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Ishimori-II equations, Davey-Stewartson-II equations,  
vortices, lumps, spectral decompositions
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\begin{document}



\title{Spectral decomposition theorem for the Ishimori--II equations}



\author{\sc Dmitry Pelinovsky

\\

{\small\it Department of Mathematics, McMaster University,}

\\

{\small\it Hamilton, Ontario, Canada, L8S 4K1}

\\

\sc and

\\

\sc Derchyi Wu \\

{\small \it Institute of Mathematics, Academia Sinica, 
Taipei, Taiwan, R.O.C.}

}



\date{\today}

\maketitle



\begin{abstract}

We study the gauge transformation between spectral problems

and their adjoints for the Ishimori-II and Davey--Stewartson II

equations. We prove the spectral decomposition theorem for

eigenfunctions of a linear operator associated with the

Ishimori--II equations.

\end{abstract}



\section{Introduction}



Spectral decomposition theorem is a fundamental result

in spectral theory of self-adjoint operators.

Many nonlinear partial differential equations

in the inverse scattering transform method

are associated with non-self-adjoint linear

operators \cite{AC}. Since the regular direct and

inverse scattering transforms are mutually reversible,

non-self-adjoint linear operators must possess

complete sets of eigenfunctions which are orthogonal

with respect to eigenfunctions of the adjoint operators.

If initial data evolve into algebraically decaying lump 

solutions of nonlinear equations, the spectral

decomposition involves both the continuous

and discrete spectrum of linear operators.

Potentials of linear operators associated

with lump solutions

are defined in a weaker function space, compared

to that required for a regular inverse scattering

transform. Nevertheless, spectral decomposition theorems for

time-dependent Schr\"{o}dinger and Dirac operators

with lump potentials were proved in \cite{PS1,PS2}

by methods of complex analysis based on the

Riemann--Hilbert and $\bar{\partial}$ problems.

It was shown that lumps of the

Kadomtsev--Petviashvili-I (KP-I) equation are

structurally stable in the spectral decomposition

\cite{PS1}, while lumps of the

Davey--Stewartson-II (DS-II) equations are

structurally unstable and decay into continuous

spectrum under generic perturbations of the initial data

\cite{PS2}.



In this paper, we study a complete set of

eigenfunctions associated with

the Ishimori-II (IS-II) equations,

\begin{eqnarray}

\nonumber

{\bf S}_t + \frac 12 {\bf S} \times

({\bf S}_{xx}- {\bf S}_{yy})+\phi_x {\bf S}_y

+\phi_y {\bf S}_x & = & 0,\\

\label{ISII}

\phi_{xx}+\phi_{yy} - {\bf S} \cdot ({\bf S}_x\times

{\bf S}_y) & = & 0,

\end{eqnarray}

where ${\bf S} = (S_1,S_2,S_3)^T = {\bf S}(x,y,t)$

is a three-component unit vector on the plane $(x,y)$

such that

$$

{\bf S}\cdot {\bf S}=1, \qquad

\lim_{|x|,|y| \to \infty} {\bf S}(x,y,t) = (0,0,1)^T,

$$

while $\phi(x,y,t)$ is scalar potential function

generated by the topological charge density

${\bf S} \cdot ({\bf S}_x\times {\bf S}_y)$.



The IS-II equations describe time evolution of

the system of static spin vortices on the plane $(x,y)$

\cite{Ishimori}. Static spin vortices are classified by the integer

values of the topological charge:

\begin{equation}

Q = \frac{1}{4 \pi} \int\!\!\int_{{\Bbb R}^2}

{\bf S} \cdot ({\bf S}_x\times {\bf S}_y) dx dy.

\end{equation}

From mathematical point of view, the IS-II equations

are remarkable because of their gauge equivalence to

the DS-II equations \cite {BC2,LS}:

\begin{eqnarray}

\nonumber

i q_t + \frac 12 \left( q_{xx}- q_{yy} \right)

+ 4 \left( \varphi - |q|^2 \right) q & = & 0, \\

\label{DSII}

\varphi_{xx} + \varphi_{yy} - 2 \left( |q|^2 \right)_{xx}

& = & 0.

\end{eqnarray}

Linear non-self-adjoint operators

for the IS-II equations (\ref{ISII}) have

potentials multiplied by $x$-derivative operators,

compared to linear operators for the DS-II equations

(\ref{DSII}). As a result, a proof of the spectral

decomposition theorem

is particularly difficult for the linear operators

of the IS-II equations. We prove the orthogonality

and completeness of eigenfunctions by using

the inverse scattering transform

for both IS-II and DS-II equations,

with the gauge transformation between these equations.

The key property in our analysis is a commutativity

theorem of the gauge transformations between

the linear operators and their adjoints.



The paper is organized as follows. In Section 2 we

review the scattering data of the IS-II equations.

In Section 3 we study the gauge transformation between

the IS-II and DS-II equations.

An adjoint problem for the IS-II equations

is introduced in Section 4. In Section 5,

we study the gauge transformation

between the adjoint problems and prove

commutativity of the gauge and adjoint transformations.

Section 6 presents applications to the spectral

decomposition and surface representation theorems.



\section{Inverse scattering transform for the Ishimori-II equations}



The IS--II equations are the compatibility

conditions of the over-determined linear system,

\begin{eqnarray}

\label{Lax1}

\left( i\partial_y+P\partial_x \right) \mbox{\boldmath $\Psi$}=0, \\

\label{Lax2}

\left( 2 \partial_t - 2 i P \partial_x^2 - 

\left( i P_x-P_yP-iP\phi_x-\phi_y \right) \partial_x \right)

\mbox{\boldmath $\Psi$}=0,

\end{eqnarray}

where $\mbox{\boldmath $\Psi$} =

(\Psi_1,\Psi_2)^T = \mbox{\boldmath $\Psi$}(x,y,t)$,

$P = S_1\sigma_1+S_2\sigma_2+S_3\sigma_3 = P(x,y,t)$,

and $\sigma_1$,$\sigma_2$,$\sigma_3$ are Pauli matrices,

$$

\sigma_1= \left( \begin{array}{ccc} 0 &1\\ 1& 0 \end{array} \right),

\qquad

\sigma_2= \left( \begin{array}{ccc} 0 & -i \\ i& 0 \end{array} \right),

\qquad

\sigma_3= \left( \begin{array}{ccc} 1 & 0\\ 0& -1 \end{array} \right),

$$

such that

$$

P = \left( \begin{array}{ccc} S_3 & S_1 - i S_2 \\

S_1 + i S_2 & -S_3 \end{array} \right).

$$

We introduce complex variables for elliptic problems on

the plane $(x,y)$:

\begin{equation}

\label{coordinate}

z = \frac{1}{2} (y - i x), \qquad

\bar{z} = \frac{1}{2} (y + i x),

\end{equation}

such that the spectral problem (\ref{Lax1})

can be rewritten for $\mbox{\boldmath $\Psi$}

= \mbox{\boldmath $\Psi$}(z)$:

\begin{equation}

\label{spectrum}

\left( \begin{array}{ccc} \partial_{\bar z} & 0 \\

0 & \partial_z \end{array} \right) \mbox{\boldmath $\Psi$}

= \frac{1}{2} \left( P(z) - \sigma_3 \right)

\left( \partial_{z} - \partial_{\bar z} \right)

\mbox{\boldmath $\Psi$},

\end{equation}

on $z \in {\Bbb C}$. Here and henceforth

we suppress the dependence of $\mbox{\boldmath $\Psi$}$

from $\bar{z}$ and $t$.



The $\overline\partial$-method was used in \cite{BC2,KM}

to solve the inverse scattering transform for the linear

spectral problem (\ref{spectrum}). We introduce a fundamental

matrix of solutions of (\ref{spectrum}),

\begin{equation}

\mbox{\boldmath $\Psi$} =

\left( \mbox{\boldmath $\mu$} e^{i k z},

\mbox{\boldmath $\chi$} e^{-ik\bar{z}} \right),

\end{equation}

where the vectors

$\mbox{\boldmath $\mu$} = (\mu_1,\mu_2)^T =

\mbox{\boldmath $\mu$}(z,k)$ and

$\mbox{\boldmath $\chi$} = (\chi_1,\chi_2)^T

= \mbox{\boldmath $\chi$}(z,k)$ solve the problems,

\begin{eqnarray}

\label{mu}

\left( \begin{array}{ccc} \partial_{\bar z} & 0 \\

0 & \partial_z \end{array} \right) \mbox{\boldmath $\mu$} +

ik \left( \begin{array}{cc} 0 & 0 \\ 0 & 1 \end{array}

\right) \mbox{\boldmath $\mu$} = {\cal P}_k \mbox{\boldmath $\mu$}, \\

\label{chi}

\left( \begin{array}{ccc} \partial_{\bar z} & 0 \\

0 & \partial_z \end{array} \right) \mbox{\boldmath $\chi$} -

ik \left( \begin{array}{ccc} 1 & 0 \\

0 & 0 \end{array} \right) \mbox{\boldmath $\chi$} =

{\cal P}_k \mbox{\boldmath $\chi$},

\end{eqnarray}

where

\begin{equation}

{\cal P}_k = \frac 1{2} \left( P(z) - \sigma_3 \right)

\left(\partial_z- \partial_{\bar z}+ik \right).

\end{equation}

The eigenvectors $\mbox{\boldmath $\mu$}(z,k)$

and $\mbox{\boldmath $\chi$}(z,k)$ satisfy

the boundary conditions on the plane $(x,y)$:

\begin{equation}

\label{normalization}

\lim_{|z| \to \infty} \mbox{\boldmath $\mu$}(z,k) =

{\bf e}_1 = \left( \begin{array}{cc} 1 \\ 0 \end{array} \right),

\qquad

\lim_{|z| \to \infty} \mbox{\boldmath $\chi$}(z,k)

= {\bf e}_2 = \left( \begin{array}{cc} 0 \\ 1 \end{array} \right).

\end{equation}

If ${\bf S}(z)$ is real-valued,

the two fundamental solutions $\mbox{\boldmath $\mu$}(z,k)$

and $\mbox{\boldmath $\chi$}(z,k)$ of the

problems (\ref{mu}) and (\ref{chi}) are related

by means of the involution transformation:

\begin{equation}

\label{involution-transformation}

\mbox{\boldmath $\chi$}(z,\bar k)= \sigma

\overline{\mbox{\boldmath $\mu$}(z,k)}, \qquad

\sigma = \left( \begin{array}{ccc} 0 & -1 \\

1 & 0 \end{array} \right).

\end{equation}

In the limit $|k| \to 0$, the eigenvector

$\mbox{\boldmath $\mu$}(z,k)$ becomes a

constant vector in $z$, which is uniquely defined

by the normalization condition (\ref{normalization}) as

\begin{equation}

\label{kZero}

\lim_{|k| \to 0} \mbox{\boldmath $\mu$}(z,k) = {\bf e}_1.

\end{equation}

In the limit $|k| \to \infty$, the eigenvector

$\mbox{\boldmath $\mu$}(z,k)$ becomes a vector

$\mbox{\boldmath $\mu$}_{\infty}(z)$, which solves

the algebraic problem,

\begin{equation}

\label{algebraic_mu}

\left( \begin{array}{ccc} 0 & 0 \\ 0 & 1 \end{array} \right)

\mbox{\boldmath $\mu$}_{\infty}(z) = \frac{1}{2}

\left( P(z) - \sigma_3 \right)

\mbox{\boldmath $\mu$}_{\infty}(z).

\end{equation}

Components of the vector $\mbox{\boldmath $\mu$}_{\infty}(z)$ are

related by means of the stereographical variable,

\begin{equation}

\label{stereo}

\frac{\mu_{2 \infty}}{\mu_{1 \infty}} =

\frac{1 - S_3}{S_1 - i S_2} = \frac{S_1 + i S_2}{1 + S_3}.

\end{equation}

The stereographical variable in (\ref{stereo})

satisfies the same family of nonlinear problem as

the DS-II equations (\ref{DSII}) \cite{Kon_book}.

We define the matrix $m_{\infty} = m_{\infty}(z)$ as

\begin{equation}

\label{mInf}

m_{\infty}(z) = \left( \mbox{\boldmath $\mu$}_{\infty}(z),

\sigma \overline{\mbox{\boldmath $\mu$}_{\infty}(z)} \right) =

\left( \begin{array}{ccc} \mu_{1 \infty} &

- \overline{\mu_{2 \infty}} \\ \mu_{2 \infty} &

\overline{\mu_{1 \infty}} \end{array} \right),

\end{equation}

such that $\det(m_{\infty}) = |\mu_{1 \infty}|^2 +

|\mu_{2 \infty}|^2$. As a result,

\begin{equation}

\label{P-relation}

P(z) = m_{\infty}(z) \sigma_3 m_{\infty}^{-1}(z),

\end{equation}

or explicitly,

\begin{eqnarray}

S_1 - i S_2 = \frac{2 \mu_{1 \infty}

\overline{\mu_{2 \infty}}}{|\mu_{1 \infty}|^2+|\mu_{2 \infty}|^2},

\qquad

S_3 = \frac{|\mu_{1 \infty}|^2 - |\mu_{2 \infty}|^2}{

|\mu_{1 \infty}|^2+|\mu_{2 \infty}|^2}.

\end{eqnarray}



The differential problem (\ref{mu}) can be transformed to

the linear integral equation by means of the Green function,

\begin{eqnarray}

\nonumber

\mu_1(z,k) & = & 1 + \frac{1}{2\pi i}

\int\!\!\int \frac{dz'\wedge d\bar z'}{z'-z}({\cal P}_k

\mbox{\boldmath $\mu$})_1(z'), \\

\label{Fredholm}

\mu_2(z,k) & = & \frac{1}{2\pi i}

\int\!\!\int \frac{dz'\wedge d\bar z'}{\bar z'-\bar z}

({\cal P}_k \mbox{\boldmath $\mu$})_2(z')

e^{ik(z'-z) + i \bar{k} (\bar{z}' - \bar{z})}.

\end{eqnarray}

The same Fredholm integral equations (\ref{Fredholm})

occur in the spectral problem for the DS-II equations

\cite{PS2}. The integral equations (\ref{Fredholm})

have a unique solution if the

potential function $P(z)$ in $L^2({\Bbb R}^2) \cap

L^1({\Bbb R}^2)$ \cite{BC1}. If the potential function

$P(z)$ in $L^2({\Bbb R}^2)$ but not in $L^1({\Bbb R}^2)$,

the integral equations (\ref{Fredholm}) may

have homogeneous solutions, corresponding to

bound states \cite{Fredholm,Pogrebkov}. We denote bound

states as ${\bf \Phi}_j(z)$, as

decaying solutions of the equation,

\begin{equation}

\label{boundstate}

\left( \begin{array}{ccc}

\partial_{\bar z}&0\\0&\partial_z\end{array} \right)

{\bf \Phi}_j + ik_j

\left( \begin{array}{ccc} 0 & 0 \\ 0 & 1 \end{array} \right)

{\bf \Phi}_j = {\cal P}_{k_j} {\bf \Phi}_j,

\end{equation}

subject to the boundary conditions,

\begin{equation}

\lim_{|z| \to \infty} z {\bf \Phi}_j(z) = {\bf e}_1.

\end{equation}

Each eigenvalue $k_j$ is double, since

there exists a linearly independent

eigenfunction ${\bf \Phi}_j'(z)$ solving

the same equation (\ref{boundstate})

and satisfying the boundary conditions

\begin{equation}

\lim_{|z| \to \infty} z {\bf \Phi}_j'(z)

e^{i k_j z + i \overline{k_j z}} = {\bf e}_2.

\end{equation}

The two linearly independent eigenfunctions

are related as

\begin{equation}

{\bf \Phi}_j'(z) = \sigma \overline{\mbox{\boldmath

$\Phi$}_j(z)} e^{-i(k_jz+\overline{k_jz})}.

\end{equation}

We assume that there is a finite number (say $n$)

of eigenvalues $k_j$ on the plane $k \in {\Bbb C}$,

such that $k_j \neq 0$, $j = 1,...,n$.

It follows from (\ref{Fredholm}) that the

eigenfunction $\mbox{\boldmath $\mu$}(z,k)$

satisfies the $\bar{\partial}$ problem in the domain

${\cal D} = \left\{ k \in {\Bbb C} : k \neq k_j,

j = 1,...,n \right\}$, such that

\begin{equation}

\label{dbar}

\frac{\partial \mbox{\boldmath $\mu$}}{\partial

\bar{k}} = b(k) {\bf N}_{\mu}(z,k),

\end{equation}

where the spectral data $b(k)$ are defined as

\begin{equation}

b(k)=\frac{1}{2\pi} \int\!\!\int {dz \wedge d\bar z}

\left( {\cal P}_k \mbox{\boldmath $\mu$} \right)_2(z)

e^{i(kz + \overline{k z})}

\end{equation}

and the linear independent eigenfunction

${\bf N}_{\mu}(z,k)$ solves the same integral equation

(\ref{Fredholm}) but with a different (oscillatory)

behavior at infinity:

\begin{eqnarray}

\nonumber

N_{\mu 1}(z,k) & = & \frac{1}{2\pi i}

\int\!\!\int \frac {dz'\wedge d\bar z'}{z'-z}

({\cal P}_k {\bf N}_\mu)_1(z') \\

\label{Fredholm1}

N_{\mu 2}(z,k) & = & e^{-i(kz + \overline{kz})} +

\frac{1}{2 \pi i}

\int\!\!\int \frac {dz'\wedge d\bar z'e^{i(

k(z'-z)+\overline{k(z'- z)})}}{\bar z'-\bar z}

({\cal P}_k {\bf N}_\mu)_2(z').

\end{eqnarray}

For real-valued ${\bf S}(z)$, there exists a local relation

between two linearly independent eigenfunctions

$\mbox{\boldmath $\mu$}(z,k)$ and ${\bf N}_{\mu}(z,k)$:

\begin{equation}

\label{relation-N-mu}

{\bf N}_\mu(z,k) = \sigma \overline{\mbox{\boldmath $\mu$}(z,k)}

e^{-i(kz+\overline{kz})}.

\end{equation}

When the eigenfunction $\mbox{\boldmath $\mu$}(z,k)$

approaches to the points $k = k_j$, $j = 1,...,n$,

where the bound state

exist, it has a singular (pole) behavior

in the limit $k \to k_j$. We assume

the following limiting behavior of $\mbox{\boldmath  $\mu$}(z,k)$:

\begin{equation}

\label{limit1}

\lim_{k \to k_j} \left( \mbox{\boldmath $\mu$}(z,k)

- \frac{i}{k-k_j} {\bf \Phi}_j(z)

\right) = (z + z_j) {\bf \Phi}_j(z) + c_j {\bf \Phi}_j'(z),

\end{equation}

where $z_j$, $c_j$ are arbitrary constants.

We notice that this limiting behavior introduces

a constraint on the class of potentials

as for the DS-II equations in \cite{PS2}.

The constraint is always satisfied for

lump potentials.

No such constraints were introduced in the

original paper \cite{Pogrebkov}, where

a different limiting behavior of $\mbox{\boldmath $\mu$}(z,k)$

was suggested as $k \to k_j$:

\begin{equation}

\label{limit2}

\lim_{k \to k_j} \frac{\partial}{\partial k}

\left( (k-k_j) \mbox{\boldmath $\mu$}(z,k) \right)

= (z + z_j) {\bf \Phi}_j(z) + c_j {\bf \Phi}_j'(z),

\end{equation}

We neglect this discrepancy between

(\ref{limit1}) and (\ref{limit2}), since

the spectral decomposition and structural

instability of points $k = k_j$ does not depend

on the types of singular behavior of

$\mbox{\boldmath $\mu$}(z,k)$ as $k \to k_j$.



Solving the $\bar{\partial}$ problem (\ref{dbar})

on plane $k \in {\Bbb C}$ subject to the boundary

conditions $\lim_{|k| \to \infty} \mbox{\boldmath $\mu$}(z,k)

= \mbox{\boldmath $\mu$}_{\infty}(z)$ and

using the limiting conditions (\ref{limit1})

at the poles $k = k_j$, $j = 1,...,n$,

we derive the expansion formula,

\begin{equation}

\label{expansion-formula}

\mbox{\boldmath $\mu$}(z,k)=

\mbox{\boldmath $\mu$}_{\infty}(z) +

\sum_{j = 1}^n \frac{i}{k - k_j}

{\bf \Phi}_j(z) + \frac{1}{2 \pi i} \int\!\!\int

\frac{dk'\wedge d\bar k'}{k' - k}b(k')

{\bf N}_\mu(z,k').

\end{equation}

Using the limiting condition (\ref{kZero})

as $k \to 0$, we express the limiting

function $\mu_{\infty}(z)$ explicitly as

\begin{equation}

\mbox{\boldmath $\mu$}_{\infty}(z) =

{\bf e}_1 + \sum_{j = 1}^n \frac{i}{k_j}

{\bf \Phi}_j(z) - \frac{1}{2 \pi i} \int\!\!\int

\frac{dk'\wedge d\bar k'}{k'}b(k') {\bf N}_\mu(z,k')

\end{equation}

As a result, we find an explicit expansion

formula for $\mbox{\boldmath $\mu$}(z,k)$ as

\begin{equation}

\label{integral-mu}

\mbox{\boldmath $\mu$}(z,k) = {\bf e}_1 +

\sum_{j=1}^n\frac{i k}{k_j(k-k_j)} {\bf \Phi}_j(z) +

\frac{k}{2\pi i}

\int\!\!\int \frac{dk'\wedge d\bar k'}{k'(k'-k)}

b(k') {\bf N}_\mu(z,k').

\end{equation}

Using the limiting relations as $k \to k_j$,

we close the system (\ref{integral-mu})

with a set of algebraic equations for $j = 1,...,n$:

\begin{eqnarray}

\nonumber

\left( z+z_j - \frac{i}{k_j} \right) {\bf \Phi}_j(z)

+ c_j {\bf \Phi}'_j(z) & = & {\bf e}_1 + \sum_{l \ne j}^n

\frac{ik_j}{k_l(k_j-k_l)} {\bf \Phi}_l(z) \\

\label{algebraic-mu} 

& + & \frac{k_j}{2\pi i} \int\!\!\int \frac{dk'\wedge

d\bar k'}{k'(k'-k_j)}b(k') {\bf N}_\mu(z,k').

\end{eqnarray}



We conclude this section with an explicit

example of a single lump solution

of the IS--II equations. Let $b(k) = 0$

and $n=1$ in the equations above. Then the algebraic system

(\ref{algebraic-mu}) is decoupled from (\ref{integral-mu})

and it has a unique solution:

\begin{equation}

\label{example1}

\Phi_{1j}(z) = \frac {\bar z+\bar z_j+\frac i{\bar k_j}}{

|z+z_j-\frac i{k_j}|^2+|c_j|^2}, \qquad

\Phi_{2j}(z)= \frac {-c_j e^{-i(k_j z+\overline{k_jz})}}{

|z+z_j-\frac i{k_j}|^2+|c_j|^2},

\end{equation}

The eigenfunction $\mbox{\boldmath $\mu$}(z,k)$

has a pole decomposition representation,

\begin{equation}

\label{example2}

\mbox{\boldmath $\mu$}(z,k)

= {\bf e}_1 + \frac{ik}{k_j(k-k_j)} {\bf \Phi}_j(z), \qquad

\mbox{\boldmath $\mu$}_{\infty}(z) =

{\bf e}_1 + \frac{i}{k_j} {\bf \Phi}_j(z).

\end{equation}

The single lump solution for the vector

${\bf S}(z)$ takes the explicit form:

\begin{eqnarray}

\label{example3}

S_1 - i S_2 & = & \frac{2i \bar{c}_j \left[(z + z_j)

( \bar{z} + \bar{z}_j + \frac{i}{\bar{k}_j})+|c_j|^2 \right]

e^{i(k_j z + \overline{k_j z})}}{\bar{k}_j

(|z + z_j|^2 + |c_j|^2)(|z + z_j - \frac{i}{k_j}|^2 + |c_j|^2)} \\

\label{example4}

S_3 & = & 1 - \frac{2 |c_j|^2}{|k_j|^2

(|z + z_j|^2 + |c_j|^2)(|z + z_j - \frac{i}{k_j}|^2 + |c_j|^2)}.

\end{eqnarray}

We notice that the single lump solution of the IS-II equations

is different from the single vortex solution 

found in \cite{Ishimori} with the Hirota's method. 

However, the single vortex solution in \cite{Ishimori}

can be recovered from the single lump solution

(\ref{example1})--(\ref{example4}) in the singular limit

$k_j \to 0$, with the appropriate shifts of $z_j$ and $\bar{z}_j$.

In our paper, we will be working only with the lump solutions

which are derived in the inverse scattering transform

for a general non-singular case $k_j \neq 0$, $j = 1,...,n$.



\section{Gauge transform between the DS-II and IS-II equations}



The DS-II equations (\ref{DSII}) in complex

coordinates (\ref{coordinate}) are related

with the following linear operator,

\begin{equation}

\label{LaxDS}

\left( \begin{array}{ccc} \partial_{\bar z} & 0 \\

0 & \partial_z \end{array} \right)

\mbox{\boldmath $\Psi$}_{DS} =

Q(z) \mbox{\boldmath $\Psi$}_{DS},

\end{equation}

where

\begin{equation}

Q(z) = \left( \begin{array}{ccc} 0 & -q(z) \\

\bar{q}(z) & 0 \end{array} \right).

\end{equation}

The fundamental matrix of the linear operator

for the DS-II equations is $\mbox{\boldmath $\Psi$}_{DS}

= \left( \mbox{\boldmath $\mu$}_{DS} e^{ikz},

\mbox{\boldmath $\chi$}_{DS} e^{-ik\bar{z}} \right)$,

where $\mbox{\boldmath $\chi$}_{DS}(z,\bar{k}) =

\sigma \overline{\mbox{\boldmath $\mu$}_{DS}(z,k)}$

and $\mbox{\boldmath $\mu$}_{DS}(z,k)$ satisfies the problem:

\begin{equation}

\label{mu-DS}

\left( \begin{array}{ccc} \partial_{\bar z} & 0 \\

0 & \partial_z \end{array} \right) \mbox{\boldmath $\mu$}_{DS}

+ ik \left( \begin{array}{cc} 0 & 0 \\ 0 & 1 \end{array}

\right) \mbox{\boldmath $\mu$}_{DS} = Q(z)

\mbox{\boldmath $\mu$}_{DS}.

\end{equation}

Linear spectral problems (\ref{spectrum}) and (\ref{LaxDS})

for the IS-II and DS-II equations are related

with the gauge transformation reported in \cite{BC2,LS}.



\begin{lemma}

\label{lemma-gauge} Let $m_{\infty}(z)$ be defined by (\ref{mInf})

such that $P(z) = m_{\infty} \sigma_3 m_{\infty}^{-1}$. Solutions

of the linear system (\ref{spectrum})

are equivalent to solutions of the

linear system (\ref{LaxDS}) with the gauge transformation

$\mbox{\boldmath $\Psi$}_{DS}(z,k) = m_{\infty}^{-1}(z)

\mbox{\boldmath $\Psi$}(z,k)$, such that

\begin{equation}

\label{gaugeTR}

Q(z) = \left( {\cal D}_z m_{\infty}^{-1} \right) m_{\infty},

\qquad {\cal D}_z = \left( \begin{array}{ccc}

\partial_{\bar z} & 0 \\ 0 &\partial_z \end{array} \right)

\end{equation}

\end{lemma}



\begin{proof}

We first show that if ${\bf m} = (\mbox{\boldmath

$\mu$}(z,k),\mbox{\boldmath $\chi$}(z,k))$ with $\mbox{\boldmath

$\mu$}$, $\mbox{\boldmath $\chi$}$ satisfying (\ref{mu}),

(\ref{chi}) and ${\bf m}_{DS} = m_{\infty}^{-1} {\bf m} = 

(\mbox{\boldmath $\mu$}_{DS},\mbox{\boldmath $\chi$}_{DS})$, 

then $\mbox{\boldmath $\mu$}_{DS}$ solves the problem 

(\ref{mu-DS}), where the matrix $Q(z)$ has zero diagonal 

entries, while $\mbox{\boldmath $\chi$}_{DS}(z,\bar{k}) = 

\sigma \overline{\mbox{\boldmath $\mu$}_{DS}(z,k)}$. 

For simplicity, we consider only the case $n = 0$. We 

rewrite (\ref{expansion-formula}) in the operator form,

\begin{equation}

\label{relation1-1} 

{\bf m} = m_{\infty} + {\cal C} {\cal T} \left({\bf m}\right),

\end{equation}

where ${\cal C}$ is the Cauchy integral operator, 

\begin{equation}

{\cal C}({\bf f}) =

\frac{1}{2\pi i} \int\!\!\int \frac{dk'\wedge d\bar k'}{k'-k}

{\bf f}(k'), 

\end{equation} 

and ${\cal T}$ is the right multiplicative operator,

such that

\begin{eqnarray} 

{\cal T}(\mbox{\boldmath $f$}) =

{\cal T}\left(\begin{array}{cc} f_{11}& f_{12}\\

f_{21}&f_{22}\end{array}\right)(z,k) = 

{\mbox{\boldmath $f$}(z,\bar k)} {\bf S}_z(k).

\end{eqnarray} 

Here the scattering matrix ${\bf S}_z(k)$ is 

\begin{equation}

{\bf S}_z(k) = \left(\begin{array}{cc} 

0 & e^{i(\bar{k}z + k \bar{z})}\overline{b(\bar k)}\\ 

e^{-i(kz+\overline{kz})} b(k) & 0\end{array}\right),

\end{equation}

and we have used (\ref{involution-transformation}) 

and (\ref{relation-N-mu}). Multiplying

(\ref{relation1-1}) with $m_{\infty}^{-1}$, we find that

\begin{equation}

\label{relation1-2} 

m_{\infty}^{-1} {\bf m} = I + {\cal C}

m_{\infty}^{-1} {\cal T} \left( {\bf m} \right) 

= I + {\cal C} {\cal T} (m_{\infty}^{-1} {\bf m}),
\end{equation}

where $I$ is the 2-by-2 identity matrix. 
We apply the operator, 

\begin{equation} 

{\cal D}_k \mbox{\boldmath $f$} =

{\cal D}_z \mbox{\boldmath $f$} + k ( {\rm ad} {\cal J}

\mbox{\boldmath $f$}) = \left( \begin{array}{ccc}

\partial_{\bar z} & 0 \\ 0 &\partial_z \end{array} 

\right)\mbox{\boldmath $f$} - \frac i2k 

[\sigma_3, \mbox{\boldmath $f$}],

\end{equation} 

to both sides of (\ref{relation1-2}) and use the

commutativity property,

\begin{equation}

\left[ {\cal D}_k, {\cal T } \right] 

\mbox{\boldmath $f$}=0.

\end{equation}

As a result, we obtain

\begin{eqnarray}

\nonumber {\cal D}_k (m_\infty^{-1} {\bf m}) =  

{\cal D}_k {\cal C} {\cal T} \left( m_{\infty}^{-1} 

{\bf m}\right) = {\cal C} {\cal T} {\cal D}_k \left( 

m_{\infty}^{-1} {\bf m} \right) + Q(z), 

\end{eqnarray}

where 

\begin{eqnarray}

\nonumber Q(z) & = & \left[ {\cal D}_k, {\cal C}\right] 

{\cal T} (m_\infty^{-1} {\bf m}) = 

-\frac{1}{2\pi i} {\rm ad}{\cal J} \int\!\!\int

dk \wedge d\bar k\,\,{\cal T} (m_\infty^{-1}{\bf m})(z,k)\\

\label{proof13} & = & \frac{1}{2\pi}\int\!\!\int dk \wedge d\bar k

\left( \begin{array}{cc} 0 & ({\cal T} (m_\infty^{-1}{\bf m}))_{12}(z,k)

\\ ({\cal T} (m_\infty^{-1}{\bf m}))_{21}(z,\bar{k}) & 0 \end{array}

\right).

\end{eqnarray}

We use the relation (\ref{relation1-2}) and 

the commutativity relation,

\begin{equation}

\left[ Q(z), {\cal T} \right] {\bf f} = 0,

\end{equation}

and finish the first part of the proof,

\begin{equation}

{\cal D}_k (m_\infty^{-1} {\bf m}) =  

\left( I - {\cal C} {\cal D}_k \right)^{-1} Q(z) = 

Q(z) \left( I - {\cal C} {\cal D}_k \right)^{-1} I = 

Q(z) m_{\infty}^{-1} {\bf m}. 

\end{equation}

In the second part of the proof, we show that

if $\mbox{\boldmath $\Psi$}$ solves (\ref{spectrum})

with $P(z) = m_{\infty} \sigma_3 m_{\infty}^{-1}$

and $\mbox{\boldmath $\Psi$} = m_{\infty} \mbox{\boldmath $\Psi$}_{DS}$,

then $\mbox{\boldmath $\Psi$}_{DS}$ solves (\ref{LaxDS}),

where $Q(z)$ is related to $m_{\infty}$ by means of (\ref{gaugeTR}).

We rewrite (\ref{spectrum}) in the form:

\begin{equation}

\label{derivative1}

{\cal D}_z \left( m_\infty \mbox{\boldmath $\Psi$}_{DS}\right)

= \frac{1}{2} \left( m_{\infty} \sigma_3 m_{\infty}^{-1} -

\sigma_3 \right) \left( \partial_z - \partial_{\bar z}

\right) m_{\infty} \mbox{\boldmath $\Psi$}_{DS}.

\end{equation}

Straightforward differentiations transform (\ref{derivative1})

to the form:

\begin{equation}

\label{derivative2}

\left( \begin{array}{cc} \partial_{\bar z} \mu_{1 \infty} &

- \partial_{\bar z} \overline{\mu_{2 \infty}} \\

\partial_z \mu_{2 \infty} & \partial_z \overline{\mu_{1 \infty}}

\end{array} \right) \mbox{\boldmath $\Psi$}_{DS} +

m_{\infty} {\cal D}_z \mbox{\boldmath $\Psi$}_{DS}

= \frac{1}{{\rm det}(m_{\infty})}

\left( \begin{array}{cc} -|\mu_{2 \infty}|^2 &

\mu_{1 \infty} \overline{\mu_{2 \infty}} \\

\overline{\mu_{1 \infty}} \mu_{2 \infty} & |\mu_{2 \infty}|^2

\end{array} \right) \left(

\left( \partial_z - \partial_{\bar z}

\right) m_{\infty} \right) \mbox{\boldmath $\Psi$}_{DS},

\end{equation}

which further transforms to the form:

\begin{equation}

\label{derivative3}

m_{\infty} {\cal D}_z \mbox{\boldmath $\Psi$}_{DS}

= m_{\infty} Q(z) \mbox{\boldmath $\Psi$}_{DS},

\end{equation}

where

\begin{equation}

\label{derivative4}

Q(z) = \frac{1}{{\rm det}(m_{\infty})}

\left( \begin{array}{cc}

- \overline{\mu_{1\infty}} \partial_{\bar z} \mu_{1 \infty}

- \overline{\mu_{2\infty}} \partial_{\bar z} \mu_{2 \infty} &

\overline{\mu_{1\infty}} \partial_{\bar z} \overline{\mu_{2 \infty}}

- \overline{\mu_{2\infty}} \partial_{\bar z}

\overline{\mu_{1 \infty}} \\

\mu_{2\infty} \partial_{z} \mu_{1 \infty}

- \mu_{1\infty} \partial_{z} \mu_{2 \infty} &

- \mu_{2\infty} \partial_{z} \overline{\mu_{2 \infty}}

- \mu_{1\infty} \partial_{z} \overline{\mu_{1 \infty}}

\end{array} \right).

\end{equation}

Explicit differentiations in (\ref{gaugeTR})

shows that representations (\ref{gaugeTR}) and

(\ref{derivative4}) are the same.

\end{proof}



\begin{corollary}

Components of the transformation matrix $m_{\infty}(z)$

in (\ref{mInf}) satisfy the constraints:

\begin{eqnarray}

\label{constraint1}

q(z) & = & \frac{1}{{\rm det}(m_{\infty})}

\left( \overline{\mu_{2 \infty}}

\partial_{\bar z} \overline{\mu_{1 \infty}}

- \overline{\mu_{1 \infty}} \partial_{\bar z}

\overline{\mu_{2 \infty}} \right), \\

\label{constraint2}

0 & = & \mu_{1 \infty} \partial_{z}

\overline{\mu_{1 \infty}} + \mu_{2 \infty}

\partial_{z} \overline{\mu_{2\infty}}.

\end{eqnarray}

\end{corollary}



We continue the explicit example (\ref{example1})--(\ref{example4})

and compute the gauge transformation between the single lump

solutions of the IS-II and DS-II equations.

It follows from (\ref{example2}) that

\begin{equation}

\label{example44}

\mu_{1 \infty} = 1 + \frac{i (\bar{z} + \bar{z}_j +

\frac{i}{\bar{k}_j})}{

k_j (|z + z_j - \frac{i}{k_j}|^2 + |c_j|^2)}, \qquad

\mu_{2 \infty} = \frac{- i c_j e^{-i(k_j z +

\overline{k_j z})}}{

k_j (|z + z_j - \frac{i}{k_j}|^2 + |c_j|^2)},

\end{equation}

such that

\begin{equation}

\label{example5}

{\rm det}(m_{\infty}) = \frac{|z + z_j|^2 + |c_j|^2}{

|z + z_j - \frac{i}{k_j}|^2 + |c_j|^2}.

\end{equation}

Constraints (\ref{constraint1}) and

(\ref{constraint2}) are satisfied with

\begin{equation}

\label{example6}

q = \frac {\bar c_j e^{i(k_jz+\overline{k_jz})}}{|z+z_j|^2+|c_j|^2},

\end{equation}

Therefore, the gauge transformation (\ref{gaugeTR})

generates a single lump solution of the DS-II equations (\ref{DSII})

from a single lump solution of the IS-II equations (\ref{ISII}).

The eigenfunctions of the Lax operator for the DS-II equations

are generated by the gauge transformation

$\mbox{\boldmath $\mu$}_{DS} = m_{\infty}^{-1} \mbox{\boldmath

$\mu$}$ to the form:

\begin{equation}

\label{example8}

\mbox{\boldmath $\mu$}_{DS}(z,k) = {\bf e}_1 + \frac{i}{k-k_j}

\mbox{\boldmath $\Phi$}_{DS}(z)

\end{equation}

and

\begin{equation}

\label{example7}

\Phi_{DS 1}(z) = \frac{\bar z+\bar z_j}{|z+z_j|^2+|c_j|^2}, \quad

\Phi_{DS 2}(z) = \frac{-c_j e^{-i(k_jz+\overline{k_jz})}}{

|z+z_j|^2+|c_j|^2}.

\end{equation}



\section{The adjoint linear problem}



Let ${\cal L}$ be a linear operator that maps $L^2({\Bbb R}^2)$

to $L^2({\Bbb R}^2)$. We define the adjoint operator ${\cal L}^a$

by the inner product for two vector functions

$f,g \in L^2({\Bbb R}^2)$:

\begin{equation}

\label{adjoint}

\langle {\bf f} | {\cal L} {\bf g} \rangle =

\langle {\cal L}^+ {\bf f} | {\bf g} \rangle, 

\qquad {\cal L}^+ = \overline{{\cal L}^a},

\end{equation}

where the standard inner product is

\begin{equation}

\label{inner-product}

\langle {\bf f} | {\bf g} \rangle =

\int\!\!\int \left( \bar{f}_1 g_1 +

\bar{f}_2 g_2 \right) dz \wedge d\bar z.

\end{equation}

If ${\cal L} \mbox{\boldmath $\Psi$} = {\bf 0}$ is a linear

problem with an eigenfunction $\mbox{\boldmath $\Psi$} \in

L^2({\Bbb R}^2)$,

then ${\cal L}^a \mbox{\boldmath $\Psi$}^a = {\bf 0}$ is

the adjoint problem with an adjoint eigenfunction

$\mbox{\boldmath $\Psi$}^a \in L^2({\Bbb R}^2)$,

which takes the form:

\begin{equation}

\label{spectrum-adjoint} \left( \begin{array}{ccc} \partial_{z} &

0 \\ 0 & \partial_{\bar z} \end{array} \right) \mbox{\boldmath

$\Psi$}^a = - \frac{1}{2} \left(\partial_z - \partial_{\bar z}

\right) \left( P^+(z) - \sigma_3 \right) \mbox{\boldmath

$\Psi$}^a,

\end{equation}

where $P^+(z) = P(z)$.

The fundamental matrix of

adjoint eigenfunctions for the IS-II equations is

\begin{equation}

\mbox{\boldmath $\Psi$}^a(z) =

\left( \mbox{\boldmath $\chi$}^a(z,k) e^{i k \overline{z}},

\mbox{\boldmath $\mu$}^a(z,k) e^{-ik z} \right),

\end{equation}

where $\mbox{\boldmath $\chi$}^a(z,k)$ and

$\mbox{\boldmath $\mu$}^a(z,k)$ solves the

adjoint problems to (\ref{mu}) and (\ref{chi}), respectively:

\begin{eqnarray}

\label{mu-adjoint}

\left( \begin{array}{ccc} \partial_{z} & 0 \\

0 & \partial_{\bar z} \end{array} \right) \mbox{\boldmath $\chi$}^a

+ ik \left( \begin{array}{cc} 0 & 0 \\ 0 & 1 \end{array}

\right) \mbox{\boldmath $\chi$}^a = -{\cal P}_k^a

\mbox{\boldmath $\chi$}^a, \\

\label{chi-adjoint}

\left( \begin{array}{ccc} \partial_{z} & 0 \\

0 & \partial_{\bar z} \end{array} \right) \mbox{\boldmath $\mu$}^a

- ik \left( \begin{array}{ccc} 1 & 0 \\

0 & 0 \end{array} \right) \mbox{\boldmath $\mu$}^a =

- {\cal P}_k^a \mbox{\boldmath $\mu$}^a,

\end{eqnarray}

where

\begin{equation}

{\cal P}_k^a = \frac{1}{2} \left(\partial_z

- \partial_{\bar z} - ik \right)

\left( P^+(z) - \sigma_3 \right).

\end{equation}

The eigenvectors

$\mbox{\boldmath $\chi$}^a(z,k)$ and

$\mbox{\boldmath $\mu$}^a(z,k)$ satisfy

the boundary conditions on the plane $(x,y)$:

\begin{equation}

\label{normalization-adjoint}

\lim_{|z| \to \infty} \mbox{\boldmath $\chi$}^a(z,k) =

{\bf e}_1,  \qquad

\lim_{|z| \to \infty} \mbox{\boldmath $\mu$}^a(z,k)

= {\bf e}_2.

\end{equation}

If ${\bf S}(z)$ is real-valued,

the two fundamental solutions $\mbox{\boldmath $\chi$}^a(z,k)$

and $\mbox{\boldmath $\mu$}^a(z,k)$ of the

problems (\ref{mu-adjoint}) and (\ref{chi-adjoint}) are related

by means of the involution transformation:

\begin{equation}

\mbox{\boldmath $\chi$}^a(z,\bar k)= -\sigma

\overline{\mbox{\boldmath $\mu$}^a(z,k)}.

\end{equation}



In the limit $|k| \to \infty$, the eigenvector

$\mbox{\boldmath $\mu$}^a(z,k)$ becomes a vector

$\mbox{\boldmath $\mu$}_{\infty}^a(z)$, which solves

the algebraic problem,

\begin{equation}

\label{algebraic_mu-adjoint}

\left( \begin{array}{ccc} 1 & 0 \\ 0 & 0 \end{array} \right)

\mbox{\boldmath $\mu$}_{\infty}^a(z) = - \frac{1}{2}

\left( P(z) - \sigma_3 \right)

\mbox{\boldmath $\mu$}_{\infty}^a(z).

\end{equation}

Components of the vector $\mbox{\boldmath $\mu$}_{\infty}^a(z)$ are

related by means of the adjoint stereographical variable,

\begin{equation}

\label{stereo-adjoint}

\frac{\mu_{2 \infty}^a}{\mu_{1 \infty}^a} = -

\frac{1 + S_3}{S_1 - i S_2} = - \frac{S_1 + i S_2}{1 - S_3}.

\end{equation}

We define the adjoint matrix $m_{\infty}^a = m_{\infty}^a(z)$ as

\begin{equation}

\label{mInf-adjoint}

m_{\infty}^a(z) = \left( -\sigma

\overline{\mbox{\boldmath $\mu$}_{\infty}^a(z)},

\mbox{\boldmath $\mu$}_{\infty}^a(z) \right) =

\left( \begin{array}{ccc} \overline{\mu_{2 \infty}^a} &

\mu_{1 \infty}^a \\ -\overline{\mu_{1 \infty}^a} &

\mu_{2 \infty}^a \end{array} \right),

\end{equation}

such that $\det(m_{\infty}^a) = |\mu_{1 \infty}^a|^2 +

|\mu_{2 \infty}^a|^2$. As a result,

\begin{equation}

\label{P-adjoint-representation}

P^+(z) = m_{\infty}^a(z) \sigma_3 (m_{\infty}^{a})^{-1}(z),

\end{equation}

or explicitly,

\begin{eqnarray}

S_1 - i S_2 = \frac{-2 \mu_{1 \infty}^a

\overline{\mu_{2 \infty}^a}}{|\mu_{1 \infty}^a|^2+|\mu_{2 \infty}^a|^2},

\qquad

S_3 = \frac{|\mu_{2 \infty}^a|^2 - |\mu_{1 \infty}^a|^2}{

|\mu_{1 \infty}^a|^2+|\mu_{2 \infty}^a|^2}.

\end{eqnarray}



The differential problem (\ref{chi-adjoint}) can be transformed to

the linear integral equation by means of the Green function,

\begin{eqnarray}

\nonumber

\mu_1^a(z,k) & = & - \frac{1}{2\pi i}

\int\!\!\int \frac{dz'\wedge d\bar z'}{\bar{z}'-\bar{z}}({\cal P}_k^a

\mbox{\boldmath $\mu$}^a)_1(z')

e^{-ik(z'-z) - i \bar{k} (\bar{z}' - \bar{z})}, \\

\label{Fredholm-adjoint}

\mu_2^a(z,k) & = & 1 - \frac{1}{2\pi i}

\int\!\!\int \frac{dz'\wedge d\bar z'}{z'-z}

({\cal P}_k^a \mbox{\boldmath $\mu$}^a)_2(z').

\end{eqnarray}

If the potential function $P(z)$ in $L^2({\Bbb R}^2)$

but not in $L^1({\Bbb R}^2)$,

the integral equations (\ref{Fredholm-adjoint}) may

have homogeneous solutions, corresponding to

bound states. The bound

states ${\bf \Phi}_j^a(z)$ are

decaying solutions of the equation,

\begin{equation}

\label{boundstate-adjoint}

\left( \begin{array}{ccc}

\partial_{z}&0\\0&\partial_{\bar z}\end{array} \right)

{\bf \Phi}_j^a - ik_j^a

\left( \begin{array}{ccc} 1 & 0 \\ 0 & 0 \end{array} \right)

{\bf \Phi}_j^a = - {\cal P}_{k_j^a}^a {\bf \Phi}_j^a,

\end{equation}

subject to the boundary conditions,

\begin{equation}

\lim_{|z| \to \infty} z {\bf \Phi}_j^a(z) = {\bf e}_2.

\end{equation}

Each eigenvalue $k_j^a$ of the adjoint problem

is double, since there exists a linearly independent

eigenfunction ${\bf \Phi}_j^{a \prime}(z)$ solving

the same equation (\ref{boundstate-adjoint})

and related to ${\bf \Phi}_j^a(z)$ by means of the involution:

\begin{equation}

{\bf \Phi}_j^{a \prime}(z) = - \sigma \overline{\mbox{\boldmath

$\Phi$}_j^a(z)} e^{i(k_j^a z+\overline{k_j^a z})}.

\end{equation}



It follows from (\ref{Fredholm-adjoint}) that the

eigenfunction $\mbox{\boldmath $\mu$}^a(z,k)$

satisfies the $\bar{\partial}$ problem in the domain

${\cal D}^a = \left\{ k \in {\Bbb C} : k \neq k_j^a,

j = 1,...,n \right\}$, such that

\begin{equation}

\label{dbar-adjoint}

\frac{\partial \mbox{\boldmath $\mu$}^a}{\partial

\bar{k}} = - b^a(k) {\bf N}_{\mu}^a(z,k),

\end{equation}

where the spectral data $b^a(k)$ are defined as

\begin{equation}

b^a(k)=\frac{1}{2\pi} \int\!\!\int {dz \wedge d\bar z}

\left( {\cal P}_k^a \mbox{\boldmath $\mu$}^a \right)_1(z)

e^{-i(kz+\overline{kz})}

\end{equation}

and the linear independent eigenfunction

${\bf N}_{\mu}^a(z,k)$ solves the same integral equation

(\ref{Fredholm-adjoint}) but with a different (oscillatory)

behavior at infinity,

\begin{eqnarray}

\nonumber

N_{\mu 1}^a(z,k) & = & e^{i(kz + \overline{kz})}

- \frac{1}{2\pi i}

\int\!\!\int \frac {dz'\wedge d\bar z'}{\bar{z}'-\bar{z}}

({\cal P}_k^a {\bf N}_{\mu}^a)_1(z')

e^{- i(k (z'-z) + \overline {k (z'- z)})}, \\

\label{Fredholm1-adjoint}

N_{\mu 2}^a(z,k) & = & -\frac{1}{2 \pi i}

\int\!\!\int \frac {dz'\wedge d\bar z'}{z'-z}

({\cal P}_k^a {\bf N}_\mu^a)_2(z').

\end{eqnarray}

For real-valued ${\bf S}(z)$, there exists a local relation

between two linearly independent eigenfunctions

$\mbox{\boldmath $\mu$}^a(z,k)$ and ${\bf N}_{\mu}^a(z,k)$:

\begin{equation}

{\bf N}_\mu^a(z,k) = - \sigma

\overline{\mbox{\boldmath $\mu$}^a(z,k)}

e^{i(kz + \overline{kz})}.

\end{equation}

When the eigenfunction $\mbox{\boldmath $\mu$}(z,k)$

approaches to the points $k = k_j^a$, $j = 1,...,n$,

where the bound states exist, it has a singular (pole) behavior

in the limit $k \to k_j^a$. We assume

the following limiting behavior of

$\mbox{\boldmath  $\mu$}^a(z,k)$:

\begin{equation}

\label{limit1-adjoint}

\lim_{k \to k_j^a} \left( \mbox{\boldmath $\mu$}^a(z,k)

+ \frac{i}{k-k_j^a} {\bf \Phi}_j^a(z)

\right) = (z + z_j^a) {\bf \Phi}_j^a(z) + c_j^a

{\bf \Phi}_j^{a\prime}(z),

\end{equation}

where $z_j^a$, $c_j^a$ are arbitrary constants.



Solving the $\bar{\partial}$ problem (\ref{dbar-adjoint})

on plane $k \in \mathbb{C}$ subject to the boundary

conditions $\lim_{|k| \to \infty} \mbox{\boldmath $\mu$}^a(z,k)

= \mbox{\boldmath $\mu$}_{\infty}^a(z)$ and

using the limiting conditions (\ref{limit1-adjoint})

at the poles $k = k_j^a$, $j = 1,...,n$,

we derive the expansion formula,

\begin{equation}

\label{integral-mu-adjoint}

\mbox{\boldmath $\mu$}^a(z,k)=

\mbox{\boldmath $\mu$}_{\infty}^a(z) -

\sum_{j = 1}^n \frac{i}{k - k_j^a}

{\bf \Phi}_j^a(z) - \frac{1}{2 \pi i} \int\!\!\int

\frac{dk'\wedge d\bar k'}{k' - k}b^a(k')

{\bf N}_\mu^a(z,k').

\end{equation}

With the use of the limiting relations as $k \to k_j^a$,

we close the system (\ref{integral-mu-adjoint})

with a set of algebraic equations for $j = 1,...,n$:

\begin{eqnarray}

\nonumber

\left( z+z_j^a \right) {\bf \Phi}_j^a(z)

+ c_j^a {\bf \Phi}_j^{a \prime}(z) & = &

\mbox{\boldmath $\mu$}^a_{\infty}(z) - \sum_{l \ne j}^n

\frac{i}{k_j^a-k_l^a} {\bf \Phi}_l^a(z) \\

\label{algebraic-mu-adjoint}

& - & \frac{1}{2\pi i}

\int\!\!\int \frac{dk'\wedge

d\bar k'}{k'-k_j^a} b^a(k') {\bf N}_\mu^a(z,k').

\end{eqnarray}

The system of equations (\ref{integral-mu-adjoint})

and (\ref{algebraic-mu-adjoint}) is incomplete, since

$\mbox{\boldmath $\mu$}_{\infty}^a(z)$ is unknown.

The adjoint problem (\ref{chi-adjoint}) does not exhibit 

any simple solution at $k = 0$, on contrary to the 

direct problem (\ref{mu}). 

We complete the system 

(\ref{integral-mu-adjoint})--(\ref{algebraic-mu-adjoint})

by deriving the relation between $m_{\infty}^a$ 

and $m_{\infty}$. The relation

appears as the commutativity condition 

between the gauge and adjoint transformations.



\section{Commutativity between the gauge and adjoint

transformations}



The adjoint problem for the DS-II equations is

\begin{equation}

\label{LaxDS-adjoint}

\left( \begin{array}{ccc} \partial_{z} & 0 \\

0 & \partial_{\bar z} \end{array} \right)

\mbox{\boldmath $\Psi$}_{DS}^a =

- Q^+(z) \mbox{\boldmath $\Psi$}_{DS}^a,

\end{equation}

where $Q^+(z) = - Q(z)$. The fundamental matrix of the adjoint

operator for the DS-II equations is $\mbox{\boldmath

$\Psi$}^a_{DS} = \left( \mbox{\boldmath $\chi$}_{DS}^a

e^{ik\bar{z}}, \mbox{\boldmath $\mu$}_{DS}^a e^{-ikz} \right)$,

where $\mbox{\boldmath $\chi$}_{DS}^a(z,\bar{k}) = -\sigma

\overline{\mbox{\boldmath $\mu$}_{DS}^a(z,k)}$. Adjoint spectral

problems (\ref{spectrum-adjoint}) and (\ref{LaxDS-adjoint}) for

the IS-II and DS-II equations are related with the adjoint gauge

transformation.



\begin{lemma}

\label{lemma-gauge-adjoint} 
Let $m_{\infty}^a(z)$ be defined by

(\ref{mInf-adjoint}) such that $P^+(z) = m_{\infty}^a \sigma_3

(m_{\infty}^a)^{-1}$. Solutions of the linear system

(\ref{spectrum-adjoint}) are equivalent to solutions of the linear

system (\ref{LaxDS-adjoint}) with the gauge transformation

$\mbox{\boldmath $\Psi$}_{DS}^a(z,k) = (m_{\infty}^a)^{-1}(z)

\mbox{\boldmath $\Psi$}^a(z,k)$, such that

\begin{equation}

\label{gaugeRT}

Q^+(z) = (m_{\infty}^a)^{-1} \left( D_{\bar z}

(m_{\infty}^a)^T \right)^T, \qquad

{\cal D}_{\bar z} = \left( \begin{array}{ccc}

\partial_{z} & 0 \\ 0 &\partial_{\bar z} \end{array} \right).

\end{equation}

\end{lemma}



\begin{proof}

We show by direct differentiation that if

$\mbox{\boldmath $\Psi$}^a$ satisfies (\ref{spectrum-adjoint})

and $\mbox{\boldmath $\Psi$}^a = m_{\infty}^a

\mbox{\boldmath $\Psi$}_{DS}^a$, then

$\mbox{\boldmath $\Psi$}_{DS}^a$ satisfies

(\ref{LaxDS-adjoint}) with $Q^+(z)$ given in (\ref{gaugeRT}).

Using (\ref{P-adjoint-representation}), we notice that

\begin{equation}

\label{relation2-1}

- \frac{1}{2} \left( P^+(z) - \sigma_3 \right) m_{\infty}^a

= \left( \begin{array}{cc} 0 & \mu_{1 \infty}^a \\

\overline{\mu_{1 \infty}^a} & 0 \end{array} \right).

\end{equation}

Substitution $\mbox{\boldmath $\Psi$}^a = m_{\infty}^a

\mbox{\boldmath $\Psi$}^a_{DS}$ reduces (\ref{spectrum-adjoint})

to the form:

\begin{equation}

\label{derivative-adjoint-1}

{\cal D}_{\bar z} m_{\infty}^a \mbox{\boldmath

$\Psi$}^a_{DS} = \left( \partial_z - \partial_{\bar z}

\right) \left( \begin{array}{cc} 0 & \mu_{1 \infty}^a \\

\overline{\mu_{1 \infty}^a} & 0 \end{array} \right)

\mbox{\boldmath $\Psi$}^a_{DS}.

\end{equation}

Differentiation expands (\ref{derivative-adjoint-1})

to the form:

\begin{equation}

m_{\infty}^a {\cal D}_{\bar z}

\mbox{\boldmath $\Psi$}^a_{DS} =

- \left( \begin{array}{cc} \partial_z

\overline{\mu_{2 \infty}^a} & \partial_{\bar z}

\mu_{1 \infty}^a \\ - \partial_z

\overline{\mu_{1 \infty}^a} & \partial_{\bar z}

\mu_{2 \infty}^a \end{array} \right)

\mbox{\boldmath $\Psi$}_{DS}^a.

\end{equation}

Inverting $m_{\infty}^a$ and matching with (\ref{LaxDS-adjoint}),

we express $Q^+(z)$ in terms of $m_{\infty}^a$ as

\begin{equation}

\label{derivative-adjoint-2}

Q^+(z) = \frac{1}{{\rm det}(m_{\infty}^a)}

\left( \begin{array}{cc} \mu_{2 \infty}^a \partial_z

\overline{\mu_{2 \infty}^a} + \mu_{1\infty}^a

\partial_{z} \overline{\mu_{1 \infty}^a} &

\mu_{2 \infty}^a \partial_{\bar z}

\mu_{1 \infty}^a - \mu_{1\infty}^a

\partial_{\bar z} \mu_{2 \infty}^a \\

\overline{\mu_{1 \infty}^a} \partial_{z}

\overline{\mu_{2 \infty}^a} -

\overline{\mu_{2 \infty}^a}

\partial_z \overline{\mu_{1 \infty}^a} &

\overline{\mu_{1 \infty}^a} \partial_{\bar z}

\mu_{1 \infty}^a

+ \overline{\mu_{2 \infty}^a} \partial_{\bar z}

\mu_{2 \infty}^a \end{array} \right).

\end{equation}

Direct differentiation shows equivalence of representations

(\ref{gaugeRT}) and (\ref{derivative-adjoint-2}).

\end{proof}



\begin{corollary}
\label{corollary-gauge-adjoint}

Components of the transformation matrix $m_{\infty}^a(z)$

in (\ref{mInf-adjoint}) satisfy the constraints:

\begin{eqnarray}

\label{constraint1-adjoint}

q(z) & = & \frac{1}{\overline{\mu_{2 \infty}^a}}

\partial_{\bar z} \mu_{1 \infty}^a, \\

\label{constraint2-adjoint}

0 & = & \mu_{1 \infty}^a \partial_{z} \overline{\mu_{1\infty}^a}

+ \mu_{2 \infty}^a \partial_z \overline{\mu_{2 \infty}^a}.

\end{eqnarray}

if $Q^+ = - Q$. 

\end{corollary}



We plot the diagram of the gauge and adjoint

transforms for the IS-II and DS-II equations:

\begin{equation}

\label{diagram}

 \begin{array}{ccc} \mbox{IS-II (\ref{spectrum}) for  $P(z)$} & 

\rightarrow & \mbox{DS-II (\ref{LaxDS}) for $Q(z)$}\\ \downarrow

&{}& \vdots \\ \mbox{adjoint IS-II (\ref{spectrum-adjoint})

for  $P^+(z)$} & \rightarrow &

\mbox{adjoint DS-II (\ref{LaxDS-adjoint}) for $Q^+(z)$}

\end{array}

\end{equation}

The gauge transform from IS-II to DS-II is given 

by Lemma 3.1. The adjoint transform from IS-II to 

adjoint IS-II is described in Section 4 for $P^+ = P$. 

The gauge transform from adjoint IS-II to 

adjoint DS-II is given by Lemma 5.1. 

It remains to show that the system (\ref{LaxDS-adjoint}) 

is indeed the adjoint system of (\ref{LaxDS}) with 

$Q^+ = -Q$. This would serve as a commutativity 

condition between the gauge and adjoint transforms. 

We prove the commutativity condition and  

close the diagram (\ref{diagram}) 

with an important relation

between $m_{\infty}^a(z)$ and $m_{\infty}(z)$.



\begin{theorem}
\label{theorem-commutativity}

The gauge transform and the adjoint transform

are commutative if and only if

\begin{equation}

\label{relation-matrices}

m_{\infty}^a = \left( m_{\infty}^{-1} \right)^+ =

\frac{m_{\infty}}{{\rm det}(m_{\infty})},

\end{equation}

such that

\begin{equation}

\mu_{1\infty}^a = - \frac{\overline{\mu_{2\infty}}}{|\mu_{1\infty}|^2

+ |\mu_{2\infty}|^2}, \qquad

\mu_{2\infty}^a = \frac{\overline{\mu_{1\infty}}}{

|\mu_{1\infty}|^2 + |\mu_{2\infty}|^2},

\end{equation}

and ${\rm det}(m_{\infty}^a) {\rm det}(m_{\infty}) = 1$.

\end{theorem}



\begin{proof}

A relation between $m_{\infty}$ and $m_{\infty}^a$ follows

from comparison of representations (\ref{P-relation})

and (\ref{P-adjoint-representation}):

\begin{equation}

\label{relation-P}

P^+(z) = (m_{\infty}^{-1})^+ \sigma_3 m_{\infty}^+ =

m_{\infty}^a \sigma_3 (m_{\infty}^{a})^{-1}.

\end{equation}

Another relation follows from comparison of 

representations (\ref{gaugeTR}) and (\ref{gaugeRT}):

\begin{equation}

\label{relation-Q}

Q^+ = m_{\infty}^+ \left( D_{\bar z} (m_{\infty}^{-1})^{+ T}

\right)^T = (m_{\infty}^a)^{-1} \left( D_{\bar z} (m_{\infty}^{a})^{T}

\right)^T.

\end{equation}

The constraint (\ref{relation-matrices}) is a sufficient

condition for the relations (\ref{relation-P}) and (\ref{relation-Q}).

We prove that this is also a necessary condition for

commutability of the gauge and adjoint transformations.

By comparing (\ref{stereo}) and (\ref{stereo-adjoint}),

we derive a relation between $\mbox{\boldmath $\mu$}_{\infty}$

and $\mbox{\boldmath $\mu$}_{\infty}^a$:

\begin{equation}

\label{final-relation-1}

\mu_{1\infty} \overline{\mu_{1\infty}^a} +

\mu_{2 \infty} \overline{\mu_{2 \infty}^a} = 0.

\end{equation}

Using combinations of constraints 

(\ref{constraint1}), (\ref{constraint2}), 

(\ref{constraint1-adjoint}), and 

(\ref{constraint2-adjoint}), combined 

with the relation (\ref{final-relation-1}),

we also derive that

\begin{eqnarray}

\label{final-relation-2}

\mu_{1 \infty}^a \mu_{2 \infty} = \overline{\mu_{1 \infty}^a}

\overline{\mu_{2 \infty}}, \qquad
\mu_{2 \infty}^a \mu_{1 \infty} = \overline{\mu_{2 \infty}^a}

\overline{\mu_{1 \infty}}.

\end{eqnarray}

The relations (\ref{final-relation-1}) and 
(\ref{final-relation-2}) are satisfied if and only if

\begin{equation}

\label{final-relation-4}

\mu_{1 \infty}^a = - c(z) \overline{\mu_{2\infty}}, \qquad

\mu_{2 \infty}^a = c(z) \overline{\mu_{1\infty}},

\end{equation}

where $c \in \mathbb{R}$. Using constraints 

(\ref{constraint1}), (\ref{constraint1-adjoint}), 

combined with the relation (\ref{final-relation-4}), 

we find that 

\begin{equation}

c(z) = \frac{c_0}{{\rm det}(m_{\infty})},

\end{equation}

where $c_0$ is constant. It follows from 

the asymptotic limits,

\begin{equation}

\lim_{|z| \to \infty} m_{\infty}(z) = I, 

\qquad \lim_{|z| \to \infty} m_{\infty}^a(z) = I,

\end{equation} 

that $c_0 = 1$ and the formula 

(\ref{relation-matrices}) is thus proved. 

\end{proof}

We can develop an alternative proof of 
of Theorem \ref{theorem-commutativity}, without 
using Lemma \ref{lemma-gauge-adjoint} and 
Corollary \ref{corollary-gauge-adjoint}.
The diagram of the gauge and adjoint

transforms for the IS-II and DS-II equations 
can be closed as:

\begin{equation}

\label{diagram1}
\begin{array}{ccc} \mbox{IS-II (\ref{spectrum}) for  $P(z)$} & 

\rightarrow & \mbox{DS-II (\ref{LaxDS}) for $Q(z)$} \\ 
\downarrow & & \downarrow \\ 
\mbox{adjoint IS-II (\ref{spectrum-adjoint}) for  $P^+(z)$} & 
\cdots  & \mbox{adjoint DS-II (\ref{LaxDS-adjoint}) for $Q^+(z)$}

\end{array}

\end{equation}

Here we will use the adjoint system for DS-II equations 
(\ref{LaxDS-adjoint}) with $Q^+ = -Q$ and show that the system 
is related to the adjoint system for IS-II equations 
(\ref{spectrum-adjoint}) with $P^+ = P$ by means of 
the gauge transform:

\begin{equation}
\label{commutativity} 
\mbox{\boldmath $\Psi$}^a(z,k) = 
\frac{m_{\infty}(z)}{\mbox{det}(m_{\infty}(z))}
\mbox{\boldmath $\Psi$}^a_{DS}(z,k).
\end{equation}
This relation is equivalent to the gauge 
transform between the adjoint IS-II and 
DS-II equations in Lemma \ref{lemma-gauge-adjoint}, 
where $m_{\infty}^a(z)$ is given by (\ref{relation-matrices}).


The left-hand-side of (\ref{spectrum-adjoint}) is written 
explicitly in the form,
\begin{eqnarray}
\nonumber
D_{\bar z} \left(\frac {m_\infty}{\mbox{det} m_\infty}
\left(\begin{array}{c}
\Psi_{1,DS}^a\\ \Psi_{2,DS}^a\end{array}\right) \right) \\
\nonumber
= \frac 1{\det m_\infty}
\left(\begin{array}{c} (\partial_z
\mu_{1\infty})\Psi_{1,DS}^a+\mu_{1\infty}\partial_z\Psi_{1,DS}^a 
-(\partial_z \bar \mu_{2\infty})\Psi_{2,DS}^a-
\bar \mu_{2\infty}\partial_z \Psi_{2,DS}^a\\
(\partial_{\bar z}\mu_{2\infty})\Psi_{1,DS}^a+
\mu_{2\infty}\partial_{\bar z}
\Psi_{1,DS}^a+(\partial_{\bar z} \bar \mu_{1\infty})\Psi_{2,DS}^a+\bar
\mu_{1\infty}\partial_{\bar z} \Psi_{2,DS}^a\end{array}\right ) \\
\label{commu-1}
- \frac 1{(\det m_\infty)^2}
\left(\begin{array} {c}( \mu_{1\infty}\Psi_{1,DS}^a-\bar
\mu_{2\infty}\Psi_{2,DS}^a)[(\partial_z \mu_{1\infty})\bar
\mu_{1\infty}+ (\partial_z\mu_{2\infty})\bar \mu_{2\infty}]\\
(\mu_{2\infty}\Psi_{1,DS}^a+\bar \mu_{1\infty}\Psi_{2,DS}^a)[
\mu_{1\infty}\partial_{\bar z}\bar \mu_{1\infty}+
\mu_{2\infty}\partial_{\bar z}\bar \mu_{2\infty}]
\end{array}\right).
\end{eqnarray}
We apply the gauge transformation 
$Q={\cal D}_z(m_\infty^{-1})m_\infty$ to the adjoint DS-II 
system (\ref{LaxDS-adjoint}) and find that 
\begin{eqnarray}
\nonumber
D_{\bar z} \left(\frac {m_\infty}{\mbox{det}(m_\infty)}
\left(\begin{array}{c}
\Psi_{1,DS}^a\\ \Psi_{2,DS}^a\end{array}\right) \right) \\
\nonumber
= \frac{1}{\mbox{det} m_{\infty}} \left(\begin{array}{c}
(\partial_z \mu_{1\infty})\Psi_{1,DS}^a - 
(\partial_z \bar \mu_{2\infty})\Psi_{2,DS}^a 
- \bar \mu_{2\infty}\partial_z \Psi_{2,DS}^a \\
(\partial_{\bar z} \mu_{2\infty})\Psi_{1,DS}^a 
+ \mu_{2\infty}(\partial_{\bar z}
\Psi_{1,DS}^a)+(\partial_{\bar z} \bar
\mu_{1\infty})\Psi_{2,DS}^a \end{array} \right) \\
\nonumber
+ \frac{1}{(\mbox{det} m_{\infty})^2} \left(\begin{array}{c}
\mu_{1\infty}(\bar\mu_{1\infty} \partial_{\bar z} 
\bar\mu_{2\infty}- \bar\mu_{2\infty}
\partial_{\bar z}\bar\mu_{1\infty})\Psi^a_{2,DS} \\
\bar \mu_{1\infty}
 (\mu_{2\infty}\partial_{ z} 
\mu_{1\infty}-\mu_{1\infty}\partial_z
\mu_{2\infty})\Psi_{1,DS}^a \end{array} \right) \\
\nonumber
+ \frac{1}{(\mbox{det} m_{\infty})^2} 
\left( \begin{array}{c} 
-|\mu_{1\infty}|^2 (\partial_z\mu_{1\infty}) 
\Psi_{1,DS}^a 
+ \bar\mu_{1\infty} \bar \mu_{2\infty}(\partial_z\mu_{1\infty})
\Psi_{2,DS}^a \\
-|\mu_{1\infty}|^2 (\partial_{\bar z} 
\bar{\mu}_{1 \infty})\Psi_{2,DS}^a -
\mu_{2\infty} \mu_{1\infty} 
(\partial_{\bar z} \bar\mu_{1\infty})\Psi_{1,DS}^a 
\end{array} \right) \\
\label{commu-2}
+ \frac{1}{(\mbox{det} m_{\infty})^2} 
\left( \begin{array}{c} 
-\mu_{1\infty}\bar\mu_{2\infty}(\partial_z
\mu_{2\infty})\Psi^a_{1,DS} +\bar \mu^2_{2\infty}
(\partial_z\mu_{2\infty})\Phi^a_{21}  \\
- \mu_{2\infty}\bar\mu_{1\infty}(\partial_{\bar z} 
\bar \mu_{2\infty})\Psi^a_{2,DS} - 
\mu^2_{2\infty} (\partial_{\bar z} 
\bar \mu_{2\infty})\Psi^a_{1,DS} \end{array} \right).
\end{eqnarray}
On the other hand, the 
right-hand-side of (\ref{spectrum-adjoint}) 
can be rewritten as
\begin{eqnarray}
\nonumber
- \frac{1}{2} \left( \partial_z - \partial_{\bar z} \right) 
(P(z) -\sigma_3) \left(\frac {m_\infty}{\mbox{det}(m_\infty)}
\left(\begin{array}{c}
\Psi_{1,DS}^a\\\Psi_{2,DS}^a\end{array}\right) \right) \\
\nonumber
= \frac{1}{\det m_{\infty}}
\left(\begin{array}{c} 
(\partial_{\bar z} \bar \mu_{2\infty})\Psi_{2,DS}^a+\bar
\mu_{2\infty}\partial_{\bar z}\Psi_{2,DS}^a-(\partial_z \bar
\mu_{2\infty})\Psi_{2,DS}^a-\bar \mu_{2\infty}\partial_z
\Psi_{2,DS}^a \\ 
(\partial_{\bar z}
\mu_{2\infty})\Psi_{1,DS}^a+\mu_{2\infty}\partial_{\bar z}
\Psi_{1,DS}^a-(\partial_{z}  \mu_{2\infty})\Psi_{1,DS}^a-
\mu_{2\infty}\partial_{z} \Psi_{1,DS}^a \end{array}\right) \\
\label{commu-3}
- \frac{1}{(\det m_{\infty})^2}
\left(\begin{array}{c} 
\bar{\mu}_{2\infty}\Psi_{2,DS}^a \left( 
\mu_{1\infty}\partial_{\bar z}\bar
\mu_{1\infty}+\mu_{2\infty}\partial_{\bar z}\bar \mu_{2\infty} -
(\partial_z\mu_{1\infty})\bar
\mu_{1\infty}-(\partial_z\mu_{2\infty})\bar \mu_{2\infty} \right) \\ 
\mu_{2\infty}\Psi_{1,DS}^a \left(
\mu_{1\infty}\partial_{\bar z}\bar
\mu_{1\infty}+\mu_{2\infty}\partial_{\bar z}\bar \mu_{2\infty} -
(\partial_z\mu_{1\infty})\bar
\mu_{1\infty}-(\partial_z\mu_{2\infty})\bar \mu_{2\infty} \right) 
\end{array}\right).
\end{eqnarray}
We prove that the right-hand-sides of (\ref{commu-2}) 
and ~(\ref{commu-3}) are equal by showing that 
the following two expressions are equal:
\begin{eqnarray}
\nonumber
\frac{1}{\det m_\infty}
\left(\begin{array}{c} 
(\partial_z \mu_{1\infty}) \Psi_{1,DS}^a 
+ \mu_{1\infty}
\frac{(\bar{\mu}_{1\infty}
 \partial_{\bar z} 
\bar{\mu}_{2\infty} - 
\bar{\mu}_{2\infty} 
\partial_{\bar z } \bar{\mu}_{1\infty}) 
\Psi_{2,DS}^a}{\det m_\infty} \\ 
(\partial_{\bar z} \bar{\mu}_{1\infty}) 
\Psi_{2,DS}^a + \bar{\mu}_{1\infty}
\frac{(\mu_{2\infty} \partial_{z} \mu_{1\infty}
-\mu_{1\infty}\partial_{z} \mu_{2\infty})
\Psi_{1,DS}^a}{\det m_\infty} \end{array} \right) \\
\label{commu-4}
+ \frac{1}{(\det m_\infty)^2}
\left(\begin{array}{c} 
-|\mu_{1\infty}|^2 (\partial_z \mu_{1\infty}) 
\Psi_{1,DS}^a - \mu_{1\infty}
\bar{\mu}_{2\infty} (\partial_z \mu_{2\infty}) 
\Psi_{1,DS}^a \\ 
-|\mu_{1\infty}|^2 (\partial_{\bar z} 
\bar{\mu}_{1\infty}) \Psi_{2,DS}^a 
- \mu_{2\infty} \bar{\mu}_{1\infty} 
(\partial_{\bar z} \bar{\mu}_{2\infty}) 
\Psi_{2,DS}^a \end{array}\right),
\end{eqnarray}
and
\begin{eqnarray}
\nonumber
\frac{1}{\det m_\infty}
\left(\begin{array}{c} 
(\partial_{\bar z} \bar{\mu}_{2\infty}) 
\Psi_{2,DS}^a + \bar{\mu}_{2\infty} 
\frac{(\mu_{2\infty} \partial_{z}\mu_{1\infty} 
- \mu_{1\infty}\partial_{z} \mu_{2\infty}) 
\Psi_{1,DS}^a}{\det m_\infty} \\ 
-(\partial_{z} \mu_{2\infty}) \Psi_{1,DS}^a 
- \mu_{2\infty}\frac{(\bar \mu_{1\infty}
\partial_{\bar z} \bar{\mu}_{2\infty}
-\bar{\mu}_{2\infty} \partial_{\bar z} 
\bar{\mu}_{1\infty}) \Psi_{2,DS}^a}{\det m_\infty}
\end{array}\right)\\
\label{commu-5}
-\frac 1{(\det m_\infty)^2}
\left(\begin{array}{c} 
|\mu_{2\infty}|^2 (\partial_{\bar z} 
\bar{\mu}_{2\infty}) \Psi_{2,DS}^a 
+ \mu_{1\infty}\bar{\mu}_{2\infty} 
(\partial_{\bar z} \bar{\mu}_{1\infty}) \Psi_{2,DS}^a \\ 
-|\mu_{2\infty}|^2(\partial_{ z} \mu_{2\infty}) 
\Psi_{1,DS}^a - \bar{\mu}_{1\infty}
\mu_{2\infty}(\partial_{z} \mu_{1\infty})\Psi_{1,DS}^a
\end{array}\right)
\end{eqnarray} 
Therefore, the gauge and adjoint transforms are commutative 
under the condition (\ref{relation-matrices}). 


\begin{lemma}

Spectral data of the adjoint problem (\ref{spectrum-adjoint}) 

are related to the spectral data of the direct problem 

(\ref{spectrum}) as

\begin{equation}

\label{symmetry-relations}

b^a(k) = \overline{b(k)}, \qquad

k_j^a = k_j, \qquad

z_j^a = z_j, \qquad

c_j^a = - \bar{c}_j.

\end{equation}

\end{lemma}



\begin{proof}

Applying the gauge transforms of Lemma 3.1 and 5.1 

to spectral representations (\ref{expansion-formula}) 

and (\ref{integral-mu-adjoint}) for $\mbox{\boldmath $\mu$}(z,k)$ 

and $\mbox{\boldmath $\mu$}^a(z,k)$, we obtain 

the spectral representations for $\mbox{\boldmath

$\mu$}_{DS}(z,k)$ and $\mbox{\boldmath $\mu$}_{DS}^a(z,k)$: 

\begin{eqnarray}

\label{expansion-DS-formula} 

\mbox{\boldmath $\mu$}_{DS}(z,k) & = & 

{\bf e}_1 + \sum_{j = 1}^n 

\frac{i}{k - k_j}(m_\infty)^{-1}{\bf \Phi}_j(z)

+ \frac{1}{2 \pi i} \int\!\!\int \frac{dk'\wedge d\bar k'}{k'-k} 

b(k')( m_\infty)^{-1} {\bf N}_\mu(z,k'), \\

\label{integral-DS-mu-adjoint} 

\mbox{\boldmath $\mu$}_{DS}^a(z,k) & = & 

{\bf e}_2 - \sum_{j = 1}^n \frac{i}{k - k_j^a} 

(m_\infty^a)^{-1}{\bf \Phi}_j^a(z) - 

\frac{1}{2 \pi i} \int\!\!\int \frac{dk'\wedge

d\bar k'}{k' - k}b^a(k')(m_\infty^a)^{-1}{\bf N}_\mu^a(z,k').

\end{eqnarray}

Similarly, applying gauge transforms to the limiting formula

(\ref{limit1}) and (\ref{limit1-adjoint}), we obtain

\begin{eqnarray}

\label{limitDS1} 

\lim_{k \to k_j} \left( \mbox{\boldmath $\mu$}_{DS}(z,k) 

- \frac{i}{k-k_j} (m_\infty)^{-1}{\bf \Phi}_j(z)

\right) & = & (z + z_j)(m_\infty)^{-1} {\bf \Phi}_j(z) + c_j

(m_\infty)^{-1}{\bf \Phi}_j'(z), \\

\label{limitDS1-adjoint} 

\lim_{k \to k_j^a} \left( \mbox{\boldmath

$\mu$}_{DS}^a(z,k) + \frac{i}{k-k_j^a} (m_\infty^a)^{-1}{\bf

\Phi}_j^a(z) \right) & = & (z + z_j^a) (m_\infty^a)^{-1}{\bf

\Phi}_j^a(z) + c_j^a(m_\infty^a)^{-1} {\bf \Phi}_j^{a\prime}(z).

\end{eqnarray}

Since eigenfunctions of the DS-II and IS-II equations are related 

by the gauge transform as 

\begin{equation}

\label{IS-DS-corr}

\begin{array}{cc} (m_\infty)^{-1}{\bf N}_\mu={\bf N}_{\mu,

DS},&\,\,\,(m^a_\infty)^{-1}{\bf N}^a_\mu={\bf N}^a_{\mu, DS},\\

(m_\infty)^{-1}\mbox{\boldmath $\Phi$}_j=\mbox{\boldmath

$\Phi$}_{j, DS},&\,\,\,(m^a_\infty)^{-1}\mbox{\boldmath

$\Phi$}_j^a=\mbox{\boldmath $\Phi$}_{j, DS}^a,\\

(m_\infty)^{-1}\mbox{\boldmath $\Phi$}'_j=\mbox{\boldmath

$\Phi$}'_{j, DS},&\,\,\,(m^a_\infty)^{-1}\mbox{\boldmath

$\Phi$}_j^{\prime a}=\mbox{\boldmath $\Phi$}_{j, DS}^{\prime a}.

\end{array}\end{equation}

we find the representations (\ref{IS-DS-corr}) and 

(\ref{expansion-DS-formula}) and limiting formulas 

(\ref{integral-DS-mu-adjoint}) and 

(\ref{limitDS1-adjoint}) for the DS-II equations 

with the same spectral data $\{ b(k), k_j, c_j, z_j \}$ 

and $\{ b^a(k), k_j^a, c_j^a, z_j^a \}$. The symmetry 

relations (\ref{symmetry-relations}) were proved 

for the DS-II equations in \cite{PS2}. 

\end{proof}



The relation (\ref{relation-matrices}) between 

$m_{\infty}^a(z)$ and $m_{\infty}(z)$

closes the system of equations (\ref{integral-mu-adjoint})

and (\ref{algebraic-mu-adjoint}) for eigenfunctions

of the adjoint problem. For the explicit example

of the single lump solution of the IS-II equations,

we find from (\ref{example44}), (\ref{example5}),  

and (\ref{relation-matrices}) that

\begin{equation}

\label{example9}

\mu_{1 \infty}^a = \frac{- i \overline{c_j} 

e^{i(k_j z + \bar{k_j z})}}{

\bar{k}_j (|z + z_j|^2 + |c_j|^2)}, \qquad

\mu_{2 \infty}^a = 1 - \frac{i (\bar{z} + \bar{z}_j)}{

k_j (|z + z_j|^2 + |c_j|^2)}.

\end{equation}

The algebraic system (\ref{algebraic-mu-adjoint})

for $b(k) = 0$ and $n = 1$ becomes closed 

with (\ref{relation-matrices}) and (\ref{symmetry-relations}) 

and it has an exact solution:

\begin{eqnarray}

\Phi_{1j}^a(z) & = & \frac{\bar{c}_j 

e^{i(k_j z+\overline{k_jz})}

\left[ |z + z_j|^2 + |c_j|^2 + \frac{i}{\bar{k}_j} 

\left( z + z_j - \bar{z} - \bar{z}_j \right) \right]}{

(|z+z_j|^2+|c_j|^2)^2}, \\

\Phi_{2j}^a(z) & = & \frac{(\bar{z} + \bar{z}_j)(|z + z_j|^2 +

|c_j|^2) - \frac{i}{k_j} ((\bar{z} + \bar{z}_j)^2 + |c_j|^2)}{

(|z+z_j|^2+|c_j|^2)^2}.

\end{eqnarray}

Using the gauge transformation

$\mbox{\boldmath $\mu$}_{DS}^a = (m_{\infty}^a)^{-1}

\mbox{\boldmath $\mu$}^a$, we find the eigenfunction of

the adjoint problem for DS-II equations from (\ref{symmetry-relations}) 

and (\ref{integral-DS-mu-adjoint}):

\begin{equation}

\mbox{\boldmath $\mu$}_{DS}^a(z,k) = {\bf e}_2 - \frac{i}{k-k_j} 

\mbox{\boldmath $\Phi$}_{j,DS}^a(z),

\end{equation}

where the bound state of the adjoint problem 

${\bf \Phi}_{j, DS}^a(z)$ is

\begin{equation}

\Phi_{1j, DS}^a(z) = \frac{\bar{c}_j e^{i(k_jz+\overline{k_jz})}}{

|z+z_j|^2+|c_j|^2}, \qquad

\Phi_{2j, DS}^a(z) = \frac{\bar z+\bar z_j}{|z+z_j|^2+|c_j|^2}.

\end{equation}



\section{Applications}



We use the main results of our analysis to study 

the spectral decomposition and surface representation theorems 

for the Ishimori--II equations. 



The spectral decomposition theorem for the DS-II equations 

was proved in \cite{PS2}. Structural instability of lump 

solutions of the DS-II equations was studied by means of 

perturbation series expansions over a complete set of 

eigenfunctions of the linear problem (\ref{LaxDS}). 

Using orthogonality and completeness of

eigenfunctions of the linear problem for

the DS-II equations and the gauge transformation 

between DS-II and IS-II equations, we

construct a basis of the spectral decomposition

for the IS-II equations. Similar to \cite{PS2}, 

the perturbation series expansions can be used 

to prove structural instability of lump solutions 

of the IS-II equations.



\begin{theorem}

An arbitrary scalar function $f(z) \in L^2({\Bbb R}^2)$ 

can be decomposed through the set of eigenfunctions 

$S=\{ [m_{\infty}^{-1} {\bf N}_{\mu}]_2(z,k),

[m_{\infty}^{-1} {\bf \Phi}_{j}]_2, j = 1,...,n \}$

as follows

\begin{equation}

f(z)= \int\!\!\int dk\wedge d\bar k \,

\alpha(k)[m_{\infty}^{-1} {\bf N}_{\mu}]_2(z,k)

+ \sum_{j=1}^n \alpha_j [m_{\infty}^{-1}

{\bf \Phi}_{j}]_2(z),

\end{equation}

where

\begin{eqnarray}

\alpha(k) = -\frac{1}{4\pi^2} \left( [(m^a_\infty)^{-1} 

{\bf N}^a_{\mu}]_1(k),f \right), \qquad

\alpha_j = \frac{1}{2\pi i} \left( 

[(m^a_\infty)^{-1} {\bf \Phi}^a_{j}]_1, f \right),

\end{eqnarray}

and the inner product for scalar functions is defined by

\begin{equation}

\label{inner-product-scalar}

\left( g(k'), f(k) \right) =

\int\!\!\int  g(z,k') f(z,k) dz\wedge d\bar z.

\end{equation}

\end{theorem}



\begin{proof}

The proof follows from Propositions 3.3--3.5 of 

\cite{PS2}. 

\end{proof}



Orthogonality and completeness relations 

for eigenfunctions of the IS-II equations 

can be formulated explicitly as follows. 



\begin{lemma}

Eigenfunctions

$[m_{\infty}^{-1} {\bf N}_{\mu}]_2(z,k)$ and

$[m_{\infty}^{-1} {\bf \Phi}_{j}]_2$

are orthogonal with respect to the eigenfunctions

$[(m^a_\infty)^{-1} {\bf N}^a_{\mu}]_1(z,k)$ and

$[(m^a_\infty)^{-1} {\bf \Phi}^a_{j}]_1$ 

according to the orthogonality relations:

\begin{eqnarray}

\label{orthogonal1}

\left( [(m^a_\infty)^{-1} {\bf N}^a_{\mu}]_1(k'),

[m_\infty^{-1} {\bf N}_{\mu}]_2(k) \right) 

& = & - 2 \pi^2 i \delta(k'-k),\\

\left( [(m^a_\infty)^{-1} {\bf N}^a_{\mu}]_1(k'),

[m_\infty^{-1} {\bf \Phi}_{j}]_2 \right) & = & 0, \\

\left( [(m^a_\infty)^{-1} {\bf \Phi}^a_{j}]_1,

[m_\infty^{-1} {\bf N}_{\mu}]_2(k) \right) & = & 0,\\

\label{orthogonal4}

\left( [(m^a_\infty)^{-1} {\bf \Phi}^a_{l}]_1,

[m_\infty^{-1} {\bf \Phi}_{j}]_2 \right) = 

2\pi i \delta_{j,l}.

\end{eqnarray}

\end{lemma}



\begin{corollary}

The orthogonality relations for eigenfunctions of 

the IS-II equations take the form of weighted inner 

vector products defined by (\ref{inner-product}):

\begin{equation}

\label{weighted-inner-product}

\left( [(m^a_\infty)^{-1} {\bf g}^a]_1,

[m_\infty^{-1} {\bf f}]_2(k) \right) =

\langle \sigma \overline{g_1^a} {\bf e}_1 | 
f_2 {\bf e}_2 \rangle 

- \frac{1}{2} \langle \sigma \overline{{\bf g}^a} | 

( P - \sigma_3 ) {\bf f} \rangle. 

\end{equation}

\end{corollary}



\begin{remark}

The orthogonality relations 

(\ref{orthogonal1})--(\ref{orthogonal4}) for 

the IS-II equations can be proved directly 

from the linear systems (\ref{spectrum}) 

and (\ref{spectrum-adjoint}), with the use 

of weighted inner products in the 

right-hand-side of (\ref{weighted-inner-product}).

\end{remark}



\begin{lemma}

Eigenfunctions

$[m_{\infty}^{-1} {\bf N}_{\mu}]_2(z,k)$ and

$[m_{\infty}^{-1} {\bf \Phi}_{j}]_2$

are complete with respect to the adjoint eigenfunctions

$[(m^a_\infty)^{-1} {\bf N}^a_{\mu}]_1(z,k)$ and

$[(m^a_\infty)^{-1} {\bf \Phi}^a_{j}]_1$

according to the completeness relation:

\begin{eqnarray}

\nonumber

\delta(z'-z) & = & -\frac 1{2\pi^2i}

\int\!\!\int dk\wedge d\bar k 

[(m^a_\infty)^{-1} {\bf N}^a_{\mu}]_1(z',k)

[m_{\infty}^{-1} {\bf N}_{\mu}]_2(z,k) \\

\label{completeness}

& - & \frac 1\pi \sum_{j=1}^n [(m^a_\infty)^{-1}

{\bf \Phi}^a_j]_1(z')[m_\infty^{-1} {\bf \Phi}_j]_2(z).

\end{eqnarray}

\end{lemma}



The linear problem for the DS-II equations and its adjoint

was used in \cite {K1,K2} to parametrize surfaces and

integrable evolutions of surfaces in ${\Bbb R}^3$ and

${\Bbb R}^4$. Their results

generalize the classical Weierstrass-Enneper representation for

conformal parametrizations of minimal surfaces in ${\Bbb R}^3$, or

stationary Lagragian surfaces in ${\Bbb R}^4$ \cite{H,HR}.

Similarly, we formulate the surface representation theorem 

by using the gauge transformation between the DS-II and IS-II 

equations.



\begin{theorem}

Define the generalized Weierstrass formulae

\begin{eqnarray*}

& &X^1+iX^2=\int_\Gamma

(-\phi_1\phi_2dz'+\psi_1\psi_2d\bar z'),\\

& &X^1-iX^2=\int_\Gamma

(\bar\psi_1\bar\psi_2dz'-\bar \phi_1 \bar \phi_2d\bar z'),\\

& &X^3+iX^4=\int_\Gamma (\bar\psi_2\phi_1dz'+\psi_1\bar\phi_2d\bar

z'),\\

& &X^3-iX^4=\int_\Gamma (\bar\psi_1\phi_2dz'+\psi_2 \bar

\phi_1d\bar z'),\end{eqnarray*}

where

\begin{eqnarray}

(\phi_1,\psi_1)^T=m_\infty^{-1} 

\mbox{\boldmath $\mu$}(z,k) e^{ikz}, \qquad 

(\psi_2,-\phi_2)^T=(m^a_\infty)^{-1} 

\mbox{\boldmath $\mu$}^a(z,k) e^{-ikz}

\end{eqnarray}

and  $\Gamma $ is a contour in ${\Bbb C}$. 

Then, the Weierstrass formulas define a

conformal immersion of a surface into 

${\Bbb R}^4 : X^i(z,\bar z): {\Bbb C} \mapsto

{\Bbb R}^4$. The induced metric of the surface is

\begin{equation}

ds^2 = u_1 u_2 dz d \bar{z}, 

\end{equation}

where $u_\alpha=|\phi_\alpha|^2+|\psi_\alpha|^2$,

$\alpha=1,2$, and the Gaussian and mean curvatures are

respectively:

\begin{equation}

K = - \frac{2}{u_1u_2} \frac{\partial^2}{\partial z \partial 

\bar{z}} \log(u_1u_2), \qquad 

H^2=\frac{4}{u_1 u_2} \; 

\frac{|\bar\mu_{1\infty} \partial_{\bar z} \bar\mu_{2\infty}

-\bar\mu_{2\infty}\partial_{\bar z}\bar\mu_{1\infty}|^2}{

(|\mu_{1\infty}|^2+|\mu_{2\infty}|^2)^2}.

\end{equation}

\end{theorem}



\begin{proof}

Proof follows from Theorem 3 and Proposition 1 in

\cite{K2}.

\end{proof}



\section*{Acknowledgements.} D. Wu would like to thank people

at the Department of Mathematics of McMaster University

for the hospitality during her visit in the spring of 2002. 
The research is supported in part by NSERC grant No. 5-36694.




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