# microscopic conservation laws for the derivative Nonlinear Schrödinger
equation

Xingdong Tang  Xingdong Tang
School of Mathematics and Statistics,
Nanjing Univeristy of Information Science and Technology,
Nanjing, 210044, People’s Republic of China<EMAIL_ADDRESS>and  Guixiang
Xu  Guixiang Xu
Laboratory of Mathematics and Complex Systems,
Ministry of Education,
School of Mathematical Sciences,
Beijing Normal University,
Beijing, 100875, People’s Republic of China<EMAIL_ADDRESS>
###### Abstract.

Compared with macroscopic conservation law for the solution of the derivative
nonlinear Schrödingger equation (DNLS) with small mass in [21], we show the
corresponding microscopic conservation laws for the Schwartz solutions of DNLS
with small mass. The new ingredient is to make use of the logarithmic
perturbation determinant introduced in [34, 35] to show one-parameter family
of microscopic conservation laws of the $A(\kappa)$ flow and the DNLS flow,
which is motivated by [11, 19, 20].

###### Key words and phrases:

Microscopic conservation law; Derivative nonlinear Schrödinger equation;
Diagonal Green’s function; Perturbation determinant.

###### 2010 Mathematics Subject Classification:

Primary: 35L70, Secondary: 35Q55.

##  1\. Introduction

We consider the derivative nonlinear Schrödinger equation (DNLS)

${i\mkern 1.0mu}\partial_{t}q+\partial^{2}_{x}q+{i\mkern
1.0mu}\partial_{x}(|q|^{2}q)=0,$ (1.1)

where $q\colon\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{C}$. (1.1) is
$L^{2}$-critical since the dilation

$\displaystyle q(t,x)\mapsto
q_{\lambda}(t,x)=\lambda^{1/2}q(\lambda^{2}t,\lambda x)$ (1.2)

leaves both (1.1) and the $L^{2}$ norm invariant. The derivative nonlinear
Schrödinger equation appears in plasma physics [28, 29, 36], and references
therein.

Local well-posedness result for (1.1) in the energy space was worked out by N.
Hayashi and T. Ozawa [14, 31]. They combined the fixed point argument with the
$L^{4}_{t}W^{1,\infty}_{x}$ estimate to construct local-in-time solution with
arbitrary data in energy space. For other results, we can refer to [12, 13].
Since (1.1) is energy subcritical case, the maximal time interval of existence
only depends on $H^{1}$ norm of initial data. Later, local well-posedness
result for (1.1) in $H^{s},s\geq 1/2$ is due to H. Takaoka [37] by Bourgain’s
Fourier restriction method. The sharpness is shown in [38] in the sense that
nonlinear evolution $u(0)\mapsto u(t)$ fails to be $C^{3}$ or even uniformly
$C^{0}$ in this topology, even when $t$ is arbitrarily close to zero and
$H^{s}$ norm of the data is small (see also Biagioni-Linares [3]).

Global well-posedness is shown for (1.1) in the energy space in [31], under
the smallness condition

$\displaystyle\|u_{0}\|^{2}_{L^{2}}<2\pi,$ (1.3)

the argument is based on the sharp Gagliardo-Nirenberg inequality and the
energy method (conservation of mass and energy). This result is improved by H.
Takaoka [38] by Bourgain’s restriction method, who proved global well-
posedness in $H^{s}$ for $s>32/33$ under the condition (1.3). In [6, 7],
I-team make use of almost conservation law [41] to show global well-posedness
in $H^{s},s>1/2$ under (1.3). Miao, Wu and Xu [24] combine almost conservation
law and the refined resonant decomposition technique to obtain the global
well-posedness in $H^{1/2}$ under (1.3). Later, Wu use the generalized
Gagliardo-Nirenberg inequality to improve the global well-posedness of (1.1)
in the energy space under the condition

$\displaystyle\|u_{0}\|^{2}_{L^{2}}<4\pi$ (1.4)

in [42], where $4\pi$ is the mass of the solitary waves with critical
parameters of (1.1). Miao, Tang and Xu use the structure analysis and
classical variational argument to show the existence of solitary waves with
two parameters and improve the global result of (1.1) in the energy space in
[26], and further use perturbation argument, modulation analysis and Lyapunov
stability to show the orbital stability of weak interaction multi-soliton
solution with subcritical parameters in the energy space in [25]. We can also
refer to [5, 9, 23, 27, 40] for the stability analysis of the solitary waves
of the (generalized) derivative nonlinear Schrödinger equation in the energy
space and to [10] for lower regularity result of (1.1) by almost conservation
law in [41].

Since (1.1) is an integrable system in [1, 18], there are lots of global well-
posedness of (1.1) with mass restriction in the weighted Sobolev spapce based
on the inverse scattering method, please refer to [15, 16, 17, 32, 33]and
reference therein.

The conjecture about (1.1) is the following.

######  Conjecture 1.1.

Let $s>0$. (1.1) is globally well-posed for all initial data in
$H^{s}(\mathbb{R})$ in the sense that the solution map $\Phi$ extends uniquely
from Schwartz space to a jointly continuous map $\Phi\colon\mathbb{R}\times
H^{s}(\mathbb{R})\rightarrow H^{s}(\mathbb{R})$.

According to the above well known result, we need loosen the continuous
dependence of the solution on initial data to consider the solution of (1.1)
in $H^{s}(\mathbb{R})$ with $s\in(0,1/2)$. The basic question is that how to
control the uniform estimates of the solution of (1.1).

Motivated by Killip-Visan-Zhang’s argument in [20], Klaus and Schippa combine
the integrability of (1.1) with the series expansion of the perturbation
determinant [34, 35] to obtain macroscopic conservation law for the Schwartz
solution of (1.1) with small mass in [21], In this paper, we will show the
corresponding microscopic form and obtain one-parameter family of microscopic
conservation laws for the $A(\kappa)$ flows (see (4.1)) and the DNLS flow by
Harrop-Griffiths-Killip-Visan’s argument in [11, 19]. Compared with
macroscopic form, microscopic conservation law with coercivity helps to show
the local smoothing effect for (1.1) in $H^{s}(\mathbb{R})$ and can be further
applied into global wellposedness analysis. We can refer to [11, 19, 39] and
reference therein.

We now recall some Hamiltonian mechanics background. (1.1) is Hamiltonian
equation with respect to the following Poisson structure:

$\\{F,G\\}:=\int\tfrac{\delta F}{\delta r}\partial_{x}\bigl{(}\tfrac{\delta
G}{\delta q}\bigr{)}+\tfrac{\delta F}{\delta
q}\partial_{x}\bigl{(}\tfrac{\delta G}{\delta r}\bigr{)}\,{\rm{d}}x,$ (1.5)

where the operators $\tfrac{\delta}{\delta q}$ and $\tfrac{\delta}{\delta r}$
denote the functional Fréchet derivatives. Any Hamiltonian $H(q,r,t)$
generates a flow via the equation

$\displaystyle\partial_{t}\begin{bmatrix}q\\\
r\end{bmatrix}=\begin{bmatrix}0&1\\\
1&0\end{bmatrix}\partial_{x}\begin{bmatrix}\frac{\delta H}{\delta q}\\\
\frac{\delta H}{\delta r}\end{bmatrix}.$ (1.6)

Correspondingly, (1.1) is the Hamiltonian flow associated to

$H_{\mathrm{DNLS}}:=\int_{\mathbb{R}}-{i\mkern
1.0mu}qr^{\prime}+\frac{1}{2}q^{2}r^{2}\,{\rm{d}}x,$ (1.7)

where $r=-\bar{q}$. Two other important Hamiltonian quantities for (1.1) are

$M:=\int qr\,{\rm{d}}x,\quad E_{\mathrm{DNLS}}:=\int
q^{\prime}r^{\prime}-\frac{3}{2}{i\mkern
1.0mu}q^{2}rr^{\prime}+\frac{1}{2}q^{3}r^{3}\,{\rm{d}}x.$ (1.8)

Conservations of $M$, $H_{\mathrm{DNLS}}$ and $E_{\mathrm{DNLS}}$ is due to
gauge, space translation and time translation invariance of (1.1), the
commutativity of $H_{\mathrm{DNLS}}$ and $E_{\mathrm{DNLS}}$ is based on the
fact that they are completely integrable, which means the existence of an
infinite family of commuting flows for (1.1). We can refer to [1] [21] for
more details. Based on the recent breakthrough in KdV, mKdV and NLS by B.
Harrop-Griffiths, R. Killip and M. Visan in [11, 19], the commuting flow
approximation will be the robust method for showing global well-posedness
theory of (1.1) in the lower regularity space. We can also refer to [22, 30]
and reference thereein.

Let us write the Lax operator related to (1.1) and its unperturbed one

$L(\varkappa):=\begin{bmatrix}\partial+{i\mkern 1.0mu}\varkappa^{2}&-\varkappa
q\\\ -\varkappa r&\partial-{i\mkern
1.0mu}\varkappa^{2}\end{bmatrix}\text{~{}~{}and~{}~{}}L_{0}(\varkappa):=\begin{bmatrix}\partial+{i\mkern
1.0mu}\varkappa^{2}&0\\\ 0&\partial-{i\mkern
1.0mu}\varkappa^{2}\end{bmatrix}.$ (1.9)

By simple calculations, we know that

$R_{0}(\kappa):=L_{0}(\kappa)^{-1}=\begin{bmatrix}(\partial+{i\mkern
1.0mu}\kappa^{2})^{-1}&0\\\ 0&(\partial-{i\mkern
1.0mu}\kappa^{2})^{-1}\end{bmatrix}$

admits the integral kernel

$G_{0}(x,y;\kappa)=e^{-{i\mkern
1.0mu}\kappa^{2}|x-y|}\begin{bmatrix}\mathbbm{1}_{\\{y<x\\}}&0\\\
0&-\mathbbm{1}_{\\{x<y\\}}\end{bmatrix}\quad\text{for ${i\mkern
1.0mu}\kappa^{2}>0$}.$

For ${i\mkern 1.0mu}\kappa^{2}<0$, we may use
${G}_{0}(x,y;\kappa)=-G_{0}(y,x;-\bar{\kappa})$. The resolvent operator
$R(\kappa):=L(\kappa)^{-1}$ for Schwartz function $q$ with small mass also has
the integral kernel $G(x,y;\kappa)$ (See Proposition 3.1)

$\begin{bmatrix}G_{11}(x,y,\kappa)&G_{12}(x,y,\kappa)\\\
G_{21}(x,y,\kappa)&G_{22}(x,y,\kappa)\end{bmatrix}.$

Let us define three key functionals as follows.

$\displaystyle\gamma(x;\kappa)$ $\displaystyle:=\operatorname{sgn}({i\mkern
1.0mu}\kappa^{2})\bigl{[}G_{11}(x,x;\kappa)-G_{22}(x,x;\kappa)\bigr{]}-1,$
$\displaystyle g_{12}(x;\kappa)$ $\displaystyle:=\operatorname{sgn}({i\mkern
1.0mu}\kappa^{2})G_{12}(x,x;\kappa),$ $\displaystyle g_{21}(x;\kappa)$
$\displaystyle:=\operatorname{sgn}({i\mkern
1.0mu}\kappa^{2})G_{21}(x,x;\kappa),$

With these preparations, the main result in this paper is

######  Theorem 1.2.

Let $s\in(0,1/2)$, ${i\mkern 1.0mu}\varkappa^{2}\in\mathbb{R}\setminus(-1,1)$,
$q(0)\in H^{s}(\mathbb{R})\cap\mathscr{S}$ with small mass. Suppose that $q$
is a solution to (1.1), then we have

$\partial_{t}\rho(\varkappa)+\partial_{x}j_{\mathrm{DNLS}}(\varkappa)=0,$

where the density $\rho$ and the flux $j_{\mathrm{DNLS}}$ are defined as
follows

$\displaystyle\rho(\varkappa):=\,$
$\displaystyle-\varkappa\frac{qg_{21}(\varkappa)+rg_{12}(\varkappa)}{2+\gamma(\varkappa)},$
$\displaystyle j_{\mathrm{DNLS}}(\varkappa):=\,$ $\displaystyle
i\varkappa\frac{q^{\prime}g_{21}(\varkappa)-r^{\prime}g_{12}(\varkappa)}{2+\gamma(\varkappa)}-i\varkappa^{2}qr-2\varkappa^{2}\rho(\varkappa)-qr\rho(\varkappa).$

######  Remark.

We give some remarks as follows.

  1. (1)

In this paper, we consider microscopic conservation law for the solution of
(1.1). That is the reason why we consider the Schwartz solution. In addition,
mass threshold is another interesting problem. Please refer to [14, 26][42]
and reference therein. Recently, it is very interesting that Bahouri and
Perelman obtain global well-posedness for (1.1) in $H^{1/2}(\mathbb{R})$
without mass restriction by combining the profile decomposition techniques
with the integrability structure in [2].

  2. (2)

The above microscopic conservation law corresponds to macroscopic conservation
law of (1.1) in [21]. In fact, we have

$\displaystyle A(\varkappa)=$ $\displaystyle\operatorname{sgn}({i\mkern
1.0mu}\varkappa^{2})\sum_{j=1}^{\infty}\frac{(-1)^{j-1}}{j}\operatorname{tr}\left\\{\left(\sqrt{R_{0}}\left(L-L_{0}\right)\sqrt{R_{0}}\right)^{j}\right\\}$
$\displaystyle=$ $\displaystyle\int_{\mathbb{R}}\rho(\varkappa)\,{\rm{d}}x.$

Compared to macroscopic conservation law, microscopic conservation law with
coercivity can be used to show the local smoothing estimate for the solution
of (1.1) with small mass in $H^{s}\cap\mathscr{S}$, $s\in(0,1/2)$ and others.
Please refer to [19, 20].

  3. (3)

The leading order term in $\rho$ has the following form

$\left|\operatorname{Re}\int_{\mathbb{R}}\rho^{[2]}(\varkappa)dx\right|\approx|\varkappa|^{2}\|q\|^{2}_{H^{-1/2}_{\varkappa}},$

which captures $L^{2}$ norm of the part of $q$ living at frequencies
$|\xi|\lesssim|\kappa|^{2}$ (See also (2.8) and (4.2)). Combining the above
conservation law and the similar argument on Besov norm estimate in [20, 21,
Section $3$] for any Hamiltonian flow preserving $A(\varkappa)$ for all
$|\varkappa|\geq 1$, we can obtain a uniform bound of $H^{s}(\mathbb{R})$ norm
of the Schwartz solution of (1.1) with small mass

$\|q(t)\|_{H^{s}}\lesssim\|q(0)\|_{H^{s}},\text{~{}~{}for~{}~{}}s\in(0,1/2).$
(1.10)

Lastly, the paper is organized as follows. In Section 2, we recall some
notations and preliminary estimates. In Section 3, we show the existence and
some properties of the Green’s function related to the Lax operator
$L(\kappa)$. In Section 4, we introduce the invairant quantity $A(\kappa)$
from the logarithmic perturbation determinant and show its microscopic
conservation laws for the $A(\kappa)$ flow and the DNLS flow.

### Acknowledgements

X. Tang was supported by NSFC (No. 12001284), and G. Xu was supported by NSFC
(No. 11671046, and No. 11831004) and by National Key Research and Development
Program of China (No. 2020YFA0712900). The authors would like to thank
Professor Monica Visan for her valuable comments and suggestions.

##  2\. Some notation and preliminary estimates

In this paper, we take $r=-\bar{q}$ and choose $s\in(0,\tfrac{1}{2}),$ and all
implicit constants can depend on $s$. We denote $\mathscr{S}$ the Schwartz
function, and introduce the notation

$B_{\delta}:=\left\\{q\in H^{s}:\|q\|_{H^{s}}\leq\delta\right\\}.$ (2.1)

which can be ensured by scaling argument under the assumption that the mass
$\|q\|_{L^{2}}$ is small enough.

We use the inner product on $L^{2}(\mathbb{R})$ as follows

$\langle f,g\rangle=\int\overline{f(x)}g(x)\,dx,$

which also gives the dual product between $H^{s}(\mathbb{R})$ and
$H^{-s}(\mathbb{R})$. In addtion, If $F:\mathscr{S}\to\mathbb{C}$ is $C^{1}$,
we have

$\tfrac{d\ }{d\theta}\Big{|}_{\theta=0}F(q+\theta
f)=\bigl{\langle}\bar{f},\tfrac{\delta F}{\delta
q}\bigr{\rangle}-\bigl{\langle}f,\tfrac{\delta F}{\delta r}\bigr{\rangle}.$
(2.2)

The Fourier transform is defined by

$\displaystyle\hat{f}(\xi)=\tfrac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}e^{-i\xi
x}f(x)\,dx,\quad\text{whence}\quad\widehat{fg}(\xi)=\tfrac{1}{\sqrt{2\pi}}[\hat{f}*\hat{g}](\xi).$

###  2.1. Sobolev spaces

For complex $\kappa$ with $|\kappa|\geq 1$ and $\sigma\in\mathbb{R}$ we define
the norm

$\|q\|_{H^{\sigma}_{\kappa}}^{2}:=\int_{\mathbb{R}}\left(4|\kappa|^{4}+\xi^{2}\right)^{\sigma}|\hat{q}(\xi)|^{2}\,d\xi$

and write $H^{\sigma}:=H^{\sigma}_{1}$.

For $0<s<\frac{1}{2}$, simple calculation yields the Sobolev inequality

$\displaystyle\|f\|_{L^{\infty}}\lesssim\|\hat{f}\|_{L^{1}}\leq\|f\|_{H^{s+\frac{1}{2}}_{\kappa}}\bigl{\|}(|\xi|^{2}+4|\kappa|^{4})^{-\frac{2s+1}{4}}\bigr{\|}_{L^{2}}\lesssim|\kappa|^{-2s}\|f\|_{H^{s+\frac{1}{2}}_{\kappa}}.$
(2.3)

Consequently, we have the following algebra property of
$H^{s+\frac{1}{2}}_{\kappa}$ space:

$\|fg\|_{H^{s+\frac{1}{2}}_{\kappa}}\lesssim|\kappa|^{-2s}\|f\|_{H^{s+\frac{1}{2}}_{\kappa}}\|g\|_{H^{s+\frac{1}{2}}_{\kappa}}.$
(2.4)

By duality and the fractional product rule in [4], Sobolev embedding, and
(2.3), we obtain

$\displaystyle\|qf\|_{H^{s-1/2}}\lesssim$
$\displaystyle|\kappa|^{-2s}\|q\|_{H^{s-1/2}}\|f\|_{H^{s+\frac{1}{2}}_{\kappa}}.$
(2.5)

###  2.2. Operator estimates and Trace

For $0<\sigma<1$ and $i\kappa^{2}\in\mathbb{R}$, $|\kappa|\geq 1$ we define
the operator $(i\kappa^{2}\mp\partial)^{-\sigma}$ using the Fourier multiplier
$(i\kappa^{2}\mp i\xi)^{-\sigma}$ where, for $\arg z\in(-\pi,\pi]$, we define

$z^{-\sigma}=|z|^{-\sigma}e^{-i\sigma\arg z}.$ (2.6)

Therefore for all ${i\mkern 1.0mu}\kappa^{2}\in\mathbb{R}$ with $|\kappa|\geq
1$ we have

$\left((i\kappa^{2}\mp\partial)^{-\sigma}\right)^{*}=(i\kappa^{2}\pm\partial)^{-\sigma},$

and

$\left\|(i\kappa^{2}\mp\partial)^{-\sigma}\right\|_{\mathrm{op}}\leq|\kappa|^{-2\sigma}.$

We denote $\mathfrak{I}_{p}$ the Schatten class of compact operators on
$L^{2}(\mathbb{R})$ whose singular numbers are $l^{p}$ summable.
$\mathfrak{I}_{p}$ is complete and is an embedded subalgebra of bounded
operators on $L^{2}(\mathbb{R})$. Moreover, we have

$\mathfrak{I}_{p}\subset\mathfrak{I}_{q},\text{~{}~{}for~{}~{}}p\leq q.$ (2.7)

Let us recall some facts about the class $\mathfrak{I}_{p}$ that we will use
repeatedly in Section 3: An operator A on $L^{2}(\mathbb{R})$ is
Hilbert–Schmidt class ($\mathfrak{I}_{2}$) if and only if it admits an
integral kernel $a(x,y)\in L^{2}(\mathbb{R}\times\mathbb{R})$, and

$\|A\|^{2}_{\mathrm{op}}\leq\|A\|_{\mathfrak{I}_{2}}^{2}=\iint_{\mathbb{R}\times\mathbb{R}}|a(x,y)|^{2}\,\,{\rm{d}}x\,{\rm{d}}y.$

The product of two Hilbert–Schmidt operators is trace class,; Moreover, we
have

$\displaystyle\operatorname{tr}(AB):=$
$\displaystyle\iint_{\mathbb{R}\times\mathbb{R}}a(x,y)b(y,x)\,{\rm{d}}y\,{\rm{d}}x=\operatorname{tr}(BA),$
$\displaystyle|\operatorname{tr}(AB)|$
$\displaystyle\leq\|AB\|_{\mathfrak{I}_{1}}\leq\|A\|_{\mathfrak{I}_{2}}\|B\|_{\mathfrak{I}_{2}}$

The class $\mathfrak{I}_{p}$ forms a two-sided ideal in the algebra of bounded
operators on $L^{2}$; indeed, for any bounded operators $B,C$ on
$L^{2}(\mathbb{R})$, we have

$\|BAC\|_{\mathfrak{I}_{p}}\leq\|B\|_{\mathrm{op}}\|A\|_{\mathfrak{I}_{p}}\|C\|_{\mathrm{op}}.$

We can refer to [8, 35] for more details.

The following estimates are the elementary estimates in this paper.

######  Lemma 2.1.

Let $s\in(0,1/2)$, ${i\mkern 1.0mu}\kappa^{2}\in\mathbb{R}$, we have

$\displaystyle\|(\partial+{i\mkern 1.0mu}\kappa^{2})^{-\frac{1}{2}}\kappa
q(\partial-{i\mkern 1.0mu}\kappa^{2})^{-\frac{1}{2}}\|_{\mathfrak{I}_{2}}$
$\displaystyle\lesssim\|q\|_{L^{2}},$ (2.8) $\displaystyle\|(\partial+{i\mkern
1.0mu}\kappa^{2})^{s-\frac{1}{2}}\kappa q(\partial-{i\mkern
1.0mu}\kappa^{2})^{-\frac{1}{2}}\|_{\mathfrak{I}_{2}}$
$\displaystyle\lesssim|\kappa|\|q\|_{H^{s-\frac{1}{2}}_{\kappa}},$ (2.9)

and

$\displaystyle\|(\partial+i\kappa^{2})^{-s-\frac{1}{2}}f(\partial-i\kappa^{2})^{-s-\frac{1}{2}}\|_{\mathrm{op}}$
$\displaystyle\lesssim|\kappa|^{-2s}\|f\|_{H^{-s-\frac{1}{2}}_{\kappa}}.$
(2.10)

###### Proof.

The estimate (2.8) can be proved by [20, Lemma 4.1]. For (2.9), it suffices to
consider the case ${i\mkern 1.0mu}\kappa^{2}=1$ by scaling argument. By
Plancherel’s Theorem, we have

$\displaystyle\|(1-\partial)^{s-\frac{1}{2}}q(1+\partial)^{-\frac{1}{2}}\|_{\mathfrak{I}_{2}}^{2}$
$\displaystyle=\operatorname{tr}\left\\{(1-\partial^{2})^{-\alpha}q(1-\partial^{2})^{-\beta}\bar{q}\right\\}$
$\displaystyle=\tfrac{1}{2\pi}\iint_{\mathbb{R}^{2}}\frac{|\hat{q}(\xi-\eta)|^{2}\,}{\left(1+\xi^{2}\right)^{\frac{1}{2}-s}\left(1+\eta^{2}\right)^{\frac{1}{2}}}d\eta\,d\xi,$

Note that

$\int_{\mathbb{R}}\frac{1}{\left(1+(\xi+\eta)^{2}\right)^{\frac{1}{2}-s}\left(1+\eta^{2}\right)^{\frac{1}{2}}}\,d\eta\lesssim(4+\xi^{2})^{s-\frac{1}{2}},$

we can obtain the estimate (2.9).

Lastly, we can obtain (2.10) by duality and (2.4) as follows

$\biggl{|}\int_{\mathbb{R}}fgh\,dx\biggr{|}\lesssim\|f\|_{H^{-s-\frac{1}{2}}_{\kappa}}\|gh\|_{H^{s+\frac{1}{2}}_{\kappa}}\lesssim|\kappa|^{-2s}\|f\|_{H^{-s-\frac{1}{2}}_{\kappa}}\|g\|_{H^{s+\frac{1}{2}}_{\kappa}}\|h\|_{H^{s+\frac{1}{2}}_{\kappa}}.$

This completes the proof. ∎

##  3\. The diagonal Green’s functions

In this Section, motivated by the ideas in [11, 19], we introduce three key
quantities $g_{12}$, $g_{21}$, and $\gamma$ from the diagonal Green’s function
related to the Lax operator $L(\kappa)$ for (1.1), and establish some
elementary estimates about them. Recall that

$L(\kappa)=L_{0}(\kappa)+\begin{bmatrix}0&-\kappa q\\\ -\kappa
r&0\end{bmatrix}\text{~{}~{}where~{}~{}}L_{0}(\kappa):=\begin{bmatrix}\partial+i\kappa^{2}&0\\\
0&\partial-i\kappa^{2}\end{bmatrix}.$ (3.1)

Since we only consider the case ${i\mkern 1.0mu}\kappa^{2}\in\mathbb{R}$ with
$|\kappa|\geq 1$, we have

$\displaystyle L(\kappa)^{*}=\begin{bmatrix}-\partial-{i\mkern
1.0mu}\bar{k}^{2}&-\bar{\kappa}\bar{r}\\\
-\bar{\kappa}\bar{q}&-\partial+i\bar{\kappa}^{2}\end{bmatrix}=-\begin{bmatrix}1&0\\\
0&-1\end{bmatrix}L(-\bar{\kappa})\begin{bmatrix}1&0\\\ 0&-1\end{bmatrix}.$
(3.2)

We now construct the Green’s function associated to $L_{0}(\kappa)$ and
$L(\kappa)$, respectively. By the Fourier transformation, the resolvent
operator

$R_{0}(\kappa):=L_{0}(\kappa)^{-1}=\begin{bmatrix}(\partial+{i\mkern
1.0mu}\kappa^{2})^{-1}&0\\\ 0&(\partial-{i\mkern
1.0mu}\kappa^{2})^{-1}\end{bmatrix}$

admits the integral kernel

$G_{0}(x,y;\kappa)=e^{-{i\mkern
1.0mu}\kappa^{2}|x-y|}\begin{bmatrix}\mathbbm{1}_{\\{y<x\\}}&0\\\
0&-\mathbbm{1}_{\\{x<y\\}}\end{bmatrix}\quad\text{for ${i\mkern
1.0mu}\kappa^{2}>0$}.$ (3.3)

For ${i\mkern 1.0mu}\kappa^{2}<0$, we may use
${G}_{0}(x,y;\kappa)=-G_{0}(y,x;-\bar{\kappa})$ by (3.2).

By the perturbation theory and the resolvent identity, the resolvent operator
$R(\kappa):=L(\kappa)^{-1}$ can be formally expressed as

$\displaystyle R=$ $\displaystyle
R_{0}+\sum_{\ell=1}^{\infty}(-1)^{\ell}\sqrt{R_{0}}\left(\sqrt{R_{0}}(L-L_{0})\sqrt{R_{0}}\right)^{\ell}\sqrt{R_{0}},$
(3.4)

where

$\displaystyle\sqrt{R_{0}}(L-L_{0})\sqrt{R_{0}}=$
$\displaystyle-\begin{bmatrix}0&\Lambda\\\ \Gamma&0\end{bmatrix},$ (3.5)
$\displaystyle\Lambda:=(\partial+{i\mkern
1.0mu}\kappa^{2})^{-\tfrac{1}{2}}\kappa q(\partial-{i\mkern
1.0mu}\kappa^{2})^{-\tfrac{1}{2}}\text{~{}and~{}}$
$\displaystyle\Gamma:=(\partial-{i\mkern
1.0mu}\kappa^{2})^{-\tfrac{1}{2}}\kappa r(\partial+{i\mkern
1.0mu}\kappa^{2})^{-\tfrac{1}{2}},$ (3.6)

and fractional powers of $R_{0}$ are defined via (2.6). By (2.8), we have

$\|\Lambda\|_{\mathfrak{I}_{2}}=\|\Gamma\|_{\mathfrak{I}_{2}}\lesssim\|q\|_{L^{2}}\lesssim\delta.$
(3.7)

Now we have the convergence of series (3.4) as part of the following result.

######  Proposition 3.1 (Existence of the Green’s function).

There exists $\delta>0$ such that $L(\kappa)$ is invertible as an operator on
$L^{2}(\mathbb{R})$, for all $q\in B_{\delta}$ and all ${i\mkern
1.0mu}\kappa^{2}\in\mathbb{R}$ with $|\kappa|\geq 1$. The resolvent operator
$R(\kappa):=L(\kappa)^{-1}$ admits an integral kernel $G(x,y;\kappa)$
satisfying

$\begin{bmatrix}G_{11}(x,y,\kappa)&G_{12}(x,y,\kappa)\\\
G_{21}(x,y,\kappa)&G_{22}(x,y,\kappa)\\\
\end{bmatrix}=-\begin{bmatrix}\bar{G}_{11}(y,x,-\bar{\kappa})&\bar{G}_{21}(y,x,-\bar{\kappa})\\\
\bar{G}_{12}(y,x,-\bar{\kappa})&\bar{G}_{22}(y,x,-\bar{\kappa})\\\
\end{bmatrix}$ (3.8)

such that the mapping

$H^{s}_{\kappa}(\mathbb{R})\ni q\mapsto G-G_{0}\in
H^{s+\frac{1}{2}}_{\kappa}\otimes H^{s+\frac{1}{2}}_{\kappa}$ (3.9)

is continuous. Moreover, $G-G_{0}$ is continuous as a function of
$(x,y)\in\mathbb{R}\times\mathbb{R}$. Lastly, we have

$\displaystyle\partial_{x}G(x,y;\kappa)$
$\displaystyle=\begin{bmatrix}-{i\mkern 1.0mu}\kappa^{2}&\kappa q(x)\\\ \kappa
r(x)&{i\mkern
1.0mu}\kappa^{2}\end{bmatrix}G(x,y;\kappa)+\begin{bmatrix}\delta(x-y)&0\\\
0&\delta(x-y)\end{bmatrix},$ (3.10) $\displaystyle\partial_{y}G(x,y;\kappa)$
$\displaystyle=G(x,y;\kappa)\begin{bmatrix}{i\mkern 1.0mu}\kappa^{2}&-\kappa
q(y)\\\ -\kappa r(y)&-{i\mkern
1.0mu}\kappa^{2}\end{bmatrix}-\begin{bmatrix}\delta(x-y)&0\\\
0&\delta(x-y)\end{bmatrix}$ (3.11)

in the sense of distributions.

###### Proof.

The proof is similar as those in [11, Proposition $3.1$]. We sketch the proof
for completeness. From (3.7), we have111Since the convergence of the tails of
the series (3.4) is key to the existence of the Green’s function, we can pay
some regularity $s>s_{0}>0$ with $s_{0}\in(0,1/2)$, use the $\mathfrak{I}_{p}$
estimate instead of the $\mathfrak{I}_{2}$ estimate to remove small mass
assumption. Here we pay attention to the regularity problem of the solution
for (1.1) in this paper.

$\bigl{\|}\sqrt{R_{0}}(L-L_{0})\sqrt{R_{0}}\,\bigr{\|}_{\mathfrak{I}_{2}}\leq\sqrt{2}\|\Lambda\|_{\mathfrak{I}_{2}}\lesssim\|q\|_{L^{2}}\lesssim\delta$

uniformly for $|\kappa|\geq 1$. Thus, for $\delta>0$ sufficiently small, the
series (3.4) converges in operator norm uniformly for $|\kappa|\geq 1$. It is
easy to verify that the sum acts as an inverse to $L(\kappa)$.

From (3.7), we also have $R-R_{0}\in\mathfrak{I}_{2}$. In particular, the
operator $R-R_{0}$ admits an integral kernel $G-G_{0}$ in
$L^{2}(\mathbb{R}\times\mathbb{R})$. Moreover, by (2.9), we know that
$R-R_{0}$ converges in the sense of Hilbert–Schmidt operators from
$H^{-s-\frac{1}{2}}_{\kappa}$ to $H^{s+\frac{1}{2}}_{\kappa}$, which implies
(3.9).

By (3.9) , we obtain that the kernel function $G-G_{0}$ is continuous in
$(x,y)$ since $s+\frac{1}{2}>\frac{1}{2}$.

For regular $q$, the identities (3.10) and (3.11) precisely express the fact
that $G$ is an integral kernel for $R(\kappa)$. They also hold for irregular
$q$ by (3.9). ∎

Let us define $\gamma$, $g_{12}$ and $g_{21}$ as follows:

$\displaystyle\gamma(x;\kappa)$ $\displaystyle:=\operatorname{sgn}({i\mkern
1.0mu}\kappa^{2})\bigl{[}G_{11}(x,x;\kappa)-G_{22}(x,x;\kappa)\bigr{]}-1,$
(3.12) $\displaystyle g_{12}(x;\kappa)$
$\displaystyle:=\operatorname{sgn}({i\mkern
1.0mu}\kappa^{2})G_{12}(x,x;\kappa),$ (3.13) $\displaystyle g_{21}(x;\kappa)$
$\displaystyle:=\operatorname{sgn}({i\mkern
1.0mu}\kappa^{2})G_{21}(x,x;\kappa),$ (3.14)

where $G_{ij}(x,y,\kappa)$, $1\leq i,j\leq 2$, are the entries of the integral
kernel $G(x,y,\kappa)$. If $q\in B_{\delta}\cap\mathscr{S}$, we may use (3.10)
and (3.11) to obtain

$\displaystyle\gamma^{\prime}$
$\displaystyle=2\kappa\left(qg_{21}-rg_{12}\right),$ (3.15) $\displaystyle
g_{12}^{\prime}$ $\displaystyle=-2i\kappa^{2}g_{12}-\kappa q[\gamma+1],$
(3.16) $\displaystyle g_{21}^{\prime}$
$\displaystyle=2i\kappa^{2}g_{21}+\kappa r[\gamma+1].$ (3.17)

By (3.8), we have

$\displaystyle\gamma(\kappa)$
$\displaystyle=\bar{\gamma}(-\bar{\kappa})\quad\text{and}\quad
g_{12}(\kappa)=-\bar{g}_{21}(-\bar{\kappa}).$ (3.18)

Moreover, by (3.15), (3.16), and (3.17) , we have the following identity

$\displaystyle\frac{\kappa^{2}-\varkappa^{2}}{\kappa^{2}}[g_{12}^{\prime}(\kappa)g_{21}(\varkappa)+g_{21}^{\prime}(\kappa)g_{12}(\varkappa)]$
$\displaystyle=[g_{12}(\kappa)g_{21}(\varkappa)+g_{21}(\kappa)g_{12}(\varkappa)]^{\prime}+\frac{\varkappa}{2\kappa}[(\gamma(\kappa)+1)(\gamma(\varkappa)+1)]^{\prime},$
(3.19)

which is closely connected to the commutativity of the $A(\kappa)$’s flows
under the Poisson bracket (1.5) .

From the series representation (3.4) of $R(\kappa)$ in $q$ and $r$, we can
deduce the corresponding series representations of $g_{12}$, $g_{21}$, and
$\gamma$ in $q$ and $r$. We use the square brackets notation as follows

$\displaystyle g_{12}^{[2m+1]}(\kappa)$
$\displaystyle:=\operatorname{sgn}({i\mkern
1.0mu}\kappa^{2})\Big{\langle}\delta_{x},\
(\partial+i\kappa^{2})^{-\frac{1}{2}}\Lambda\left(\Gamma\Lambda\right)^{m}(\partial-i\kappa^{2})^{-\frac{1}{2}}\delta_{x}\Big{\rangle},$
(3.20) $\displaystyle g_{21}^{[2m+1]}(\kappa)$
$\displaystyle:=\operatorname{sgn}({i\mkern
1.0mu}\kappa^{2})\Big{\langle}\delta_{x},\
(\partial-i\kappa^{2})^{-\frac{1}{2}}\Gamma\left(\Lambda\Gamma\right)^{m}(\partial+i\kappa^{2})^{-\frac{1}{2}}\delta_{x}\Big{\rangle},$
(3.21)

with $g_{12}^{[2m]}(\kappa)=g_{21}^{[2m]}(\kappa):=0$, and similarly,
$\gamma^{[2m+1]}(\kappa):=0$ and

$\displaystyle\gamma^{[2m]}(\kappa):=$
$\displaystyle\operatorname{sgn}({i\mkern
1.0mu}\kappa^{2})\Big{\langle}\delta_{x},\
(\partial+i\kappa^{2})^{-\frac{1}{2}}(\Lambda\Gamma)^{m}(\partial+i\kappa^{2})^{-\frac{1}{2}}\delta_{x}\Big{\rangle}$
$\displaystyle-\operatorname{sgn}({i\mkern
1.0mu}\kappa^{2})\Big{\langle}\delta_{x},\
(\partial-i\kappa^{2})^{-\frac{1}{2}}(\Gamma\Lambda)^{m}(\partial-i\kappa^{2})^{-\frac{1}{2}}\delta_{x}\Big{\rangle}.$
(3.22)

then by (3.4), we have

$g_{12}(\kappa)=\sum_{\ell=1}^{\infty}g_{12}^{[\ell]}(\kappa),\quad
g_{21}(\kappa)=\sum_{\ell=1}^{\infty}g_{21}^{[\ell]}(\kappa),\text{~{}and~{}}\gamma(\kappa)=\sum_{\ell=2}^{\infty}\gamma^{[\ell]}(\kappa).$
(3.23)

We also write the tails of these series as

$\displaystyle g_{12}^{[\geq m]}(\kappa):=$
$\displaystyle\sum_{\ell=m}^{\infty}g_{12}^{[\ell]}(\kappa),\quad
g_{21}^{[\geq m]}(\kappa):=\sum_{\ell=m}^{\infty}g_{21}^{[\ell]}(\kappa),$
$\displaystyle\gamma^{[\geq
m]}(\kappa):=\sum_{\ell=m}^{\infty}\gamma^{[\ell]}(\kappa).$

By (3.16) (3.17) and (3.36), we obtain the identities

$g_{12}=-(2i\kappa^{2}+\partial)^{-1}[\kappa q+\kappa\gamma q],\quad
g_{21}=-(2i\kappa^{2}-\partial)^{-1}[\kappa r+\kappa\gamma r],$

and

$\gamma=-2g_{12}g_{21}-\tfrac{1}{2}\gamma^{2},$

from which we have the explicit expressions for the leading order terms

$\displaystyle g_{12}^{[1]}(\kappa)=-\tfrac{\kappa
q}{2i\kappa^{2}+\partial},\qquad
g_{12}^{[3]}(\kappa)=\tfrac{2}{2i\kappa^{2}+\partial}\big{(}\kappa
q\cdot\tfrac{\kappa r}{2i\kappa^{2}-\partial}\cdot\tfrac{\kappa
q}{2i\kappa^{2}+\partial}\big{)},$ (3.24) $\displaystyle
g_{21}^{[1]}(\kappa)=-\tfrac{\kappa
r}{2i\kappa^{2}-\partial},\qquad\hphantom{-}g_{21}^{[3]}(\kappa)=\tfrac{2}{2i\kappa^{2}-\partial}\big{(}\kappa
r\cdot\tfrac{\kappa q}{2i\kappa^{2}+\partial}\cdot\tfrac{\kappa
r}{2i\kappa^{2}-\partial}\big{)},$ (3.25)

and

$\displaystyle\gamma^{[2]}(\kappa)$ $\displaystyle=-2\,\tfrac{\kappa
q}{2i\kappa^{2}+\partial}\cdot\tfrac{\kappa r}{2i\kappa^{2}-\partial},$ (3.26)
$\displaystyle\gamma^{[4]}(\kappa)$ $\displaystyle=\tfrac{\kappa
q}{2i\kappa^{2}+\partial}\cdot\tfrac{4}{2i\kappa^{2}-\partial}\big{(}\kappa
r\cdot\tfrac{\kappa q}{2i\kappa^{2}+\partial}\cdot\tfrac{\kappa
r}{2i\kappa^{2}-\partial}\big{)}$
$\displaystyle\quad+\tfrac{4}{2i\kappa^{2}+\partial}\big{(}\kappa
q\cdot\tfrac{\kappa r}{2i\kappa^{2}-\partial}\cdot\tfrac{\kappa
q}{2i\kappa^{2}+\partial}\big{)}\cdot\tfrac{\kappa r}{2i\kappa^{2}-\partial}$
$\displaystyle\quad-2\,\tfrac{\kappa
q}{2i\kappa^{2}+\partial}\cdot\tfrac{\kappa
r}{2i\kappa^{2}-\partial}\cdot\tfrac{\kappa
q}{2i\kappa^{2}+\partial}\cdot\tfrac{\kappa r}{2i\kappa^{2}-\partial}.$ (3.27)

We are now ready to obtain some basic estimates on $g_{12}$, $g_{21}$.

######  Proposition 3.2 (Properties of $g_{12}$ and $g_{21}$).

There exists $\delta>0$ such that for all ${i\mkern
1.0mu}\kappa^{2}\in\mathbb{R}$ with $|\kappa|\geq 1$ the maps $q\mapsto
g_{12}(\kappa)$ and $q\mapsto g_{21}(\kappa)$ are (real analytic)
diffeomorphisms of $B_{\delta}$ into $H^{s+\frac{1}{2}}$ satisfying the
estimates

$\|g_{12}(\kappa)\|_{H^{s+\frac{1}{2}}_{\kappa}}+\|g_{21}(\kappa)\|_{H^{s+\frac{1}{2}}_{\kappa}}\lesssim|\kappa|\,\|q\|_{H^{s-\frac{1}{2}}_{\kappa}}.$
(3.28)

Further, the remainders satisfy the estimate

$\|g_{12}^{[\geq 3]}(\kappa)\|_{H^{s+\frac{1}{2}}_{\kappa}}+\|g_{21}^{[\geq
3]}(\kappa)\|_{H^{s+\frac{1}{2}}_{\kappa}}\lesssim|\kappa|\|q\|_{H^{s-\frac{1}{2}}_{\kappa}}\|q\|^{2}_{L^{2}},$
(3.29)

uniformly in $\kappa$. Finally, if $q$ is Schwartz then so are
$g_{12}(\kappa)$ and $g_{21}(\kappa)$.

###### Proof.

It suffices to consider the case ${i\mkern 1.0mu}\kappa^{2}\geq 1$ as the case
${i\mkern 1.0mu}\kappa^{2}\leq-1$ is similar, and by (3.18), it suffices to
consider $g_{12}(\kappa)$. Recalling (3.24), we obtain

$\|g_{12}^{[1]}(\kappa)\|_{H^{s+\frac{1}{2}}_{\kappa}}=|\kappa|\,\|q\|_{H^{s-\frac{1}{2}}_{\kappa}}.$
(3.30)

To bound the remainder terms in the series222If we use $\mathfrak{I}_{p}$ with
$p>2$ instead of $\mathfrak{I}_{2}$ once again, we may pay some regularity on
$s>s_{0}>0$ to remove the small mass assumption., we employ duality and Lemma
2.1:

$\displaystyle\bigl{|}\langle f,g_{12}^{[\geq 3]}(\kappa)\rangle\bigr{|}$
$\displaystyle\leq\|(\partial-i\kappa^{2})^{-s-\frac{1}{2}}\bar{f}(\partial+i\kappa^{2})^{-s-\frac{1}{2}}\|_{\mathrm{op}}$
$\displaystyle\qquad\qquad\times\sum_{\ell=1}^{\infty}\|(\partial+i\kappa^{2})^{s-\frac{1}{2}}kq(\partial-i\kappa^{2})^{-\frac{1}{2}}\|_{\mathfrak{I}_{2}}^{2\ell+1}|\kappa|^{-2s(2\ell-1)}$
$\displaystyle\lesssim|\kappa|^{-2s}\|f\|_{H^{-s-\frac{1}{2}}_{\kappa}}\sum_{\ell=1}^{\infty}\left(|\kappa|\|q\|_{H_{\kappa}^{s-\frac{1}{2}}}\right)^{2\ell+1}|\kappa|^{-2s(2\ell-1)}$
$\displaystyle\lesssim\|f\|_{H^{-s-\frac{1}{2}}_{\kappa}}|\kappa|\,\|q\|_{H_{\kappa}^{s-\frac{1}{2}}}\sum_{\ell=1}^{\infty}\left(|\kappa|^{1-2s}\|q\|_{H_{\kappa}^{s-\frac{1}{2}}}\right)^{2\ell}$
$\displaystyle\lesssim\|f\|_{H^{-s-\frac{1}{2}}_{\kappa}}|\kappa|\,\|q\|_{H_{\kappa}^{s-\frac{1}{2}}}\|q\|^{2}_{L^{2}}$
(3.31)

provided $\delta>0$ is sufficiently small. This proves (3.29) and completes
the proof of (3.28).

In addtion, we have

$\tfrac{\delta g_{12}}{\delta
q}(\kappa)\bigr{|}_{q=0}=-\frac{\kappa}{2i\kappa^{2}+\partial}\quad\text{and}\quad\tfrac{\delta
g_{12}}{\delta r}(\kappa)\bigr{|}_{q=0}=0$

which is an isomorphism, as noted already in (3.30). Furthermore, for any
$f\in\mathscr{S}$, we have

$\displaystyle\left.\frac{d}{d\varepsilon}\right|_{\varepsilon=0}G(x,z;q+\varepsilon
f)$ $\displaystyle=-\int G(x,y;q)\begin{bmatrix}0&f(y)\\\
-\bar{f}(y)&0\end{bmatrix}G(y,z;q)\,dy.$

By similar analysis as those to prove (3.29), we have

$\displaystyle\bigl{\|}\tfrac{\delta g_{12}}{\delta
r}(\kappa)\bigr{\|}_{H^{s-\frac{1}{2}}_{\kappa}\rightarrow
H^{s+\frac{1}{2}}_{\kappa}}+\bigl{\|}\tfrac{\delta g_{12}}{\delta
q}(\kappa)+\frac{\kappa}{2i\kappa^{2}+\partial}\bigr{\|}_{H^{s-\frac{1}{2}}_{\kappa}\rightarrow
H^{s+\frac{1}{2}}_{\kappa}}\lesssim\|q\|^{2}_{L^{2}}\lesssim\delta^{2},$

and so the inverse function theorem implies the diffeomorphism property for
sufficiently small $\delta$ .

The regulairty result can be easily obtained in a similar argument as those in
[19, Proposition $2.2$] and [11, Proposition 3.2]. ∎

We also have some estimates on $\gamma$ as follows.

######  Proposition 3.3 (Properties of $\gamma$).

There exists $\delta>0$ such that for all ${i\mkern
1.0mu}\kappa^{2}\in\mathbb{R}$ with $|\kappa|\geq 1$ the map
$q\mapsto\gamma(\kappa)$ is bounded from $B_{\delta}$ to $L^{1}\cap
H^{s+\frac{1}{2}}$ and we have the estimates

$\displaystyle\|\gamma(\kappa)\|_{H^{s+\frac{1}{2}}_{\kappa}}$
$\displaystyle\lesssim|\kappa|^{2-2s}\|q\|_{H^{s-\frac{1}{2}}_{\kappa}}^{2},$
(3.32) $\displaystyle\|\gamma(\kappa)\|_{L^{\infty}}$
$\displaystyle\lesssim|\kappa|^{2-4s}\|q\|_{H^{s-\frac{1}{2}}_{\kappa}}^{2},$
(3.33) $\displaystyle\|\gamma(\kappa)\|_{L^{1}}$
$\displaystyle\lesssim|\kappa|^{2}\|q\|_{H^{-1}_{\kappa}}^{2}+|\kappa|^{-2(4s-1)}\|q\|_{H^{s-\frac{1}{2}}_{\kappa}}^{4},$
(3.34) $\displaystyle\|\gamma^{[\geq 4]}(\kappa)\|_{L^{1}}$
$\displaystyle\lesssim|\kappa|^{-2(4s-1)}\|q\|_{H^{s-\frac{1}{2}}_{\kappa}}^{4},$
(3.35)

uniformly in $\kappa$. Further, we have the quadratic identity

$\gamma+\frac{1}{2}\gamma^{2}=-2g_{12}g_{21},$ (3.36)

and if $q$ is Schwartz, then so is $\gamma(\kappa)$.

###### Proof.

Once again it suffices to consider the case ${i\mkern 1.0mu}\kappa^{2}\geq 1$.
Using (3.26) and (2.4), we obtain

$\displaystyle\|\gamma^{[2]}\|_{H_{\kappa}^{s+\frac{1}{2}}}\lesssim\|\tfrac{\kappa
q}{2i\kappa^{2}+\partial}\cdot\tfrac{\kappa
r}{2i\kappa^{2}-\partial}\|_{H_{\kappa}^{s+\frac{1}{2}}}\lesssim|\kappa|^{2-2s}\|q\|_{H^{s-\frac{1}{2}}_{\kappa}}^{2}.$

To handle $\gamma^{[\geq 4]}$ we use the series representation (3.23) and the
same dual argument used to prove (3.29). The estimate (3.33) then follows from
(3.32) via (2.3).

Choosing $\varkappa=\kappa$ in (3.19), we obtain

$\partial_{x}\bigl{\\{}2g_{12}(x;\kappa)g_{21}(x;\kappa)+\tfrac{1}{2}\gamma(x;\kappa)^{2}+\gamma(x;\kappa)\bigr{\\}}=0.$

By (3.28) and (3.32), we know that the term in the braces vanishes as
$|x|\to\infty$. Thus the identity (3.36) follows by integration.

By using this quadratic identity, we may write

$\gamma^{[\geq 4]}=-\tfrac{1}{2}\gamma^{2}-2g_{12}^{[\geq 3]}\cdot
g_{21}-2g_{12}^{[1]}\cdot g_{21}^{[\geq 3]}.$ (3.37)

By Proposition 3.2 and (3.32), we have

$\displaystyle\|g_{12}^{[\geq 3]}\|_{L^{2}}+\|g_{21}^{[\geq 3]}\|_{L^{2}}$
$\displaystyle\lesssim|\kappa|^{-2(s+\frac{1}{2})}\bigl{[}\|g_{12}^{[\geq
3]}\|_{H^{s+\frac{1}{2}}_{\kappa}}+\|g_{21}^{[\geq
3]}\|_{H^{s+\frac{1}{2}}_{\kappa}}\bigr{]}\lesssim|\kappa|^{-6s+2}\|q\|_{H^{s-\frac{1}{2}}_{\kappa}}^{3},$
$\displaystyle\|g_{12}\|_{L^{2}}+\|g_{21}\|_{L^{2}}$
$\displaystyle\lesssim|\kappa|^{-2(s+\frac{1}{2})}\bigl{(}\|g_{12}\|_{H_{\kappa}^{s+\frac{1}{2}}}+\|g_{21}\|_{H_{\kappa}^{s+\frac{1}{2}}}\bigr{)}\lesssim|\kappa|^{-2s}\|q\|_{H^{s-\frac{1}{2}}_{\kappa}},$

and

$\displaystyle\|\gamma\|_{L^{2}}$
$\displaystyle\lesssim|\kappa|^{-2(s+\frac{1}{2})}\|\gamma\|_{H^{s+\frac{1}{2}}_{\kappa}}\lesssim|\kappa|^{-4s+1}\|q\|_{H^{s-\frac{1}{2}}_{\kappa}}^{2},$

which together with Hölder’s inequality imply that

$\displaystyle\|\gamma^{[\geq 4]}\|_{L^{1}}$
$\displaystyle\lesssim\|\gamma\|_{L^{2}}^{2}+\|g_{12}^{[\geq
3]}\|_{L^{2}}\|g_{21}\|_{L^{2}}+\|g_{12}^{[1]}\|_{L^{2}}\|g_{21}^{[\geq
3]}\|_{L^{2}}\lesssim|\kappa|^{-2(4s-1)}\|q\|_{H^{s-\frac{1}{2}}_{\kappa}}^{4},$

thus we obtain (3.35). The estimate (3.34) then follows from applying the
Cauchy-Schwarz inequality to (3.26).

The regulairty result can also be deduced by a similar argument as those in
[19, Proposition $2.2$] and [11, Proposition 3.3]. ∎

Due to the structure of microscopic conservation law in (4.13), the
combination function $\frac{g_{12}(\kappa)}{2+\gamma(\kappa)}$ will be also
used later. We now give the analogue estimates. Firstly, we denote:

$\tfrac{g_{12}}{2+\gamma}=\big{(}\tfrac{g_{12}}{2+\gamma}\big{)}^{[1]}+\big{(}\tfrac{g_{12}}{2+\gamma}\big{)}^{[3]}+\big{(}\tfrac{g_{12}}{2+\gamma}\big{)}^{[\geq
5]},$

where the leading order terms are given by

$\big{(}\tfrac{g_{12}}{2+\gamma}\big{)}^{[1]}=\tfrac{1}{2}g_{12}^{[1]}\quad\text{and}\quad\big{(}\tfrac{g_{12}}{2+\gamma}\big{)}^{[3]}=\tfrac{1}{2}g_{12}^{[3]}-\tfrac{1}{4}g_{12}^{[1]}\gamma^{[2]},$
(3.38)

and the remainder term is given by

$\displaystyle\big{(}\tfrac{g_{12}}{2+\gamma}\big{)}^{[\geq 3]}$
$\displaystyle=\tfrac{1}{2}g_{12}^{[\geq
3]}-\tfrac{g_{12}\gamma}{2(2+\gamma)}.$ (3.39)

We can now show the following estimates about
$\tfrac{g_{12}(\kappa)}{2+\gamma(\kappa)}$.

######  Corollary 3.4.

Let $s\in(0,1/2)$ and $q\in B_{\delta}$. there exists $\delta>0$ such that for
all ${i\mkern 1.0mu}\kappa^{2}\in\mathbb{R}$ with $|\kappa|\geq 1$, we have
the estimates

$\displaystyle|\kappa|^{2}\bigl{\|}\tfrac{g_{12}(\kappa)}{2+\gamma(\kappa)}\bigr{\|}_{H^{s-1/2}}+\bigl{\|}\tfrac{g_{12}(\kappa)}{2+\gamma(\kappa)}\bigr{\|}_{H^{s+1/2}}$
$\displaystyle\lesssim|\kappa|\|q\|_{H^{s-1/2}},$ (3.40)
$\displaystyle|\kappa|^{2}\bigl{\|}\big{(}\tfrac{g_{12}(\kappa)}{2+\gamma(\kappa)}\big{)}^{[\geq
3]}\bigr{\|}_{H^{s-1/2}}+\bigl{\|}\big{(}\tfrac{g_{12}(\kappa)}{2+\gamma(\kappa)}\big{)}^{[\geq
3]}\bigr{\|}_{H^{s+1/2}}$
$\displaystyle\lesssim|\kappa|\|q\|_{H^{s-1/2}}\|q\|_{L^{2}}^{2}.$ (3.41)

###### Proof.

From (3.38) and (3.24), we see that

$|\kappa|^{2}\bigl{\|}\big{(}\tfrac{g_{12}}{2+\gamma}\big{)}^{[1]}\bigr{\|}_{H^{s-1/2}}+\bigl{\|}\big{(}\tfrac{g_{12}}{2+\gamma}\big{)}^{[1]}\bigr{\|}_{H^{s+1/2}}\approx\bigl{\|}(2i\kappa^{2}+\partial)\big{(}\tfrac{g_{12}(\varkappa)}{2+\gamma(\varkappa)}\big{)}^{[1]}\bigr{\|}_{H^{s}}\approx|\kappa|\|q\|_{H^{s-1/2}}.$

Thus it suffice for (3.40) to show (3.41). Moreover, by (3.16), we have

$(2i\kappa^{2}+\partial)\big{(}\tfrac{g_{12}}{2+\gamma}\big{)}^{[\geq
3]}=-\kappa\tfrac{\gamma}{2(2+\gamma)}q+\tfrac{g_{12}}{(2+\gamma)^{2}}\gamma^{\prime},$

and therefore, we have

LHS of (3.41)
$\displaystyle\lesssim|\kappa|\bigl{\|}\tfrac{\gamma}{2+\gamma}q\bigr{\|}_{H^{s-1/2}}+\bigl{\|}\tfrac{g_{12}}{(2+\gamma)^{2}}\gamma^{\prime}\bigr{\|}_{H^{s-1/2}}$
$\displaystyle\lesssim|\kappa|^{1-2s}\|q\|_{H^{s-1/2}}\bigl{\|}\tfrac{\gamma}{2+\gamma}\bigr{\|}_{H^{s+1/2}_{\kappa}}+|\kappa|^{-2s}\|\gamma^{\prime}\|_{H^{s-1/2}}\bigl{\|}\tfrac{g_{12}}{(2+\gamma)^{2}}\bigr{\|}_{H^{s+1/2}_{\kappa}}$
$\displaystyle\lesssim|\kappa|^{1-2s}\|q\|_{H^{s-1/2}}\bigl{\|}\gamma\bigr{\|}_{H^{s+1/2}_{\kappa}}$
$\displaystyle\lesssim|\kappa|\|q\|_{H^{s-1/2}}\|q\|^{2}_{L^{2}},$

where the second step we use (2.5) and (3.32), and the third step we expand as
series and employ the algebra property (2.4), together with (3.28) and (3.32).
This yields (3.40) for sufficiently small $\delta$. ∎

##  4\. Conservation laws and dynamics

In this section, we will firstly introduce the invariant quantity $A(\kappa)$
from the logrithmic perturbation determinant, which is related to the
integrability and spectral invariance of (1.1), then we show the dynamics and
microscopic conservation laws of the $A(\kappa)$’s flow and the DNLS (i.e.
$H_{\mathrm{DNLS}}$) flow, respectively.

###  4.1. Conservation Laws

Inspired by [11, 19, 20, 34], we formally define the logarithmic perturbation
determinant $\operatorname{sgn}({i\mkern
1.0mu}\kappa^{2})\log\det(L_{0}^{-1}L)$ as follows:

$A(\kappa;q,r):=\operatorname{sgn}({i\mkern
1.0mu}\kappa^{2})\sum_{j=1}^{\infty}\frac{(-1)^{j-1}}{j}\operatorname{tr}\left\\{\left(\sqrt{R_{0}}\left(L-L_{0}\right)\sqrt{R_{0}}\right)^{j}\right\\}.$
(4.1)

By (3.5), simple calculations deduce that for $q,r\in\mathscr{S}$, we have

$A(\kappa;q,r)=-\operatorname{sgn}({i\mkern
1.0mu}\kappa^{2})\sum_{m=1}^{\infty}\tfrac{1}{m}\operatorname{tr}\left\\{(\Lambda\Gamma)^{m}\right\\}.$
(4.2)

In the following, we will use (4.2) as the definition of $A(\kappa;q,r)$. For
the sake of simplicity, we write

$A(\kappa;q,r)=\sum\limits_{m=1}^{\infty}A_{m}(\kappa;q,r),\;A_{m}(\kappa;q,r):=-\operatorname{sgn}({i\mkern
1.0mu}\kappa^{2})\tfrac{1}{m}\operatorname{tr}\left\\{(\Lambda\Gamma)^{m}\right\\}.$
(4.3)

Firstly, we have

######  Lemma 4.1 (Properties of $A$).

There exists $\delta>0$ such that for all $q\in B_{\delta}\cap\mathscr{S}$ and
$i\kappa^{2}\in\mathbb{R}$ with $|\kappa|\geq 1$, the series (4.2) converges
absolutely. Moreover, we have

$\displaystyle\tfrac{\delta\,}{\delta q}A(\kappa)=-\kappa
g_{21}(\kappa),\quad\tfrac{\delta\,}{\delta r}A(\kappa)=-\kappa
g_{12}(\kappa),$ (4.4)
$\displaystyle\;\gamma^{\prime}(\kappa)=2\left(-q\tfrac{\delta\,}{\delta
q}A(\kappa)+r\tfrac{\delta\,}{\delta r}A(\kappa)\right).$ (4.5)

###### Proof.

By (3.7), we know that $A(\kappa;q,r)$ converges absolutely for all $q\in
B_{\delta}\cap\mathscr{S}$. By simple calculations, we have

$\tfrac{\delta\,\,}{\delta q}A_{m}=-\kappa
g_{21}^{[2m-1]}\quad\text{and}\quad\tfrac{\delta\,\,}{\delta r}A_{m}=-\kappa
g_{21}^{[2m-1]},$ (4.6)

which together with (3.15), (3.23) implies (4.5). ∎

Next, we show that the mass, the Hamiltonian and the energy defined by (1.7)
and (1.8) respectively, arise as the coefficients in the asymptotic expansion
of $A(\kappa)$ as $|\kappa|\to\infty$. More precisely, we have.

######  Lemma 4.2 (Asymptotic expansion of $A(\kappa)$).

For $q\in B_{\delta}\cap\mathscr{S}$, we have, as $|\kappa|\to\infty$,

$A(\kappa)=(-{i\mkern 1.0mu})\tfrac{M}{2}-\tfrac{(-{i\mkern
1.0mu})^{2}}{2i\kappa^{2}}\frac{H_{\mathrm{DNLS}}}{2}+\frac{(-i)^{3}}{(2i\kappa^{2})^{2}}\frac{E_{\mathrm{DNLS}}}{2}+O(|\kappa|^{-6}).$
(4.7)

###### Proof.

By (3.16) and (3.17), and (4.5), we have

$\begin{split}2{i\mkern 1.0mu}\kappa^{2}\tfrac{\delta A}{\delta
q}=&\;\partial\tfrac{\delta A}{\delta
q}+2\kappa^{2}r\cdot\partial^{-1}\left(-q\tfrac{\delta A}{\delta
q}+r\tfrac{\delta A}{\delta r}\right)+\kappa^{2}r,\\\ 2{i\mkern
1.0mu}\kappa^{2}\tfrac{\delta A}{\delta r}=&-\partial\tfrac{\delta A}{\delta
r}+2\kappa^{2}q\cdot\partial^{-1}\left(-q\tfrac{\delta A}{\delta
q}+r\tfrac{\delta A}{\delta r}\right)+\kappa^{2}q.\end{split}$

By the asymptotic analysis, we have

$\begin{split}\tfrac{\delta A}{\delta q}&=-\kappa g_{21}\\\ &=-{i\mkern
1.0mu}\frac{1}{2}r+\frac{1}{2{i\mkern 1.0mu}\kappa^{2}}\bigl{(}-\frac{{i\mkern
1.0mu}}{2}r^{\prime}+\frac{1}{2}qr^{2}\bigr{)}+\frac{1}{(2{i\mkern
1.0mu}\kappa^{2})^{2}}\left(-\frac{{i\mkern
1.0mu}}{2}r^{\prime\prime}+\frac{3}{2}qrr^{\prime}+\frac{3}{4}{i\mkern
1.0mu}q^{2}r^{3}\right)+O(|\kappa|^{-6}),\\\ \tfrac{\delta A}{\delta
r}&=-\kappa g_{12}\\\ &=-{i\mkern 1.0mu}\frac{1}{2}q+\frac{1}{2{i\mkern
1.0mu}\kappa^{2}}\bigl{(}\frac{{i\mkern
1.0mu}}{2}q^{\prime}+\frac{1}{2}q^{2}r\bigr{)}+\frac{1}{(2{i\mkern
1.0mu}\kappa^{2})^{2}}\left(-\frac{{i\mkern
1.0mu}}{2}q^{\prime\prime}-\frac{3}{2}qq^{\prime}r+\frac{3}{4}{i\mkern
1.0mu}q^{3}r^{2}\right)+O(|\kappa|^{-6}),\end{split}$ (4.8)

which together with the fact that

$A(q,r)=\int_{0}^{1}\partial_{\theta}A(\theta q,\theta r)\,{\rm{d}}\theta$
(4.9)

imply (4.7). ∎

######  Remark.

By computing $\gamma=2\partial^{-1}\left(-q\tfrac{\delta A}{\delta
q}+r\tfrac{\delta A}{\delta r}\right)$, we can obtain asymptotic expansion for
$\gamma$,

$\displaystyle\gamma=\frac{1}{2\kappa^{2}}qr-\frac{1}{4\kappa^{4}}\bigl{(}{i\mkern
1.0mu}q^{\prime}r-{i\mkern
1.0mu}qr^{\prime}+\frac{3}{2}q^{2}r^{2}\bigr{)}+O(|\kappa|^{-6}).$ (4.10)

######  Lemma 4.3 (Density function of $A(\kappa)$).

For all $q\in B_{\delta}\cap\mathscr{S}$ and $i\kappa^{2}\in\mathbb{R}$ with
$|\kappa|\geq 1$, we have

$\displaystyle A(\kappa)=-\bar{A}(-\bar{\kappa}),$ (4.11)
$\displaystyle\tfrac{\partial\,}{\partial\kappa}A(\kappa)=\int_{\mathbb{R}}\Bigl{[}2i\kappa\cdot\gamma(\kappa)-\bigl{(}qg_{21}(\kappa)+rg_{12}(\kappa)\bigr{)}\Bigr{]}\,{\rm{d}}x,$
(4.12) $\displaystyle
A(\kappa)=\int_{\mathbb{R}}\rho(\kappa)\,{\rm{d}}x,\quad\text{where}\quad\rho(\kappa)=-\kappa\frac{qg_{21}(\kappa)+rg_{12}(\kappa)}{2+\gamma(\kappa)}.$
(4.13)

###### Proof.

Firstly, we show that (4.12). In fact, simple calculations imply that

$\frac{\partial\ }{\partial\kappa}A_{m}=\int_{\mathbb{R}}\Bigl{[}2{i\mkern
1.0mu}\kappa\gamma^{[2m]}-\bigl{(}qg_{21}^{[2m-1]}+rg_{12}^{[2m-1]}\bigr{)}\Bigr{]}\,{\rm{d}}x.$
(4.14)

By summation with respect to $m$, we can obtain (4.12).

We now estimate (4.13). On one hand, by differentiating (3.16), (3.17), and
(3.36) with respect to $\kappa$, we obtain that,

$\displaystyle\partial_{x}\Bigl{(}g_{12}\tfrac{\partial
g_{21}}{\partial\kappa}-\tfrac{\partial
g_{12}}{\partial\kappa}g_{21}\Bigr{)}=$
$\displaystyle\kappa(qg_{21}+rg_{12})\tfrac{\partial\
}{\partial\kappa}(\gamma+1)-\kappa(\gamma+1)\tfrac{\partial\
}{\partial\kappa}(qg_{21}+rg_{12})$ $\displaystyle-2{i\mkern
1.0mu}\kappa\gamma(\gamma+2)+(\gamma+1)(qg_{21}+rg_{12}).$

Using (3.15), we have

$\displaystyle-\bigl{(}g_{12}\tfrac{\partial
g_{21}}{\partial\kappa}-\tfrac{\partial
g_{12}}{\partial\kappa}g_{21}\bigr{)}\gamma^{\prime}=-\gamma(2+\gamma)\tfrac{\partial\
}{\partial\kappa}\bigl{(}qg_{21}-rg_{12}\bigr{)}+\bigl{(}qg_{21}-rg_{12}\bigr{)}(1+\gamma)\tfrac{\partial\gamma}{\partial\kappa}.$

Combining the above two identities, we get

$\displaystyle\partial_{x}\frac{g_{12}\tfrac{\partial
g_{21}}{\partial\kappa}-\tfrac{\partial
g_{12}}{\partial\kappa}g_{21}}{2+\gamma}$
$\displaystyle=\bigl{(}qg_{21}+rg_{12}-2{i\mkern
1.0mu}\kappa\gamma\bigr{)}-\frac{\partial\
}{\partial\kappa}\bigl{(}\kappa\frac{qg_{21}+rg_{12}}{2+\gamma}\bigr{)},$

which can be integrated in $x$ to yield

$\displaystyle\frac{\partial\
}{\partial\kappa}\int\kappa\frac{qg_{21}+rg_{12}}{2+\gamma}\,{\rm{d}}x=\int
qg_{21}+rg_{12}-2{i\mkern 1.0mu}\kappa\gamma\,{\rm{d}}x=-\frac{\partial
A}{\partial\kappa}.$ (4.15)

On the other hand, by (4.7), we have

$A(\kappa)=-i\tfrac{1}{2}M+O(|\kappa|^{-2}),\quad\text{as}\quad\kappa\to\infty.$
(4.16)

and by (4.8) and (4.10), we obtain

$\rho(\kappa)=-{i\mkern
1.0mu}\frac{1}{2}qr+O(|\kappa|^{-2}),\quad\text{as}\quad\kappa\to\infty.$
(4.17)

Combining (4.15), (4.16) and (4.17), we can obtain (4.13).

Lastly, (4.11) is obvious from (3.18) and (4.13). ∎

Next, we show the commutation of $A(\kappa)$’s under the Poisson bracket
(1.5), which implies that $A(\varkappa)$ is an invariant quantity under the
$A(\kappa)$’s flows.

######  Lemma 4.4 (Poisson brackets).

There exists $\delta>0$ such that for any $\kappa$ and $\varkappa$ with
${i\mkern 1.0mu}\kappa^{2},\,{i\mkern
1.0mu}\varkappa^{2}\in\mathbb{R}\setminus(-1,1)$ with
$\kappa^{2}\neq\varkappa^{2}$ and any $q\in B_{\delta}\cap\mathscr{S}$ we have

$\\{A(\kappa),A(\varkappa)\\}=0.$ (4.18)

###### Proof.

By (4.5) and (1.5), we have

$\displaystyle\\{A(\kappa),A(\varkappa)\\}$ $\displaystyle=\kappa\varkappa\int
g_{12}(\kappa)g_{21}^{\prime}(\varkappa)+g_{21}(\kappa)g_{12}^{\prime}(\varkappa)\,{\rm{d}}x,$

which together with (3.19) implies that (4.18) holds. . ∎

Next we will exhibit the dynamics of $g_{12}$, $g_{21}$ and $\gamma$ defined
by (3.12), (3.13) and (3.14) along the $A(\kappa)$ flow and DNLS flow
respectively.

###  4.2. Dynamics I: the $A(\kappa)$ flow.

Firstly, we have

######  Lemma 4.5 (Dynamics of $A(\kappa)$ flow).

Let $i\kappa^{2}\in\mathbb{R}$ with $|\kappa|>1$. Under the $A(\kappa)$ flow,
we have

$\displaystyle\frac{\,{\rm{d}}\ }{\,{\rm{d}}t}q=-\kappa
g_{12}^{\prime}(\kappa)\quad\text{and}\quad\frac{\,{\rm{d}}\
}{\,{\rm{d}}t}r=-\kappa g_{21}^{\prime}(\kappa).$ (4.19)

###### Proof.

It is obvious from (4.5) and (1.6). ∎

######  Lemma 4.6 (Lax pair for the $A(\kappa)$ flow).

Let ${i\mkern 1.0mu}\kappa^{2},{i\mkern
1.0mu}\varkappa^{2}\in\mathbb{R}\setminus(-1,1)$ with
$\kappa^{2}\neq\varkappa^{2}$, and $L(\varkappa)$ be defined by (1.9). Under
the $A(\kappa)$ flow, we have

$\displaystyle\frac{\,{\rm{d}}\
}{\,{\rm{d}}t}L(\varkappa)=[P_{A(\kappa)},L(\varkappa)],$ (4.20)

where

$P_{A(\kappa)}=\begin{bmatrix}-\frac{1}{2}\Xi(\gamma(\kappa)+1)&-\Theta
g_{12}(\kappa)\\\ -\Theta
g_{21}(\kappa)&\frac{1}{2}\Xi(\gamma(\kappa)+1)\end{bmatrix}\text{~{}~{}with~{}~{}}\Theta=\tfrac{\varkappa\kappa^{3}}{\kappa^{2}-\varkappa^{2}},~{}~{}\Xi=\tfrac{\varkappa^{2}\kappa^{2}}{\kappa^{2}-\varkappa^{2}}.$
(4.21)

###### Proof.

Firstly, by (3.15) and the fact that

$\displaystyle\gamma^{\prime}(\kappa)=$ $\displaystyle(\partial+{i\mkern
1.0mu}\varkappa^{2})\bigl{(}\gamma(\kappa)+1\bigr{)}-\bigl{(}\gamma(\kappa)+1\bigr{)}(\partial+{i\mkern
1.0mu}\varkappa^{2})$ $\displaystyle=$ $\displaystyle(\partial-{i\mkern
1.0mu}\varkappa^{2})\bigl{(}\gamma(\kappa)+1\bigr{)}-\bigl{(}\gamma(\kappa)+1\bigr{)}(\partial-{i\mkern
1.0mu}\varkappa^{2}),$

we have

$\displaystyle(\partial+{i\mkern
1.0mu}\varkappa^{2})\bigl{(}\gamma(\kappa)+1\bigr{)}-\bigl{(}\gamma(\kappa)+1\bigr{)}(\partial+{i\mkern
1.0mu}\varkappa^{2})-2\kappa\bigl{(}qg_{21}(\kappa)-rg_{12}(\kappa)\bigr{)}=0.$
(4.22) $\displaystyle(\partial-{i\mkern
1.0mu}\varkappa^{2})\bigl{(}\gamma(\kappa)+1\bigr{)}-\bigl{(}\gamma(\kappa)+1\bigr{)}(\partial-{i\mkern
1.0mu}\varkappa^{2})-2\kappa\bigl{(}qg_{21}(\kappa)-rg_{12}(\kappa)\bigr{)}=0.$
(4.23)

Next, by (3.16) and the fact that

$g_{12}^{\prime}(\kappa)+2{i\mkern
1.0mu}\varkappa^{2}g_{12}(\kappa)=(\partial+{i\mkern
1.0mu}\varkappa^{2})g_{12}(\kappa)-g_{12}(\kappa)(\partial-{i\mkern
1.0mu}\varkappa^{2}),$

we get

$\displaystyle(\kappa^{2}-\varkappa^{2})g_{12}^{\prime}(\kappa)=\kappa^{2}\bigl{[}(\partial+{i\mkern
1.0mu}\varkappa^{2})g_{12}(\kappa)-g_{12}(\kappa)(\partial-{i\mkern
1.0mu}\varkappa^{2})\bigr{]}+\kappa\varkappa^{2}q\bigl{(}\gamma(\kappa)+1\bigr{)}.$
(4.24)

Finally, by (3.17) and the fact that

$g_{21}^{\prime}(\kappa)-2{i\mkern
1.0mu}\varkappa^{2}g_{21}(\kappa)=(\partial-{i\mkern
1.0mu}\varkappa^{2})g_{21}(\kappa)-g_{21}(\kappa)(\partial+{i\mkern
1.0mu}\varkappa^{2}),$

we obtain

$\displaystyle(\kappa^{2}-\varkappa^{2})g_{21}^{\prime}(\kappa)=\kappa^{2}\bigl{[}(\partial-{i\mkern
1.0mu}\varkappa^{2})g_{21}(\kappa)-g_{21}(\kappa)(\partial+{i\mkern
1.0mu}\varkappa^{2})\bigr{]}-\kappa\varkappa^{2}r\bigl{(}\gamma(\kappa)+1\bigr{)}.$
(4.25)

Combining (1.9), (4.19) with (4.22), (4.23), (4.24), (4.25), we can obtain
(4.20). ∎

######  Proposition 4.7 (Microscopic conservation law for the $A(\kappa)$
flow).

Let ${i\mkern 1.0mu}\varkappa^{2},{i\mkern
1.0mu}\kappa^{2}\in\mathbb{R}\setminus(-1,1)$ with
$\kappa^{2}\neq\varkappa^{2}$. Under the $A(\kappa)$ flow, we have

$\displaystyle\frac{\,{\rm{d}}\ }{\,{\rm{d}}t}g_{12}(\varkappa)$
$\displaystyle=-\Xi g_{12}(\varkappa)\left[\gamma(\kappa)+1\right]+\Theta
g_{12}(\kappa)\left[\gamma(\varkappa)+1\right],$ (4.26)
$\displaystyle\frac{\,{\rm{d}}\ }{\,{\rm{d}}t}g_{21}(\varkappa)$
$\displaystyle=\Xi g_{21}(\varkappa)\left[\gamma(\kappa)+1\right]-\Theta
g_{21}(\kappa)\left[\gamma(\varkappa)+1\right],$ (4.27)
$\displaystyle\frac{\,{\rm{d}}\ }{\,{\rm{d}}t}\gamma(\varkappa)$
$\displaystyle=-2\Theta\left[g_{12}(\kappa)g_{21}(\varkappa)-g_{21}(\kappa)g_{12}(\varkappa)\right],$
(4.28)

and the following microscopic conservation laws

$\displaystyle\partial_{t}\bigl{\\{}2i\varkappa\cdot\gamma(\varkappa)-\left(qg_{21}(\varkappa)+rg_{12}(\varkappa)\right)\bigr{\\}}+$
$\displaystyle\partial_{x}j_{\gamma}(\varkappa,\kappa)=0,$ (4.29)
$\displaystyle\partial_{t}\rho(\varkappa)+\partial_{x}j_{A(\kappa)}(\varkappa,\kappa)=0,$
(4.30)

where the flux functions $j_{\gamma}$ and $j_{A(\kappa)}$ are determined by

$\displaystyle j_{\gamma}(\varkappa,\kappa):=$
$\displaystyle-\tfrac{\kappa^{3}(\kappa^{2}+\varkappa^{2})}{(\kappa^{2}-\varkappa^{2})^{2}}\bigl{[}g_{12}(\kappa)g_{21}(\varkappa)+g_{21}(\kappa)g_{12}(\varkappa)\bigr{]}$
$\displaystyle-\tfrac{\kappa^{4}\varkappa}{(\kappa^{2}-\varkappa^{2})^{2}}\bigl{[}(\gamma(\kappa)+1)(\gamma(\varkappa)+1)\bigr{]},$
(4.31) $\displaystyle j_{A(\kappa)}(\varkappa,\kappa):=$
$\displaystyle-\Theta\tfrac{g_{12}(\kappa)g_{21}(\varkappa)+g_{12}(\varkappa)g_{21}(\kappa)}{2+\gamma(\varkappa)}-\Theta\frac{\varkappa}{2\kappa}\gamma(\kappa).$
(4.32)

######  Remark.

Due to the coercivity of the quadratic term of the flux, the conservation law
(4.30) is more useful than (4.29).

###### Proof.

Firstly, we show the dynamics (4.26), (4.27) and (4.28). By (1.6), Proposition
3.1 and Lemma 4.6, we have

$\displaystyle\frac{\,{\rm{d}}\ }{\,{\rm{d}}t}G(x,z;\varkappa)$
$\displaystyle=-\int G(x,y;\varkappa)\frac{\,{\rm{d}}\
}{\,{\rm{d}}t}L(\varkappa)G(y,z;\varkappa)\,{\rm{d}}{y}$ $\displaystyle=\int
G(x,y;\varkappa)\begin{bmatrix}0&-\kappa^{2}g_{12}^{\prime}(\kappa)\\\
-\kappa^{2}g_{21}^{\prime}(\kappa)&0\end{bmatrix}G(y,z;\varkappa)\,{\rm{d}}{y}$
$\displaystyle=P_{A(\kappa)}(x;\kappa,\varkappa)G(x,z;\varkappa)-G(x,z;\varkappa)P_{A(\kappa)}(z;\kappa,\varkappa).$

By choosing $x=z$ and (4.20), we have

$\displaystyle\frac{\,{\rm{d}}\ }{\,{\rm{d}}t}G(x,x;\varkappa)=$
$\displaystyle-\Xi\begin{bmatrix}0&g_{12}(\varkappa)\bigl{(}\gamma(\kappa)+1\bigr{)}\\\
-g_{21}(\varkappa)\bigl{(}\gamma(\kappa)+1\bigr{)}&0\end{bmatrix}$
$\displaystyle-\Theta\begin{bmatrix}g_{12}(\kappa)g_{21}(\varkappa)-g_{21}(\kappa)g_{12}(\varkappa)&-g_{12}(\kappa)\bigl{(}\gamma(\varkappa)+1\bigr{)}\\\
g_{21}(\kappa)\bigl{(}\gamma(\varkappa)+1\bigr{)}&-g_{12}(\kappa)g_{21}(\varkappa)+g_{21}(\kappa)g_{12}(\varkappa)\end{bmatrix},$

which implies (4.26), (4.27) and (4.28).

Next, we prove (4.29). A direct computation implies that

$\displaystyle\partial_{t}\bigl{\\{}2i\varkappa\gamma(\varkappa)-qg_{21}(\varkappa)-rg_{12}(\varkappa)\bigr{\\}}$
$\displaystyle=$ $\displaystyle-g_{21}(\varkappa)\frac{\,{\rm{d}}\
}{\,{\rm{d}}t}q-g_{12}(\varkappa)\frac{\,{\rm{d}}\ }{\,{\rm{d}}t}r+2{i\mkern
1.0mu}\varkappa\frac{\,{\rm{d}}\
}{\,{\rm{d}}t}\gamma(\varkappa)-q\frac{\,{\rm{d}}\
}{\,{\rm{d}}t}g_{21}(\varkappa)-r\frac{\,{\rm{d}}\
}{\,{\rm{d}}t}g_{12}(\varkappa).$ (4.33)

On the one hand, by Lemma 4.5, we have

$-g_{21}(\varkappa)\frac{\,{\rm{d}}\
}{\,{\rm{d}}t}q-g_{12}(\varkappa)\frac{\,{\rm{d}}\
}{\,{\rm{d}}t}r=\kappa\left[g_{21}(\varkappa)g_{12}^{\prime}(\kappa)+g_{12}(\varkappa)g_{21}^{\prime}(\kappa)\right].$
(4.34)

On the other hand, by (4.26), (4.27), and (4.28), we obtain that

$\displaystyle 2{i\mkern 1.0mu}\varkappa\frac{\,{\rm{d}}\
}{\,{\rm{d}}t}\gamma(\varkappa)-q\frac{\,{\rm{d}}\
}{\,{\rm{d}}t}g_{21}(\varkappa)-r\frac{\,{\rm{d}}\
}{\,{\rm{d}}t}g_{12}(\varkappa)$ $\displaystyle=$
$\displaystyle-4i\varkappa\Theta g_{12}(\kappa)g_{21}(\varkappa)-\Xi
g_{21}(\varkappa)q\left[\gamma(\kappa)+1\right]+\Theta
g_{21}(\kappa)q\left[\gamma(\varkappa)+1\right]$
$\displaystyle+4i\varkappa\Theta g_{12}(\kappa)g_{21}(\varkappa)+\Xi
g_{12}(\varkappa)r\left[\gamma(\kappa)+1\right]-\Theta
g_{12}(\kappa)r\left[\gamma(\varkappa)+1\right]$ $\displaystyle=$
$\displaystyle\frac{\varkappa\Theta}{\kappa^{2}}\left[g_{21}(\varkappa)g_{12}^{\prime}(\kappa)+g_{12}(\varkappa)g_{21}^{\prime}(\kappa)\right]-\frac{\Theta}{\varkappa}\left[g_{21}(\kappa)g_{12}^{\prime}(\varkappa)+g_{12}(\kappa)g_{21}^{\prime}(\varkappa)\right].$
(4.35)

Combining (4.33), (4.34), (4.35) with (3.19), we can obtain (4.7).

Finally, let us show that (4.32) holds. A direct calculation implies that

$\displaystyle\;\bigl{(}\gamma(\varkappa)+2\bigr{)}^{2}\partial_{t}\rho(\varkappa)$
$\displaystyle=$
$\displaystyle-\varkappa\bigl{[}g_{21}(\varkappa)\frac{\,{\rm{d}}\
}{\,{\rm{d}}t}q+g_{12}(\varkappa)\frac{\,{\rm{d}}\
}{\,{\rm{d}}t}r\bigr{]}\bigl{(}\gamma(\varkappa)+2\bigr{)}$ (4.36)
$\displaystyle-\varkappa\bigl{[}q\frac{\,{\rm{d}}\
}{\,{\rm{d}}t}g_{21}(\varkappa)+r\frac{\,{\rm{d}}\
}{\,{\rm{d}}t}g_{12}(\varkappa)\bigr{]}\bigl{(}\gamma(\varkappa)+2\bigr{)}$
(4.37)
$\displaystyle+\varkappa\bigl{[}qg_{21}(\varkappa)+rg_{12}(\varkappa)\bigr{]}\partial_{t}\gamma(\varkappa).$
(4.38)

By Lemma 4.5 and (4.26), (4.27), we have

$\displaystyle\eqref{j sub A 1}+\eqref{j sub A 2}$ $\displaystyle=$
$\displaystyle\varkappa\kappa\left[g_{21}(\varkappa)g_{12}^{\prime}(\kappa)+g_{12}(\varkappa)g_{21}^{\prime}(\kappa)\right]\bigl{(}\gamma(\varkappa)+2\bigr{)}$
$\displaystyle+\frac{\varkappa\Theta}{2\kappa}\gamma^{\prime}(\kappa)\bigl{(}\gamma(\varkappa)+1\bigr{)}\bigl{(}\gamma(\varkappa)+2\bigr{)}-\frac{\Xi}{2}\gamma^{\prime}(\varkappa)\bigl{(}\gamma(\kappa)+1\bigr{)}\bigl{(}\gamma(\varkappa)+2\bigr{)}$
$\displaystyle=$
$\displaystyle\varkappa\kappa\left[g_{21}(\varkappa)g_{12}^{\prime}(\kappa)+g_{12}(\varkappa)g_{21}^{\prime}(\kappa)\right]\bigl{(}\gamma(\varkappa)+2\bigr{)}$
$\displaystyle-\frac{\Xi}{2}\left[\bigl{(}\gamma(\kappa)+1\bigr{)}\bigl{(}\gamma(\varkappa)+1\bigr{)}\right]^{\prime}\bigl{(}\gamma(\varkappa)+2\bigr{)}+\Xi\gamma^{\prime}(\kappa)\bigl{(}\gamma(\varkappa)+1\bigr{)}\bigl{(}\gamma(\varkappa)+2\bigr{)}.$
(4.39)

By (4.28), we have

$\displaystyle\eqref{j sub A 3}=$
$\displaystyle-2\varkappa\Theta\left[qg_{21}(\varkappa)+rg_{12}(\varkappa)\right]\left[g_{12}(\kappa)g_{21}(\varkappa)-g_{21}(\kappa)g_{12}(\varkappa)\right]$
$\displaystyle=$
$\displaystyle-2\varkappa\Theta\left[qg_{21}(\varkappa)-rg_{12}(\varkappa)\right]\left[g_{12}(\kappa)g_{21}(\varkappa)+g_{21}(\kappa)g_{12}(\varkappa)\right]$
$\displaystyle+4\Theta\varkappa
g_{12}(\varkappa)g_{21}(\varkappa)\left[qg_{21}(\kappa)-rg_{12}(\kappa)\right]$
$\displaystyle=$
$\displaystyle-\Theta\gamma^{\prime}(\varkappa)\left[g_{12}(\kappa)g_{21}(\varkappa)+g_{21}(\kappa)g_{12}(\varkappa)\right]-\frac{\varkappa\Theta}{2\kappa}\gamma^{\prime}(\kappa)\gamma(\varkappa)\left[\gamma(\varkappa)+2\right].$
(4.40)

Combining (4.2), (4.2) and (3.19), we can deduce (4.32) . ∎

###  4.3. Dynamics II: the DNLS flow:.

Now we turn to the $H_{\mathrm{DNLS}}$ flow. We firstly recall the Lax
representation for (1.1) as follows.

######  Lemma 4.8 (Lax Pair for the DNLS flow, [1, 18]).

Let ${i\mkern 1.0mu}\varkappa^{2}\in\mathbb{R}\setminus(-1,1)$, $L(\varkappa)$
be defined by (1.9). Under the DNLS flow, we have

$\displaystyle\frac{\,{\rm{d}}\
}{\,{\rm{d}}t}L(\varkappa)=[P_{H_{\mathrm{DNLS}}},L(\varkappa)],$ (4.41)

where
$P_{H_{\mathrm{DNLS}}}=\begin{bmatrix}-2i\varkappa^{4}-i\varkappa^{2}qr&2\varkappa^{3}q+i\varkappa
q^{\prime}+\varkappa q^{2}r\\\ 2\varkappa^{3}r-i\varkappa r^{\prime}+\varkappa
qr^{2}&2i\varkappa^{4}+i\varkappa^{2}qr\end{bmatrix}$.

Following the analogue argument as those in Proposition 4.7, we can obtain
main result in this paper.

######  Theorem 4.9 (Microscopic conservation law for the DNLS flow).

Let ${i\mkern 1.0mu}\varkappa^{2}\in\mathbb{R}\setminus(-1,1)$. Under the DNLS
flow, we have

$\displaystyle\frac{\,{\rm{d}}\
}{\,{\rm{d}}t}q=iq^{\prime\prime}+(q^{2}r)^{\prime}\quad\text{and}\quad\frac{\,{\rm{d}}\
}{\,{\rm{d}}t}r=-ir^{\prime\prime}+(qr^{2})^{\prime}.$ (4.42)

and

$\displaystyle\frac{\,{\rm{d}}\ }{\,{\rm{d}}t}g_{12}(\varkappa)$
$\displaystyle=-2\left(2{i\mkern
1.0mu}\varkappa^{4}+i\varkappa^{2}qr\right)g_{12}(\varkappa)-\left(2\varkappa^{3}q+i\varkappa
q^{\prime}+\varkappa q^{2}r\right)\left(\gamma(\varkappa)+1\right),$ (4.43)
$\displaystyle\frac{\,{\rm{d}}\ }{\,{\rm{d}}t}g_{21}(\varkappa)$
$\displaystyle=2\left(2{i\mkern
1.0mu}\varkappa^{4}+i\varkappa^{2}qr\right)g_{21}(\varkappa)+\left(2\varkappa^{3}r-i\varkappa
r^{\prime}+\varkappa qr^{2}\right)\left(\gamma(\varkappa)+1\right),$
$\displaystyle\frac{\,{\rm{d}}\ }{\,{\rm{d}}t}\gamma(\varkappa)$
$\displaystyle=2\varkappa^{2}\gamma^{\prime}(\varkappa)+2i\varkappa(q^{\prime}g_{21}(\varkappa)+r^{\prime}g_{12}(\varkappa))+qr\gamma^{\prime}(\varkappa).$

Moreover, we have the following microscopic conservation law

$\partial_{t}\rho(\varkappa)+\partial_{x}j_{\mathrm{DNLS}}(\varkappa)=0,$
(4.44)

where the density $\rho$ and the flux $j_{\mathrm{DNLS}}$ are defined by
(4.13) and

$\displaystyle j_{\mathrm{DNLS}}(\varkappa):=\,$ $\displaystyle
i\varkappa\frac{q^{\prime}g_{21}(\varkappa)-r^{\prime}g_{12}(\varkappa)}{2+\gamma(\varkappa)}-i\varkappa^{2}qr-2\varkappa^{2}\rho(\varkappa)-qr\rho(\varkappa).$
(4.45)

###### Proof.

By (1.6), Proposition 3.1 and Lemma 4.8, it is easy to obtain the dynamics of
$q$, $r$ and $g_{12}$, $g_{21}$ and $\gamma$ under the $H_{\mathrm{DNLS}}$
flow. Combining the dynamics of $q$, $r$, $g_{12}$, $g_{21}$, $\gamma$ and
(3.19), we can complete the proof of the microscopic conservation law (4.44).
∎

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