# Space-based gravitational wave signal detection and extraction with deep
neural network

Tianyu Zhao
Department of Astronomy
Beijing Normal University
Beijing, China 100875
&Ruoxi Lyu11footnotemark: 1
Department of Statistics
University of Auckland
Auckland, New Zealand 1142
Zhixiang Ren
Peng Cheng Laboratory
Shenzhen,China 518066
<EMAIL_ADDRESS>
&He Wang
CAS Key Laboratory of Theoretical Physics
Institute of Theoretical Physics
Chinese Academy of Sciences
Beijing, China 100190
&Zhoujian Cao22footnotemark: 2
Department of Astronomy
Beijing Normal University
Beijing, China 100875
<EMAIL_ADDRESS>
These authors contributed equally to this work.Corresponding author(s)

###### Abstract

Space-based gravitational wave (GW) detectors will be able to observe signals
from sources that are otherwise nearly impossible from current ground-based
detection. Consequently, the well established signal detection method, matched
filtering, will require a complex template bank, leading to a computational
cost that is too expensive in practice. Here, we develop a high-accuracy GW
signal detection and extraction method for all space-based GW sources. As a
proof of concept, we show that a science-driven and uniform multi-stage deep
neural network can identify synthetic signals that are submerged in Gaussian
noise. Our method has more than 99% accuracy for signal detection of various
sources while obtaining at least 95% similarity compared with target signals.
We further demonstrate the interpretability and strong generalization behavior
for several extended scenarios.

_Keywords_ Gravitational wave $\cdot$ Transformer $\cdot$ Detection $\cdot$
Extraction

##  1 Introduction

the first direct detection of GWs coming from coalescing binary black holes
(BBHs) was made by the LIGO/Virgo Collaboration [1], which verifies Einstein’s
General Relativity. As detectors become more sensitive, more and more GW
events are discovered and enable a new era of multi-messenger astronomy. A
total of 93 events have been reported in the three observations [2]. GWs have
become a new probe allowing cross-validation with a variety of fundamental
physical theories [3, 4, 5, 6].

Ground-based GW detectors such as LIGO, Virgo, and KAGRA cannot detect GWs at
lower frequencies due to seismic noise [7], therefore space-based detectors
are being developed. Laser Interferometer Space Antenna (LISA) will be
launched around 2034 [8], and Taiji [9] and TianQin [10] are also in progress.
LISA is expected to observe a variety of GW sources [11], including Galactic
binaries (GB), massive black hole binaries (MBHB), and extreme-mass-ratio-
inspirals (EMRI). The most common GB sources are binary white dwarf (BWD)
systems, which will populate the whole frequency band of the LISA detector.
Massive black holes (MBH) exist in most galactic centers, and the MBHs merge
along with the galaxies, which happens regularly in the Universe [12]. EMRI
system formed when the MBH capture compact objects (CO) surrounding them.
Unlike stars, COs can avoid tidal disruption and approach the central MBH,
radiating a significant amount of energy in GWs at low frequencies. Beyond
these resolvable sources, a huge number of unresolvable events will sum up
incoherently, forming a stochastic GW background (SGWB). The detection of
those GWs in the LISA mission enables us to gain a better understanding of
black holes and galaxies [13].

GW data processing is complicated due to the the overwhelming non-stationary
and non-Gaussian noise[14, 15], and sudden temporary glitches [14, 15, 16].
Earlier GW detection methods are divided into two categories:  a) theoretical
template-based algorithms like matched filtering,  b) template-independent
algorithms.  In principle, the most accurate results can be achieved by using
a matched filtering algorithm to detect signals buried in Gaussian noise [17].
This is currently the most widely used algorithm in the detection of GWs. The
additional complexity of space-based detection over ground-based detection can
be attributed to the different sources of various types. The optimal template
for matched filtering would have to include all the GW sources in the data.
However this is not practical because of the high parameter space dimension to
be explored. Moreover, the typical duration of the signal detected by LISA is
longer than that by ground-based detectors, resulting in even a larger
computational effort for the matched filtering algorithm. For ground-based
detection, a series of template-independent signal extraction and detection
algorithms have been developed, such as CWb [18], and BayesWave [19], both
based on the wavelet transformation. Ref. [20] propose a total-variation-based
method, and a novel approach based on the Hilbert-Huang transform was
developed recently [21]. The advantage of these algorithms is that they are
not limited by theoretical template banks and can extract the signal from
noisy data. The disadvantage of these algorithms is that it is only available
for burst signals, which is not suitable for space-based GW signals.

Deep learning has already been successful in various GW data analysis, such as
signal detection [22, 23, 24, 25, 26, 27, 28], parameter estimation [29, 30],
glitch classification [31, 32, 33], noise reduction [15, 16], and signal
extraction [34, 35, 36, 37]. Several deep neural network techniques have been
used in signal extraction, including dictionary learning [34], WaveNet [35],
denoising autencoder [36], and LSTM [37]. Most of these existing methods for
detection[22, 24, 25, 26] are designed for ground-based GWs, and for space-
based signal, [28] and [38] only achieve single source detection of EMRI and
MBHB signals, respectively. However, there is no unified treatment for all
sources of the space-based GW signal detection and extraction method.

In this article, we develop a uniformed deep learning-based model for space-
based GW signal detection and extraction for the four main GW sources of LISA.
Our model is based on a self-attention sequence-to-sequence architecture that
performs well when dealing with time series data. We have integrated long-term
and short-term feature extraction blocks in our model to catch the dependency
of GW signal in high-dimensional latent space. To our knowledge, this is the
first study to achieve near-perfect detection accuracy and high precision
extraction for all potential GW signals from space-based detection. The
model’s intermediate results can be interpreted as the encoded signal
waveform, revealing a strong correlation between what needs to be learned and
what has been learned by the neural network. In our test results analysis, we
obtained average overlaps (see equation (14)) of 95% of our test samples are
greater than $0.95$. It takes less than $10^{-2}$ second to perform extraction
and detection, which is a huge improvement over traditional approaches.
Finally, our method can also achieve considerable extraction effects for some
signals generated by other models that are not in the training dataset,
demonstrating the strong generalization ability of the model.

##  2 Results

###  2.1 Deep neural network

We extend the mask-based prediction method for speech enhancement in Conv-
TasNet[39] with a self-attention-based network to our task. As shown in Fig.
1, the network consists of four processing stages, encoder, extraction net,
decoder and classifier. First, the encoder maps the signal from detector to a
high-dimensional representation and splits it into short chunks. This encoded
representation is used to estimate a mask network for signal extraction. The
extraction network produces a mask matrix with Transformer blocks catching
both short-term and long-term dependency from chunks. The decoder uses a
transposed convolution layer for the element-wise multiplying of the mask
matrix and encoded representation to reconstruct the extracted signal.
Finally, the extracted signal is sent to a multi-layer linear perception
classifier for detection. The classifier gives the predicted probability that
the input data contains a true GW signal.

Figure 1: The overall architecture of our Transformer based deep neural
network, includes a convolutional network-based encoder and decoder, a
Transformer-based extraction network, and a multi-layer perception-based
classifier.

###  2.2 Generate space-based GWs dataset

Due to the large differences in signals of GWs from different sources, we
generate training and testing datasets for each source. we choose a universal
sampling rate of $0.1{\rm Hz}$ for all datasets. The parameter space of signal
generation is sampled with a uniform grid. Then, the parameters on each grid
point are used to generate the corresponding GW signals. For noise generation,
we use the noise power spectral density (PSD) of LISA [40] to simulate
Gaussian noise. To simulate the different signal-to-noise ratio (SNR) levels
in signal extraction, we set the SNR equal to 50, 40, and 30 following
equation (12).

The SGWB data is directly generated by the PSD given by equation (4), in which
the parameter $\alpha$ characterizes the amplitude of the SGWB signal. We set
$\alpha$ equal to -11.35, -11.55, and -11.75 to generate the signal in the
test datasets. To train the model for the GW detection task, we generate that
half of the samples contain signals and half are pure noise. Fig. 2 shows some
sample cases for our generated data. Specific details for each dataset will be
presented in Method.

(a)

(b)

(c)

(d)

(e)

Figure 2: Training data samples of each GW source in the frequency domain and
time domain. (a), an example of EMRI, MBHB, and BWD signal power spectra along
with SGWB PSD and LISA sensitivity curve in the frequency domain. (b)-(e),
each sub-figure shows an example of a GW signal from a specific source. Here
we plot the whitened $h_{I}(t)$ (orange) and $h_{I}(t)$ with noise (purple).
The signal-to-noise ratio (SNR) of EMRI, MBHB, and BWD signals is $50$, and
the SGWB signal has $n_{t}=2/3$ and $\alpha=-11.35$ (see equation (4)).

###  2.3 Interpretability of the network

(a)

(b)

(c)

(d)

Figure 3: The interpretability of the deep neural network is demonstrated by
the intermediate results, where the corresponding salient features represent
the physical signal in the time domain. Each sub-figure shows the attention
map (top panel), the average attention received by each token from all tokens
(middle panel), and the corresponding embedded signal contained in the input
data (bottom panel).

To better understand what information the self-attention-based neural network
learned, we explore the corresponding relation between the attention mechanism
and the embedded input signal. We calculate the attention maps of various
layers of the network. The attention map presents average output of attention
heads in each layer, between each pairs of tokens. The output of each
attention head is a weighted sum of the embedded tokens of signal, which we
will give the detail definition in Methods.

The core of our network is the extraction net, consisting of several STTBs
(Short-Time Transformer Block) and LTTBs (Long-Time Transformer Block) stacked
together. Both LTTB and STTB contain multi-head attention layers, which enable
our model to have universal learning capability for different GW sources. In
STTB, attention is only calculated between tokens in the same chunk, which
indicates that all tokens exchange information within the corresponding chunk.
The attention map of STTB is a diagonal line consisting of squares in chunk
size, implying STTBs are only interested in local structures. And the top
panel of each sub-figure in Fig. 3 shows the attention maps between all
embedded tokens in LTTB. We can see that for different GW sources, the model
weights show different patterns. We sum each column of the attention map to
obtain the attention received by each token, which called attention weights
(middle panel). As seen for the EMRI, MBHB and BWD model, the attention
weights and the signal after sliding average follow the same pattern. We also
applied the ADF test to the summed attention map matrix for both the BWD and
SGWB models owing to the stationary nature of the signals in the data. These
results were -6.84 and -15.33, respectively, indicating that the attention
maps are also stationary. This means that our LTTB can learn the global
dependency of the data. The above experiments show that our self-attention-
based model has the ability to learn both local and global structures for
different physical scenarios.

###  2.4 GW signal extraction

The most straightforward test is to use our model to extract a piece of data
with Gaussian noise and see whether it is able to extract the signal. In Fig.
4 we show some examples of the signal extraction effect of our model for
different types of GW signals. Three subplots a,b, and c show the signal
extraction effect of our model for EMRI, MBHB, and BWD signals, and the
overlap between the model output and the template is calculated. The overlap
shows a very good signal extraction effect. Since SGWB does not have a
specific waveform, we do not do this test here for the corresponding model.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 4: The signal extraction examples and the overlapping (between the
target and extracted signals) distributions for different GW sources. (a) and
(b), EMRI. (c) and (d), MBHB. (e) and (f), BWD. The extracted signal is
compared with whitened templates. Only the middle part of the BWD waveform is
presented to show the details of the waveform. The overlap between extracted
data and waveform templates is shown on the top. The high values indicate the
strong performance of our method on signal extraction for different GW
sources. Test on different signal SNR also shows our models’ generalization
ability of our method.

Then, we perform some statistical tests on the model. We generate the test
data in the same way as we generate the training dataset. We use a coarser
grid and create test datasets of $10,000$ samples containing signals with SNR
equal to 30, 40, and 50 respectively for each type of GW. In the left column
of Fig. 4, the signal extraction effect of the three models for each of the
three types of GWs is depicted. The showcase is selected according to the 15
percent quantile of the testing overlap of each GW source. In the right
column, it can be seen that for the case of ${\rm SNR}=50$, the signal
extraction effect is almost perfect for MBHB signals, the overlap is higher
than $0.99$ for all test samples. The overlap for the BWD signal is greater
than $0.97$ for $95\%$ of the test samples. Although it is slightly less
effective in extracting the EMRI signal due to its complexity, $92\%$ of
samples have overlap greater than $0.95$.

###  2.5 GW signal detection

The signals extracted by the neural network are used to build up four new
datasets to test the ability of our model to detect GW signals. These four
test datasets each included 10,000 samples containing signals and 10,000
samples containing pure noise. Here we utilize the detection rate, another
term of true positive rate (TPR), to measure the probability of correctly
identifying signals. For each signal type and a SNR of (30, 40, 50) and a
false alarm rate at 1%, the detection rates are (98.20%, 99.70%, 99.71%) for
EMRI, (99.99%, 100%, 100%) for MBHB, (99.37%, 99.97%, 99.98%) for BWD,
respectively. For SGWB signals with $\alpha$ of (-11.75, -11.55, -11.35) and a
false alarm rate at 1%, the detection rates are (95.05%, 99.97%, 100%),
respectively. To quantify the performance of our model in detection tasks, we
use the receiver operating characteristic (ROC) curve as shown in Fig. 5. In
ROC analysis, the true positive rate (TPR) and the false positive rate (FPR)
are plotted as the probability threshold for classify a candidate as positive
(i.e. signal) is altered. The area under the ROC curve (AUC) has been used to
evaluate the classifier performance, which is a single scalar value between 0
and 1. In general, the higher the AUC, the better the classifier. We calculate
AUC using Scikit-Learn library [41]. For all 4 signal types, the AUC are all
close to $1$, which indicates a fairly high sensitivity for classifying
(signal detection).

(a)

(b)

(c)

(d)

Figure 5: The signal detection performance from our classification
perspective. (a)-(c), each sub-figure shows the ROC curve of a model aimed at
detecting EMRI, MBHB, and BWD signals with test data SNR equal to 30, 40, and
50 respectively. (d), the sub-figure shows the ROC curve of a model aimed at
detecting SGWB with 3 different amplitudes. The near-perfect ROC performance
indicates the strong ability of our method for multiple GW source detection.

###  2.6 Model generalization behavior

Next, we evaluate the generalization ability of the model. Results is shown in
Fig. 6. The parameter space of our EMRI training dataset is only 4-dimensional
with a fixed initial semi-latus rectum $p_{0}=20M$. Fig. 6(a) shows the result
of testing our model using a signal with $p_{0}=30M$, which indicates a strong
generalization capability beyond the training parameter space. As mentioned
before, the training dataset of MBHB only contains GW signals from spin-
aligned MBHB systems with quasi-circular orbits, without considering the case
of orbital eccentricity. Here we generate a GW signal with initial orbital
eccentricity $e_{0}=0.5$ using the SEOBNRE model [42]. In Fig. 6(b), we can
see the extracted effect, which demonstrates that our model has good
generalizability. To test the generalization ability of the BWD model, our
test signal is generated using an evolving BWD system that considers mass
transfer, tidal forces, and gravitational radiation effects [43]. The
extracted result is shown in Fig. 6(c). The output detection probabilities of
these three test signals are all equal to $1$. The final test case is the SGWB
model, here we consider broken power law signals follow equation (5) with
parameters $\alpha=-11.18$, $n_{t1}=2/3$, $n_{t2}=-1/3$, and $f_{T}=0.002$.
This spectral shape might arise from the combination of two physically
distinct sources. The classification test obtaining ${\rm AUC}=0.99997$.

Then we evaluate our models’ generalization ability to weaker signals. For
EMRI, MBHB, and BWD models, we test them using data with lower SNR, for the
SGWB model, we use test data with smaller amplitude. The histogram of overlap
between extracted signal and template is shown in Fig. 4, at the same time the
ROC curve of signal detection results is shown in Fig. 5. Throughout this
series of tests, we have demonstrated the generalizability of our models in a
wide variety of scenarios.

From an astrophysical perspective, first, LISA will observe MBHBs with very
high SNR, typically bigger than 100, out to very high redshift [11]. Then for
EMRI signals, due to its physical complexity, the detection SNR threshold is
$\sim 20$ [11]. Next, for the LISA verification binaries almost half of them
reach a ${\rm SNR}\geq 30$ [44]. Finally, the upper limit of the SGWB is
$\Omega_{GW}(25{\rm Hz})\leq 3.4\times 10^{-9}$ for the case of $n_{t}=2/3$,
corresponding to $\alpha=-11.74$ [45]. In summary, the generalization ability
of the model shows the potential of our model to be applied in practical
situations.

(a)

(b)

(c)

Figure 6: Showcases of the generalization behaviour of our method for EMRI,
MBHB and BWD as shown in sub-figure (a), (b) and (c), respectively. The
extracted waveform is compared with whitened templates. Only the middle part
of the BWD waveform is presented to show the details of the waveform. The
overlap between extracted data and waveform templates is shown on the top.

##  3 Discussion

With the results presented above, we show the efficacy of our Transformer-
based deep neural network for space-based GW signal extraction and detection
of multiple sources. Our method is reusable for either simulated or future
observational data and gives an almost real-time analysis with low
computational cost.

One potential limitation of our method is the generation behavior. We show the
generalization performance for multiple signals outside the training dataset,
those cases are still simpler than realistic astrophysical conditions. For
example, the time delay interferometry technique is normally required to
suppress the laser frequency noise in the space-based GW data analysis, which
introduces further complexity in the detector response to the GW signal.
Therefore, from a deep learning perspective, the patterns in the time domain
might be different and a re-training of our neural network is required.

As the first proof-of-principle study of a comprehensive space-based GW signal
detection and extraction, the primary target of our neural network is not to
replace the conventional match filtering, but is more about giving a efficient
way to pre-process the potential large amount data from the space-based
detectors. Our results demonstrate the potential of DNNs for GW data analysis
in a more automated and real-time fashion.

##  4 Method

###  4.1 GW sources in space-based detection

As mentioned in the section 1, space-based GW detectors are being developed to
detect GW signals at frequencies of $[10^{-4},0.1]{\rm Hz}$. The main GW
sources in this frequency band are EMRI, MBHB, BWD, and SGWB. Next, we will
describe the details of the signals that come from each GW source.

#### EMRI

The MBHs in the centers of galaxies are typically surrounded by clusters of
stars. These stars eventually evolved into COs, which will be a black holes,
neutron stars or white dwarfs. Some of those COs can get captured onto orbits
bound to the central MBH and then gradually inspiral into the MBH via emission
of GWs. Typically the ratio of the mass of the CO that is falling into the MBH
to the mass of the MBH is $\sim 10^{-5}$, so these events are called EMRIs.

EMRI signal waveforms are characterized by the complex time domain strain
$h(t)$, in the source frame $h(t)$ is given by [46]:

$h(t)=\frac{\mu}{d_{L}}\sum_{lmkn}A_{lmkn}(t)S_{lmkn}(t,\theta)e^{im\phi}e^{-i\Phi_{mkn}(t)},$
(1)

where $\mu$ is the mass of the small black hole, $t$ is the time of arrival of
the GW, $\theta$ is the source-frame polar angle, $\phi$ is the source-frame
azimuthal angle, $d_{L}$ is the luminosity distance, and {l; m; k; n} are the
indices describing harmonic modes. The indices l, m, k, and n label the
orbital angular momentum, azimuthal, polar, and radial modes, respectively.
$\Phi_{mkn}=m\Phi_{\varphi}+k\Phi_{\theta}+n\Phi_{r}$ is the summation of
phases for each mode. $A_{lmkn}$ is the amplitude of GW. $S_{lmkn}(t,\theta)$
is spin-weighted spheroidal harmonic function.

In the detector frame the EMRI signal waveform is determined by 17 parameters:
$\\{M,\mu,a,{\vec{a}}_{2},p_{0},e_{0},x_{I,0},d_{L},\theta_{S},\phi_{S},\theta_{K},\phi_{K},\Phi_{\varphi,0},\Phi_{\theta,0},\Phi_{r,0}\\}$.
$M$ is the mass of the MBH, $\theta_{S}$, and $\phi_{S}$ are the polar and
azimuthal sky location angles. $\theta_{K}$ and $\phi_{K}$ are the azimuthal
and polar angles describing the orientation of the spin angular momentum
vector of the MBH. ${\vec{a}}_{2}$ is the spin vector of the CO, which doesn’t
considered in the waveform model. $p$ is semi-latus rectum, $e$ is
eccentricity, $I$ is orbit’s inclination angle from the equatorial plane, and
$x_{I}\equiv\cos I$.

#### MBHB

Most galaxies appear to host black holes at their centers. Galaxies and MBHBs
coevolved during the evolution of the Universe. So the observation of GWs from
the MBHB system can improve our understanding of important astronomical
phenomena such as the formation of the MBH and the merging of galaxies [11].

In this paper we just consider the GW from spin-aligned MBHB system, which
characterized by $\\{M_{tot},q,s^{z}_{1},s^{z}_{2}\\}$, where
$M_{tot}=m_{1}+m_{2}$, $m_{1}$ and $m_{2}$ are the mass of two black holes
respectively. $q=m_{2}/m_{1}$ ($m_{1}>m_{2}$) is the mass ratio. $s^{z}_{1}$
and $s^{z}_{2}$ are spin parameters of two black holes, and $z$ represent the
direction of orbital angular momentum.

#### BWD

The Milky Way contains a large population of compact binaries, most of which
are BWD with a period of $\sim 1hour$. This is right in the LISA’s sensitive
frequency band. The signal of BWD in the source frame is quite simple:

$\displaystyle h_{+}(t)$
$\displaystyle=\mathcal{A}(1+\cos^{2}\iota)\cos\Phi(t),$ (2) $\displaystyle
h_{\times}(t)$ $\displaystyle=2\mathcal{A}\sin\iota\sin\Phi(t),$
$\displaystyle\Phi(t)$ $\displaystyle=\phi_{0}+2\pi ft+\pi\dot{f}t^{2}.$

$\mathcal{A}$ is the overall amplitude, $\phi_{0}$ is the initial phase at the
start of the observation, $\iota$ is the inclination of the BWD orbit to the
line of sight from the origin of the Solar System Barycentric (SSB) frame. The
intrinsic parameter is the frequency of the signal $f$ and its derivative
$\dot{f}$.

Frequency evolves slowly and some binaries will be chirping to higher
frequency due to the decay of the orbit through the emission of GWs, but other
binaries will be moving to lower frequency as a result of mass transfer
between the binary components driving an increase in the orbital separation
[11].

#### SGWB

There are many resolvable sources, but there is also a large number of events
that cannot be resolved individually, resulting in SGWB. Astrophysical
background components are guaranteed in the LISA band, originating from
unresolved Galactic Binaries (GB) and stellar-originated black hole mergers.
SGWBs that are Gaussian, isotropic, and stationary can be fully described by
their spectrum [47]:

$\Omega_{GW}(f)=\frac{f}{\rho_{c}}\frac{{\rm d}\,\rho_{GW}}{{\rm d}\,f},$ (3)

where $\rho_{GW}$ is the energy density of gravitational radiation contained
in the frequency range $f$ to $f+df$, $\rho_{c}=\frac{3H_{0}^{2}c^{2}}{8\pi
G}$ is the critical density of the universe, where $c$ is the speed of light,
and $G$ is Newton’s constant, $H_{0}=67.9{\rm km\ s^{-1\ }Mpc^{-1}}$ is the
Hubble constant.

We intend to follow the simplified assumption that the signal can be well
described by a power law, defined as amplitude and slope, as most studies have
done previously. Then the signal is described by [48]:

$h^{2}\Omega_{GW}(f)=10^{\alpha}\left(\frac{f}{f_{*}}\right)^{n_{t}},$ (4)

where $h$ is dimensionless Hubble constant, $f_{*}$ is pivot frequency,
$\alpha$ characterize its amplitude at $f_{*}$ and $n_{t}$ is the slope of
spectrum.

An other formalism used to test our model’s generalization ability is broken
power law which defined:

$h^{2}\Omega_{GW}(f)=10^{\alpha}\left[H(f_{T}-f)\left(\frac{f}{f_{*}}\right)^{n_{t1}}+H(f-f_{T})\left(\frac{f_{T}}{f_{*}}\right)^{n_{t1}}\left(\frac{f}{f_{T}}\right)^{n_{t2}}\right],$
(5)

where $n_{t1}$ and $n_{t2}$ are the slopes of two spectrum segments, $H(f)$ is
Heaviside step function.

###  4.2 Data Curation

First, we simulate the noise data from the LISA sensitivity curve:

$S_{n}(f)=\frac{1}{L^{2}\mathcal{R}(f)}\left(P_{\rm
OMS}+2(1+\cos^{2}(f/f_{*}))\frac{P_{\rm acc}}{(2\pi f)^{4}}\right),$ (6)

where

$\displaystyle P_{\rm OMS}$ $\displaystyle=(1.5\times 10^{-11}\,{\rm
m})^{2}\left(1+\left(\frac{2\,{\rm mHz}}{f}\right)^{4}\right)\,{\rm Hz^{-1}}$
(7)

and

$\displaystyle P_{\rm acc}$ $\displaystyle=(3\times 10^{-15}\,{\rm
m})^{2}\left(1+\left(\frac{0.4\,{\rm
mHz}}{f}\right)^{2}\right)\left(1+\left(\frac{f}{8\,{\rm
mHz}}\right)^{4}\right)\,{\rm Hz^{-1}}$ (8)

are the optical metrology noise and acceleration noise respectively.
$L=2.5\times 10^{9}\,{\rm m}$, $f_{*}=19.09\,{\rm mHz}$, $\mathcal{R}(f)$ is
the transfer function. The full expression for $\mathcal{R}(f)$ used here is
computed numerically [40]. Next we specify the generation of each dataset.

We use the augmented analytic kludge (AAK) [49, 50, 46] model to generate the
EMRI signal. It is because the AAK model combines the accuracy of the
numerical kludge (NK) model and the computational speed of the analytic kludge
(AK) model quite well. Note that the parameter $\iota$ in the AAK model is a
orbital parameter: $\cos\iota=L_{z}/\sqrt{L_{z}^{2}+Q}$, where $Q$ is Carter
constant, $L_{z}$ is $z$ component of the specific angular momentum. For
simplicity, we just consider a small parameter space to generate the training
data. Detailed parameters range is shown in the Table 1, and the lower bound
of $a$ and $e_{0}$ is limited by the FastEMRIWaveform toolkit we used [51].

Table 1: Summary of parameter setups in EMRI signal simulation. Parameter | Lower bound | Upper bound
---|---|---
$M$ | $10^{5}M_{\odot}$ | $10^{7}M_{\odot}$
$a$ | $10^{-3}$ | $0.99$
$e_{0}$ | $10^{-3}$ | $0.5$
$\iota$ | $0$ | $\pi$

For MBHB signal generation, we used SEOBNRv4_opt, which is a version of the
SEOBNRv4 code [52] with significant optimizations. It could produce signals
for a high spin, high mass ratio MBHB system. We adopted the log-uniform
distribution for the parameter $M_{tot}$ from Ref. [53]. Detailed parameters
range is shown in Table 2.

Table 2: Summary of parameter setups in MBHB signal simulation. Parameter | Lower bound | Upper bound
---|---|---
$M_{tot}$ | $10^{6}M_{\odot}$ | $10^{8}M_{\odot}$
$q$ | $0.01$ | $1$
$s_{1}^{z}$ | $-0.99$ | $0.99$
$s_{2}^{z}$ | $-0.99$ | $0.99$

For the BWD dataset, we generate the signal directly using equations (2). We
follow the parameter setting of the LISA Data Challenge (LDC) 1-4 dataset
[54], but we focus only on the intrinsic parameters $f$ and $\dot{f}$, see
Table 3 for details.

Table 3: Summary of parameter setups in BWD signal simulation. Parameter | $f$ | $\dot{f}$
---|---|---
range-1 | $[0.1,4]{\rm mHz}$ | $[-3\times 10^{-17},6\times 10^{-16}]{\rm Hz^{2}}$
range-2 | $[4,15]{\rm mHz}$ | $[-3\times 10^{-15},4\times 10^{-14}]{\rm Hz^{2}}$

After the generation of signal, we project it to LISA detector as follows:

$\displaystyle
h_{I,II}(t)=\frac{\sqrt{3}}{2}\left(h_{+}(t)F^{+}_{I,II}(\theta_{S},\phi_{S},\psi_{S})\right.$
(9)
$\displaystyle\left.+h_{\times}(t)F^{\times}_{I,II}(\theta_{S},\phi_{S},\psi_{S})\right),$

where $F^{+}_{I,II}$ and $F^{\times}_{I,II}$ are the antenna pattern
functions:

$\displaystyle
F^{+}_{I}(\theta_{S},\phi_{S},\psi_{S})=\frac{1}{2}(1+\cos^{2}\theta_{S})(\cos
2\phi_{S}\cos 2\psi_{S}$ (10) $\displaystyle-\cos\theta_{S}\sin 2\phi_{S}\sin
2\psi_{S}),$ $\displaystyle
F^{\times}_{I}(\theta_{S},\phi_{S},\psi_{S})=\frac{1}{2}(1+\cos^{2}\theta_{S})(\cos
2\phi_{S}\sin 2\psi_{S}$ $\displaystyle+\cos\theta_{S}\sin 2\phi_{S}\cos
2\psi_{S}),$

and

$\displaystyle
F^{+}_{II}(\theta_{S},\phi_{S},\psi_{S})=F^{+}_{I}(\theta_{S},\phi_{S},\psi_{S}-\pi/4),$
(11) $\displaystyle
F^{\times}_{II}(\theta_{S},\phi_{S},\psi_{S})=F^{\times}_{I}(\theta_{S},\phi_{S},\psi_{S}-\pi/4),$

where $\psi_{S}$ is the polarisation angle. Furthermore, this antenna pattern
function varies with time as a result of the motion of the LISA detector. In
this paper, we only use data from the channel $I$ for extraction and
classification.

For those 3 datasets, we inject the signal to the noise with specific optimal
SNR as:

$\displaystyle{\rm SNR}=\left(s\mid s\right)^{-1/2},$ (12)

where $s$ represent the signal template, the inner product $(h\mid s)$ is:

$(h\mid
s)=2\int_{f_{min}}^{f_{max}}\frac{\tilde{h}^{*}(f)\tilde{s}(f)+\tilde{h}(f)\tilde{s}^{*}(f)}{S_{n}(f)}\,df,$
(13)

where $f_{min}=3\times 10^{-5}{\rm Hz}$ and $f_{max}=0.05{\rm Hz}$.
$\tilde{h}(f)$ and $\tilde{s}(f)$ are frequency domain signals and the
superscript $*$ means complex conjugate, $S_{n}(f)$ is the noise PSD. Here
following the setting of the LDC dataset, we set the SNR of the training data
to $50$. Then the data was whitened and normalized to $[-1,1]$. During the
whitening procedure, we applied the Tukey window with $\alpha=1/8$.

This inner product can also be used to measure how well the output of our
model matches the signal waveform template, we calculate the overlap between
them, which defined as:

$\displaystyle\mathcal{O}(h,s)=\left(\hat{h}\mid\hat{s}\right)$ (14)

with

$\displaystyle\hat{h}=h\left(h\mid h\right)^{-1/2},$ (15)

where $h$ represent the model output and $s$ represent the template.

The SGWB dataset is generated in a very different way than several other
datasets, We just need to simulate SGWB data based on its PSD which is defined
by:

$\displaystyle S_{h}(f)=\frac{3H_{0}}{4\pi^{2}}\frac{\Omega_{GW}(f)}{f^{3}}.$
(16)

We choose $n_{t}=2/3$ $\alpha=-11.35$ and $f_{*}=10^{-3}{\rm Hz}$ (see
equation (4)) according to LDC configuration and Ref. [48] , which represent
SGWB formed by compact binary coalescences. We could generate the SGWB signal
by $S_{h}(f)$ directly. We then combine the signal and noise and perform the
whitening and normalization operations as described above. Lastly, we present
the pure signals with noise PSD and SGWB PSD in Fourier domain and the data
generated to train our model in Fig. 2.

###  4.3 Transformer

Transformer[55] is a kind of deep neural network (DNN) proposed for machine
translation, and it soon achieved superior performance in various tasks in
natural language processing[56] and computer vision[57]. Based entirely on
attention, Transformer has a great ability to capture both long-range and
short-range dependency in sequence data. In this section, we introduce key
structures of a general Transformer.

#### Self-Attention

Self-Attention can be described as a function with an input vector Query and
an output vector pair Key-Value. The Key-Value pair is a weighted sum, which
returns the information of the Query with the corresponding Key. In the
Transformer network block, all query vectors and key-values pairs have the
same dimension $d$. Given a sequence of queries $Q\in\mathbb{R}^{l\times d}$
with length $l$, keys $K\in\mathbb{R}^{l\times d}$ and values
$V\in\mathbb{R}^{l\times d}$, the Transformer compute scaled dot-product
attention as:

$\displaystyle{\rm Attention}{(Q,K,V)}={\rm
softmax}(\frac{QK^{T}}{\sqrt{d}})V.$ (17)

#### Multi-Head Attention

Instead of applying a single attention function with $d_{model}$-dimensional
queries, keys, and values. The Transformer uses multi-head attention to
combine information from different linear projections of original queries,
keys, and values. The structure of multi-head attention shows in Fig.1b.

If the Transformer has $H$ heads, the sequence of attention output $head_{i}$
is:

$\displaystyle head_{i}={\rm Attention}{(Q_{i},K_{i},V_{i})}={\rm
softmax}(\frac{Q_{i}K_{i}^{T}}{\sqrt{d}})V_{i},$ (18)

where $Q_{i}=QW_{i}^{Q}$, $K_{i}=KW_{i}^{K}$, $V_{i}=VW_{i}^{V}$, are
projected queries, keys and values, corresponding to head $i\in\\{1,...,H\\}$
with learning parameters $W_{i}^{Q},W_{i}^{K},W_{i}^{V}$, respectively. Here,

$\displaystyle A=\frac{1}{H}\sum_{i=1}^{H}{\rm
softmax}(\frac{Q_{i}K_{i}^{T}}{\sqrt{d}})$ (19)

is also called attention map. $A\in\mathbb{R}^{l\times l}$, in which each
element $A_{qk}$ indicates how much attention token $q$ puts on token $k$.
With collection of all parameters $W^{H}\in\mathbb{R}^{Hd_{model}\times d}$,
the multi-head attention (MHA) of these H heads can be written as:

$\displaystyle{\rm MHA}{(Q,K,V)}={\rm Concat}(head_{1},...,head_{H})W^{H}.$
(20)

The multi-head attention mechanism can be computed in parallel for each head,
which leads to high efficiency. Moreover, multi-head attention connects the
information from different projection subspaces directly, helping the
Transformer learn the long-term dependencies of the input sequence easier.

#### Feed Forward and Residual Connection

In addition to attention layers, Transformer block have a fully connected feed
forward network that operates separately and identically on each position:

$\displaystyle{\rm FFN}(H^{\prime})={\rm
ReLU}(H^{\prime}{W}_{1}+b_{1})W_{2}+b_{2},$ (21)

where $H^{\prime}$ is the output of last layer, and $W_{1},b_{1},W_{2},b_{2}$
are trainable parameters. The dimension of input and output is equal to
model’s dimension $d_{model}$, and the inner-layer dimension $d_{ffn}$ should
be larger than $d_{model}$. In a deeper Transformer model, a residual
connection module is inserted followed by a Layer Normalization Module. The
output of the Transformer block can be written as:

$\displaystyle H^{\prime}={\rm LayerNorm}({\rm Attention}(X)+X),$ (22)
$\displaystyle H={\rm LayerNorm}({\rm FFN}(H^{\prime})+H^{\prime}).$ (23)

###  4.4 Network structure

Let the T-length observation $x\in\mathbb{R}^{T}$ be a time-domain signal we
receive. $x$ is a mixture of a target GW signal $s$ and noise $n$ as $x=s+n$,
where the noise is from the environment and instruments. Our goal is to
recover $s$ from $x$. The recovered signal $\hat{s}\in\mathbb{R}^{T}$ can be
written as:

$\displaystyle\hat{s}={\rm Dec}({\rm Enc}(x)\otimes\mathcal{M}).$ (24)

The decoder reconstructs the signal with encoded input $x$ element-wise
multiplication by the mask $\mathcal{M}$ predicted by the masking net. After
recovering the signals, we add a multi-layer linear perception to detect
whether it is a GW signal or a pure noise.

#### Encoder and Decoder

We use a CNN as an encoder because it can extract local features from a long
time-domain sequence, which could compress information. With time-domain input
$x\in\mathbb{R}^{T}$, the encoded ${\rm Enc}(x)\in\mathbb{R}^{L\times
T^{\prime}}$. Since the output estimated signal has the same length as input
$x\in\mathbb{R}^{T}$, the decoder for reconstruction uses a transposed
convolution layer.

#### Masking Extraction Net

The masking network is built by following a basic structure in SepFormer[58].
We employ two Transformer blocks the STTB (Short-Time Transformer Block) and
the LTTB (Long-Time Transformer Block) in the masking net. The masking network
is fed by the encoded input. We split the input signal into chunks and
concatenated them to be a tensor $g\in\mathbb{R}^{L\times C\times N}$, where
$C$ is the length of each chunk and $N$ is the number of chunks.

The tensor $g$ is processed by Transformer blocks. The STTB computes the
multi-head attention in each chunk respectively, which catches the short-time
dependency in the chunk. Then the LTTB focuses on another dimension of tensor
$g$, modeling the long-time dependency by the attention across chunks. The
output from Transformer blocks passes through a PReLU non-linearity layer and
a 2-D convolution layer for matching the output size of the decoder. Then a
two-path convolution with different linear functions is used to get the mask.

#### Multi-Layer Perception

The extracted signal recovered by the decoder feeds to the MLP for detection.
We use two fully connected layers to classify GW signals and noise. The first
linear layer has 512 dimension, and the second linear layer outputs the vector
to a probability of true GW signal.

###  4.5 Loss function

Our loss function combines both the extraction loss and the detection loss.
The extraction loss is based on the scale invariant signal-to-distortion
ratio[59] in audio enhancement, which is defined as:

$\displaystyle s_{target}$ $\displaystyle=\frac{\hat{x}^{T}x}{\|x\|^{2}},$
(25) $\displaystyle\mathcal{L}_{EX}$
$\displaystyle=10\log_{10}\frac{s_{target}}{\hat{x}-s_{target}},$

where $\hat{x}$ is the estimated output and the $x$ is the target.

The detection loss is the binary cross-entropy. Suppose that the data set has
$N$ samples with label $y$, and the $\hat{y}$ is the predicted probability of
the sample. The BCE loss is defined as:

$\displaystyle\mathcal{L}_{DE}=-\frac{1}{N}\sum_{i=1}^{N}y_{i}\log(\hat{y_{i}})+(1-y_{i})\log(1-\hat{y_{i}}).$
(26)

Therefore, the total loss of the deep neural network is:

$\displaystyle\mathcal{L}=\mathcal{L}_{EX}+\mathcal{L}_{DE}.$ (27)

###  4.6 Implementation details

Our extraction network repeats both STTB and LTTB twice ($M=2$), with 4
parallel attention heads and a 512-dimensional feed-forward layer in each
block. We set the length of split chunks $C=25$. In the training stage, the
model is trained for 100 epochs. We set initial learning rate as
${lr}_{max}=1e^{-3}$. After 35 epochs, the learning rate is annealed by
halving if the validation performance does not improve for two generations.
Adam[60] is used as the optimizer with $\beta_{1}=0.9,\beta_{2}=0.98$. The
network is trained on a single NVIDIA V100 GPU. All of our code is implemented
in Python within the SpeechBrain [61] toolkit.

##  5 Acknowledgements

The research was supported by the Peng Cheng Laboratory Cloud Brain and by
grant No.PCL2021A13. This work was also supported in part by the National Key
Research and Development Program of China Grant No. 2021YFC2203001 and in part
by the NSFC (No. 11920101003 and No. 12021003). Z.C was supported by “the
Interdiscipline Research Funds of Beijing Normal University" and CAS Project
for Young Scientists in Basic Research YSBR-006.

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On affine Toda field theories related to ${\bf D}_{r}$
algebras and their real Hamiltonian forms

Vladimir S. Gerdjikova,b,111E-mail<EMAIL_ADDRESS>and Georgi G.
Grahovskic,222E-mail<EMAIL_ADDRESS>
a Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 8
Acad. G. Bonchev str., 1113 Sofia, Bulgaria
bInstitute for Advanced Physical Studies, 111 Tsarigradsko chaussee, Sofia
1784, Bulgaria
c Department of Mathematical Sciences, University of Essex, Wivenhoe Park,
Colchester CO4 3SQ, UK

To the memory of our colleague and
friend A. B. Yanovski (1953–2023)

###### Abstract

The paper deals with affine 2-dimensional Toda field theories related to
simple Lie algebras of the classical series ${\bf D}_{r}$. We demonstrate that
the complexification procedure followed by a restriction to a specified real
Hamiltonian form commutes with the external automorphisms of $\mathfrak{g}$.
This is illustrated on the examples ${\bf D}_{r+1}^{(1)}\to{\bf B}_{r}^{(1)}$
and ${\bf D}_{4}^{(1)}\to{\bf G}_{2}^{(1)}$ using external automorphisms of
the corresponding extended Dynkin diagrams.

##  1 Introduction

It is well known that with each simple Lie algebra $\mathfrak{g}$ of rank $r$
one can relate a 2-dimensional affine Toda field theory (ATFT) [43, 8, 9, 45,
46, 3, 4]. Their Lagrangian densities are given by:

${\cal L}[{\bf q}]={1\over 2}\left((\partial_{x_{0}}{\bf
q}\cdot\partial^{x_{0}}{\bf q})-(\partial_{x_{1}}{\bf
q}\cdot\partial^{x_{1}}{\bf q}\right)-\sum_{k=0}^{n}n_{k}\left({\rm
e}^{-2(\alpha_{k}\cdot{\bf q})}-1\right),$ (1)

where the field ${\bf q}(x,t)$ is an $r$-dimensional vector. The equations of
motion are therefore:

${\partial^{2}{\bf q}\over\partial x\partial
t}=\sum_{j=0}^{r}n_{j}\alpha_{j}e^{-2(\alpha_{j},{\bf q}(x,t))},$ (2)

where $t=\frac{1}{2}(x_{1}-x_{0})$ and $x=\frac{1}{2}(x_{1}+x_{0})$. The
important results that mattered for the derivation of ATFT were: 1) the work
of A. Zamolodchikov [51] on deformation of conformal field theories preserving
integrability and 2) the discovery of Mikhailov’s reduction group which
allowed one to derive their Lax representation [42, 43]. Here $r$ is the rank
of ${\mathfrak{g}}$, $\alpha_{k}$’s ($k=1,\dots,n$) are the simple roots of
${\mathfrak{g}}$ and $\alpha_{0}$ is the minimal root.

The notion of real Hamiltonian forms was introduced in [28] and used to study
reductions of ATFTs in [23, 24, 27]. Real Hamiltonian forms (RHF) are another
type of “reductions” of Hamiltonian systems. First one complexifies the ATFT
and then applies an involution $\mathcal{C}$ which is compatible with the
Poisson structure of the initial system. This is similar to the obtaining a
real forms of a semi-simple Lie algebra. that is why the RHF have indefinite
kinetic energy quadratic form.

The structure of the paper is as follows. Section 2 contains some facts about
the ATFT and simple Lie algebras. Section 3 describes the complexification
procedure and Section 4 specifies the RHF of ATFT for which the involution is
the external automorphism of $\mathfrak{g}$. Thus we demonstrate that the
complexification procedure commutes with the external automorphisms of
$\mathfrak{g}$. This is illustrated on the examples ${\bf
D}_{r+1}^{(1)}\to{\bf B}_{r}^{(1)}$ and ${\bf D}_{4}^{(1)}\to{\bf
G}_{2}^{(1)}$.

##  2 Preliminaries

The Lax representations of the ATFT models widely discussed in the literature
(see e.g. [42, 43, 45, 40, 41] and the references therein). Most of the
results on it are related mostly to the normal real form of the Lie algebra
$\mathfrak{g}$, see [38, 7, 10, 39, 48, 49, 22]. Below we use the following
Lax pair:

$\displaystyle L\psi\equiv\left(i{d\over dx}-iq_{x}(x,t)-\lambda
J_{0}\right)\psi(x,t,\lambda)=0,$ (3) $\displaystyle M\psi\equiv\left(i{d\over
dt}-{1\over\lambda}I(x,t)\right)\psi(x,t,\lambda)=0,$ (4)

whose potentials take values in $\mathfrak{g}$. Here $q(x,t)\in\mathfrak{h}$
\- the Cartan subalgebra of $\mathfrak{g}$, and
${\boldsymbol{q}}(x,t)=(q_{1},\dots,q_{r})$ is its dual $r$-component vector.
The potentials of the Lax operators take the form

$J_{0}=\sum_{\alpha\in\pi}E_{\alpha},\qquad
I(x,t)=\sum_{\alpha\in\pi}e^{-(\alpha,{\boldsymbol{q}}(x,t))}E_{-\alpha}.$ (5)

Here $\pi_{\mathfrak{g}}$ stands for the set of admissible roots of
$\mathfrak{g}$, i.e.
$\pi_{\mathfrak{g}}=\\{\alpha_{0},\alpha_{1},\dots,\alpha_{r}\\}$, with
$\alpha_{1},\dots,\alpha_{r}$ being the simple roots of $\mathfrak{g}$ and
$\alpha_{0}$ being the minimal root of $\mathfrak{g}$. The Dynkin graph
corresponding to the set of admissible roots
$\pi_{\mathfrak{g}}=\\{\alpha_{0},\alpha_{1},\dots,\alpha_{r}\\}$ of
${\mathfrak{g}}$ is called extended Dynkin diagram (EDD). The equations of
motion are of the form (2) where $n_{j}$ are the minimal positive integer
coefficients $n_{k}$ that provide the decomposition of $\alpha_{0}$ over the
simple roots of $\mathfrak{g}$:

$-\alpha_{0}=\sum_{k=1}^{r}n_{k}\alpha_{k}.$ (6)

It is well known that ATFT models are an infinite-dimensional Hamiltonian
system. The (canonical) Hamiltonian structure is given by:

$\displaystyle H_{\mathfrak{g}}$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}dx\,\mathcal{H}_{\mathfrak{g}}(x,t),\qquad\mathcal{H}_{\mathfrak{g}}(x,t)={1\over
2}({\bf p}(x,t),{\bf p}(x,t))+\sum_{k=0}^{r}n_{k}(e^{-({\bf
q}(x,t),\alpha_{k})}-1),$ (7) $\displaystyle\Omega_{\mathfrak{g}}$
$\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}dx\,\omega_{\mathfrak{g}}(x,t),\qquad\omega_{\mathfrak{g}}(x,t)=(\delta{\bf
p}(x,t)\wedge\delta{\bf q}(x,t)),$ (8)

where $H_{\mathfrak{g}}$ is the canonical Hamilton function and
$\Omega_{\mathfrak{g}}$ is the canonical symplectic structure. Here also ${\bf
p}=d{\bf q}/dx$ are the canonical momenta and coordinates satisfying canonical
Poisson brackets:

$\\{q_{k}(x,t),p_{j}(y,t)\\}=\delta_{jk}\delta(x-y).$ (9)

The infinite-dimensional phase space $\mathcal{M}=\\{{\bf q}(x,t),{\bf
p}(x,t)\\}$ is spanned by the canonical coordinates and momenta.

######  Remark 1.

In what follows we shall need the coefficients $n_{k}$ for the ${\bf
D}_{r+1}^{(1)}$ and ${\bf B}_{r}^{(1)}$ algebras, which are equal to [38, 7,
10]:

$\displaystyle{\bf D}_{r+1}^{(1)}:$ $\displaystyle\qquad n_{1}$
$\displaystyle=1,\;n_{2}=n_{3}=\cdots=n_{r-1}=2,$ $\displaystyle\quad n_{r}$
$\displaystyle=n_{r+1}=1,$ (10) $\displaystyle{\bf B}_{r}^{(1)}:$
$\displaystyle\qquad n_{1}$
$\displaystyle=1,\;n_{2}=n_{3}=\cdots=n_{r-1}=n_{r}=2.$

##  3 Complexification of ATFT

The starting point in the construction of real Hamiltonian forms (RHF) is the
complexification of the field functions ${\bf q}(x,t)$ and ${\bf p}(x,t)$
involved in the Hamiltonian (7):

${\bf q}^{\mathbb{C}}={\bf q}^{0}+i{\bf q}^{1},\qquad{\bf p}^{\mathbb{C}}={\bf
p}^{0}+i{\bf p}^{1}.$

Next we introduce an involution $\mathcal{C}$ acting on the phase space
$\mathcal{M}\equiv\\{q_{k}(x),p_{k}(x)\\}_{k=1}^{n}$ as follows:

$\displaystyle\mbox{1)}\qquad\mathcal{C}(F(p_{k},q_{k}))=F(\mathcal{C}(p_{k}),\mathcal{C}(q_{k})),$
$\displaystyle\mbox{2)}\qquad\mathcal{C}\left(\\{F(p_{k},q_{k}),G(p_{k},q_{k})\\}\right)=\left\\{\mathcal{C}(F),\mathcal{C}(G)\right\\},$
(11) $\displaystyle\mbox{3)}\qquad\mathcal{C}(H(p_{k},q_{k}))=H(p_{k},q_{k}).$

Here $F(p_{k},q_{k})$, $G(p_{k},q_{k})$ and the Hamiltonian $H(p_{k},q_{k})$
are functionals on $\mathcal{M}$ depending analytically on the fields
$q_{k}(x,t)$ and $p_{k}(x,t)$.

The complexification of the ATFT is rather straightforward. The resulting
complex ATFT (CATFT) can be written down as standard Hamiltonian system with
twice as many fields ${\bf q}^{a}(x,t)$, ${\bf p}^{a}(x,t)$, $a=0,1$:

${\bf p}^{\mathbb{C}}(x,t)={\bf p}^{0}(x,t)+i{\bf p}^{1}(x,t),\qquad{\bf
q}^{\mathbb{C}}(x,t)={\bf q}^{0}(x,t)+i{\bf q}^{1}(x,t),$ (12)
$\\{{q}_{k}^{0}(x,t),{p}_{j}^{0}(y,t)\\}=-\\{{q}_{k}^{1}(x,t),{p}_{j}^{1}(y,t)\\}=\delta_{kj}\delta(x-y).$
(13)

The densities of the corresponding Hamiltonian and symplectic form equal

$\displaystyle\mathcal{H}_{\rm ATFT}^{\mathbb{C}}$ $\displaystyle\equiv$
$\displaystyle\mathrm{Re\,}\mathcal{H}_{\rm ATFT}({\bf p}^{0}+i{\bf
p}^{1},{\bf q}^{0}+i{\bf q}^{1})$ (14) $\displaystyle=$ $\displaystyle{1\over
2}({\bf p}^{0},{\bf p}^{0})-{1\over 2}({\bf p}^{1},{\bf
p}^{1})+\sum_{k=1}^{r}n_{k}e^{-({\bf q}^{0},\alpha_{k})}\cos(({\bf
q}^{1},\alpha_{k})),$ $\displaystyle\omega^{\mathbb{C}}$ $\displaystyle=$
$\displaystyle(d{\bf p}^{0}\wedge id{\bf q}^{0})-(d{\bf p}^{1}\wedge d{\bf
q}^{1}).$ (15)

The family of RHF then are obtained from the CATFT by imposing an invariance
condition with respect to the involution
$\tilde{\mathcal{C}}\equiv\mathcal{C}\circ\ast$ where by $\ast$ we denote the
complex conjugation. The involution $\tilde{\mathcal{C}}$ splits the phase
space $\mathcal{M}^{\mathbb{C}}$ into a direct sum
$\mathcal{M}^{\mathbb{C}}\equiv{\cal
M}_{+}^{\mathbb{C}}\oplus\mathcal{M}_{-}^{\mathbb{C}}$ where

$\displaystyle\mathcal{M}_{+}^{\mathbb{C}}$
$\displaystyle=\mathcal{M}_{0}\oplus i\mathcal{M}_{1},$
$\displaystyle\quad\mathcal{M}_{-}^{\mathbb{C}}$
$\displaystyle=\mathcal{M}_{0}\oplus-i\mathcal{M}_{1},$ (16)
$\displaystyle\mathcal{C}({\boldsymbol{q}}^{+}+i{\boldsymbol{q}}^{-})$
$\displaystyle=({\boldsymbol{q}}^{+}-i{\boldsymbol{q}}^{-}),$
$\displaystyle\quad\mathcal{C}({\boldsymbol{p}}^{+}+i{\boldsymbol{p}}^{-})$
$\displaystyle=({\boldsymbol{p}}^{+}-i{\boldsymbol{p}}^{-}).$

Each involution $\mathcal{C}$ on ${\cal M}$ induces an involution
$\mathcal{C}^{\\#}$ on $\mathfrak{g}$. Thus to each involution $\mathcal{C}$
one can relate a RHF of the ATFT. Due to Property 3), $\mathcal{C}^{\\#}$
preserves the system of admissible roots of $\mathfrak{g}$ (and thus the
extended Dynkin diagrams of $\mathfrak{g}$).

Indeed, the condition 3) above requires that:

$(\mathcal{C}({\boldsymbol{q}}),\alpha)=({\boldsymbol{q}},\mathcal{C}^{\\#}(\alpha)),\qquad\alpha\in\pi_{\mathfrak{g}},$
(17)

and therefore we must have
$\mathcal{C}(\pi_{\mathfrak{g}})=\pi_{\mathfrak{g}}$.

The relation (17) defines uniquely the relation between $\mathcal{C}$ and
$\mathcal{C}^{\\#}$. Using $\mathcal{C}^{\\#}$ we can split the root space
${\mathbb{E}}^{n}$ into direct sum
${\mathbb{E}}^{n}={\mathbb{E}}_{+}\oplus{\mathbb{E}}_{-}$ of two
eigensubspaces of $\mathcal{C}^{\\#}$. Taking the average of the roots
$\alpha_{j}$ with respect to $\mathcal{C}^{\\#}$ we get:

$\beta_{j}={1\over 2}(\alpha_{j}+\mathcal{C}^{\\#}(\alpha_{j})),\qquad
j=0,\dots,n_{+}=\dim{\mathbb{E}}_{+}.$ (18)

By construction the set $\\{\beta_{0},\beta_{1},\dots,\beta_{n_{+}}\\}$ will
be a set of admissible roots for some Kac-Moody algebra with rank $n_{+}$.
Graphically each set of admissible roots can be represented by an extended
Dynkin diagrams. Therefore one can relate an automorphism $\mathcal{C}^{\\#}$
to each ${\mathbb{Z}}_{2}$ symmetry of the extended Dynkin diagram.

The splitting of ${\mathbb{E}}^{n}$ naturally leads to the splittings of the
fields:

$\displaystyle{\bf p}={\bf p}^{+}+{\bf p}^{-},\qquad{\bf q}={\bf q}^{+}+{\bf
q}^{-},$ (19)

where ${\bf p}^{+},{\bf q}^{+}\in{\mathbb{E}}_{+}$ and ${\bf p}^{-},{\bf
q}^{-}\in{\mathbb{E}}_{-}$. If we also introduce:

$\gamma_{j}={1\over 2}(\alpha_{j}-\mathcal{C}^{\\#}(\alpha_{j})),\qquad
j=0,\dots,n_{-}=\dim{\mathbb{E}}_{-}.$ (20)

The Hamiltonian along with the terms related to the simple roots, contains
also the minimal root $\alpha_{0}$ given in (6). The RHF of ATFT are more
general integrable systems than the models described in [14, 15, 16, 40, 41]
which involve only the fields ${\bf q}^{+}$, ${\bf p}^{+}$ invariant with
respect to $\mathcal{C}$.

##  4 RHF of ATFT from external automorphisms

Let us now outline the RHF obtained from the algebras ${\bf D}_{r+1}$ using
the external automorphisms. The set of admissible roots ${\bf D}_{r+1}$ are:

$\displaystyle\alpha_{1}$ $\displaystyle=$ $\displaystyle
e_{1}-e_{2},\quad\alpha_{2}=e_{2}-e_{3},\quad\dots,\quad\alpha_{r}=e_{r}-e_{r+1},\quad\alpha_{r+1}=e_{r}+e_{r+1},\quad\alpha_{0}=e_{1}+e_{2}.$

Here $\alpha_{1},\alpha_{2}\dots,\alpha_{r+1}$ form the set of simple roots of
${\bf D}_{r+1}^{(1)}$ and $\alpha_{0}$ is the minimal root of the algebra. The
extended Dynkin diagram of ${\bf D}_{r+1}^{(1)}$ is shown on the upper panel
of Figure 1. Here also $e_{k}$, $k=1,\dots,r+1$ are the basic vectors of the
dual Euclidean space ${\mathbb{E}}^{r+1}$, i.e. any vector ${\bf
q}\in{\mathbb{E}}^{r+1}$ can be written as

${\bf q}=\sum_{k=1}^{r+1}q_{k}e_{k}.$

The components $q_{k}$, $k=1,\dots r+1$ are the coordinates in the phase
space.

The fundamental weights of ${\bf D}_{r}^{(1)}$ are

$\displaystyle\omega_{k}$ $\displaystyle=$ $\displaystyle
e_{1}+\dots+e_{k},\qquad 1\leq k\leq r-1,$ $\displaystyle\omega_{r}$
$\displaystyle=$ $\displaystyle{1\over
2}(e_{1}+e_{2}+\dots+e_{r-1}+e_{r}-e_{r+1}),$ $\displaystyle\omega_{r+1}$
$\displaystyle=$ $\displaystyle{1\over
2}(e_{1}+e_{2}+\dots+e_{r-1}+e_{r}+e_{r+1}).$

The corresponding Hamiltonian has the form

$H_{{\bf D}_{r}^{(1)}}=\int_{-\infty}^{\infty}{\rm
d}x\,\left(\sum_{k=1}^{r}{p_{k}^{2}\over 2}+\sum_{k=1}^{r}n_{k}{\rm
e}^{2(q_{k}-q_{k-1})}+{\rm e}^{2(q_{r+1}-q_{r})}+{\rm
e}^{-2(q_{r+1}+q_{r})}+{\rm e}^{2(q_{1}+q_{2})}\right)$ (21)

$\dots$$\alpha_{1}$$\alpha_{0}$$\alpha_{2}$$\alpha_{3}$$\alpha_{r-2}$$\alpha_{r-1}$$\alpha_{r}$$\alpha_{r+1}$${\bf
D}_{r+1}^{(1)}:$${\cal C}^{\\#}$

$\dots$$\beta_{1}$$\beta_{0}$$\beta_{2}$$\beta_{3}$$\beta_{r-2}$$\beta_{r-1}$$\beta_{r}$${\bf
B}_{r}^{(1)}:$

Figure 1: Extended Dynkin diagrams of the complex untwisted affine Kac-Moody
algebras ${\bf D}_{r+1}^{(1)}$ and ${\bf B}_{r}^{(1)}$ (upper and lower panels
respectively).

###  4.1 From ${\bf D}_{r+1}^{(1)}$ to ${\bf B}_{r}^{(1)}$

Let us choose ${\mathfrak{g}}\simeq{\bf D}_{r+1}^{(1)}$ and fix up the
involution $\mathcal{C}$ acting on the phase space as follows:

$\displaystyle\mathcal{C}(q_{k})=q_{k},\quad\mathcal{C}(p_{k})=p_{k},\quad
k=1,\dots,r;\quad\mathcal{C}(q_{r+1})=-q_{r+1},\quad\mathcal{C}(p_{r+1})=-p_{r+1}.$

Then introducing on $\mathcal{M}_{\pm}$ new coordinates by

$\displaystyle q_{k}^{+}=q_{k},\qquad p_{k}^{+}=p_{k},\qquad
q_{r}^{-}=q_{r},\qquad p_{r}^{-}=p_{r},\qquad k=1,...,r;$ $\displaystyle
q_{r+1}^{-}=q_{r+1},\qquad p_{r+1}^{-}=p_{r+1},$

i.e. $\dim\mathcal{M}_{+}=2r$ and $\dim\mathcal{M}_{-}=2$.

This involution induces an involution $\mathcal{C}^{\\#}$ of the Kac-Moody
algebra ${\bf D}_{r+1}^{(1)}$ which acts on the root space as follows (see the
upper panel of Fig. 1):

$\displaystyle\mathcal{C}^{\\#}(e_{k})=e_{k},\qquad
k=1,\dots,r;\qquad\mathcal{C}^{\\#}(e_{r+1})=-e_{r+1},$
$\displaystyle\mathcal{C}^{\\#}(\alpha_{k})=\alpha_{k},\qquad\mathcal{C}^{\\#}(\alpha_{r+1})=\alpha_{r},\qquad\mathcal{C}^{\\#}(\alpha_{r})=\alpha_{r+1},$

The involution $\mathcal{C}^{\\#}$ splits the root space ${\mathbb{E}}^{r+1}$
into a direct sum of eigensubspaces:
${\mathbb{E}}^{r+1}={\mathbb{E}}_{+}\oplus{\mathbb{E}}_{-}$ with
$\dim{\mathbb{E}}_{+}=r$ and $\dim{\mathbb{E}}_{-}=1$. The restriction of
$\pi$ onto ${\mathbb{E}}_{+}$ leads to the admissible root system
$\pi^{\prime}=\\{\beta_{0},\dots,\beta_{r}\\}$ of ${\bf B}_{r}^{(1)}$:

$\beta_{k}=\alpha_{k},\qquad k=0,\dots,r-1;\qquad\beta_{r}={1\over
2}\left[\alpha_{r}+\mathcal{C}^{\\#}(\alpha_{r})\right],\qquad$

The subspace ${\mathbb{E}}_{-}$ is spanned by the only nontrivial vector

$\gamma_{r}={1\over
2}\left[\alpha_{r}-\mathcal{C}^{\\#}(\alpha_{r})\right]=-e_{r+1}.$

The reduced RHF is described by the densities $\mathcal{H}^{\mathbb{R}}$,
$\omega^{\mathbb{R}}$:

$\displaystyle\mathcal{H}^{\mathbb{R}}={1\over
2}\sum_{k=1}^{r}p_{k}^{+}{}^{2}-{1\over
2}p_{r+1}^{-}{}^{2}+\sum_{k=1}^{r}n_{k}e^{2(q_{k+1}^{+}-q_{k}^{+})}+e^{2q_{r+1}^{+}}\cos
q_{r+1}^{-},$ (22) $\displaystyle\omega^{\mathbb{R}}=\sum_{k=1}^{r}\delta
p_{k}^{+}\wedge\delta q_{k}^{+}-\delta p_{r+1}^{-}\wedge\delta q_{r+1}^{-}.$
(23)

The restriction on ${\mathbb{E}}_{+}$ (i.e. setting $q_{r+1}^{-}=0$ in (22))
gives the canonical Hamiltonian of ${\bf B}_{r}^{(1)}$ ATFT:

$H_{{\bf B}_{r}^{(1)}}=\int_{-\infty}^{\infty}{\rm
d}x\,\left(\sum_{k=1}^{r}{p_{k}^{2}\over 2}+\sum_{k=1}^{r-1}n_{k}{\rm
e}^{2(q_{k}-q_{k-1})}+2{\rm e}^{-2q_{r}}+{\rm e}^{2(q_{1}+q_{2})}\right),$
(24)

as expected.

$\alpha_{1}$$\alpha_{0}$$\alpha_{2}$$\alpha_{3}$$\alpha_{4}$(a)$\quad{\bf
D}_{4}^{(1)}\longrightarrow{\bf B}_{3}^{(1)}$:${\cal
C}^{\\#}$$\Longrightarrow$$\beta_{1}$$\beta_{0}$$\beta_{2}$$\beta_{3}$

$\alpha_{1}$$\alpha_{0}$$\alpha_{2}$$\alpha_{3}$$\alpha_{4}$(b)$\quad{\bf
D}_{4}^{(1)}\longrightarrow{\bf G}_{2}^{(1)}$:${\cal W}^{\\#}$${\cal
W}^{\\#}$${\cal
W}^{\\#}$$\qquad\Longrightarrow$$\beta_{0}$$\beta_{2}$$\beta_{1}$

Figure 2: Reductions of ${\bf D}_{4}^{(1)}$ affine Lie algebra: (a) ${\bf
D}_{4}^{(1)}\rightarrow{\bf B}_{3}^{(1)}$; (b) ${\bf
D}_{4}^{(1)}\rightarrow{\bf G}_{2}^{(1)}$.

###  4.2 From ${\bf D}_{4}^{(1)}$ to ${\bf B}_{3}^{(1)}$

If we take now a particular case od Example 1 with $r=3$, the Hamiltonian (21)
will reduce to the generic ATFT Hamiltonian related to ${\bf D}_{4}^{(1)}$:

$H_{{\bf D}_{4}^{(1)}}=\int_{-\infty}^{\infty}{\rm
d}x\,\left(\sum_{k=0}^{3}{p_{k}^{2}\over 2}++{\rm e}^{2(q_{2}-q_{1})}+2{\rm
e}^{2(q_{3}-q_{2})}+{\rm e}^{2(q_{4}-q_{3})}+{\rm e}^{-2(q_{3}+q_{4})}+{\rm
e}^{2(q_{1}+q_{2})}\right).$ (25)

The corresponding extended root system is

$\overline{\pi}\left({\bf
D}_{4}^{(1)}\right)=\\{e_{1}+e_{2},e_{1}-e_{2},e_{2}-e_{3},e_{3}-e_{4},e_{3}+e_{4}\\},$

and the extended Dynkin diagram is given on Figure 2(a). This algebra has 4
outer ${\mathbb{Z}}_{2}$ automorphisms and one ${\mathbb{Z}}_{2}$ automorphism
[7, 47]. Here we will take $C^{\\#}$ to be the outer
${\mathbb{Z}}_{2}$-automorphism, swapping $\alpha_{3}$ and $\alpha 4$ (See
Figure 2(a)). The action of $C^{\\#}$ on the root space ${\mathbb{E}}^{4}$ is
then given by:

$\displaystyle\mathcal{C}^{\\#}(e_{1})=e_{1},\qquad\mathcal{C}^{\\#}(e_{2})=e_{2},\qquad\mathcal{C}^{\\#}(e_{3})=e_{4},\qquad\mathcal{C}^{\\#}(e_{4})=e_{3};$
$\displaystyle\mathcal{C}^{\\#}(\alpha_{1})=\alpha_{1},\qquad\mathcal{C}^{\\#}(\alpha_{2})=\alpha_{2},\qquad\mathcal{C}^{\\#}(\alpha_{3})=\alpha_{4},\qquad\mathcal{C}^{\\#}(\alpha_{4})=\alpha_{3}.$

The involution $\mathcal{C}^{\\#}$ splits the root space ${\mathbb{E}}^{r+1}$
into a direct sum of eigensubspaces:
${\mathbb{E}}^{r+1}={\mathbb{E}}_{+}\oplus{\mathbb{E}}_{-}$ with
$\dim{\mathbb{E}}_{+}=3$ and $\dim{\mathbb{E}}_{-}=1$. The restriction of
$\pi$ onto ${\mathbb{E}}_{+}$ leads to the admissible root system
$\pi^{\prime}=\\{\beta_{0},\beta_{1},\beta_{2},\beta_{3}\\}$ of ${\bf
B}_{3}^{(1)}$:

$\beta_{0}=\alpha_{0},\qquad\beta_{1}=\alpha_{1},\qquad\beta_{2}=\alpha_{2},\qquad\beta_{3}={1\over
2}\left[\alpha_{3}+\mathcal{C}^{\\#}(\alpha_{3})\right]=e_{3}.$

The subspace ${\mathbb{E}}_{-}$ is spanned by the vector

$\gamma_{3}={1\over
2}\left[\alpha_{3}-\mathcal{C}^{\\#}(\alpha_{3})\right]=-e_{4}.$

As a result, the densities of the reduced RHF become:

$\displaystyle\mathcal{H}^{\mathbb{R}}={1\over
2}\left(p_{1}^{+}\right)^{2}+{1\over 2}\left(p_{2}^{+}\right)^{2}+{1\over
2}\left(p_{3}^{+}\right)^{2}-{1\over 2}\left(p_{4}^{-}\right)^{2}+{\rm
e}^{2(q_{2}^{+}-q_{1}^{+})}+2{\rm e}^{2(q_{3}^{+}-q_{2}^{+})}+2{\rm
e}^{-2q_{3}^{+}}\cos q_{4}^{-},$ (26)
$\displaystyle\omega^{\mathbb{R}}=\sum_{k=1}^{r}\delta p_{k}^{+}\wedge\delta
q_{k}^{+}-\delta p_{r+1}^{-}\wedge\delta q_{r+1}^{-}.$ (27)

The restriction on ${\mathbb{E}}_{+}$ (i.e. setting $p_{4}^{-}=q_{4}^{-}=0$ in
(26)) gives the canonical Hamiltonian of ${\bf B}_{3}^{(1)}$ ATFT:

$H_{{\bf B}_{3}^{(1)}}=\int_{-\infty}^{\infty}{\rm d}x\,\left({1\over
2}\sum_{j=1}^{3}\left(p_{j}^{+}\right)^{2}+{\rm
e}^{2(q_{2}^{+}-q_{1}^{+})}+2{\rm e}^{2(q_{3}^{+}-q_{2}^{+})}+2{\rm
e}^{-2q_{3}^{+}}\right).$ (28)

###  4.3 From ${\bf D}_{4}^{(1)}$ to ${\bf G}_{2}^{(1)}$

If we take again the ${\bf D}_{4}^{(1)}$ Hamiltonian (25) and use the
${\mathbb{Z}}_{3}$ outer automorphism of ${\bf D}_{4}^{(1)}$ sketched on
Figure 2(b), then we will get the induced action ${\cal W}^{\\#}$ on the root
space ${\mathbb{E}}^{4}$:

$\begin{split}{\cal
W}^{\\#}(e_{1})&=\frac{1}{2}(e_{1}+e_{2}+e_{3}-e_{4}),\quad{\cal
W}^{\\#}(e_{2})=\frac{1}{2}(e_{1}+e_{2}-e_{3}+e_{4}),\\\ {\cal
W}^{\\#}(e_{3})&=\frac{1}{2}(e_{1}-e_{2}+e_{3}+e_{4}),\quad{\cal
W}^{\\#}(e_{4})=\frac{1}{2}(e_{1}-e_{2}-e_{3}-e_{4}),\end{split}$ (29)

i.e.

$\begin{split}{\cal W}^{\\#}(e_{1}+e_{2})=e_{1}+e_{2},\qquad{\cal
W}^{\\#}(e_{2}-e_{3})=e_{2}-e_{3},\qquad{\cal
W}^{\\#}(e_{1}-e_{2})=e_{3}-e_{4},\\\ {\cal
W}^{\\#}(e_{3}-e_{4})=e_{3}+e_{4},\qquad{\cal
W}^{\\#}(e_{3}+e_{4})=e_{1}-e_{2},\end{split}$ (30)

and

$\displaystyle\beta_{0}$ $\displaystyle=$
$\displaystyle-(e_{1}+e_{2}),\qquad\beta_{2}=(e_{2}-e_{3}),$
$\displaystyle\beta_{1}$ $\displaystyle=$ $\displaystyle{1\over
3}\left[\alpha_{1}+{\cal W}^{\\#}(\alpha_{1})+{\cal
W}^{\\#}{}^{2}(\alpha_{1})\right]={1\over 3}(e_{1}-e_{2}+2e_{3}).$

This is the extended root system of ${\bf G}_{2}^{(1)}$ (see Figure 2(b)).
This results to a RHF related to ${\bf G}_{2}^{(1)}$ with Hamiltonian given by

$H_{{\bf G}_{2}^{(1)}}=\int_{-\infty}^{\infty}{\rm d}x\,\left({1\over
2}\left(p_{1}^{+}\right)^{2}+{1\over 2}\left(p_{2}^{+}\right)^{2}+{1\over
2}\left(p_{3}^{+}\right)^{2}+{\rm e}^{-2(\beta_{0},{\bf q})}+2{\rm
e}^{-2(\beta_{1},{\bf q})}+3{\rm e}^{-2(\beta_{2},{\bf q})}\right),$

where the three component vectors ${\bf q}$ and ${\bf p}$ are restricted by
$q_{1}+q_{2}+q_{3}=0$ and $p_{1}+p_{2}+p_{3}=0$.

The above results establish the following

######  Proposition 1.

Let us consider the RHF of ATFT using the external automorphisms of the
algebras ${\bf D}^{(1)}_{r+1}$. If we use the second order external
automorphisms, then

$\begin{split}\mathcal{H}^{\mathbb{R}}_{{\bf D}^{(1)}_{r+1}}=\mathcal{H}_{{\bf
B}^{(1)}_{r}},\qquad\mathcal{H}^{\mathbb{R}}_{{\bf
D}^{(1)}_{4}}=\mathcal{H}_{{\bf B}^{(1)}_{3}}.\end{split}$ (31)

If we use the third order external automorphism of $D_{4}$, then we obtain

$\begin{split}\mathcal{H}^{\mathbb{R}}_{{\bf D}^{(1)}_{4}}=\mathcal{H}_{{\bf
G}^{(1)}_{2}}.\end{split}$ (32)

The next step would be to analyze the more general situation when the RHF is
generated by $\mathcal{C}_{0}\circ\mathcal{C}^{\\#}$ where $\mathcal{C}_{0}$
is a generic suitable automorphism and $\mathcal{C}^{\\#}$ is an external
automorphism. Our hypothesis is that the above Proposition will hold true also
for this more general case.

##  5 Conclusions

We presented here real Hamiltonian forms of affine Toda field theories related
to the untwisted complex Kac-Moody algebra ${\bf D}_{4}^{(1)}$. We outlined
the construction of the RHF and studied ${\mathbb{Z}}_{2}$ and
${\mathbb{Z}}_{3}$ symmetries of the extended Dynkin diagrams. Thus resulted
in reductions to ${\bf B}_{3}^{(1)}$ and ${\bf G}_{2}^{(1)}$.

The spectral properties of the Lax operators of the real Hamiltonian forms of
ATFTs can be studied in the frame of the ISM [17, 44]. This will lead to the
construction of Jost solutions and scattering data for Lax operators with
complex-valued Cartan elements [5, 6, 30]. The continuous spectrum of the Lax
operators will consist of $2h$ rays intersecting at the origin and closing
angles $\pi/h$.

The interpretation of the ISM as a generalized Fourier transforms [2, 18, 30]
allows one to study all the fundamental properties of the corresponding
nonlinear evolutionary equations (NLEE’s): i) the description of the class of
NLEE related to a given Lax operator $L(\lambda)$ and solvable by the ISM; ii)
derivation of the infinite family of integrals of motion; iii) their hierarchy
of Hamiltonian structures [29]; and iv) description of the gauge equivalent
systems [25, 26, 34, 35, 36, 37].

Some additional problems are natural extensions to the results presented here:

  * •

The complete classification of all nonequivalent RHF of ATFT.

  * •

The description of the hierarchy of Hamiltonian structures of RHF of ATFT (for
a review of the infinite-dimensional cases see e.g. [13, 20] and the
references therein) and the classical $r$-matrix. It is also an open problem
to construct the RHF for ATFT using some of its higher Hamiltonian structures.

  * •

The extension of the dressing Zakharov-Shabat method [50] to the above classes
of Lax operators is also an open problem. One of the difficulties is due to
the fact that the $\mathbb{Z}_{h}$ reductions requires dressing factors with
$2h$ pole singularities [19, 21].

  * •

Another open problem is to study types of boundary conditions and boundary
effects of ATFT’s and their RHF [11, 12].

The last and more challenging problem is to prove the complete integrability
of all these models. The ideas of [2, 29] about the interpretation of the
inverse scattering method as a generalized Fourier transform holds true also
for the $\mathbb{Z}_{h}$ reduces Lax operators [30, 31, 32, 33]. This may
allow one to derive the action-angle variables for these classes of NLEE.

## Acknowledgements

One of us (V.G.) is grateful to professor P. Bamidis for useful discussions
and for his hospitality at Aristotle University, Thessaloniki, Greece, for
common participation under the Operational Programme ”Science and Education
for Smart Growth”, project BG05M2OP001-2.016-0025. This work is supported by
the Bulgarian National Science Fund, grant KP-06N42-2.

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