# Training Robust Spiking Neural Networks with ViewPoint Transform and
SpatioTemporal Stretching

###### Abstract

Neuromorphic vision sensors (event cameras) simulate biological visual
perception systems and have the advantages of high temporal resolution, less
data redundancy, low power consumption, and large dynamic range. Since both
events and spikes are modeled from neural signals, event cameras are
inherently suitable for spiking neural networks (SNNs), which are considered
promising models for artificial intelligence (AI) and theoretical
neuroscience. However, the unconventional visual signals of these cameras pose
a great challenge to the robustness of spiking neural networks. In this paper,
we propose a novel data augmentation method, ViewPoint Transform and
SpatioTemporal Stretching (VPT-STS). It improves the robustness of SNNs by
transforming the rotation centers and angles in the spatiotemporal domain to
generate samples from different viewpoints. Furthermore, we introduce the
spatiotemporal stretching to avoid potential information loss in viewpoint
transformation. Extensive experiments on prevailing neuromorphic datasets
demonstrate that VPT-STS is broadly effective on multi-event representations
and significantly outperforms pure spatial geometric transformations. Notably,
the SNNs model with VPT-STS achieves a state-of-the-art accuracy of 84.4% on
the DVS-CIFAR10 dataset.

Index Terms—  Spiking Neural Networks, Neuromorphic Data, Data Augmentation,
ViewPoint Transform and SpatioTemporal Stretching

##  1 Introduction

Inspired by the primate visual system, neuromorphic vision cameras generate
events by sampling the brightness of objects. For example, the Dynamic Vision
Sensor (DVS) [1] camera and the Vidar [2] camera are inspired by the outer
three-layer structure of the retina and the foveal three-layer structure,
respectively. Both of them have the advantages of high temporal resolution,
less data redundancy, low power consumption, and large dynamic range [3]. In
addition, spiking neural networks (SNNs) are similarly inspired by the
learning mechanisms of the mammalian brain and are considered a promising
model for artificial intelligence (AI) and theoretical neuroscience [4]. In
theory, as the third generation of neural networks, SNNs are computationally
more powerful than traditional convolutional neural networks (CNNs) [4].
Therefore, event cameras are inherently suitable for SNNs.

However, the unconventional visual signals of these cameras also pose a great
challenge to the robustness of SNNs. Most existing data augmentations are
fundamentally designed for RGB data and lack exploration of neuromorphic
events. For example, Cutout [5] artificially impedes a rectangular block in
the image to simulate the impact of occlusion on the image. Random erasing [6]
further optimizes the erased pixel value by adding noise. Mixup [7] uses the
weighted sum of two images as training samples to smooth the transition line
between classes. Since neuromorphic data have an additional temporal dimension
and differ widely in imaging principles, novel data augmentations are required
to process the spatiotemporal visual signals of these cameras.

In this paper, we propose a novel data augmentation method suitable for
events, ViewPoint Transformation and SpatioTemporal Stretching (VPT-STS).
Viewpoint transformation solves the spatiotemporal scale mismatch of samples
by introducing a balance coefficient, and generates samples from different
viewpoints by transforming the rotation centers and angles in the
spatiotemporal domain. Furthermore, we introduce spatiotemporal stretching to
avoid potential information loss in viewpoint transformation. Extensive
experiments are performed on prevailing neuromorphic datasets. It turns out
that VPT-STS is broadly effective on multiple event representations and
significantly outperforms pure spatial geometric transformations. Insightful
analysis shows that VPT-STS improves the robustness of SNNs against different
spatial locations. In particular, the SNNs model with VPT-STS achieves a
state-of-the-art accuracy of 84.4% on the DVS-CIFAR10 dataset.

Furthermore, while this work is related to EventDrop [8], NDA [9], there are
some notable differences. For example, NDA is a pure global geometric
transformation, while VPT-STS changes the viewpoint of samples in the
spatiotemporal domain. EventDrop is only experimented on CNNs, it introduces
noise by dropping events, but may cause problems with dead neurons on SNNs.
VPT-STS is applicable to both CNNs and SNNs, maintaining the continuity of
samples. In addition, EventDrop transforms both temporal and spatial domains,
but as two independent strategies, it does not combine the spatiotemporal
information of the samples. To our knowledge, VPT-STS is the first event data
augmentation that simultaneously incorporates spatiotemporal transformations.

##  2 Method

###  2.1 Event Generation Model

The event generation model [3, 4] is abstracted from dynamic vision sensors
[1]. Each pixel of the event camera responds to changes in its logarithmic
photocurrent $L=\log(I)$. Specifically, in a noise-free scenario, an event
$e_{k}=(x_{k},y_{k},t_{k},p_{k})$ is triggered at pixel $X_{k}=(y_{k},x_{k})$
and at time $t_{k}$ as soon as the brightness variation $|\Delta L|$ reaches a
temporal contrast threshold $C$ since the last event at the pixel. The event
generation model can be expressed by the following formula:

$\displaystyle\Delta L(X_{k},t_{k})=L(X_{k},t_{k})-L(X_{k},t_{k}-\Delta
t_{k})=p_{k}C$ (1)

where $C>0$, $\Delta t_{k}$ is the time elapsed since the last event at the
same pixel, and the polarity $p_{k}\in\\{+1,-1\\}$ is the sign of the
brightness change. During a period, the event camera triggers event stream
$\mathcal{E}$:

$\mathcal{E}=\left\\{e_{k}\right\\}_{k=1}^{N}=\left\\{\left(X_{k},t_{k},p_{k}\right)\right\\}_{k=1}^{N}$
(2)

where $N$ represents the number of events in the set $\mathcal{E}$.

As shown in Figure 1, an event is generated each time the brightness variances
reach the threshold, and then $|\Delta L|$ is cleared. The event stream can be
represented as a matrix:

$\displaystyle M_{\varepsilon}=\begin{pmatrix}y_{1}&x_{1}&t_{1}&1\\\
\vdots&\vdots&\vdots&\vdots\\\ y_{N}&x_{N}&t_{N}&1\\\ \end{pmatrix}_{4\times
N}$ (3)

For convenience, we omit the unconverted polarity $p$.

###  2.2 Motivation

This work stems from the observation that it is difficult to maintain absolute
frontal view between the sample and cameras, which easily leads to a slight
shift of the viewpoint. Considering this small offset distance, we use
viewpoint rotation to approximate the deformation of samples in space and
time. In addition, since events record the brightness change of samples,
especially changes of the edge, variations of the illumination angle will also
cause the effect of viewpoint transformation, which suggests that we can
enhance the robustness of SNNs by generating viewpoint-transformed samples.

Fig. 1: Event generation model.

###  2.3 The Proposed Method.

To generate viewpoint-transformed samples, we draw on the idea of spatio-
temporal rotation. For viewpoint transformation (VPT), we introduce
translation matrices $T_{b}$, $T_{a}$, which represent the translation to the
rotation center $(x_{c},y_{c},t_{c})$ and the translation back to the original
position, respectively.

$\begin{aligned} T_{b}=\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&1&0\\\
-y_{c}&-x_{c}&-t_{c}&1\\\ \end{pmatrix}\end{aligned},\begin{aligned}
T_{a}=\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&1&0\\\ y_{c}&x_{c}&t_{c}&1\\\
\end{pmatrix}\end{aligned}$ (4)

Suppose that rotate along the $y$ and $t$ planes with $x$ as the axis, we can
easily derive the rotation matrix $R_{r}^{YT}$:

$\displaystyle R_{r}^{YT}=\begin{pmatrix}cos\theta&0&sin\theta&0\\\ 0&1&0&0\\\
-sin\theta&0&cos\theta&0\\\ 0&0&0&1\\\ \end{pmatrix}$ (5)

where $\theta$ is the rotation angle. In practice, Eq 5 is an unbalanced
matrix due to the mismatch between the time and space dimensions in the
$M_{\varepsilon}$ matrix. Therefore, we introduce a balance coefficient $\tau$
to scale the space and time dimension, which results in a better visual
effects. The balanced matrix $R_{br}^{YT}$ can be formulated as:

$\displaystyle R_{br}^{YT}=\begin{pmatrix}cos\theta&0&\tau sin\theta&0\\\
0&1&0&0\\\ -\frac{1}{\tau}sin\theta&0&cos\theta&0\\\ 0&0&0&1\\\ \end{pmatrix}$
(6)

Table 1: Performance of VPT-STS on SNNs and CNNs with various representations. Datasets | Method | Accuracy (%)
---|---|---
SNNs | EventFrame | EventCount | VoxelGrid | EST
CIFAR10-DVS | Baseline | 83.20 | 78.71 | 78.85 | 77.47 | 78.81
VPT-STS | 84.40 | 79.58 | 79.12 | 79.62 | 79.37
N-Caltech101 | Baseline | 78.98 | 73.08 | 73.66 | 77.08 | 78.41
VPT-STS | 81.05 | 76.96 | 76.38 | 79.13 | 78.88
N-CARS | Baseline | 95.40 | 94.44 | 94.76 | 93.86 | 94.97
VPT-STS | 95.85 | 94.60 | 94.81 | 94.30 | 94.99

Set $x_{c}=0$, the viewpoint transformation matrix $M_{br}^{YT}$ can be
formulated by calculating $T_{b}R_{br}^{YT}T_{a}$:

$\displaystyle\begin{pmatrix}cos\theta&0&\tau sin\theta&0\\\ 0&1&0&0\\\
-\frac{1}{\tau}sin\theta&0&cos\theta&0\\\
-x_{c}cos\theta+\frac{1}{\tau}t_{c}sin\theta+x_{c}&0&-\tau x_{c}sin\theta-
t_{c}cos\theta+t_{c}&1\\\ \end{pmatrix}$ (7)

Similarly, the viewpoint transformation matrix $M_{br}^{XT}$ in the $x$ and
$t$ dimensions can be formulated as:

$\displaystyle\begin{pmatrix}1&0&0&0\\\ 0&cos\theta&\tau sin\theta&0\\\
0&-\frac{1}{\tau}sin\theta&cos\theta&0\\\
0&-x_{c}cos\theta+\frac{1}{\tau}t_{c}sin\theta+x_{c}&-\tau x_{c}sin\theta-
t_{c}cos\theta+t_{c}&1\\\ \end{pmatrix}$ (8)

Therefore, the viewpoint-transformed matrix $M_{VPT}^{YT}$ and $M_{VPT}^{XT}$
can be formulated as:

$\left.\begin{aligned} M_{VPT}^{YT}=M_{\varepsilon}M_{br}^{YT}&\\\
M_{VPT}^{XT}=M_{\varepsilon}M_{br}^{XT}&\end{aligned}\right\\}$ (9)

Furthermore, since events beyond the resolution will be discarded during the
viewpoint transformation, we introduce spatiotemporal stretching (STS) to
avoid potential information loss. STS stretches the temporal mapping in the
VPT by a coefficient $\frac{1}{cos\theta}$ while maintaining the spatial
coordinates unchanged. Therefore, by setting $t_{c}=0$, we get the transformed
$(t)_{STS}^{YT}$ and $(t)_{STS}^{XT}$ from Eq. 7 and Eq. 8:

$\displaystyle\left.\begin{aligned} (t_{k})_{VPT}^{YT}=(t_{k})-\tau
tan\theta\cdot((y_{k})-y_{c})&\\\ (t_{k})_{VPT}^{XT}=(t_{k})-\tau
tan\theta\cdot((x_{k})-x_{c})&\end{aligned}\right\\}$ (10)

The time of STS is advanced or delayed according to the distance from the
center $|x-x_{c}|$ ($|y-y_{c}|$), causing event stream to be stretched long
the time axis according to the spatial coordinates.

##  3 Experiments

(a) Baseline accuracy.

(b) VPT-STS accuracy.

(c) Difference of accuracy reduction.

Fig. 2: Performance of VPT-STS and Baseline under different perturbations.

###  3.1 Implementation

Extensive experiments are performed to demonstrate the superiority of the VPT-
STS method on prevailing neuromorphic datasets, including CIFAR10-DVS(CIF-DVS)
[10], N-Caltech101(N-Cal) [11], N-CARS [12] datasets. N-Caltech101 and
CIFAR10-DVS datasets are generated by neuromorphic vision sensors on the basis
of traditional datasets, while N-CARS is collected in the real world. For the
convenience of comparison, the model without VPT-STS with the same parameters
is used as the baseline. STBP [13] methods are used to train SNN-VGG9 network,
other parameters mainly refer to NDA [14]. For example, the Adam optimizer is
used with an initial learning rate of $1e-3$. The neuron threshold and leakage
coefficient are $1$ and $0.5$, respectively. In addition, we also evaluate the
performance of VPT-STS on various event representations with the Resnet9
network, including EST [15], VoxelGrid [16], EventFrame [17] and EventCount
[18] representations.

Table 2: Performance of VPT-STS and previous SOTAs on CIFAR10-DVS and N-CARS datasets. Methods | References | Accuracy (%)
---|---|---
CIF-DVS | N-CARS
HATS[12] | CVPR 2018 | 52.40 | 81.0
Dart[19] | TPAMI 2020 | 65.80 | -
Dspike [20] | NeurIPS 2021 | 75.40 | -
STBP [13] | AAAI 2021 | 67.80 | -
AutoSNN [21] | ICML 2022 | 72.50 | -
RecDis [22] | CVPR 2022 | 72.42 | -
DSR [23] | CVPR 2022 | 77.27 | -
NDA [9] | ECCV 2022 | 81.70 | 90.1
VPT-STS | - | 84.40 | 95.85

###  3.2 Performance on various representations

Extensive experiments are conducted to evaluate the performance of VPT-STS
method on different event representations, covering SNNs and CNNs. As shown in
Tab. 1, SNNs with VPT-STS methods achieve significant improvements on three
prevailing datasets. And VPT-STS also performs well on four representations
commonly used by CNNs. It is worth noting that EST maintains the most
spatiotemporal information from neuromorphic data and thus performs best
overall. Furthermore, since the samples of N-CARS are collected in the real
world, its initial viewpoint diversity is already enriched compared to the
other two datasets. Considering the high baseline on N-CARS, VPT-STS still
further imporves the robustness of SNNs.

Table 3: Comparison of Different Strategies. Methods | Accuracy (%)
---|---
CIF-DVS | N-Cal | N-CARS
Baseline | 83.20 | 78.98 | 95.40
Rotation | 83.90 | 80.19 | 95.46
VPT | 84.40 | 81.05 | 95.56
STS | 84.30 | 80.56 | 95.85

###  3.3 Compared with SOTAs

As shown in Tab. 2, we compare VPT-STS with recent state-of-the-art results on
neuromorphic datasets. The results show that VPT-STS achieves substantial
improvements over previous SOTAs. It is worth noting that VPT-STS
significantly outperforms NDA, which is an ensemble of six geometric
transformations. The experimental results demonstrate the superiority of
combining spatiotemporal information for data augmentation. Since VPT-STS is
orthogonal to most training algorithms, it can provide a better baseline and
improve the performance of existing models.

###  3.4 Ablation Studies on VPT-STS

As shown in Fig. 3, the performance of VPT-STS with different rotation angles
is evaluated on the N-Caltech101 dataset. It turns out that a suitable
rotation angle is important for the performance of data augmentation, which
can increase data diversity without losing features.

###  3.5 Analysis of VPT-STS

To gain further insight into the workings of VPT-STS, we add different
strategies on the baseline to analyze the effective components of VPT-STS. As
shown in Table 3, spatial rotation (Rotation) is performed as a comparative
experiment for VPT-STS. It turns out that both VPT and STS including
spatiotemporal transformations are significantly better than pure spatial
geometric transformations on all three datasets, which illustrate the
importance of spatiotemporal transformations. While VPT and STS are
implemented with operations similar to rotation, it actually improves the
robustness of SNNs to different viewpoints. Furthermore, we evaluate the
robustness of SNNs to viewpoint fluctuations by adding different degrees of
spatiotemporal rotation to the test data. Figures 2(a) and 2(b) show the
performance of the baseline model and the model trained by VPT-STS under
different disturbances, respectively. The results show that the general trend
of the accuracy change is to decrease with the increase of the perturbation
amplitude. In addition, Fig. 2(c) shows the difference in the accuracy
reduction of the VPT-STS compared to baseline. As the perturbation amplitude
increases, the difference in the accuracy reduction of the two models is less
than zero, and the absolute value grows, which illustrate that the accuracy
reduction of baseline is larger than that of VPT-STS. Experimental results
show that the model trained with VPT-STS generalize better and improves the
robustness of SNNs against spatial location variances.

Fig. 3: Performance of VPT-STS at different angles.

##  4 Conclusion

We propose a novel data augmentation method suitable for events, viewpoint
transformation and spatiotemporal stretching (VPT-STS). Extensive experiments
on prevailing neuromorphic datasets show that VPT-STS is broadly effective on
multiple event representations and significantly outperforms pure spatial
geometric transformations. It achieves substantial improvements over previous
SOTAs by improving the robustness of SNNs to different viewpoints.

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††thanks: Corresponding author<EMAIL_ADDRESS>Data Availability
Statement: The data that support the findings of this study are available from
the corresponding author upon reasonable request.

# Decomposition solutions and Bäcklund transformations of the BKP and CKP
equations

Xiazhi Hao1 and S. Y. Lou2 Corresponding<EMAIL_ADDRESS>Data
Availability Statement: The data that support the findings of this study are
available from the corresponding author upon reasonable request. 1College of
Science, Zhejiang University of Technology, Hangzhou, 310014, China
2School of Physical Science and Technology, Ningbo University, Ningbo, 315211,
China

###### Abstract

In this paper, we define the modified formal variable separation approach and
show how it determines, in a remarkably simple manner, the decomposition
solutions, the Bäcklund transformations, the Lax pair, and the linear
superposition solution of the B-type Kadomtsev-Petviashvili equation. Also,
the decomposition solutions, the Bäcklund transformation and the Lax pair
relating to the C-type Kadomtsev-Petviashvili equation is obtain by the same
technique. This indicates that the decomposition may provide a description of
integrable behavior in nonlinear systems, while, at the same time,
establishing an efficient method for determining relationships between the
particular systems.
Key words: Decomposition solution, Bäcklund transformation, Lax pair, Linear
superposition solution

###### pacs:

05.45.Yv,02.30.Ik,47.20.Ky,52.35.Mw,52.35.Sb

##  I Introduction

Various techniques are available for obtaining exact solutions of nonlinear
partial differential equations (PDEs) such as the inverse scattering method
abib14 ; rbib22 ; vbib4 , group theoretical method abib23 , the singularity
approach jbib3 ; jbib9 ; rbib3 ; sbib8 , and some direct methods xbib1 ;
ebib14 ; ybib4 ; wbib25 . In all these methods, one can identify a central
link, Bäcklund transformation (BT for short), one analytical tool for dealing
with integrability problems, which enables one to relate pairs of solutions of
certain nonlinear PDEs. Indeed the number of papers produced so far on BTs is
fairly large, but not surprisingly so since these transformations have
attracted the attention of both mathematicians and physicists, and have been
incredibly actively developed in different directions, their relevance being
well established in differential geometry and algebra as well as in nonlinear
science, and other fields of application gbib16 ; mbib2 ; mbib15 ; jbib10 .

Nevertheless, among these methods, it seems to us that the convenient one to
construct BTs is the decomposition method based on the formal variable
separation approach (FVSA) sbib42 ; sbib43 ; xbib26 . We propose such a new
decomposition through which the BTs may exist and show how it works for two
important nonlinear PDEs: the B-type and C-type Kadomtsev-Petviashvili
equations (BKP and CKP). In the authors’ opinion, this decomposition provides
a very straightforward and natural way to construct BTs, in addition, yields
some new results. It may be expected that these results can be carried over to
other equations.

The modified FVSA consists in looking for the general decomposition solution
of the (2+1)-dimensional PDE in the form

$\displaystyle w_{y}$ $\displaystyle=$ $\displaystyle
F(x,y,t,w,w_{x},w_{xx},\dots,w_{xm}),$ (1) $\displaystyle w_{t}$
$\displaystyle=$ $\displaystyle G(x,y,t,w,w_{x},w_{xx},\dots,w_{xn})$ (2)

with the integrability condition

$\displaystyle w_{yt}-w_{ty}=[F,G]=0$ (3)

preserved, here, $w_{xm}=\partial^{m}_{x}w,w_{xn}=\partial^{n}_{x}w,$ $m$ and
$n$ are integers. $F$ and $G$ depend on $x,y,t$ and derivatives of $w$ with
respect to $x$.

The successive practical steps of the method are the following.

(I) Determine the possible values of $m$ and $n$ by balancing two or more
terms of the PDE and expressing that they dominate the other terms.

(II) Substitute the decompositions (1)-(2) into the PDE and integrability
condition (3), then obtain the expressions for $w$ and its derivatives in the
form of an overdetermined systems.

(III) Determine $F$ and $G$ by solving systems obtained in step II, whose
solution yields the desired decomposition solution.

Examples are always a good starting point. Consider the BKP equation

$\displaystyle
u_{xt}+(u_{x4}+15uu_{x2}+15u^{3}-15uv-5u_{xy})_{xx}-5u_{yy}=0,v_{x}=u_{y},$
(4)

where
$u_{x}=\partial_{x}u,u_{x2}=\partial_{x}^{2}u,u_{x3}=\partial_{x}^{3}u,\dots,$
which is a (2+1)-dimensional generalization of the Sawada-Kotera (SK) equation
and exhibits many elegant integrable behavior ybib17 ; rbib21 . It is
convenient to deal analytically with a potential function $w$, introduced by
setting $u=w_{x}$, and it follows equation (4) that $w$ can be taken to
satisfy the equation

$\displaystyle
w_{xt}=5w_{yy}-(w_{x5}+15w_{x}w_{x3}+15w_{x}^{3}-15w_{x}w_{y}-5w_{xxy})_{x}.$
(5)

The CKP equation is defined as

$\displaystyle
9u_{xt}+(u_{x4}+15uu_{x2}+15u^{3}+\frac{45}{4}u_{x}^{2}-15uv-5u_{xy})_{xx}-5u_{yy}=0,v_{x}=u_{y},$
(6) $\displaystyle
9w_{xt}={5w_{yy}}-({w_{x5}}+15w_{x}w_{x3}+{15w_{x}^{3}}+\frac{45w_{xx}^{2}}{4}-{15w_{x}w_{y}}-{5w_{xxy}})_{x},$
(7)

in which $u$ denotes the conservative field and $w$ the potential one. When
$u_{y}$ tends to zero, the reduced equation of the CKP equation (6) is the
well-known Kaup–Kupershmidt (KK) equation ibib4 . The close connection between
these two completely integrable fifth order equations is well documented
jbib13 , nevertheless, despite their evident duality, equations (4) and (6)
are fundamentally different. There is no scaling which reduces one to the
other.

The application of the decomposition method to the potential BKP equation (5)
and potential CKP equation (7) is described in sections II and III. At the
same time, the BTs and the Lax pairs for the potential BKP and potential CKP
equations are found in a remarkably natural manner. Finally, some conclusions
are given in part IV. Some lengthy and technical calculations have been moved
to the Appendix.

##  II The decomposition solutions, Bäcklund transformation, Lax pair and
linear superposition solutions of the potential BKP equation

Analysis of the potential BKP equation (5) shows the leading orders $m=3,n=5$,
in this case, $w_{y}$ and $w_{t}$ are expressible in terms of

$\displaystyle w_{y}$ $\displaystyle=$ $\displaystyle
F(x,y,t,w,w_{x},w_{xx},w_{xxx}),$ (8) $\displaystyle w_{t}$ $\displaystyle=$
$\displaystyle G(x,y,t,w,w_{x},w_{xx},w_{xxx},w_{xxxx},w_{xxxxx}),$ (9)

and we therefore require the equations (8) and (9) are subject to the
integrability condition

$\displaystyle F_{t}=G_{y},$ (10)

whenever equation (5) is satisfied. The derivation of $F$ and $G$ is
elementary but somewhat lengthy and can be found in the Appendix. The result
of calculation leads directly to the following theorems.

Theorem 1. If $w$ is a solution of the consistent variable coefficient
potential Korteweg-de Vries (KdV) decomposition system

$\displaystyle v_{y}$ $\displaystyle=$ $\displaystyle
v_{xxx}-\frac{a}{2}v_{x}^{2}+\frac{m_{0}}{a}x+n_{y},$ (11) $\displaystyle
v_{t}$ $\displaystyle=$ $\displaystyle
9v_{xxxxx}-15av_{x}v_{xxx}-\frac{15a}{2}v_{xx}^{2}+\frac{5a^{2}}{2}v_{x}^{3}+[15m_{2}+15m_{1}y-(5m_{0}^{2}+\frac{1}{2}m_{0t})y^{2}-5m_{0}x]v_{x}$
(12) $\displaystyle-$ $\displaystyle
5m_{0}v+[(10m_{0}^{2}+m_{0t})y-15m_{1}]\frac{x}{a}+\frac{y^{3}}{6a}m_{0}(10m_{0}^{2}+m_{0t})-\frac{15y^{2}}{2a}m_{0}m_{1}-\frac{15y}{a}m_{0}m_{2}$
$\displaystyle+$ $\displaystyle 5m_{0}n+n_{t}+m_{3},$ $\displaystyle w_{y}$
$\displaystyle=$ $\displaystyle
w_{xxx}+3w_{x}^{2}+av_{x}w_{x}+\frac{a^{2}}{6}v_{x}^{2}-\frac{1}{3}m_{0}(x+m_{0}y^{2})-\frac{m_{0t}y^{2}}{30}+m_{1}y+m_{2},$
(13) $\displaystyle w_{t}$ $\displaystyle=$ $\displaystyle
9w_{xxxxx}+15(av+6w)_{x}w_{xxx}+5a(av+3w)_{x}v_{xxx}+15(av+3w)_{xx}w_{xx}+\frac{15}{2}w_{x}[(av+3w)_{x}^{2}+3w_{x}^{2}]$
(14) $\displaystyle+$
$\displaystyle\frac{5a^{2}}{2}v_{xx}^{2}+w_{x}(15m_{2}+15m_{1}y-5m_{0}^{2}y^{2}-5m_{0}x-\frac{1}{2}m_{0t}y^{2})-5m_{0}w-\frac{xy}{3}(m_{0t}+10m_{0}^{2})+5m_{1}x$
$\displaystyle-$
$\displaystyle\frac{y^{3}}{90}(m_{0tt}+30m_{0}m_{0t}+100m_{0}^{3})+\frac{y^{2}}{2}(m_{1t}+10m_{0}m_{1})+(m_{2t}+10m_{0}m_{2})_{y}+m_{4},$

where $m_{i},i=0,1,\dots,4$ are arbitrary functions of $t,n$ is an arbitrary
function of $y,t$ and $a$ is an arbitrary constant, then $w$ is a solution of
the potential BKP equation (5).

The arbitrary parameter $a$ determines that $v$ defined by equations (11)-(12)
may be a decomposition solution of potential BKP equation (5) as well. The
reader may care to show $v$ may indeed be transformed to a solution of
potential BKP equation (5) when $a=-3$ and $a=-6$. Generalizing, we can state
the following.

Corollary 1. Under the decomposition of Theorem 1, assume that the constant
$a$ and the function $n$ be as

$\displaystyle
a=-3,n=m_{2}y+\frac{1}{2}m_{1}y^{2}-\frac{1}{90}(10m_{0}^{2}+m_{0t})y^{3}+n_{0}(t),$
(15)

then, both $v$ and $w$ given by (11)-(14) are solutions of the potential BKP
equation (5).

We record another corollary of Theorem 1.

Corollary 2. The functions $v$ and $w$ fixed by (11)-(14) in the statement of
Theorem 1 with the additional condition

$\displaystyle a=-6,n=m_{2}y+n_{0}(t),m_{0}=m_{1}=0$ (16)

satisfy the potential BKP equation (5) as well.

Note that the results in Corollaries 1 and 2 indicate that potential BKP
equation (5) possesses the variable coefficient KdV decomposition solutions
(11)-(12) with conditions (15) and/or (16) xbib24 . On the other hand, the
variable coefficient KdV decompositions (13)-(14) with (15) and/or (16) form
BTs which connect two solutions $v$ and $w$ of the potential BKP equation (5).

A further consequence of decomposition is that we can establish a general BT
associated with potential BKP (5). This result is formulated as the following
theorem.

Theorem 2. (Bäcklund Transformation) The decomposition relation

$\displaystyle p_{y}$ $\displaystyle=$ $\displaystyle
p_{xxx}+\frac{3}{2}(w+v)_{xx}p+\frac{3}{2}(w_{x}^{2}-v_{x}^{2})+\frac{1}{4}(p^{3})_{x},p\equiv
w-v,$ (17) $\displaystyle p_{t}$ $\displaystyle=$ $\displaystyle
9p_{xxxxx}+\frac{9}{16}(p^{5})_{x}+\frac{45}{8}(w+v)_{xx}p^{3}-\frac{45}{8}(3v_{x}^{2}-3w_{x}^{2}-2p_{xxx})p^{2}+\frac{15}{4}(9v_{x}^{2}+9w_{x}^{2}-6v_{x}w_{x}$
(18) $\displaystyle+$ $\displaystyle
4v_{xxx}+4w_{xxx}+2v_{y}+2w_{y})_{x}p-15w_{xxx}(v-4w)_{x}-15v_{xxx}(4v-w)_{x}+45w_{xx}^{2}-45v_{xx}^{2}$
$\displaystyle+$
$\displaystyle\frac{45}{2}p_{x}(v_{x}^{2}+w_{x}^{2})+\frac{15}{2}(w+v)_{y}p_{x},$

constitutes a BT between two solutions $w$ and $v$ of the potential BKP
equation (5).

It is easily verified by direct calculation that this system (17)-(18) is
integrable and that $v$ satisfies equation (5). It is worth noting that the BT
has the form of a conservation law

$\displaystyle p_{y}$ $\displaystyle=$
$\displaystyle[p_{xx}+\frac{3}{2}p(w_{x}+v_{x})+\frac{p^{3}}{4}]_{x},$
$\displaystyle p_{t}$ $\displaystyle=$
$\displaystyle[9p_{xxxx}+\frac{9p^{5}}{16}+\frac{45}{8}(w+v)_{x}p^{3}+\frac{45}{4}p^{2}p_{xx}+(15(w+v)_{xxx}+\frac{45}{2}(w_{x}^{2}+v_{x}^{2})+\frac{15}{2}(w+v)_{y})p$
$\displaystyle+$ $\displaystyle\frac{45}{2}(p_{x}(w+v)_{x})_{x}]_{x}.$

Finding the BT of a PDE is equivalent to finding its Lax pair mbib16 ; mbib17
. Therefore, further consideration of the BT for the construction of the Lax
pair seems warranted. With appeal to Theorem 2, we can obtain the well-known
Lax pair bbib2

$\displaystyle\psi_{y}$ $\displaystyle=$
$\displaystyle\psi_{xxx}+3w_{x}\psi_{x},$ (19) $\displaystyle\psi_{t}$
$\displaystyle=$ $\displaystyle
9\psi_{xxxxx}+45w_{x}\psi_{xxx}+45w_{xx}\psi_{xx}+15(2w_{xxx}-3w_{x}^{2}+w_{y})\psi_{x}$
(20)

of the potential BKP equation (5) by substitution of the transformation
$v=w+\ln(\psi^{2})_{x}$ into the BT (17)-(18) as may be verified by direct
calculation. The elimination of $w$ is made between Lax pair, and results in
the Schwarzian form of the potential BKP equation (5)

$\displaystyle S_{xxx}+C_{x}+4S^{2}_{x}-5K^{2}_{x}+5(S_{x}K-S_{y}-K_{y})=0$
(21)

with Schwarzian derivatives
$S=\frac{\psi_{xxx}}{\psi_{x}}-\frac{3\psi_{xx}^{2}}{2\psi_{x}^{2}},C=\frac{\psi_{t}}{\psi_{x}}$
and $K=\frac{\psi_{y}}{\psi_{x}}$.

Notice that 1. Each of two Corollaries and Theorem 2 takes the form of a
system of coupled differential equations for two unknown functions, and all
imply that if one of the two functions is a solution of the potential BKP (5),
then also the second function solves the same equation. 2. The BT (17)-(18)
implies a type of the Sharma–Tasso–Olver (STO) decomposition solution of the
potential BKP equation (5) when $v=0$ xbib24 , also presents a means of
finding the Lax pair, then Schwarzian equation.

Theorem 3. Let the function $w$ be a solution to the consistent variable
coefficient Svinolupov Sokolov (SS) system

$\displaystyle w_{y}$ $\displaystyle=$ $\displaystyle
w_{xxx}+\frac{3w_{xx}^{2}}{2W}+\frac{3}{2}w_{x}^{2}-\frac{27}{4}M^{2}+\frac{3}{2}m_{1}x+\frac{3y}{20}(m_{1}y+2m_{2})_{t}+m_{3},$
(22) $\displaystyle w_{t}$ $\displaystyle=$ $\displaystyle
9w_{xxxxx}+45w_{x}w_{xxx}+\frac{45}{2}w_{x}^{3}+\frac{3}{4}[30m_{1}x-135M^{2}+3y(m_{1}y+2m_{2})_{t}+20m_{3}]w_{x}+\frac{45}{2}m_{1}w$
(23) $\displaystyle-$
$\displaystyle\frac{3}{2}(45m_{1}M-m_{1t}y-m_{2t})x+\frac{y^{2}}{20}(m_{1}y+3m_{2})_{tt}-\frac{27}{4}y(m_{1}y+2m_{2})_{t}M+y(m_{3t}-45m_{1}m_{3})+m_{4}$
$\displaystyle+$
$\displaystyle\frac{45}{2W}[2w_{xx}w_{xxxx}+w_{xxx}^{2}+(3M+w_{x})w_{xx}^{2}]+\frac{180}{W^{2}}w_{xx}^{2}w_{xxx}+\frac{315}{2W^{3}}w_{xx}^{4}+\frac{405}{4}M^{3},$

where $M\equiv m_{1}y+m_{2},W\equiv 3M-2w_{x}$ and $m_{i},i=1,2,3,4$ are
arbitrary functions of $t$, then $w$ is a solution of the potential BKP
equation (5).

Theorem 4. The potential BKP equation (5) possesses the following variable
coefficient KdV decomposition solution

$\displaystyle w_{y}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}w_{xxx}-\frac{3}{2}w_{x}^{2}+6Mw_{x}+\frac{y}{10}(m_{1}y+2m_{2})_{t}+m_{1}x-6M^{2}+m_{3},M\equiv
m_{1}y+m_{2},$ (24) $\displaystyle w_{t}$ $\displaystyle=$
$\displaystyle-\frac{9}{4}w_{xxxxx}-\frac{45}{4}(2w_{x}w_{xxx}+w_{xx}^{2}+2w_{x}^{3})+15m_{1}w+\frac{3}{2}y(m_{1}y+2m_{2})_{t}w_{x}+15(6M^{2}+m_{1}x+m_{3})w_{x}$
(25) $\displaystyle+$
$\displaystyle\frac{y^{2}}{30}(m_{1}y+3m_{2})_{tt}-3(y^{2}m_{1t}+10m_{1}x+2ym_{2t})M+M_{t}x+(m_{3t}-30m_{1}m_{3})y-60M^{3}+m_{4},$

where $m_{i},i=1,2,3,4$ are arbitrary functions of $t$.

Theorem 5. The variable coefficient potential SK decomposition solution of the
potential BKP equation (5) possesses the form

$\displaystyle w_{y}$ $\displaystyle=$ $\displaystyle
Mw_{x}+\frac{y}{30}(M+m_{2})_{t}+\frac{1}{3}m_{1}x-\frac{1}{2}M^{2}+m_{3},M\equiv
m_{1}y+m_{2},$ (26) $\displaystyle w_{t}$ $\displaystyle=$ $\displaystyle-
w_{xxxxx}+5(M-3w_{x})w_{xxx}+15(M-w_{x})w_{x}^{2}+(\frac{1}{2}y^{2}m_{1t}+ym_{2t}+5m_{1}x-\frac{5}{2}M^{2}+15m_{3})w_{x}$
(27) $\displaystyle+$ $\displaystyle
5m_{1}w+\frac{y^{2}}{90}(m_{1}y+3m_{2})_{tt}-\frac{y}{3}(My-x)m_{1t}+\frac{1}{3}(x-2M_{y})m_{2t}-\frac{10}{3}m_{1}(Mx+3m_{3}y)$
$\displaystyle+$ $\displaystyle m_{3t}y+m_{4}$

with $m_{i},i=1,2,\dots,4$ being arbitrary functions of $t$.

Suppose that arbitrary functions $m_{i}=0,i=1,2,\dots,4$, then by Theorems 3
to 5, three special decompositions correspond to constant coefficient SS , KdV
and SK decomposition solutions of potential BKP equation (5), respectively
xbib24 .

If a BT represents one of the different aspects of the property of
integrability for a PDE, the existence of a linear combination principle
allows further to build explicitly some classes of solutions depending on
decomposition system (11)-(12). The linear combination solution of the
potential BKP equation (5) from the result of the decomposition is
established.

Theorem 6. If $v_{1}$ and $v_{2}$ are solutions of the variable coefficient
potential KdV decompositions

$\displaystyle v_{1y}$ $\displaystyle=$ $\displaystyle
v_{1xxx}-\frac{1}{2}a_{1}v_{1x}^{2}+\frac{1}{a_{1}}m_{0}x+n_{y},M\equiv
m_{1}y+m_{2},$ (28) $\displaystyle v_{1t}$ $\displaystyle=$ $\displaystyle
9v_{1xxxxx}+\frac{5}{2}a_{1}^{2}v_{1x}^{3}-\frac{15}{2}a_{1}(2v_{1x}v_{1xxx}+v_{1xx}^{2})+[15M-5m_{0}x-\frac{1}{2}(10m_{0}^{2}+m_{0t})y^{2}])v_{1x}$
(29) $\displaystyle-$ $\displaystyle
5m_{0}v_{1}+5m_{0}n+n_{t}+m_{3}+\frac{1}{a_{1}}\\{[(10m_{0}^{2}+m_{0t})y-15m_{1}]x+\frac{5}{3}y^{3}m_{0}^{3}$
$\displaystyle+$
$\displaystyle\frac{1}{6}m_{0}m_{0t}y^{3}-\frac{15}{2}m_{0}(m_{1}y+2m_{2})y\\},$

and

$\displaystyle v_{2y}$ $\displaystyle=$ $\displaystyle
v_{2xxx}-\frac{1}{2}a_{2}v_{2x}^{2}+\frac{1}{a_{2}}(\frac{1}{5}m_{0t}y^{2}-a_{1}n_{y}+m_{0}x+2y^{2}m_{0}^{2}-6M),$
(30) $\displaystyle v_{2t}$ $\displaystyle=$ $\displaystyle
9v_{2xxxxx}+\frac{5}{2}a_{2}^{2}v_{2x}^{3}-\frac{15}{2}a_{2}(2v_{2x}v_{2xxx}+v_{2xx}^{2})-\frac{1}{2}y^{2}v_{2x}m_{0t}+(15M-5m_{0}^{2}y^{2}-5m_{0}x)v_{2x}-5m_{0}v_{2}$
(31) $\displaystyle+$ $\displaystyle
m_{4}+\frac{1}{a_{2}}[\frac{1}{15}m_{0tt}y^{3}+\frac{1}{6}(6xy+11m_{0}y^{3})m_{0t}-3m_{1t}y^{2}-6m_{2t}y-a_{1}n_{t}+5(2m_{0}^{2}y-3m_{1})x$
$\displaystyle+$ $\displaystyle
5y^{3}m_{0}^{3}-\frac{45}{2}m_{0}m_{1}y^{2}-45m_{0}m_{2}y-5m_{0}a_{1}n],$

respectively, then their linear combination

$\displaystyle w=-\frac{1}{6}(a_{1}v_{1}+a_{2}v_{2})$ (32)

is a solution of the potential BKP equation (5).

Generally, the functions $v_{1}$ and $v_{2}$ in Theorem 6 are not the
solutions of the potential BKP equation (5) except in certain special cases
mentioned in Corollaries 1 and 2, whereas the linear combination (32) of
$v_{1}$ and $v_{2}$ solves the potential BKP equation (5). The linear
superposition solutions with fixed $a_{1}$ and $a_{2}$ in xbib24  are
particular cases of Theorem 6.

##  III The Decomposition solutions, Bäcklund transformation and Lax pair of
the potential CKP equation

As a second illustration, we repeat the entire procedure in the previous
section to find the decomposition solutions and Bäcklund transformation of the
potential CKP equation (7). Similarly, a straightforward calculation
determines the following theorems.

Theorem 7. (Bäcklund Transformation) Let $v$ be any solution of the potential
CKP equation (7), a different solution $w$ of equation (7) is then defined by
the Bäcklund transformation

$\displaystyle p_{y}$ $\displaystyle=$
$\displaystyle[p_{xx}-\frac{3p_{x}^{2}}{4p}+\frac{3}{2}p(w_{x}+v_{x})+\frac{p^{3}}{4}]_{x},\
p=w-v,$ (33) $\displaystyle p_{t}$ $\displaystyle=$
$\displaystyle[p_{xxxx}-\frac{5}{2p}p_{x}p_{xxx}+\frac{5p}{3}(w+v)_{xxx}-\frac{5}{4p}p_{xx}^{2}+(\frac{5p^{2}}{4}+\frac{5p_{x}^{2}}{p^{2}}+\frac{5}{2}(w+v)_{x})p_{xx}+\frac{5}{4}p_{x}(w+v)_{xx}$
(34) $\displaystyle-$
$\displaystyle\frac{35p_{x}^{4}}{16p^{3}}-\frac{15(w+v)_{x}}{8p}p_{x}^{2}+\frac{15}{8}p(w_{x}^{2}+v_{x}^{2})+\frac{p^{5}}{16}+\frac{5p}{6}(w+v)_{y}+\frac{5p^{3}}{8}(w+v)_{x}+\frac{5}{4}pw_{x}v_{x}]_{x}.$

This BT (33) and (34) appears to be a new result and is again in a
conservation form. The Lax pair arises from the BT, which results in two
equations

$\displaystyle\psi_{y}$ $\displaystyle=$
$\displaystyle\psi_{xxx}+3w_{x}\psi_{x}+\frac{3}{2}w_{xx}\psi,$ (35)
$\displaystyle\psi_{t}$ $\displaystyle=$
$\displaystyle\psi_{xxxxx}+5w_{x}\psi_{xxx}+\frac{15}{2}w_{xx}\psi_{xx}+5(\frac{7}{6}w_{xxx}+w_{x}^{2}+\frac{w_{y}}{3})\psi_{x}+5(\frac{w_{xxxx}}{3}+w_{x}w_{xx}+\frac{w_{xy}}{6})\psi$
(36)

followed by the relation $v=w+\ln(\partial^{-1}_{x}\psi^{2})_{x}$.

When $v=0$, the BT is reduced to the modified STO decomposition solution

$\displaystyle w_{y}$ $\displaystyle=$
$\displaystyle[w_{xx}-\frac{3w_{x}^{2}}{4w}+\frac{3}{2}ww_{x}+\frac{w^{3}}{4}]_{x},$
(37) $\displaystyle w_{t}$ $\displaystyle=$
$\displaystyle[w_{xxxx}-\frac{5}{2w}w_{x}w_{xxx}+\frac{5}{3}ww_{xxx}-\frac{5}{4w}w_{xx}^{2}+5(\frac{w^{2}}{4}+\frac{w_{x}^{2}}{w^{2}}+\frac{3}{4}w_{x})w_{xx}-\frac{35w_{x}^{4}}{16w^{3}}-\frac{15w_{x}^{3}}{8w}$
(38) $\displaystyle+$
$\displaystyle\frac{15ww_{x}^{2}}{8}+\frac{w^{5}}{16}+\frac{5ww_{y}}{6}+\frac{5w^{3}w_{x}}{8}]_{x}$

of the potential CKP equation (7). This very special modified STO
decomposition is linked to the STO decomposition

$\displaystyle f_{y}$ $\displaystyle=$
$\displaystyle[f_{xx}+\frac{1}{4}f^{3}+\frac{3}{2}(f_{x}f)]_{x},$ (39)
$\displaystyle f_{t}$ $\displaystyle=$
$\displaystyle[f_{xxxx}+5f_{x}f_{xx}+5(\frac{ff_{xxx}}{2}+\frac{f^{2}f_{xx}}{2}+\frac{f^{3}f_{x}}{4}+\frac{3ff_{x}^{2}}{4})+\frac{f^{5}}{16}]_{x}$
(40)

by transformation $w_{x}=fw-w^{2}$, which is exactly the same as the STO
decomposition of the potential BKP equation (5).

Theorem 8.  The function $w$ is a solution of the potential CKP equation (7)
provided that $w$ satisfies the consistent variable coefficient SS system

$\displaystyle w_{y}$ $\displaystyle=$ $\displaystyle
w_{xxx}+\frac{3w_{xx}^{2}}{2W}+{3}w_{x}^{2}-\frac{9Mw_{x}}{2}-\frac{27}{8}M^{2}+\frac{3}{2}m_{1}x+\frac{27y}{20}(m_{1}y+2m_{2})_{t}+\frac{27m_{2}^{2}}{8}+m_{3},$
(41) $\displaystyle w_{t}$ $\displaystyle=$ $\displaystyle
w_{xxxxx}+\frac{5w_{xx}w_{xxxx}}{W}+\frac{5w_{xxx}^{2}}{2W}+(\frac{20w_{xx}^{2}}{W^{2}}+10w_{x}-\frac{15M}{2})w_{xxx}+\frac{35w_{xx}^{4}}{2W^{3}}+\frac{5(3M+4w_{x})w_{xx}^{2}}{4W}$
(42) $\displaystyle+$ $\displaystyle
10w_{x}^{3}-\frac{45M}{2}w_{x}^{2}+(\frac{45}{8}M^{2}+\frac{45m_{2}^{2}}{8}+\frac{5m_{1}x}{2}+\frac{5m_{3}}{3})w_{x}+\frac{(9w_{x}y^{2}+6xy-27y^{2}M)m_{1t}}{4}$
$\displaystyle+$
$\displaystyle\frac{(18w_{x}y+6x-27y(2m_{1}y+m2))m_{2t}}{4}+\frac{9y^{2}(m_{1}y+3m_{2})_{tt}}{20}-\frac{15m_{1}Mx}{2}+\frac{45M^{3}}{4}-\frac{135m_{1}m_{2}^{2}y}{8}$
$\displaystyle+$ $\displaystyle\frac{5m_{1}w}{2}-5m_{1}m_{3}y+m_{3t}y+m_{4},$

where $M\equiv m_{1}y+m_{2},W\equiv 3M-2w_{x}$ and $m_{i},i=1,2,3,4$ are
arbitrary functions of $t$ .

Theorem 9. The potential CKP equation (7) possesses the following variable
coefficient KdV decomposition solution

$\displaystyle w_{y}$ $\displaystyle=$
$\displaystyle\frac{1}{4}w_{xxx}+\frac{3}{2}w_{x}^{2}+\frac{3m_{0}x}{5}+\frac{27y^{2}(m_{0t}-2m_{0}^{2})}{50}+M,M\equiv
m_{1}y+m_{2},$ (43) $\displaystyle w_{t}$ $\displaystyle=$
$\displaystyle\frac{1}{16}w_{xxxxx}+\frac{5}{8}(2w_{x}w_{xxx}+w_{xx}^{2}+4w_{x}^{3})+(m_{0}x-\frac{9m_{0}^{2}y^{2}}{5}+\frac{5}{3}M+\frac{9m_{0t}y^{2}}{10})w_{x}+m_{0}w$
(44) $\displaystyle+$
$\displaystyle((\frac{3m_{0t}}{5}-\frac{6m_{0}^{2}}{5})y+\frac{5m_{1}}{9})x+(\frac{18m_{0}^{3}}{25}-\frac{27m_{0}m_{0t}}{25}+\frac{9m_{0tt}}{50})y^{3}+(\frac{m_{1t}}{2}-m_{0}m_{1})y^{2}$
$\displaystyle+$ $\displaystyle(m_{2t}-2m_{0}m_{2})y+m_{3},$

where $m_{i},i=1,2,3,4$ are arbitrary functions of $t$.

Theorem 10. The variable coefficient potential KK decomposition solution of
the potential CKP equation (7) possesses the form

$\displaystyle w_{y}$ $\displaystyle=$ $\displaystyle
Mw_{x}+\frac{3y}{10}(M+m_{2})_{t}+\frac{1}{3}m_{1}x-\frac{1}{2}M^{2}+\frac{m_{2}^{2}}{2}+m_{3},M\equiv
m_{1}y+m_{2},$ (45) $\displaystyle w_{t}$ $\displaystyle=$
$\displaystyle-\frac{w_{xxxxx}}{9}+\frac{5}{9}(M-3w_{x})w_{xxx}-\frac{5w_{xx}^{2}}{4}-\frac{5w_{x}^{3}}{3}+\frac{5M}{3}w_{x}^{2}+(\frac{5m_{1}x}{9}-\frac{5M^{2}}{18}+\frac{5m_{2}^{2}}{6}+\frac{5m_{3}}{3}$
(46) $\displaystyle+$
$\displaystyle\frac{m_{1t}y^{2}}{2}+m_{2t}y)w_{x}+\frac{5m_{1}w}{9}+\frac{y^{2}}{10}(m_{1}y+3m_{2})_{tt}-\frac{m_{1t}y^{2}M}{3}+\frac{xM_{t}}{3}+\frac{m_{2t}y(m_{2}-2m_{1}y)}{3}$
$\displaystyle+$ $\displaystyle
m_{3t}y-\frac{10m_{1}xM}{27}-\frac{5m_{1}y(m_{2}^{2}+2m_{3})}{9}+m_{4}$

with $m_{i},i=1,2,\dots,4$ being arbitrary functions of $t$.

The proof of Theorems 7-10 is of the same form as that presented in the
Appendix for the potential BKP equation (5). The calculation is omitted here.

We have attempted but failed to provide the linear superposition solution of
the potential CKP equation (7) from the decomposition solutions.

##  IV Conclusions

In this study, which extends the work of a previous paper xbib24 , we have
presented a modified formal variable separation approach for deriving
decomposition solutions of the potential BKP and CKP equations. This method is
heuristic, it gives decomposition solutions of the (2+1)-dimensional PDEs in a
unified way. The result makes clear that we establish the connection between
the potential BKP and CKP equations with several classic integrable systems.
One might therefore hope to deduce the solutions of the potential BKP and CKP
equations from solutions of the classical integrable systems.

Specifically, we are able to construct the BTs from decomposition. It is
widely accepted that the existence of a BT serves as a sign to the
integrability of a PDE. While the BTs for the (1+1)-dimensional SK and KK
equations have been known for some time mbib16 , the BTs for the
(2+1)-dimensional SK (4) and KK (6) equations have not previously been
reported. The BTs of the potential BKP and CKP equations are given here in
explicit forms for the first time. Of course, other standard analytic
techniques for obtaining BTs of nonlinear PDEs are available, but, the
construction of the BTs from decomposition is a remarkably straightforward
way.

In the case of the potential BKP and CKP equations, the system defining the
BTs found has two properties. First, it is linearizable since it results in
the Lax pair. Second, it has a conservation form. Observe that the solution to
the potential BKP equation in Theorem 6 also has a unique property: the well-
known linear superposition principle holds if the parameters are taken on some
special values.

It is note that yet despite the close resemblance of the potential BKP and CKP
equations, they are fundamentally different. There is no scaling which
transformations one equation into the other. Whereas in view of the system
(39)-(40), it is seen that the STO decomposition links the potential BKP and
CKP equations.

As an elementary tool for constructing solutions of PDEs, decomposition is as
straightforward to apply as us believe. We hope to expand on some of these
theoretical aspects in future work.

## Acknowledgement

The work was sponsored by the National Natural Science Foundations of China
(Nos. 11975131, 11435005), K. C. Wong Magna Fund in Ningbo University, the
Natural Science Foundation of Zhejiang Province No. LQ20A010009.

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## Appendix

We illustrate the decomposition procedure for the case of the potential BKP
equation (5).

###### Proof.

Substituting (8) and (9) with $m>3$ into the potential BKP equation (5), plus
the decomposition compatibility condition (10), one can find that the
decomposition does not hold generally for $m>3$. Therefore, we take $m=3$ and
then $n=5$ in the decomposition relations (8) and (9). Substituting (8) and
(9) into (5) gives

$\displaystyle w_{x6}(1-5F_{x_{3}}^{2}-5F_{x_{3}}+G_{x_{5}})+W=0,$ (47)

where $F=F(x,y,t,w,w_{x},w_{x2},w_{x3})\equiv
F(x,y,t,x_{0},x_{1},x_{2},x_{3})$,
$G=G(x,y,t,w,w_{x},w_{x2},w_{x3},w_{x4},w_{x5})\equiv
G(x,y,t,x_{0},x_{1},x_{2},x_{3},x_{4},x_{5})$, and
$W=W(x,y,t,x_{0},x_{1},\ldots,x_{5})$ is a complicated expression of
$x,y,t,x_{0},x_{1},\ldots,x_{5}$. Vanishing coefficient of $w_{x6}$, we have

$\displaystyle G=(5F_{x_{3}}+5F_{x_{3}}^{2}-1)x_{5}+G_{1},$ (48)

where $G_{1}=G_{1}(x,y,t,x_{0},x_{1},\ldots,x_{4})$ is a function of
$\\{x,y,t,x_{0},x_{1},\ldots,x_{4}\\}$. By using the relation (48), (47) is
changed to

$\displaystyle
w_{x5}[G_{1x_{4}}-5(F_{x_{3}}+2)(x_{1}F_{x_{0}x_{3}}+x_{2}F_{x_{1}x_{3}}+x_{3}F_{x_{2}x_{3}}+x_{4}F_{x_{3}x_{3}})-5F_{x_{2}}(1+2F_{x_{3}})-5F_{xx_{3}}(2+F_{x_{3}})]$
$\displaystyle+W_{1}=0,$ (49)

where $W_{1}=W_{1}(x,y,t,x_{0},x_{1},\ldots,x_{4})$ is $w_{x5}$ independent.
Eliminating the coefficient of $w_{x5}$ yields

$\displaystyle G_{1}$ $\displaystyle=$ $\displaystyle
5(F_{x_{3}}+2)(\frac{1}{2}x_{4}F_{x_{3}x_{3}}+x_{3}F_{x_{2}x_{3}}+x_{2}F_{x_{1}x_{3}}+x_{1}F_{x_{0}x_{3}})x_{4}+5x_{4}F_{x_{2}}(1+2F_{x_{3}})$
(50) $\displaystyle+$ $\displaystyle 5x_{4}F_{xx_{3}}(2+F_{x_{3}})+G_{2},$

with $G_{2}\equiv G_{2}(x,y,t,x_{0},x_{1},x_{2},x_{3})$ being a function of
$\\{x,y,t,x_{0},x_{1},x_{2},x_{3}\\}$.

Similarly, substituting the decomposition (8) and (9) with $m=3,\ n=5$, the
results (48) and (50) into the consistent condition (10), we have

$\displaystyle
5w_{x7}(1-F_{x_{3}})^{2}(x_{1}F_{x_{0}x_{3}}+x_{2}F_{x_{1}x_{3}}+x_{3}F_{x_{2}x_{3}}+{x_{4}}F_{x_{3}x_{3}}+F_{xx_{3}})+\Gamma=0,$
(51)

where $\Gamma=\Gamma(x,y,t,x_{0},\ldots,x_{6})$ is a $w_{x7}$ independent
function of the lower-order differentiations of $w$ with respect to $x$.
Vanishing the coefficient of $w_{x7}w_{x4}$ in (51), we get

$\displaystyle
F=F_{1}(x,y,t,x_{0},x_{1},x_{2})x_{3}+H(x,y,t,x_{0},x_{1},x_{2}).$ (52)

Substituting (52) into (51) and requiring the coefficient of $w_{x7}$ being
zero result $F_{1}(x_{0},\ x_{1},\ x_{2})=H_{1}(y,t)$. Thus, we have

$\displaystyle F=H_{1}(y,t)x_{3}+H(x,y,t,x_{0},x_{1},x_{2}).$ (53)

Taking account of (53), (49) becomes

$\displaystyle
w_{x4}\big{[}G_{2x_{3}}-5(H_{1}+2)x_{3}H_{x_{2}x_{2}}-5(H_{1}+2)x_{2}H_{x_{1}x_{2}}+5x_{1}(3-H_{1}H_{x_{0}x_{2}}-3H_{1}-2H_{x_{0}x_{2}})$
$\displaystyle-10H_{1}H_{x_{1}}-5H_{x_{2}}^{2}-5H_{x_{1}}-5H_{1}H_{xx_{2}}-10H_{xx_{2}}\big{]}+W_{2}=0$
(54)

with $W_{2}=W_{2}(x,y,t,x_{0},x_{1},x_{2},x_{3})$. Vanishing the coefficient
of $w_{x4}$ in (54) leads to

$\displaystyle G_{2}$ $\displaystyle=$ $\displaystyle
5\big{[}\frac{H_{1}+2}{2}x_{3}H_{x_{2}x_{2}}+(H_{1}+2)x_{2}H_{x_{1}x_{2}}-x_{1}(3-H_{1}H_{x_{0}x_{2}}-3H_{1}-2H_{x_{0}x_{2}})+2H_{1}H_{x_{1}}$
(55)
$\displaystyle+H_{x_{2}}^{2}+H_{x_{1}}+H_{1}H_{xx_{2}}+2H_{xx_{2}}\big{]}x_{3}+J,$

where $J=J(x,y,t,x_{0},x_{1},x_{2})$. Up to now, the decomposition relation is
simplified to

$\displaystyle w_{y}$ $\displaystyle=H_{1}w_{x3}+H,$ (56) $\displaystyle
w_{t}$
$\displaystyle=(5H_{1}+5H_{1}^{2}-1)w_{x_{5}}+5H_{x_{2}}(1+2H_{1})w_{x_{4}}+5\big{[}\frac{H_{1}+2}{2}w_{x_{3}}H_{x_{2}x_{2}}+(H_{1}+2)w_{x_{2}}H_{x_{1}x_{2}}$
(57) $\displaystyle-
w_{x_{1}}(3-H_{1}H_{x_{0}x_{2}}-3H_{1}-2H_{x_{0}x_{2}})+2H_{1}H_{x_{1}}+H_{x_{2}}^{2}+H_{x_{1}}+H_{1}H_{xx_{2}}+2H_{xx_{2}}\big{]}w_{x_{3}}+J$

with three undetermined functions
$H_{1}=H_{1}(y,t),H=H(x,y,t,x_{0},x_{1},x_{2})$ and
$J=J(x,y,t,x_{0},x_{1},x_{2})$.

Inserting (56) and (57) into the potential BKP equation (5) and the consistent
condition (10), then, vanishing the coefficients of $w_{xk}$ for $k\geq 3$
leaves the set of determining equations on $\\{H,H_{1},J\\}$. Solving this set
of equations, for nontrivial solutions we shall have the following several
cases for the unknowns.

Case 1. When $H_{1}=0$, further calculation then leads to the expression

$\displaystyle\left\\{\begin{aligned}
H&=Mw_{x}+\frac{y}{30}(M+m_{2})_{t}+\frac{1}{3}m_{1}x-\frac{1}{2}M^{2}+m_{3},M\equiv
m_{1}y+m_{2},\\\
J&=15(M-w_{x})w_{x}^{2}+(\frac{1}{2}y^{2}m_{1t}+ym_{2t}+5m_{1}x-\frac{5}{2}M^{2}+15m_{3})w_{x}+5m_{1}w+\frac{y^{2}}{90}(m_{1}y+3m_{2})_{tt}\\\
&-\frac{y}{3}(My-x)m_{1t}+\frac{1}{3}(x-2M_{y})m_{2t}-\frac{10}{3}m_{1}(Mx+3m_{3}y)+m_{3t}y+m_{4}\end{aligned}\right.$
(58)

with $m_{i},i=1,2,\dots,4$ being arbitrary functions of $t$.

Case 2. Taking $H_{1}=1$ in the result of this case, then one can find three
different solutions.

The first one is

$\displaystyle\left\\{\begin{aligned}
H&=\frac{3w_{xx}^{2}}{2W}+\frac{3}{2}w_{x}^{2}-\frac{27}{4}M^{2}+\frac{3}{2}m_{1}x+\frac{3y}{20}(m_{1}y+2m_{2})_{t}+m_{3},\\\
J&=\frac{45}{2}w_{x}^{3}+\frac{3}{4}[30m_{1}x-135M^{2}+3y(m_{1}y+2m_{2})_{t}+20m_{3}]w_{x}+\frac{45}{2}m_{1}w-\frac{3}{2}(45m_{1}M-m_{1t}y\\\
&-m_{2t})x+\frac{y^{2}}{20}(m_{1}y+3m_{2})_{tt}-\frac{27}{4}y(m_{1}y+2m_{2})_{t}M+y(m_{3t}-45m_{1}m_{3})+m_{4}\\\
&+\frac{45}{2W}[(3M+w_{x})w_{xx}^{2}]+\frac{315}{2W^{3}}w_{xx}^{4}+\frac{405}{4}M^{3},\end{aligned}\right.$
(59)

where $M\equiv m_{1}y+m_{2},W\equiv 3M-2w_{x}$ and $m_{i},i=1,2,3,4$ are
arbitrary functions of $t$.

The second one is

$\left\\{\begin{aligned}
H&=3w_{x}^{2}+av_{x}w_{x}+\frac{a^{2}}{6}v_{x}^{2}-\frac{1}{3}m_{0}(x+m_{0}y^{2})-\frac{m_{0t}y^{2}}{30}+m_{1}y+m_{2},\\\
J&=5a(av+3w)_{x}v_{xxx}+15(av+3w)_{xx}w_{xx}+\frac{15}{2}w_{x}[(av+3w)_{x}^{2}+3w_{x}^{2}]+\frac{5a^{2}}{2}v_{xx}^{2}+w_{x}(15m_{2}\\\
&+15m_{1}y-5m_{0}^{2}y^{2}-5m_{0}x-\frac{1}{2}m_{0t}y^{2})-5m_{0}w-\frac{xy}{3}(m_{0t}+10m_{0}^{2})+5m_{1}x-\frac{y^{3}}{90}(m_{0tt}\\\
&+30m_{0}m_{0t}+100m_{0}^{3})+\frac{y^{2}}{2}(m_{1t}+10m_{0}m_{1})+(m_{2t}+10m_{0}m_{2})_{y}+m_{4}\end{aligned}\right.$
(60)

with $v$ satisfying

$\left\\{\begin{aligned}
v_{y}=&v_{xxx}-\frac{a}{2}v_{x}^{2}+\frac{m_{0}}{a}x+n_{y},\\\
v_{t}=&9v_{xxxxx}-15av_{x}v_{xxx}-\frac{15a}{2}v_{xx}^{2}+\frac{5a^{2}}{2}v_{x}^{3}+[15m_{2}+15m_{1}y-(5m_{0}^{2}+\frac{1}{2}m_{0t})y^{2}-5m_{0}x]v_{x}\\\
&-5m_{0}v+[(10m_{0}^{2}+m_{0t})y-15m_{1}]\frac{x}{a}+\frac{y^{3}}{6a}m_{0}(10m_{0}^{2}+m_{0t})-\frac{15y^{2}}{2a}m_{0}m_{1}-\frac{15y}{a}m_{0}m_{2}\\\
&+5m_{0}n+n_{t}+m_{3}.\end{aligned}\right.$ (61)

The third one is

$\left\\{\begin{aligned}
H&=-v_{xxx}+v_{y}+\frac{3}{2}(w+v)_{xx}(w-v)+\frac{3}{2}(w_{x}^{2}-v_{x}^{2})+\frac{1}{4}[(w-v)^{3}]_{x},p\equiv
w-v,\\\
J&=v_{t}-9v_{xxxxx}+\frac{15}{2}pv_{xxxx}+(\frac{15}{2}w_{x}-\frac{105}{2}v_{x})v_{xxx}-45v_{xx}^{2}+\frac{45}{2}pv_{x}v_{xx}+15pv_{xy}+45w_{xx}^{2}\\\
&+(\frac{225}{2}pw_{x}-45pv_{x}+\frac{45}{4}p^{3})w_{xx}+\frac{45}{4}v_{x}^{2}p_{x}+\frac{135}{4}p^{2}w_{x}p_{x}+15p_{x}v_{y}+\frac{135}{4}w_{x}^{2}p_{x}+\frac{45}{16}p^{4}p_{x},\end{aligned}\right.$
(62)

in which case, the decomposition solution depends on another solution $v$ of
the potential BKP equation (5).

Case 3. $H_{1}=-\frac{1}{2}$. In this case, the functions $H$ and $J$ are
fixed as

$\left\\{\begin{aligned}
H&=-\frac{3}{2}w_{x}^{2}+6Mw_{x}+\frac{y}{10}(m_{1}y+2m_{2})_{t}+m_{1}x-6M^{2}+m_{3},M\equiv
m_{1}y+m_{2},\\\
J&=-\frac{45}{4}(w_{xx}^{2}+2w_{x}^{3})+15m_{1}w+\frac{3}{2}y(m_{1}y+2m_{2})_{t}w_{x}+15(6M^{2}+m_{1}x+m_{3})w_{x}+\frac{y^{2}}{30}(m_{1}y\\\
&+3m_{2})_{tt}-3(y^{2}m_{1t}+10m_{1}x+2ym_{2t})M+M_{t}x+(m_{3t}-30m_{1}m_{3})y-60M^{3}+m_{4},\end{aligned}\right.$
(63)

where $m_{i},i=1,2,3,4$ are arbitrary functions of $t$.

By substituting these solutions (58)–(63) into the decomposition relations
(56) and (57), Theorems 1-5 are proved. ∎


---|---|---
| S | G&S | M | FE | BS | T | G&H | S | G&S | M | FE | Sc | T | H&G | L | So
Mower et al. (2011), (2011) | x | x |  |  |  | x |  | x |  |  |  | x |  |  |  |
Pop et al. (2013), (2013) | x | x |  |  |  |  |  | x |  |  | x |  |  | x |  |
Soleiman et al. (2014), (2014) | x | x | x |  |  |  |  | x | x |  |  |  | x |  |  |
Khosla et al. (2015), (2015) | x |  |  | x |  | x |  | x |  | x |  | x |  |  |  |
Shimaya et al. (2016), (2016) | x |  |  |  |  |  | x | x |  |  |  |  |  | x |  |
Bekele et al. (2016), (2016) | x |  |  |  | x |  | x | x | x |  | x |  |  | x |  |
Amanatiadis et al. (2017), (2017) | x | x |  |  |  |  |  | x | x |  |  |  |  |  |  |
Tanaka et al. (2017), (2017) | x |  |  | x |  |  |  | x |  |  |  | x |  |  |  |
Desideri et al. (2018), (2018) | x | x |  |  |  |  |  | x | x | x |  |  |  |  |  | x
Axelsson et al. (2019), (2019) | x | x |  |  |  |  |  | x | x |  |  | x |  |  | x | x
Cao et al. (2019), (2019) | x |  |  |  |  |  |  | x | x | x |  |  |  | x |  |
So et al. (2019), (2019) | x | x |  |  |  |  |  | x | x |  |  | x |  |  |  |
Shimaya et al. (2019), (2019) | x |  |  |  |  |  |  | x | x |  |  |  |  | x |  |
Zhang et al. (2019), (2019) | x | x |  |  |  |  |  | x | x | x |  |  |  | x |  |
Cervera et al. (2019), (2019) | x |  |  |  |  | x |  | x | x |  | x |  |  |  |  | x
So et al. (2020), (2020) | x |  |  |  |  |  |  | x | x | x |  |  |  |  |  |
Fachantidis et al. (2020), (2020) | x | x |  |  |  | x | x | x |  |  | x |  |  | x |  |
Marino et al. (2020), (2020) | x |  |  |  |  |  |  | x | x | x |  |  |  | x |  |
Chung (2021), (2021) | x | x |  |  |  |  | x | x | x | x |  |  |  | x |  |
van Otterdijk et al. (2020), (2020) | x | x |  |  |  |  | x | x | x |  |  |  |  |  |  |
Spitale et al. (2020b), (2020b) | x |  |  |  |  |  |  | x |  |  | x |  |  |  |  |
Huijnen et al. (2021), (2021) | x | x |  |  |  |  |  | x | x | x |  |  |  |  |  | x
Smith et al. (2021), (2021) | x |  |  |  |  |  |  | x |  |  |  |  |  |  |  | x
Catania et al. (2021), (2021) | x |  |  |  |  |  |  | x |  |  |  |  |  |  |  | x
Count | 24 | 12 | 1 | 2 | 1 | 4 | 5 | 24 | 15 | 8 | 5 | 5 | 1 | 9 | 1 | 6

Table 6. Demographics of the participants involved in the surveyed study
including their age, gender, and diagnosis. If a row has not checks it means
that we did not retrieve any information about that specific variable in the
corresponding paper.
Legend: neurodevelopmental disorder (NDD), typically-developed (TD), female
(F), male (M), autism spectrum disorder (ASD), intellectual disability (ID),
developmental language disorder (DLD).

Ref |  Number |  Age |  Gender |  Diagnosis
---|---|---|---|---
| NDD | TD | Pre | Grade | Teen | Young | Adult | F - NDD | F - TD | M - NDD | M - TD | ASD | ID | DLD
|  |  | schooler | schooler |  | adult |  |  |  |  |  |  |  |
Mower et al. (2011), (2011) | 2 |  |  | x |  |  |  |  |  | 2 |  | x |  |
Pop et al. (2013), (2013) | 20 |  | x | x |  |  |  | 10 |  | 10 |  | x |  |
Soleiman et al. (2014), (2014) | 3 | 2 | x |  |  |  |  |  |  |  |  | x |  |
Khosla et al. (2015), (2015) | 2 |  |  |  |  |  | x | 1 |  | 1 |  | x |  |
Shimaya et al. (2016), (2016) | 3 |  |  |  | x |  |  | 2 |  | 1 |  | x |  |
Bekele et al. (2016), (2016) | 12 |  |  |  | x |  |  |  |  | 12 |  | x |  |
Amanatiadis et al. (2017), (2017) | 2 |  |  | x |  |  |  |  |  |  |  | x |  |
Tanaka et al. (2017), (2017) | 10 | 18 |  | x | x | x |  |  | 3 | 10 | 15 | x |  |
Desideri et al. (2018), (2018) | 2 |  |  | x |  |  |  |  |  | 2 |  | x | x |
Axelsson et al. (2019), (2019) | 10 |  |  |  |  |  |  |  |  |  |  | x |  |
Cao et al. (2019), (2019) | 15 | 15 | x |  |  |  |  | 2 | 3 | 13 | 12 | x |  |
So et al. (2019), (2019) | 23 |  |  | x |  |  |  | 3 |  | 20 |  | x |  |
Shimaya et al. (2019), (2019) | 2 | 32 |  |  | x | x | x | 2 | 16 | 16 |  | x |  |
Zhang et al. (2019), (2019) | 20 | 20 |  | x |  |  |  | 2 | 3 | 18 | 17 | x |  |
Cervera et al. (2019), (2019) | 14 |  | x |  |  |  |  |  |  |  |  | x |  |
So et al. (2020), (2020) | 23 |  | x |  |  |  |  | 3 |  | 20 |  | x |  |
Fachantidis et al. (2020), (2020) | 4 |  |  | x |  |  |  |  |  | 4 |  | x |  |
Marino et al. (2020), (2020) | 14 |  | x | x |  |  |  | 2 |  | 12 |  | x |  |
Chung (2021), (2021) | 15 |  |  | x |  |  |  | 2 |  | 13 |  | x |  |
van Otterdijk et al. (2020), (2020) | 6 |  | x | x |  |  |  | 1 |  | 5 |  | x |  |
Spitale et al. (2020b), (2020b) | 3 | 14 |  | x |  |  |  |  | 8 | 3 | 6 |  |  | x
Huijnen et al. (2021), (2021) | 9 |  |  | x |  |  |  | 1 |  | 8 |  | x |  |
Smith et al. (2021), (2021) | 43 |  |  |  |  |  | x |  |  |  |  |  | x |
Catania et al. (2021), (2021) | 9 |  | x | x | x |  |  | 2 |  | 7 |  | x |  |
Count | 24 | 6 | 8 | 14 | 5 | 2 | 3 | 13 | 5 | 19 | 4 | 22 | 2 | 1
Average | 11.08 | 16.83 |  |  |  |  |  | 2.54 | 6.60 | 9.32 | 12.50 |  |  |
St. Deviation | 9.59 | 8.88 |  |  |  |  |  | 2.24 | 5.08 | 6.20 | 4.15 |  |  |

Table 7. Agent’s goals in terms of assessing and training and the
corresponding skills addressed, such as communication and social, emotional,
joint attention, touch confidence, and sensorial stimulation. If a row has not
checks it means that we did not retrieve any information about that specific
variable in the corresponding paper.

Ref |  Agent’s goal |  Skills addressed
---|---|---
| Assessing | Training | Communication | Emotional | Joint | Touch | Sensorial
|  |  | and social | control | attention | confidence | stimulation
Mower et al. (2011), (2011) |  | x | x | x |  |  |
Pop et al. (2013), (2013) |  | x | x |  |  |  |
Soleiman et al. (2014), (2014) |  | x | x |  |  |  |
Khosla et al. (2015), (2015) | x |  | x |  |  |  |
Shimaya et al. (2016), (2016) |  | x | x |  |  |  |
Bekele et al. (2016), (2016) |  | x | x | x |  |  |
Amanatiadis et al. (2017), (2017) |  | x | x |  | x |  |
Tanaka et al. (2017), (2017) |  | x | x |  |  |  |
Desideri et al. (2018), (2018) |  | x | x |  |  |  |
Axelsson et al. (2019), (2019) |  | x | x |  | x |  |
Cao et al. (2019), (2019) | x |  | x |  | x |  |
So et al. (2019), (2019) |  | x | x |  |  |  |
Shimaya et al. (2019), (2019) |  | x | x |  |  |  |
Zhang et al. (2019), (2019) |  | x | x |  |  |  |
Cervera et al. (2019), (2019) |  | x | x |  |  |  |
So et al. (2020), (2020) |  | x | x |  | x |  |
Fachantidis et al. (2020), (2020) | x |  | x | x |  |  |
Marino et al. (2020), (2020) |  | x |  | x |  |  |
Chung (2021), (2021) |  | x | x |  |  |  |
van Otterdijk et al. (2020), (2020) |  | x | x |  | x |  |
Spitale et al. (2020b), (2020b) | x | x | x |  |  |  |
Huijnen et al. (2021), (2021) | x |  | x |  |  | x |
Smith et al. (2021), (2021) |  | x | x |  |  |  |
Catania et al. (2021), (2021) | x |  | x |  |  |  | x
Count | 6 | 19 | 23 | 4 | 5 | 1 | 1

Table 8. Study designs - such as random control trail, within-subject,
between-subject studies - and duration of the empirical study reported in the
surveyed papers. If a row has not checks it means that we did not retrieve any
information about that specific variable in the corresponding paper.
Legend: typically-developed (TD), neurodevelopmental disorder (NDD), random
control trail (RCT).

Ref |  Study Design |  Study Duration
---|---|---
| TD vs. | RCT | Within- | Between- | Single- | Longitudinal | Exploratory | Pre- | Number | Duration
| NDD |  | subject | subject | case |  |  | Post- | sessions | (day)
Mower et al. (2011), (2011) |  |  |  |  |  |  | x |  | 4 |
Pop et al. (2013), (2013) |  | x |  | x |  |  |  |  | 6 | 6
Soleiman et al. (2014), (2014) | x |  |  |  | x |  | x |  | 1 | 1
Khosla et al. (2015), (2015) |  |  |  |  | x | x |  |  |  | 270
Shimaya et al. (2016), (2016) |  |  |  |  | x |  |  | x | 4 | 4
Bekele et al. (2016), (2016) |  | x |  | x |  |  |  | x | 5 |
Amanatiadis et al. (2017), (2017) |  |  |  |  | x |  |  |  | 1 | 1
Tanaka et al. (2017), (2017) | x |  |  |  |  |  | x | x | 1 | 1
Desideri et al. (2018), (2018) |  |  |  |  | x |  |  | x | 12 | 28
Axelsson et al. (2019), (2019) |  |  | x |  |  |  |  |  | 1 | 1
Cao et al. (2019), (2019) | x | x | x |  |  |  |  |  | 1 | 1
So et al. (2019), (2019) |  | x |  | x |  |  |  | x | 9 | 63
Shimaya et al. (2019), (2019) |  |  |  |  | x |  |  |  | 4 | 4
Zhang et al. (2019), (2019) | x |  |  |  |  |  | x |  | 1 | 1
Cervera et al. (2019), (2019) |  | x |  | x |  | x |  | x | 40 | 270
So et al. (2020), (2020) |  | x |  | x |  | x |  | x | 9 | 63
Fachantidis et al. (2020), (2020) |  |  | x |  | x |  |  |  | 8 | 150
Marino et al. (2020), (2020) |  | x |  | x |  |  |  | x | 6 | 84
Chung (2021), (2021) |  |  |  |  | x | x |  | x | 12 | 84
van Otterdijk et al. (2020), (2020) |  | x |  |  |  | x |  |  | 20 | 140
Spitale et al. (2020b), (2020b) | x | x | x |  | x |  |  |  | 3 | 21
Huijnen et al. (2021), (2021) |  | x | x |  |  |  |  |  | 4 | 28
Smith et al. (2021), (2021) |  | x |  | x |  | x |  | x | 12 | 84
Catania et al. (2021), (2021) |  |  |  |  | x |  | x |  | 2 | 21
Count | 5 | 11 | 5 | 7 | 10 | 6 | 5 | 10 | 23 | 22
Average |  |  |  |  |  |  |  |  | 7.22 | 60.27
St. Deviation |  |  |  |  |  |  |  |  | 8.44 | 79.66

Table 9. Study intervention type, defined as the type of interaction for the
therapeutic treatment (i.e., task-based or free interaction) and the data
collection methodology adopted by the authors in the surveyed papers. If a row
has not checks it means that we did not retrieve any information about that
specific variable in the corresponding paper.

Ref |  Intervention type |  Data collection
---|---|---
| Task-based | Free | Interview | Questionnaire | Real-time | In batch | Real-time | In batch
|  | interaction |  |  | automated | automated | manual | manual
|  |  |  |  | data logging | data logging | data logging | data logging
Mower et al. (2011), (2011) | x |  | x |  | x |  |  | x
Pop et al. (2013), (2013) | x |  |  | x |  |  |  | x
Soleiman et al. (2014), (2014) |  | x |  |  |  |  |  | x
Khosla et al. (2015), (2015) |  | x |  |  | x |  |  | x
Shimaya et al. (2016), (2016) |  | x |  |  |  | x |  |
Bekele et al. (2016), (2016) | x |  |  |  | x |  |  |
Amanatiadis et al. (2017), (2017) | x |  |  | x |  |  | x |
Tanaka et al. (2017), (2017) | x |  |  |  | x |  |  |
Desideri et al. (2018), (2018) | x |  |  |  | x |  |  | x
Axelsson et al. (2019), (2019) | x |  |  | x |  |  |  | x
Cao et al. (2019), (2019) | x |  |  |  | x |  |  |
So et al. (2019), (2019) | x |  |  |  |  |  |  | x
Shimaya et al. (2019), (2019) |  | x |  | x |  | x |  |
Zhang et al. (2019), (2019) | x |  | x |  | x |  |  |
Cervera et al. (2019), (2019) | x |  |  |  |  |  |  | x
So et al. (2020), (2020) | x |  |  | x |  |  |  |
Fachantidis et al. (2020), (2020) | x |  | x |  |  |  | x |
Marino et al. (2020), (2020) | x |  |  | x |  |  |  |
Chung (2021), (2021) | x |  |  |  |  |  |  | x
van Otterdijk et al. (2020), (2020) | x |  |  |  |  |  |  | x
Spitale et al. (2020b), (2020b) | x |  |  | x |  |  |  | x
Huijnen et al. (2021), (2021) |  | x | x |  |  |  |  | x
Smith et al. (2021), (2021) |  | x |  |  |  |  |  | x
Catania et al. (2021), (2021) | x |  | x | x | x |  |  |
Count | 18 | 6 | 5 | 8 | 8 | 2 | 2 | 13