 2 naturally incorporates the Occam factor:

$\Omega=(2\pi)^{m/2}\frac{\sqrt{\text{det}(\bf{Cov_{p}})}}{\prod_{i=1}^{m}\Delta
p_{i}}$ (7)

that accounts for parameter correlations as well as parameter ranges and
provides a penalty for a model with significant parameter correlations and/or
large initial parameter ranges (parameter uncertainty). However, since values
of $BFI$ can vary drastically, a more convenient way of evaluation is through
comparison of $\ln(BFI)$ of the corresponding models. In such a case model
evaluation is reduced to calculation of the $\ln(BFI)$ ratios — designated in
this paper as $\ln(BF)$ — with the following scale16 26 for $\ln(BF)$ values
that differentiate between the models:

  * •

$<1$ – barely worth considering,

  * •

$1-2$ – substantial,

  * •

$2-5$ – strong evidence,

  * •

$>5$ – decisive.

We proceed below by testing this approach on a reference data set (crystalline
Ge, c-Ge) before applying the procedure to the Magic Size clusters of CdS.

##  3 Results and Discussion

###  3.1 The Case of Crystalline Germanium

To verify the utility of BFI we first used the XAS data 27 for Germanium
collected at 12K (the x-ray absorption spectrum is shown in Figure 2). The
data were selected on the account of their high quality. The structure of Ge
at this temperature is well-established and has been verified by previous
publications 28293027.The data analysis was performed in Larch 11 for
background removal, normalisation and the actual fitting of the models to the
experimental data since we found Larch to provide the most comprehensive
fitting statistics. Figure 1 shows the EXAFS equation used to fit data where
$N$ is number of nearest neighbours, $R$ is absorber-scatterer atom distance,
$S_{0}^{2}$ is an amplitude reduction factor, $\sigma^{2}$ is Debye-Waller
factor, $F(k)$ is photoelectron scattering amplitude, $\lambda(k)$ is
photoelectron mean free path, and $\phi(k)$ is the phase shift. The latter
three parameters ($F(k)$, $\lambda(k)$ and $\phi(k)$) are calculated using the
FEFF 9 code 31 32 and therefore are not refined.

(a) Ge spectrum in E-space.

(b) Ge spectrum in k-space.

(c) Ge spectrum in R-space.

Fig. 2: The Ge 12K XAS spectrum used for fits.

Three structural models have been selected for comparison: (i) the actual
structure of crystalline Ge at 12 K known to be a 4-coordinated face-centered
diamond cubic type 29 33 (Figure LABEL:fig:n4cif, $Fd3m$, Model 1); (ii)
6-coordinated high-pressure Ge phase VI structure ($Cmma$, Model 2); (iii)
6-coordinated high pressure $\beta$-Sn structure of Ge 34 ($I4/amd$, Models
3a, 3b). For Model 3 two different refinements were carried out: one (Model
3a) was for a single shell of 6 nearest neighbours, while for the other (Model
3b) 3 shells of 2 atoms were used to reflect the actual nearest neighbour
configuration in the $\beta$-Sn structure. This was also used to gauge the
effect on the BFI of increasing the number of model parameters.

To enable a fair model comparison, for each model we only looked at the first
peak in the $R$-space (corresponding to the Ge-Ge bond length of 2.45 Å; the
atomic shell structure beyond the first shell is very different in the three
selected models) and we used single-scattering paths only (see Figure 2). The
data were fitted over the range of $2.00<k<22.93\AA^{-1}$ in $k$-space (see
Fig. 2). This ensured that only the first-shell EXAFS were fitted. Parameter
ranges are given in Table 1 and are defined as follows. The amplitude
reduction factor $S_{0}^{2}$ corrects for inelastic effects in the absorbing
atom 35. This is empirically established to be in the range 0.8-1, and is
well-covered by 0.5 range. The shift in the edge position $E_{0}$ accounts for
errors in experimental calibration and for empirical convention in
determination of the absorption edge position 19 7 — the range typically does
not exceed 10 eV. Relative change in the nearest-neighbor interatomic distance
$\Delta R$ is not expected to exceed 10% as the interatomic distances are
determined by the covalent radii of elements and the pressure-temperature
conditions (as an example, 10% bond length variation is well above that
expected on melting or under pressures as high as 10 GPa in a typical
semiconductor material such as Ge 30). Mean squared relative displacements of
atoms due to atomic vibrations $\sigma^{2}$ (and static disorder, if any)
accounts for damping effects on $\chi(k)$. The initial value can be calculated
using e.g. correlated Debye or Einstein approximations 3637 38 and for c-Ge at
12 K this is around 0.003 Å2 27 hence the range of $\pm 0.003$ is selected to
make it positive-definite. The number of nearest neighbours $N$ was set
according to the structural models and was not refined.

Parameter | Initial value | Range
---|---|---
$S0^{2}$ | 0.9 | 0.5
E0 | 0 | 10
$\Delta R$ | 0 | 0.1
$\sigma^{2}$ | 0.003 | 0.006
Table 1: Parameter ranges for all models.

For Model 1 (zinc-blende structure), one single-scattering single-shell path
was used to fit the spectrum. For Model 2 (the high-pressure $Cmma$
structure), the spectrum was fit with 3 single-scattering single-shell paths
between (in total) 6 atoms in the first shell to describe the signal. For
($\beta$-Sn) Model 3a, one single-scattering single-shell path was used at
first, and then 3 single-scattering first shell paths were used (model 3b).

Model | Ln(BFI)
---|---
1 | -6.65
2 | -10.07
3a | -7.32
3b | -10.22
Table 2: Models and Ln(BF) values for the different Ge models. Fig. 3: Ln(BF)
values for the Ge models.

The summary of the results for the $\ln(BF)$ (the difference between
$\ln(BFI)$ values) are shown in Fig. 3. One can see that Model 1 is favoured
over all other models except for 3a (the single path $\beta$-Sn fit): the
$\ln(BF)$ between Model 1 and all other models (except 3a) are found to be $>$
3 providing strong evidence for Model 1 being the preferred structure. The
$\ln BF$ value between Model 1 and 3a is 0.67 is slightly in favour of Model 1
but not statistically significant according to the criteria outlined at the
end of the previous section. However, the currently available fitting
statistics FoMs found in the corresponding tables† favour other models: Model
2 has lowest $\chi^{2}$ and $\chi_{\nu}^{2}$, Model 3b has the lowest value of
$R$-factor, while AIC and BIC favour Model 2 over Model 1. This shows that
reliance only on the currently used FoMs in EXAFS analysis can lead to an
incorrect atomic structure model as the best solution. At the same time, we
see that the $BFI$ is able to deliver the correct result in this relatively
complicated case – after all we used a single peak only in the EXAFS FT
magnitude in order to differentiate between the models. Having verified the
utility of the proposed $BFI$-based FoM in case of the reference system, in
the next section we apply the Bayes approach to analysis of EXAFS of MSCs.

###  3.2 Bulk CdS $k$-space Fitting

Before proceeding on CdS MSCs we further tested the utility of the new FoM in
$k$-space fitting of the bulk crystalline CdS. As reference data, bulk
crystalline CdS EXAFS data at 90K at Cd K-edge were fit in Larch 11. The first
shell (Cd-S scattering paths) in k-space was fit using several different
structures respectively: a zinc-blende, wurtzite 39, NaCl-like 40 and cmcm
sturcutres 41, latter two being high-pressure derived structures.

Parameter | Initial value | Range
---|---|---
$\Delta R$ | 0 | 0.1
E0 | 0 | 10
$S0^{2}$ | 0.9 | 0.5
$\sigma^{2}$ | 0.006 | 0.006

Table 3: Parameter values and ranges for all MSCs. Model | Ln(BFI)
---|---
ZB | -0.738
WZ | -1.31
NaCl-Like | -2.04
cmcm | -2.31
Table 4: Ln(BFI) values for each model in the bulk CdS EXAFS fitting (1st
shell).

K-space noise ($\epsilon_{k}$) was was evaluated from the signal between
6.50<k<18.30 Å and the fit to the EXAFS data was carried out in the region
2.50<k<15.0 Å, parameter ranges for the BFI calculations are shown in Table 3.
The results of the fit are shown in the Table 4. The $BFI$-based FoMs support
zinc-blende and wurtzite structures significantly over the NaCl-like and cmcm
models. This is consistent with our previous results where XPDF analysis of
bulk CdS (and of regular CdS quantum dots) has shown CdS to be a mix of zinc-
blende and wurtzite structures 5.

###  3.3 Magic Sized Clusters

Magic Sized Clusters (MSCs) are ultra-small ($<$3nm) colloidal semiconductor
systems 4. They are materials of interest due to their monodisperse nature 3
that suggests one can deliver atomic-level control of system size using
colloidal synthesis route. Their atomic structure is still under debate as are
the methods of their structural verification. One of the key challenges for
the latter is the possibility for stable and multiple meta-stable) atomic
arrangements that are size- and temperature-dependent 9, 5, 6.

The MSCs under investigation in this work are CdS. These MSCs exhibit a sharp
UV-vis absorption peak at 311 nm (MSC 311) but when heated to $60^{\circ}$C 4
this peak shifts to 322 nm (MSC 322) 6 and the shift is accompanied by atomic
structure rearrangement as indicated by x-ray pair distribution function
(xPDF) and XAS analysis 5, 6. Due to their small size leading to the lack of
long-range order, establishing the atomic-level structure of MSCs is
challenging 42 and in this work we examine the sensitivity of our new EXAFS
$BFI$-based FoM to the structural model selection. To this end we compared 4
models as candidates of possible structures of MSCs 311 and 322: i) Cd40S19
with zinc-blende structure, ii) Cd40S20 with wurtzite structure, iii) Cd33S32
$\beta$-Sn-like structure and ii) Cd37S20 with an InP-like structure. The
rationale for the model selection is as follows:

  1. i)

Bulk CdS can possess wurtzite or zinc-blende structure (while regular CdS
quantum dots are known to exhibit both characters 5). Hence, when constructing
a model for an unknown atomic structure of MSCs, the atomic arrangement found
in the bulk can be a starting point if there is no other information (this is
frequently the case).

  2. ii)

It has been observed 30 that average interatomic distances in small
nanoparticles can be reduced compared to their bulk counterparts. This can be
interpreted as an effective pressure on these systems. Such compression may
results in distortion towards the $\beta$-Sn structure 43, hence it is
reasonable to use it as one of the structural models.

  3. iii)

It has recently been shown that an InP-like structure 44 provides the best fit
to PDF 5 data in CdS MSCs.

All clusters have been cut as spherical regions of appropriate size from the
corresponding bulk crystalline structure and were terminated with oxygen
(except for InP-like cluster where the structure from our recent work 5, 6 was
used). The bond length was set to that found in zinc-blende and wurtzite
structure at ambient condition (2.55 Å). After fitting with the initial
structures they were all relaxed over 500 steps using Avogadro 45 Universal
Force Field 46 and fitting process has been repeated. Data analysis has been
carried out in Larch 11 and Mathematica 13.0 (for $BFI$ calculations). The
initial parameter values and their ranges for all models are given in Table 3
(if the standard errors returned by Larch exceed the half range set for
$\Delta p_{i}$ terms, the range is corrected to 2 $\times$ standard error).

For each model the five highest-importance single-scattering paths were fitted
up to the second cumulant ($\sigma^{2}$), we then fitted the third cumulant
for the highest-importance Cd-O single scattering path of those included in
the fit already. While at low temperatures (90 K in our case) $3^{rd}$
cumulants for the atoms in the bulk of a nanoparticle are expected not to be
significant, this may not be the case for the surface Cd-O coordination shell.
The $\ln(BFI)$s for and $\ln(BF)$s between the models were calculated using
the same method as in the Ge model comparisons with the fits carried out in
the range of $2-17$ Å-1 for the the coordination shell $1-3$ Å (see Fig. 4).
For the noise value used in EXAFS chi-squared calculations we used the
standard deviation of the $k$-space spectra for the MSC 311 and 322 data at
$14.50-15$ Å-1 and $13-14$ Å-1 respectively.

(a) R-space EXAFS of the MSCs.

(b) $k^{2}$-space EXAFS of the MSCs.

Fig. 4: MSC 311 and 322 data used for fits.

The results for $\ln(BFI)$ for as-prepared and relaxed model clusters for 311
and 322 MSCs are given in the Tables 5 and 6 below and summarised in Figs.
5(a) and 5(b).

Model | Ln(BFI)
---|---
zinc-blende | -5.71
$\beta$-Sn | -7.36
InP | -5.63
Wurtzite | -7.19
Table 5: $\ln(BFI)$ values for optimized models, MSC 311. Model | Ln(BFI)
---|---
zinc-blende | -10.78
$\beta$-Sn | -11.97
InP | -6.39
Wurtzite | -15.11
Table 6: ln(BFI) values for optimized models, MSC 322.

These results show that in the case of MSCs 311 InP and Zinc-Blende models are
almost equally favoured over the $\beta$-Sn and Wurtzite structures. The
results for $\ln(BFI)$ of 322 MSCs show that InP-like structure is
significantly favored over all other models. Thus, our findings here indicate
that the new FoM: i) is pointing to the InP-like model as the most probable
structure; ii) is detecting the difference between the EXAFS data of the two
MSCs (as reflected in changing values of FoM and model preferences). This is a
very significant finding considering the differences between the EXAFS signals
for MSC 311 and 322 are very small (see Figs. 4) and the corresponding
$\chi^{2}$-values are also quite close (see Supporting Information). The
preference for InP-like structure is certainly in line with previous work 6 5
where it was shown to provide the best model to fit xPDF, EXAFS and XANES
data. All this provided very strong support to using the new Bayesian FoM we
introduced in this work as a universal metric for model comparison and
selection.

(a) Ln(BF) values for MSC 311.

(b) Ln(BF) values for MSC 322.

Fig. 5: Charts of the results for each MSC.

##  4 Conclusions

In this work we introduced the Bayesian-based statistical metric, the Bayes
Factor Integral, for model comparison in EXAFS analysis. We showed for the
first time that the new FoM provides a superior tool for model comparison in
EXAFS by quantifying the intuitions about the parameter ranges and
correlations through the Occam factor. We tested the new Bayesian FoM against
reference EXAFS data for Ge and demonstrated that the it is superior to the
FoMs typically used in EXAFS analysis and reliably predicts the correct
structure. In the process we showed that ignoring model parameter correlations
may result in a selection of an incorrect structural model. We then applied
the new FoM for model comparison in analysis of MSCs where we demonstrated
that it is sensitive to the differences between EXAFS signals of MSCs 311 and
322 and can point to the most probable structural models for these systems.

So far we utilised identical parameter ranges for all models of interest in
all our tests (except for the range correction when the standard error $\geq
1/2\Delta p_{i}$). Imposing stricter model-specific parameter ranges in the
calculation of model $BFI$ values should provide better results in model
selection. Such model-specific parameter ranges can be obtained, for example,
from molecular-dynamics or ab-initio simulations.

We note that the current approach is so far limited by the requirement of
providing initial guess structures, while the ultimate goal of structural
analysis of MSCs (and of non-periodic systems in general) is producing
structural models for materials with unknown structures. We believe that newly
developed FoM can be the crucial aid on that pathway, for example by combining
analysis described in this paper with machine learning/AI methods.

## Author Contributions

## Conflicts of interest

There are no conflicts to declare.

## Acknowledgements

LH is grateful to Diamond Light Source and Queen Mary University of London for
the joint studentship and funding to support this work.

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