# On the Perturbed Second Painlevé Equation

Joshua Holroyd  School of Mathematics and Statistics F07, University of
Sydney, NSW 2006 Australia<EMAIL_ADDRESS>and  Nalini Joshi
School of Mathematics and Statistics F07, University of Sydney, NSW 2006
Australia<EMAIL_ADDRESS>
###### Abstract.

We consider a perturbed version of the second Painlevé equation
($\textrm{P}_{\textrm{II}}$), which arises in applications, and show that it
possesses solutions analogous to the celebrated Hastings-McLeod and
tritronquée solutions of $\textrm{P}_{\textrm{II}}$. The Hastings-McLeod-type
solution of the perturbed equation is holomorphic, real-valued and positive on
the whole real-line, while the tritronquée-type solution is holomorphic in a
large sector of the complex plane. These properties also characterise the
corresponding solutions of $\textrm{P}_{\textrm{II}}$ and are surprising
because the perturbed equation does not possess additional distinctive
properties that characterise $\textrm{P}_{\textrm{II}}$, particularly the
Painlevé property.

###### 2020 Mathematics Subject Classification:

33E17,34E10,41A60

JH’s ORCID ID is 0000-0003-1354-1271.

NJ’s ORCID ID is 0000-0001-7504-4444. NJ’s research was supported by an
Australian Research Council Discovery Project #DP210100129.

_Dedicated to Sir Michael V. Berry on the occasion of his
$\textit{80}^{\textit{th}}$ birthday._

##  1\. Introduction

The second Painlevé Equation ($\textrm{P}_{\textrm{II}}$) has a rich history
as a mathematical model, arising in a very wide range of applications (see,
for example,
clarkson2006painleve,forrester2015painleve,tracy1994level,tracy1999random). In
many of these applications, the particular case of $\textrm{P}_{\textrm{II}}$
that arises is the case $\alpha=0$ of

$\textrm{P}_{\textrm{II}}:\qquad y^{\prime\prime}(x)=2y(x)^{3}+xy(x)+\alpha,$

where primes denote differentiation in $x$ and $\alpha$ is a given parameter.
In this paper, we consider a perturbed version of $\textrm{P}_{\textrm{II}}$
with $\alpha=0$:

$\textrm{P}_{\textrm{II}}(\mu):\qquad
y^{\prime\prime}(x)=2y(x)^{3}+x^{\mu}y(x),\qquad
y:\mathbb{C}\rightarrow\mathbb{C},\qquad\mu\in\mathbb{Z}^{+}.$

The main results of this paper show that there exists (i) a holomorphic, real-
valued, positive solution of $\textrm{P}_{\textrm{II}}(\mu)$ on the real-line
$x\in\mathbb{R}$; and (ii) a solution that is asymptotic to a power-series
expansion in $x$, as $|x|\to\infty$, in a sector of angular extent
$4\pi/(\mu+2)$ in $\mathbb{C}$. These properties mirror two well-known
solutions of $\textrm{P}_{\textrm{II}}$, called the Hastings-McLeod solution
[19] and Boutroux’s tritronquée solution [7] respectively, which are
distinguished by their global properties.

$\textrm{P}_{\textrm{II}}$ belongs to a class of six ordinary differential
equations (ODEs), called the Painlevé equations, whose solutions are known to
have only poles as movable singularities conte2012painleve,i:56. Every non-
rational solution of $\textrm{P}_{\textrm{II}}$ is known to possess an
infinite number of movable poles [20], which are typically distributed across
the interior of each sector $S_{n}$ of the complex plane given by [7]

$S_{n}=\bigl{\\{}x\in\mathbb{C}\bigm{|}(n-1)\pi/3\leq\arg{x}\leq
n\pi/3\bigr{\\}},\ n\in\mathbb{Z},\mod{6}.$ (1.1)

(Here, the word “movable” refers to the location of each pole depending on
initial conditions.) Therefore, solutions that are free of movable poles
across one or more sectors $S_{n}$ are uncommon, while those free of poles in
a domain containing the real line are of particular interest for physical
applications. The tritronquée solution is free of poles in a wide sector of
the complex plane, given by four contiguous sectors $S_{n}$, while the
Hastings-McLeod solution is free of poles and analytic on the whole real-line.

The Hastings-McLeod solution now plays a critical role in modern random matrix
theory, due to its appearance in the the Tracy-Widom distribution, describing
the distribution of the largest eigenvalue of Gaussian unitary ensembles of
random matrices, as the size of each matrix approaches infinity. The Hastings-
McLeod solution also arises in the analogous distribution functions for the
Gaussian orthogonal and symplectic ensembles [30, 9, 13].

There are many more physical models in which $\textrm{P}_{\textrm{II}}$
arises. We mention, in particular, fluid dynamics [14, 18, 9, 26, 1, 8],
mathematical physics [27, 29, 31, 28, 15], and electrodynamics [5, 4]. In
particular, the second Painlevé Equation arises in a model of steady one-
dimensional two-ion electro-diffusion [4]. A review may be found in [11]. In
connection with plasma physics, de Boer and Ludford [12] posed the question of
the existence of a solution of a generalised form of
$\textrm{P}_{\textrm{II}}$ with specific boundary conditions. Hastings and
McLeod [19] not only showed the existence of a solution to this problem, but
they also proved it to be unique; this is the now-famous Hastings-McLeod
solution of $\textrm{P}_{\textrm{II}}$. It is natural in many of the physical
contexts to consider small perturbations of the model of interest. We describe
such a perturbation of Bass’ electro-diffusion model in Appendix A and show
how $\textrm{P}_{\textrm{II}}(\mu)$ arises naturally.

A perturbed version of the first Painlevé equation has been previously
considered joshi2003PImu,oldedaalhuis22. Tronquée and tritronquée solutions of
perturbed $\textrm{P}_{\textrm{I}}$ were shown to exist (and be unique in the
tritronquée case) in joshi2003PImu. Meanwhile, oldedaalhuis22 studies the full
exponentially-improved asymptotic expansion corresponding to a formal series
solution of perturbed $\textrm{P}_{\textrm{I}}$. These tronquée solutions are
uniquely characterised by the Stokes multiplier, which is an arbitrary
constant multiplying the exponentially small terms. Asymptotic approximations
for the locations of singularities along boundaries of validity are deduced in
terms of the Stokes multipliers oldedaalhuis22.

In the case of $\textrm{P}_{\textrm{II}}(\mu)$, tronquée solutions are
associated with a formal power-series solution $y_{f}\sim(-x^{\mu}/2)^{1/2}$
as $|x|\rightarrow\infty$ (given in full in Section 3), we mention that the
(singly) exponentially small perturbation is given by

$y\sim
y_{f}+kx^{-\mu/4}\exp{\left(\frac{2}{\mu+2}\left(-2x^{\mu+2}\right)^{1/2}+\textit{o}(1)\right)},$

where the square root branch depends on the sector of the complex plane, and
$k$ is the free constant multiplying exponentially small behaviour. The
tritronquée solution, which we study in Section 3, corresponds to the unique
$k=0$ case of this expansion.

###  1.1. Main results

We identify a tritronquée solution of $\textrm{P}_{\textrm{II}}(\mu)$, through
asymptotic expansions of solutions as $|x|\to\infty$; showing that a true
solution asymptotic to such a series in the sector $|\arg(x)|\leq
2\pi/(\mu+2)$ exists and is unique. These results are given in Propositions
3.4 and 3.7. We also consider a boundary-value problem on the real-line; as
$x\to-\infty$, the solution is asymptotic to a power-series expansion, and as
$x\to+\infty$, the solution decays to zero exponentially fast. We show that
this problem has a unique solution in the case of $\mu=2k+1$,
$k\in\mathbb{N}$, which generalises the Hastings-McLeod solution. We give this
result in Theorem 4.2.

###  1.2. Background

The Painlevé equations were found in a search for ODEs that defined new higher
transcendental functions a century ago. Here the word “new” refers to these
functions not being able to be expressed in terms of any classical operations
applied to previously known such functions (we refer to definitions of these
terms in [32]). Both the Hastings-McLeod and tritronquée solutions are
examples of such transcendental functions.

For special values of the parameter $\alpha$, $\textrm{P}_{\textrm{II}}{}$
admits hierarchies of rational solutions [2, 3, 16, 10] and special solutions
related to the classical Airy functions [17]. Beyond special solutions,
transcendental solutions exist that vanish as $x\to+\infty$ on the real line,
providing another relation to the Airy functions in the case when $\alpha=0$.
The Hastings-McLeod solution first attracted attention in the asymptotic
analysis of such solutions.

The asymptotic analysis of the Painlevé equations was initiated by Boutroux
[7], who also briefly considered the generalised equation
$\textrm{P}_{\textrm{II}}(\mu)$. Boutroux showed that the transformed version
of $\textrm{P}_{\textrm{II}}$

$u^{\prime\prime}(z)=2u^{3}(z)+u(z)+\frac{2\alpha}{3z}-\frac{u^{\prime}(z)}{z}+\frac{u(z)}{9z^{2}},$

is more amenable to asymptotic analysis, where

$y(x)=x^{1/2}u(z)\qquad\text{and}\qquad z=\frac{2}{3}x^{3/2}.$

The limit $|x|\to\infty$ now becomes $|z|\to\infty$, and direct observation
shows that a (Jacobi) elliptic function describes the leading-order behaviour,
solving the ODE

$w^{\prime\prime}(z)=2w^{3}(z)+w(z).$

Boutroux went on to describe how lines of movable poles curve around to
approach the boundaries of the sectors $S_{n}$ described by Equation (1.1).

###  1.3. Outline

In Section 2, we describe decaying solutions of
$\textrm{P}_{\textrm{II}}(\mu)$ as $|x|\rightarrow\infty$ on the real-line. We
prove the existence and uniqueness of the analogue of Boutroux’s tritronquée
solution for $\textrm{P}_{\textrm{II}}(\mu)$ and its asymptotic validity in a
broad sector of the complex plane in Section 3. In Section 4, we prove the
existence and uniqueness of a natural generalisation of the Hastings-McLeod
solution. The paper’s results are summarised in Section 5. Finally, we show
how $\textrm{P}_{\textrm{II}}(\mu)$ arises in a physical setting in Appendix
A.

##  2\. Asymptotics of decaying solutions on the real-line

Since $\mu>0$, the decaying solutions of $\textrm{P}_{\textrm{II}}(\mu)$
satisfy the leading order equation

$y^{\prime\prime}(x)\sim x^{\mu}y(x),$

as $x\to\pm\infty$ on the real-line. We analyse the asymptotic behaviours of
$y(x)$, which satisfy this leading-order equation.

The linear ODE $\phi^{\prime\prime}(x)=x^{\mu}\phi(x)$ is a generalised Airy
equation and is satisfied by two linearly independent functions

$\begin{split}f(x)=\sqrt{\frac{2x}{\pi\alpha}}K_{1/(2\alpha)}\left(\frac{x^{\alpha}}{\alpha}\right)\qquad\text{and}\qquad
g(x)=\sqrt{\frac{2\pi
x}{\alpha}}I_{1/(2\alpha)}\left(\frac{x^{\alpha}}{\alpha}\right),\end{split}$
(2.1)

where $\alpha=(\mu+2)/2$, and $I_{\nu}$ and $K_{\nu}$ are the modified Bessel
functions of the first and second kind respectively. The functions $f(x)$ and
$g(x)$ are analogues of the well known Airy functions Ai$(x)$ and Bi$(x)$,
reducing to scaled versions of these when $\alpha=3/2$ (or equivalently
$\mu=1$).

Recalling that $\mu$ is positive and real, we find that

$y(x)\sim
kf(x)=kx^{-\mu/4}\exp\left(-\frac{2}{\mu+2}x^{(\mu+2)/2}+\textit{o}(1)\right),\
{\rm as}\ x\to+\infty,$ (2.2)

for some arbitrary constant $k$. As is well known for Airy functions, a
solution $y(x)$ of $\textrm{P}_{\textrm{II}}(\mu)$ that decays to zero as
$|x|\rightarrow\infty$ may not be exponentially small in other directions. In
particular, for every odd integer $\mu$, the negative real semi-axis is an
anti-Stokes line [6], meaning that the exponential contribution is purely
oscillatory as $|x|\rightarrow\infty$. Given an odd positive integer $\mu>1$,
standard asymptotic techniques lead to

$y(x)\sim
c_{1}|x|^{-\mu/4}\sin\left(\frac{2}{\mu+2}|x|^{(\mu+2)/2}-c_{2}+\textit{o}(1)\right)\qquad\text{as}\qquad
x\rightarrow-\infty,$ (2.3)

for arbitrary constants $c_{1}$ and $c_{2}$. In the case $\mu=1$, more
detailed results were described by Hastings and McLeod [19].

##  3\. Tritronquée Solutions

The main object of study in this section is an asymptotic series, which solves
$\textrm{P}_{\textrm{II}}(\mu)$ given in Equation (3.2). We prove the
existence and uniqueness of a true solution that is asymptotic to this series
for $|\arg(x)|\leq 2\pi/(\mu+2)$ as $|x|\rightarrow\infty$. This solution
corresponds to the tritronquée solution in the case $\mu=1$.

We refer the reader to Boutroux [7] for detailed descriptions of the solutions
he called intégrales tronquées, which are pole-free (for sufficiently large
$x$) within two adjacent sectors of the form $S_{n}$ defined in Equation
(1.1). The region $S_{n}\cup S_{n+1}$ ($n\in\mathbb{Z}$ mod 6) described by
such a pair of sectors has a bisector given by a ray $\arg x=n\pi/3$. The term
tronquée arises from the fact that any line of poles that originally lies on
such a ray must be truncated as $|x|$ increases (due to the pole-free nature
of the solution).

There are three such rays in four contiguous sectors $S_{n}\cup S_{n+1}\cup
S_{n+2}\cup S_{n+3}$ ($n\in\mathbb{Z}$ mod 6). Boutroux showed that, for each
choice of such a region, there exists a unique solution he called tritronquée
(triply-truncated), which is asymptotically pole-free in the region. For
$\textrm{P}_{\textrm{II}}(\mu)$, these special rays are given by
arg$(x)=n\pi/(\mu+2)$ for $n\in\mathbb{Z}$.

We deduce an asymptotic series expansion of a tritronquée solution of
$\textrm{P}_{\textrm{II}}(\mu)$ in Proposition 3.2. In Propositions 3.4 and
3.7 we prove the existence and uniqueness of such a solution for arbitrary
$\mu\in\mathbb{Z}^{+}$. The proof of existence relies on Wasow’s theorem. The
further step required for uniqueness is analogous to the proof given by Joshi
and Kitaev in [22] for the first Painlevé equation. We start by giving a
generalisation of Boutroux’s transformation of variables.

######  Definition 3.1 (Boutroux variables).

Let $y(x)$ be a solution of $\textrm{P}_{\textrm{II}}(\mu)$. We define new
variables $u(z)$ and $z$ by

$y(x)=x^{\mu/2}u(z)\qquad\text{and}\qquad z=\frac{2}{\mu+2}x^{(\mu+2)/2}.$

Substitution into $\textrm{P}_{\textrm{II}}(\mu)$ shows that the function
$u(z)$ satisfies

$u^{\prime\prime}(z)=2u(z)^{3}+u(z)-\frac{3\mu}{\mu+2}\frac{u^{\prime}(z)}{z}-\frac{\mu(\mu-2)}{(\mu+2)^{2}}\frac{u(z)}{z^{2}}.$
(3.1)

In the following proposition, we consider its asymptotic expansion as
$|z|\to\infty$.

######  Proposition 3.2.

The formal series

$u_{f}(z)=\sum_{n=0}^{\infty}\frac{a_{n}}{z^{2n}},$

where $a_{0}=i/\sqrt{2}$, $a_{1}=-a_{0}\frac{\mu(\mu-2)}{2(\mu+2)^{2}}$, and
the coefficients $a_{n}$, for all $n>1$, are given by the recurrence relation

$\begin{split}a_{n}&=\sigma(n)\,a_{n-1}+a_{0}\sum_{k=1}^{n-1}a_{k}a_{n-k}+\sum_{j=1}^{n-1}\sum_{k=0}^{n-j}a_{j}a_{k}a_{n-j-k},\\\
\text{where}\qquad\sigma(n)&=-(2n-1)(n-1)+\frac{3\mu}{\mu+2}(n-1)-\frac{\mu(\mu-2)}{2(\mu+2)^{2}},\end{split}$

is a formal solution of Equation (3.1).

###### Proof.

Substitution of the given series into Equation (3.1). ∎

This gives rise to a formal series solution of $\textrm{P}_{\textrm{II}}(\mu)$
in the original variables:

$y_{f}(x)=\sqrt{x^{\mu}}\sum_{n=0}^{\infty}\frac{\left(\frac{\mu+2}{2}\right)^{2n}a_{n}}{x^{(\mu+2)n}},$
(3.2)

with coefficients $a_{n}$ described in Proposition 3.2.

######  Remark 3.3.

Recalling that $a_{0}$ is purely imaginary, we note that for $x$ on the
negative real semi-axis, $y_{f}(x)$ may only be real-valued for the case when
$\mu$ is odd.

###  3.1. Existence

We now prove the existence of true solutions asymptotic to the given power-
series in a given sector of angular width up to $2\pi/(\mu+2)$ in the
following proposition.

######  Proposition 3.4 (tronquée solutions).

Given $\mu\in\mathbb{Z}^{+}$ and a sector $\mathcal{S}$ of angle less than
$2\pi/(\mu+2)$, there exists a solution of $\textrm{P}_{\textrm{II}}(\mu)$
whose asymptotic behaviour as $|x|\rightarrow\infty$ in $\mathcal{S}$ is given
by the asymptotic series (3.2).

###### Proof.

Let $y(x)$ be a solution of $\textrm{P}_{\textrm{II}}(\mu)$. We use the change
of variables given in Definition 3.1 and further define

$w(z)=u(z)-a_{0}=u(z)-i/\sqrt{2}.$

Then a straightforward calculation shows that the function $w(z)$ satisfies

$w^{\prime\prime}(z)=2w(z)^{3}+3i\sqrt{2}w(z)^{2}-2w(z)-\frac{3\mu}{\mu+2}\frac{w^{\prime}(z)}{z}-\frac{\mu(\mu-2)}{(\mu+2)^{2}}\frac{(w(z)+i/\sqrt{2})}{z^{2}},$
(3.3)

and this ODE admits the formal power-series solution

$w_{f}(z)=\sum_{n=1}^{\infty}\frac{a_{n}}{z^{2n}}.$

Now, by defining variables $w_{1}=w$ and $w_{2}=w^{\prime}$, Equation (3.3)
becomes the system

$\begin{split}w_{1}^{\prime}&=w_{2},\\\
w_{2}^{\prime}&=2w_{1}^{3}+3i\sqrt{2}w_{1}^{2}-2w_{1}-\frac{3\mu}{\mu+2}\frac{w_{2}}{z}-\frac{\mu(\mu-2)}{(\mu+2)^{2}}\frac{(w_{1}+i/\sqrt{2})}{z^{2}}.\end{split}$

Let $f_{1}=w_{2}$ and $f_{2}$ be the right side of the second equation. This
system is, therefore, in the form

$z^{-q}\frac{d\boldsymbol{w}}{dz}=\boldsymbol{f}(z,w_{1},w_{2}),$

where $\boldsymbol{w}=(w_{1},w_{2})^{T}$ $\boldsymbol{f}=(f_{1},f_{2})^{T}$
and $q=0$. The Jacobian of $\boldsymbol{f}$, evaluated at
$\boldsymbol{w}=\boldsymbol{0}$ and $|z|\rightarrow\infty$, has nonzero
eigenvalues $\lambda_{\pm}=\pm i\sqrt{2}$. These results allow us to apply
Wasow’s theorem [33, Theorem 12.1], showing the existence of a true solution
with the desired asymptotic behaviour. ∎

######  Definition 3.5.

The solutions defined by Proposition 3.4 are called tronquée solutions.

######  Corollary 3.6.

Let $\mu\in\mathbb{Z}^{+}$ and $\mathcal{S}\in\mathbb{C}$ be a given sector of
angle less than $2\pi/(\mu+2)$. Suppose $y(x)$ is a tronquée solution of
$\textrm{P}_{\textrm{II}}(\mu)$ in $\mathcal{S}$. Then the $k$th derivative,
$y^{(k)}(x)$, for $k\in\\{0,1,2,\ldots\\}$, has asymptotic behaviour given by
the $k$th term-by-term differentiation of series (3.2), i.e.,
$y_{f}^{(k)}(x)$, as $|x|\rightarrow\infty$ in $\mathcal{S}$.

###### Proof.

The asymptotic behaviour of $y(x)$ implies that it is holomorphic in
$\mathcal{S}$ for sufficiently large $x$. The proof follows from Wasow’s
theorem [33, Theorem 8.8]. ∎

###  3.2. Uniqueness

In the following proposition, we prove that there is a unique true solution
asymptotic to the given power-series in $|\text{arg}(x)|\leq 2\pi/(\mu+2)$.

######  Proposition 3.7 (tritronquée solution).

Given $\mu\in\mathbb{Z}^{+}$, there exists a unique solution $Y(x)$ of
$\textrm{P}_{\textrm{II}}(\mu)$ that has the asymptotic expansion $y_{f}(x)$
in the sector $|\arg(x)|\leq 2\pi/(\mu+2)$ as $|x|\rightarrow\infty$.

###### Proof.

Let $0<\epsilon\in\mathbb{R}$. Let $y_{1}(x)$ and $y_{2}(x)$ be tronquée
solutions as defined by Proposition 3.4 in the respective sectors

$S_{1}:-\epsilon/2<\text{arg}(x)<\theta-\epsilon\qquad\text{and}\qquad
S_{2}:-\theta+\epsilon<\text{arg}(x)<\epsilon/2,$

where $\theta=2\pi/(\mu+2)$. Consider solutions $y_{1}(x)$ and $y_{2}(x)$ in
the overlapping sector of angular width $\epsilon$, centered about the
positive real axis. Since the two tronquée solutions have the same asymptotic
expansion, we get

$v(x):=y_{1}(x)-y_{2}(x)\underset{|\text{arg}(x)|<\epsilon/2}{\underset{|x|\rightarrow\infty}{=}}\textit{o}\left(\frac{\sqrt{x^{\mu}}}{x^{(\mu+2)n}}\right),$
(3.4)

for every $n\in\\{0,1,2,\ldots\\}$. Furthermore, we have
$v^{\prime\prime}(x)=y_{1}^{\prime\prime}(x)-y_{2}^{\prime\prime}(x)$, and by
applying $\textrm{P}_{\textrm{II}}(\mu)$, we see that $v(x)$ satisfies the
linear ODE

$v^{\prime\prime}(x)=(2y_{1}(x)^{2}+2y_{1}(x)y_{2}(x)+2y_{2}(x)^{2}+x^{\mu})v=(-f(x)+g(x))v,$

where

$f(x)=2x^{\mu}\qquad\text{and}\qquad
g(x)=2y_{1}(x)^{2}+2y_{1}(x)y_{2}(x)+2y_{2}(x)^{2}+3x^{\mu}.$

The functions $f$ and $g$ are holomorphic for sufficiently large $x$ in the
sector $S_{1}\cap S_{2}$. It follows from asymptotic series (3.2) that $g$ is
$\mathcal{O}(1/x^{2})$, and thus the integral

$\int_{x}^{\infty}\left|f^{-1/4}\left(f^{-1/4}\right)^{\prime\prime}-gf^{-1/2}\right|dx,$

taken along the real axis converges. Therefore, we may apply Olver’s theorem
[25, Theorem 2.2] to obtain

$\displaystyle v(x)=$ $\displaystyle
c_{1}f^{-1/4}\exp\left(i\int_{x}^{\infty}f^{1/2}dx\right)\bigl{(}1+\textit{o}(1)\bigr{)}$
$\displaystyle+c_{2}f^{-1/4}\exp\left(-i\int_{x}^{\infty}f^{1/2}dx\right)\bigl{(}1+\textit{o}(1)\bigr{)},$

where $c_{1}$ and $c_{2}$ are some constants. For real $x$, this contradicts
the asymptotic behaviour (3.4) unless $c_{1}=c_{2}=0$. So it follows that
$y_{1}(x)=y_{2}(x)$ for sufficiently large real $x$. ∎

######  Definition 3.8.

The solution defined by Proposition 3.7 is called the tritronquée solution
$Y(x)$.

######  Remark 3.9.

Note that the sector in which $Y$ is asymptotically free of poles is symmetric
around the positive real semi-axis.

Given $Y(x)$, defined above, we may generate other tritronquée solutions by
using the discrete symmetry of $\textrm{P}_{\textrm{II}}(\mu)$. The functions

$Y_{n}(x)=\exp{\left(-\frac{2\pi\,i\,n}{\mu+2}\right)}Y\left(\exp{\left(-\frac{2\pi\,i\,n}{\mu+2}\right)}x\right),$
(3.5)

for $n\in\mathbb{Z}$, also satisfy $\textrm{P}_{\textrm{II}}(\mu)$ and are
tritronquée in rotated sectors of width $4\pi/(\mu+2)$. This transformation
gives a tritronquée solution corresponding to each ray of angle $2\pi
n/(\mu+2)$.

##  4\. The Hastings-McLeod-Type Solution

In this section, we define the generalised Hastings-McLeod solution via a
boundary-value problem on the real-line. We then prove that the existence of
this solution fails when $\mu$ is even while it exists and is unique when
$\mu$ is odd.

######  Definition 4.1 (Hastings-McLeod-type solution).

Given $\mu\in\mathbb{Z}^{+}$, for $x\in\mathbb{R}$, we call a solution $y(x)$
of $\textrm{P}_{\textrm{II}}(\mu)$ a Hastings-McLeod-type solution if and only
if $y(x)\sim y_{f}(x)$ as $x\rightarrow-\infty$ (see Equation (3.2)), and
$y(x)\sim kf(x)$ as $x\rightarrow+\infty$ (see Equations (2.1)) for some
$k\in\mathbb{R}^{+}$.

We now prove the main result of this section.

######  Theorem 4.2.

For every odd integer $\mu>1$, there exists a unique, Hastings-McLeod-type
solution $y(x)$ of $\textrm{P}_{\textrm{II}}(\mu)$. On the other hand, for
every even integer $\mu>1$, no Hastings-McLeod-type solution of
$\textrm{P}_{\textrm{II}}(\mu)$ exists.

######  Remark 4.3.

As discussed in Remark 3.3, the formal solution $y_{f}(x)$, for real $x<0$,
may only be real-valued for odd $\mu$. For even values of $\mu$, the
corresponding Hastings-Mcleod-type solution would not be real-valued and thus
not mirror the distinctive properties of the original Hastings-McLeod
solution.

Hastings and McLeod originally tackled this problem while considering a
different generalisation of $\textrm{P}_{\textrm{II}}$, that is

$y^{\prime\prime}(x)=2y(x)^{\alpha+1}+xy(x),$

which coincides with $\textrm{P}_{\textrm{II}}(\mu)$ when $\alpha=2$ and
$\mu=1$ (i.e. $\textrm{P}_{\textrm{II}}$). We will prove Theorem 4.2 by
extending the approach taken by Hastings and McLeod to our case with
$\alpha=2$ and more general $\mu$.

###  4.1. Existence

The proof of the existence (and non-existence when $\mu$ is even) part of
Theorem 4.2 follows from a series of lemmas.

######  Lemma 4.4.

Given $\mu\in\mathbb{Z}^{+}$ and $k\in\mathbb{R}^{+}$, there exists a unique
solution $y_{k}(x)$ of $\textrm{P}_{\textrm{II}}(\mu)$ which is asymptotic to
$kf(x)$ as $x\rightarrow+\infty$. At each $x$ for which the solution continues
to exist as $x$ decreases, the solution and its derivatives are continuous
functions of $k$.

###### Proof.

By using $f(x)$ and $g(x)$ as integrating factors, we find that $y_{k}(x)$
must satisfy the integral equation

$y_{k}(x)=kf(x)+\int_{x}^{\infty}\left(f(x)g(t)-f(t)g(x)\right)y_{k}(t)^{3}dt,$
(4.1)

where $f$ and $g$ are defined in Equations (2.1). By application of a standard
contraction mapping argument, in an appropriate space of decaying, continuous
functions equipped with the uniform norm, Equation (4.1) may be shown to have
a unique solution, giving $y_{k}$ and its continuous dependence on $k$. ∎

######  Lemma 4.5.

For every even $\mu\in\mathbb{Z}^{+}$, and every $k\in\mathbb{R}^{+}$,
$y_{k}(x)$ remains positive as $x$ decreases and becomes singular at some
finite value of $x$.

###### Proof.

Let $k\in\mathbb{R}^{+}$ and assume $\mu\in\mathbb{Z}^{+}$ is even. For
sufficiently large $x\in\mathbb{R}^{+}$, $y_{k}(x)$ behaves like $kf(x)$.
Therefore, there exists $x_{0}\in\mathbb{R}^{+}$ so that $y_{k}(x)>0$ and
$y_{k}^{\prime}(x)<0$ for all $x\in[x_{0},\infty)$. Since $\mu$ is even, we
immediately deduce from $\textrm{P}_{\textrm{II}}(\mu)$ that $y_{k}(x)$ is
convex and positive for all $x\in\mathbb{R}$.

The remainder of the proof proceeds by contradiction. Assume that $y_{k}(x)$
remains bounded for all $x\in(-\infty,x_{0})$, i.e., it is not singular in
this interval. For all $x\in(-\infty,x_{0})$, we have
$y_{k}^{\prime\prime}(x)>y_{k}(x)^{3}$ and thus
$y_{k}^{\prime}(x)y_{k}^{\prime\prime}(x)<y_{k}^{\prime}(x)y_{k}(x)^{3}$
(since $y_{k}^{\prime}$ is negative). We integrate the latter autonomous
inequality between $x$ and $x_{0}$, which after rearranging terms gives

$y_{k}^{\prime}(x)^{2}>\frac{1}{2}y_{k}(x)^{4}+y_{k}^{\prime}(x_{0})^{2}-\frac{1}{2}y_{k}(x_{0})^{4}.$
(4.2)

We now show that $y_{k}(x)$ is monotonically increasing to $+\infty$ as $x$
decreases. Suppose instead that $y_{k}^{\prime}(x_{1})$ vanishes at some
$x_{1}<x_{0}$, while $y_{k}^{\prime}(x)<0$ for all $x\in(x_{1},x_{0}]$. The
inequality (4.2) would then imply that $y_{k}(x_{1})<y_{k}(x_{0})$, which
contradicts the fact that $y_{k}(x)$ is still increasing as we move from
$x_{0}$ to $x_{1}$.

Therefore, we can choose $x_{1}\in(-\infty,x_{0})$ such that
$\frac{1}{4}y_{k}(x_{1})^{4}+y_{k}^{\prime}(x_{0})^{2}-\frac{1}{2}y_{k}(x_{0})^{4}>0$,
hence $y_{k}^{\prime}(x)^{2}>\frac{1}{4}y_{k}(x)^{4}$ for all
$x\in(-\infty,x_{1})$. Therefore,
$y_{k}^{\prime}(x)<-\frac{1}{2}y_{k}(x)^{2}$, and another integration gives

$\frac{1}{y_{k}(x_{1})}-\frac{1}{y_{k}(x)}>\frac{x_{1}-x}{2},$

for all $x<x_{1}$, which contradicts the assumption that $y_{k}(x)$ is bounded
and non-singular on the interval. ∎

As a direct result of Lemma 4.5, we have proven the non-existence of the
Hastings-McLeod-type solution in the case of even $\mu$. We now continue our
argument in the case of odd $\mu$.

######  Lemma 4.6.

For every odd $\mu\in\mathbb{Z}^{+}$, the set $S_{1}\subset\mathbb{R}^{+}$ of
$k$ values, for which $y_{k}(x)$ remains positive as $x$ decreases and becomes
singular at some finite value of $x$, is an open set.

###### Proof.

Suppose we have $k_{0}\in\mathbb{R}^{+}$ and $x_{0}\in\mathbb{R}$ such that
$y_{k_{0}}(x)$ blows up at $x=x_{0}$. We now show that there exists a
neighbourhood of $k$ values (sufficiently close to $k_{0}$) such that the
solution $y_{k}$ still becomes singular. Since $y_{k_{0}}(x_{0})$ is
unbounded, we may choose $x_{1}>x_{0}$ to make $y_{k_{0}}(x_{1})$ arbitrarily
large. We choose $x_{1}$ near $x_{0}$ such that

$y_{k_{0}}(x_{1})^{2}>(|x_{1}|+1)^{\mu},$ (4.3)

noting then that $y_{k_{0}}(x)$ remains convex on $(x_{1}-1,x_{1})$ while it
continues to exist. Based on the continuous dependence of $y_{k}(x)$ on $k$,
we may replace $k_{0}$ with sufficiently close $k$ so that the inequality
(4.3) continues to hold. Furthermore, on the interval $(x_{1}-1,x_{1})$, we
have $|x|<|x_{1}|+1$, i.e. $|x|^{\mu}<(|x_{1}|+1)^{\mu}$. Hence

$|x|^{\mu}<(|x_{1}|+1)^{\mu}<y_{k}(x_{1})^{2}<y_{k}(x)^{2}.$

So if $x<0$, we have

$y_{k}^{\prime\prime}(x)=2y_{k}(x)^{3}+x^{\mu}y_{k}(x)=2y_{k}(x)^{3}-|x|^{\mu}y_{k}(x)>2y_{k}(x)^{3}-y_{k}(x)^{3}=y_{k}(x)^{3}.$
(4.4)

Note that if $x\geq 0$ we still have $y_{k}^{\prime\prime}(x)>y_{k}(x)^{3}$.

We have therefore shown that $y_{k}^{\prime\prime}(x)>y_{k}(x)^{3}$ on
$x\in(x_{1}-1,x_{1})$. Then from the integration of this autonomous
inequality, the proof of Lemma 4.5 shows that $y_{k}(x)$ will become unbounded
in this interval given sufficiently large $|y_{k}(x_{1})|$ and
$|y_{k}^{\prime}(x_{1})|$. As previously noted, we may choose $x_{1}$ so that
these are arbitrarily large and so the desired result follows. ∎

######  Lemma 4.7.

For every odd $\mu\in\mathbb{Z}^{+}$, the set $S_{2}\subset\mathbb{R}^{+}$ of
$k$ values, for which $y_{k}(x)$ becomes negative at some point as $x$
decreases, is an open set.

###### Proof.

Suppose there exist $k_{0}$ and $x_{0}$ such that $y_{k_{0}}(x_{0})<0$, then
by the continuity of $y_{k}$ in $k$, we may deduce that $y_{k}(x_{0})<0$ for
all $k$ in some neighborhood of $k_{0}$. ∎

######  Lemma 4.8.

For every odd $\mu\in\mathbb{Z}^{+}$, the set $S_{1}$ is non-empty.

###### Proof.

It follows from Equation (4.1) that $y_{k}(x)>kf(x)$ for sufficiently large
$x$. Differentiation of Equation (4.1) shows similarly that
$y_{k}^{\prime}(x)<kf^{\prime}(x)<0$. Using the same argument as in the proof
of Lemma 4.6, we may choose sufficiently large $x_{1}$ and $k$ so that
$y_{k}(x)$ blows up in $(x_{1}-1,x_{1})$. ∎

######  Lemma 4.9.

For every odd $\mu\in\mathbb{Z}^{+}$, the set $S_{2}$ is non-empty.

###### Proof.

Let $\mu>1$ be an odd integer. Consider the graph of $p(x)$ given by the
positive real root of $2p(x)^{2}=-x^{\mu}$ on the interval $x\in(-\infty,0]$.
Let $x_{0}<0$. Recall that $y_{0}(x)$ corresponds to the $k=0$ case in Lemma
4.4 and is the identically zero solution of $\textrm{P}_{\textrm{II}}(\mu)$.
By the continuity of $y_{k}(x_{0})$ in $k$, we may choose $k$ sufficiently
small so that $y_{k}(x_{0})<p(x_{0})$ and
$p^{\prime}(x_{0})<y_{k}^{\prime}(x_{0})<0$. Examination of
$\textrm{P}_{\textrm{II}}(\mu)$ then shows that $y_{k}^{\prime\prime}(x)<0$
for $x\leq x_{0}$. Therefore, $y_{k}(x)$ vanishes in this interval and becomes
negative to the left of $x_{0}$. ∎

We now conclude the proof of existence. We have two non-empty, open sets
$S_{1}$ and $S_{2}$, which are disjoint. So there must exist at least one
positive value of $k$, which lies in neither $S_{1}$ nor $S_{2}$. The solution
$y_{k}(x)$ corresponding to such a $k$ is then finite and positive for all
$x$. If such a solution were to decay to zero as $x\rightarrow-\infty$, then
it would be oscillatory in this asymptotic limit (see Equation (2.3)), thus
vanishing at some point and contradicting the fact that $k\notin S_{2}$.
Therefore, we conclude that $y(x)\sim y_{f}(x)$ as $x\rightarrow-\infty$. So
$y_{k}(x)$ is a Hastings-McLeod-type solution.

###  4.2. Uniqueness

We now give the argument for uniqueness to complete the proof of Theorem 4.2.

######  Lemma 4.10.

Let $\mu$ be an odd integer. Suppose $y_{k_{1}}(x)$ and $y_{k_{2}}(x)$ are
Hastings-Mcleod-type solutions with $k_{1}>k_{2}$, then
$y_{k_{1}}(x)>y_{k_{2}}(x)$ for all $x$.

###### Proof.

Note that the asymptotic behaviour as $x\to+\infty$ implies that
$y_{k_{1}}(x)>y_{k_{2}}(x)$ for sufficiently large $x$. If the assertion of
the lemma fails to hold, then there exists $x_{0}\in\mathbb{R}$ such that
$y_{k_{1}}(x_{0})=y_{k_{2}}(x_{0})$, while $y_{k_{1}}(x)>y_{k_{2}}(x)$ on
$(x_{0},\infty)$. For this to occur, we must have
$y_{k_{1}}^{\prime}(x_{0})^{2}\leq y_{k_{2}}^{\prime}(x_{0})^{2}$.

Define the energy function

$V_{k}(x)=y_{k}^{\prime}(x)^{2}-x^{\mu}y_{k}(x)^{2}-y_{k}(x)^{4}.$ (4.5)

From $\textrm{P}_{\textrm{II}}(\mu)$, it follows that

$V_{k}^{\prime}(x)=-\mu x^{\mu-1}y_{k}(x)^{2}.$ (4.6)

From Equation (4.5), we have

$V_{k_{1}}(x_{0})-V_{k_{2}}(x_{0})=y_{k_{1}}^{\prime}(x_{0})^{2}-y_{k_{2}}^{\prime}(x_{0})^{2}\leq
0,$ (4.7)

which implies $V_{k_{1}}(x_{0})\leq V_{k_{2}}(x_{0})$.

Using Equation (4.6), $y_{k_{1}}(x)>y_{k_{2}}(x)$ implies that
$V_{k_{1}}^{\prime}(x)<V_{k_{2}}^{\prime}(x)$. We proceed to integrate the
autonomous inequality $V_{k_{1}}^{\prime}(x)-V_{k_{2}}^{\prime}(x)<0$ over the
interval $(x_{0},N)$, where $N\gg 1$, giving

$\displaystyle V_{k_{1}}(N)-V_{k_{2}}(N)<V_{k_{1}}(x_{0})-V_{k_{2}}(x_{0}).$
(4.8)

However, for sufficiently large $N$ we have $V_{k_{1}}(N)-V_{k_{2}}(N)>0$, and
we already deduced that $V_{k_{1}}(x_{0})-V_{k_{2}}(x_{0})\leq 0$. These
observations contradict the inequality (4.8). Therefore, $x_{0}$ cannot exist
and the desired result follows. ∎

######  Lemma 4.11.

Let $\mu$ be an odd integer. Suppose $y_{k_{1}}(x)$ and $y_{k_{2}}(x)$ are
Hastings-Mcleod-type solutions. Then $k_{1}=k_{2}$.

###### Proof.

Suppose $k_{1}\not=k_{2}$. Without loss of generality, we assume
$k_{1}>k_{2}$. As in the proof of Proposition 3.7, we can appeal to another
theorem of Olver [25, Theorem 2.1, p.193] to show that
$p(x)=y_{k_{1}}(x)-y_{k_{2}}(x)$ satisfies

$\displaystyle p(x)=$ $\displaystyle
b_{1}f^{-1/4}\exp\left(\int_{-\infty}^{x}f^{1/2}dx\right)\bigl{(}1+\textit{o}(1)\bigr{)}$
$\displaystyle+b_{2}f^{-1/4}\exp\left(-\int_{-\infty}^{x}f^{1/2}dx\right)\bigl{(}1+\textit{o}(1)\bigr{)},$
(4.9)

for some constants $b_{1}$ and $b_{2}$, where $f=-2\,x^{\mu}$, for
sufficiently large $|x|$, $x<0$. Olver’s theorem also states that the
behaviour (4.2) is twice differentiable. However, this contradicts the assumed
asymptotic behaviour $y_{k_{j}}\sim y_{f}$, $k=1,2$ given in (3.2) unless
$b_{1}=0$.

Suppose $b_{2}\not=0$. Then we use the differentiability of the result (4.2)
to obtain

$\displaystyle V_{k_{1}}(x)-V_{k_{2}}(x)$
$\displaystyle=-2\left(2y_{f}(x)^{3}+x^{\mu}\,y_{f}(x)\right)p(x)\bigl{(}1+\textit{o}(1)\bigr{)}$
$\displaystyle=-2y_{f}^{\prime\prime}(x)p(x)\bigl{(}1+\textit{o}(1)\bigr{)}.$

For sufficiently large $|x|$, $x<0$, recalling that $\mu>1$ is an odd integer
and $p(x)>0$, this asymptotic behaviour is negative. However, we showed in the
previous lemma that $V_{k_{1}}(x)-V_{k_{2}}(x)$ is positive for sufficiently
large positive $x$ and that $V_{k_{1}}^{\prime}(x)-V_{k_{2}}^{\prime}(x)<0$
for all $x$, which is a contradiction. Therefore, we must have $b_{2}=0$.

It follows that $y_{k_{1}}(x)=y_{k_{2}}(x)$ for sufficiently large real $|x|$,
with $x<0$. This contradicts the result of Lemma 4.10 unless $k_{1}=k_{2}$. ∎

This completes our proof of uniqueness of the Hastings-McLeod type solution.

######  Remark 4.12.

Consider the behaviours of $y_{k}(x)$ depending on the parameter
$k\in\mathbb{R}^{+}$. For odd $\mu$, suppose $k^{*}$ is the unique value for
which $y_{k*}(x)$ is the Hastings-Mcleod-type solution. Using the same
argument as in Lemma 4.10, we may show that $0<k<k^{*}$ implies
$|y_{k}(x)|<y_{k^{*}}(x)$ for all $x$. So for $0<k<k^{*}$, we have a family of
bounded solutions on the real-line; these solutions have oscillatory
asymptotic behaviour given by Equation (2.3). On the other hand, when
$k^{*}<k$, we have solutions which are positive and convex for all $x$,
becoming singular at some finite point as $x$ decreases.

##  5\. Conclusion

In this paper, we investigated a family of nonlinear ODEs, parameterised by
$\mu$, which are perturbations of the second Painlevé equation. We showed how
such equations arise in an application, specifically, in a mathematical model
of electro-diffusion. Our main results show that the perturbed equation
possesses solutions that are natural generalisations of two celebrated
solutions of $\textrm{P}_{\textrm{II}}$.

We found that the perturbed equation always has solutions which we call
tritronqueé, first described by Boutroux for $\textrm{P}_{\textrm{II}}$. These
solutions are asymptotic to a power-series expansion in a surprisingly broad
annular region of the complex plane, near infinity. Furthermore, if our
integer parameter $\mu$ is odd, $\textrm{P}_{\textrm{II}}(\mu)$ has a solution
analogous to the famous Hastings-McLeod solution of
$\textrm{P}_{\textrm{II}}$. The Hastings-McLeod-type solution is holomorphic,
real-valued and positive on the entire real-line, with known asymptotic
behaviour in either direction.

These results naturally give rise to similar questions about perturbations of
other Painlevé equations. They suggest that certain behaviours, important in
applications, are preserved by classes of perturbations.

##  Appendix A Physical application of the perturbed equation

This section shows how $\textrm{P}_{\textrm{II}}(\mu)$ arises in Bass’
electro-diffusion model [4] under a small change in the physical setting.
Specifically, we allow particle flux to be non-constant.

Assume that $c_{+}(x)$ and $c_{-}(x)$ represent the respective (dimensionless)
concentrations of two ionic species with equal and opposite charge, while
$E(x)$ represents the (dimensionless) induced electric field. The following
coupled nonlinear ODEs govern the model of electro-diffusion:

$\displaystyle c^{\prime}_{+}(x)$ $\displaystyle=E(x)c_{+}(x)+A_{+},$ (A.1)
$\displaystyle c^{\prime}_{-}(x)$ $\displaystyle=-E(x)c_{-}(x)+A_{-},$ (A.2)
$\displaystyle\lambda^{2}E^{\prime}(x)$ $\displaystyle=c_{+}(x)-c_{-}(x),$
(A.3)

where $\lambda$ is a constant, the terms $A_{\pm}$ arise due to Fick’s law’s
standard passive diffusion process, and we note that $A_{\pm}$ is proportional
to the ratio between the species flux in the $x$-direction, $\phi_{\pm}$, and
the species diffusivity, $D_{\pm}$.

In the original model, $\phi_{\pm}$ is constant in $x$ due to the conservation
of mass in a strictly one-dimensional setting. However, we consider the
possibility that $\phi_{\pm}$ may undergo small amounts of variation, as the
species may fluctuate in other spatial dimensions. However, we assume
$D_{\pm}$ is constant.

Consider $A_{\pm}$ as a perturbative series, whereby $A_{\pm}$ is polynomial
in $x$ and constant at leading order. For now, consider a linear correction
term. Furthermore, we shall consider a somewhat simplified case whereby
$A_{+}=A_{-}$, so we have

$A_{+}=A_{-}=A_{0}+\epsilon x.$ (A.4)

Then by adding Equations (A.1) and (A.2), and applying Equation (A.3), the
result is

$c^{\prime}_{+}(x)+c^{\prime}_{-}(x)=\lambda^{2}E(x)E^{\prime}(x)+2A_{0}+2\epsilon
x.$ (A.5)

Upon integrating Equation (A.5) and disregarding the constant of integration,
we have

$c_{+}(x)+c_{-}(x)=\frac{\lambda^{2}}{2}E(x)^{2}+\epsilon x^{2}+2A_{0}x.$
(A.6)

Meanwhile, by differentiating Equation (A.3), and then applying Equations
(A.1) and (A.2), we obtain

$\lambda^{2}E^{\prime\prime}(x)=E(x)(c_{+}(x)+c_{-}(x)).$ (A.7)

Hence, by combining Equations (A.6) and (A.7), we obtain

$\lambda^{2}E^{\prime\prime}(x)=\frac{\lambda^{2}}{2}E(x)^{3}+(\epsilon
x^{2}+2A_{0}x)E(x).$ (A.8)

Equation (A.8) is a nonlinear ODE which governs the induced electric field as
a function of $x$ and is similar in form to $\textrm{P}_{\textrm{II}}(\mu)$
with $\mu=2$. Adding higher degree correction terms in Equation (A.4) would
correspond to a higher degree polynomial $p(x)$ in the coefficient of $E$,
that is,

$\lambda^{2}E^{\prime\prime}(x)=\frac{\lambda^{2}}{2}E(x)^{3}+p(x)E(x).$

The presence of such higher degree ($>1$) terms in the coefficient function
$p(x)$ motivates the study of our perturbed form of
$\textrm{P}_{\textrm{II}}{}$.

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# Multi-head Uncertainty Inference for Adversarial Attack Detection

###### Abstract

Deep neural networks (DNNs) are sensitive and susceptible to tiny perturbation
by adversarial attacks which causes erroneous predictions. Various methods,
including adversarial defense and uncertainty inference (UI), have been
developed in recent years to overcome the adversarial attacks. In this paper,
we propose a multi-head uncertainty inference (MH-UI) framework for detecting
adversarial attack examples. We adopt a multi-head architecture with multiple
prediction heads (_i.e._ , classifiers) to obtain predictions from different
depths in the DNNs and introduce shallow information for the UI. Using
independent heads at different depths, the normalized predictions are assumed
to follow the same Dirichlet distribution, and we estimate distribution
parameter of it by moment matching. Cognitive uncertainty brought by the
adversarial attacks will be reflected and amplified on the distribution.
Experimental results show that the proposed MH-UI framework can outperform all
the referred UI methods in the adversarial attack detection task with
different settings.

Index Terms—  Uncertainty inference, adversarial attack detection, image
recognition, Dirichlet distribution

##  1 Introduction

Deep neural networks (DNNs) are powerful learning tools in computer vision
such as image recognition [1] and image segmentation [2]. However, DNNs have a
inevitable problem which concerns with the their stability with respect to
tiny perturbations on their input images that may be perceived by human eyes
[3]. Their predictions may be arbitrarily affected by the random or targeted
imperceptible perturbation.

Adversarial attack task [4] was proposed to study the best way on confusing
the DNNs by generating adversarial attack examples with the aforementioned
perturbations,  _e.g._ , fast gradient sign method (FGSM) [5]. These
perturbations are commonly found by optimizing the input images to maximize
the prediction errors and minimize additional perturbations simultaneously
[6],  _i.e._ , white-box attacks. The adversarial attack examples can greatly
affect the security and reliability of the DNNs.

In recent years, in order to overcome the adversarial attacks, adversarial
defense methods usually concentrate on random smoothing [7], adversarial
training [8], and Jacobian regularization [9]. Other work [10, 11, 12, 13]
introduced uncertainty inference (UI) for misclassification and out-of-domain
detection and estimated the prediction uncertainty of a DNN, which can be
treated as defense techniques for the adversarial attacks. Compared with the
original adversarial defense methods that were proposed for one specific
adversarial attack method, the UI methods can be commonly adapted to dispose
different adversarial attack methods [14, 15].

Most of the recently proposed UI methods can be divided into Monte-Carlo (MC)
sampling based methods [11] and direct estimation based ones [10, 12, 13]. The
former utilized dropout [16] to generate multiple subnetworks and obtained
samples of predictions by them to compute variance as the uncertainty, which
is computational expensive. The latter enhanced the training procedure by
additional loss functions [10, 12, 13] and/or modules [10], and directly
inferred the uncertainty with the one-shot predictions, which is highly
dependent on the learning ability of a DNN itself [10] and may be affected by
the quality of extracted features. Therefore, how to construct a UI model that
overcomes both of the drawbacks should be carefully investigated.

Fig. 1: Structure of the proposed multi-head uncertainty inference (MH-UI). We
take a DNN with $5$ layers for image recognition as an example. Images are
input into the multi-head architecture-based model and obtain multiple
predictions from each head. For the clean data input, they usually gather in
one specific class, while those of the adversarial data respectively focus on
distinct classes. The predictions are treat as samples for Dirichlet
distribution estimation.

On the basis of the previous work [10, 11, 12, 13], we propose to perform
uncertainty analysis on the output of the DNNs to determine whether the
corresponding input image is an adversarial attack example. In this paper, we
propose a Multi-head Uncertainty Inference (MH-UI) framework for detecting the
adversarial attack examples. As the shallower layers extract texture
information and the adversarial noises usually reflect in the texture, we
adapt a multi-head architecture, which is similar to the GoogLeNet-style one
with multiple prediction heads [17], to introduce shallow information and
obtain predictions from different depths in a DNN. Independent classifiers
(_i.e._ , fully connected (FC) layers) are utilized in the different depths
and the predictions of each are assumed to be identically Dirichlet
distributed samples where distribution parameters are estimated by moment
matching. Experimental results show that the proposed MH-UI framework can
outperform all the referred UI methods in the adversarial attack detection
task with different settings.

##  2 Multi-head Uncertainty Inference

###  2.1 Multi-head Architecture

In this section, we introduce the multi-head architecture for the MH-UI
framework. Figure1 shows that a DNN stacked by $N$ layers (blocks) for image
recognition can be represented as

$\displaystyle\boldsymbol{y}_{N}$ $\displaystyle=g_{N}\odot f(\boldsymbol{x})$
$\displaystyle=g_{N}\odot f_{N}\odot f_{N-1}\odot\cdots\odot f_{2}\odot
f_{1}(\boldsymbol{x}),$ (1)

where $\boldsymbol{x}\in R^{3\times W\times H}$ ($W$ and $H$ are width and
height, respectively) and $\boldsymbol{y}_{N}\in R^{C}$ ($C$ is class number)
are an input image and the corresponding normalized predictions, respectively,
$f_{n},n=1,\cdots,N$ is the $n^{th}$ layer, and $\odot$ is the stack
operation. And $g_{N}$ is the classifier of the DNN, which is constructed by
multiple FC layers with a Softmax layer behind.

Recalled that the shallower layers extract texture information and the
adversarial noises usually reflect in the texture. The texture information in
the shallower layers should be helpful for resisting the adversarial noises
and detecting the adversarial attack samples. Thus, we introduce auxiliary
classifier, namely head, for each layer in order to access information from
different semantic levels. Here, we define their corresponding normalized
predictions
$\left\\{\boldsymbol{y}_{1},\boldsymbol{y}_{2},\cdots,\boldsymbol{y}_{N-1}\right\\}$
($\boldsymbol{y}_{n}\in R^{C},n=1,\cdots,N$) as

$\displaystyle\boldsymbol{y}_{n}=g_{n}\odot f_{n}\odot f_{n-1}\odot\cdots\odot
f_{2}\odot f_{1}(\boldsymbol{x}),$ (2)

where $g_{n}$ is the head for the $n^{th}$ layer. Note that $f_{n}$ can be set
as any architecture, not only inception block [17].

Since $\boldsymbol{y}_{n}$ is normalized, we roughly assume that it follows
Dirichlet distribution, which is a common assumption [12, 13]. Here, we
further define all the normalized predictions of the heads ($g_{N}$ can be
also considered as a head) as samples generated from an identical Dirichlet
distribution $Dir(\boldsymbol{\alpha})$ with $C$-dimensional distribution
parameter vector $\boldsymbol{\alpha}$. By further analyzing all the
predictions, we can obtain the uncertainty of $\boldsymbol{x}$ and apply the
uncertainty for final adversarial attack detection.

###  2.2 Uncertainty Inference

In this section, we introduce the way to infer the uncertainty of the input
image $\boldsymbol{x}$. After training the whole model $f$ with the heads
$\\{g_{1},\cdots,g_{N}\\}$, the identically Dirichlet distributed predictions
$\boldsymbol{y}=\left\\{\boldsymbol{y}_{1},\boldsymbol{y}_{2},\cdots,\boldsymbol{y}_{N-1}\right\\}$
are obtained. As defining $Dir(\boldsymbol{\alpha})$ for $\boldsymbol{y}_{n}$,
we then estimate the parameter
$\boldsymbol{\alpha}=\left\\{\alpha_{1},\alpha_{2},\cdots,\alpha_{C}\right\\}$
by the aforementioned samples.

Table 1: Means of AUROC (%) on CIFAR-$10$ dataset of different head combinations under FGSM attack. The leftmost column of the table represents the combined sequence of the different heads as $1$: head $\left\\{1-3,10\right\\}$; $2$: head $\left\\{7-10\right\\}$; $3$: head $\left\\{4-10\right\\}$; $4$: head $\left\\{1-6,10\right\\}$; $5$: head $\left\\{1-10\right\\}$. The numbers in the brackets represent the heads involved in the uncertainty calculation. Head $1$ is the head for the first inception (closest to input). Head $10$ is the final output of the model.  | $\boldsymbol{\epsilon}=0.05$ | $\boldsymbol{\epsilon}=0.1$ | $\boldsymbol{\epsilon}=0.25$ | $\boldsymbol{\epsilon}=0.5$ | $\boldsymbol{\epsilon}=1.0$
---|---|---|---|---|---
Max.P. | Ent. | Max.P. | Ent. | Max.P. | Ent. | Max.P. | Ent. | Max.P. | Ent.
$1$ | 69.03 | 69.77 | 74.29 | 75.29 | 83.17 | 84.36 | 91.13 | 90.80 | 89.35 | 91.29
$2$ | 77.52 | 77.56 | 79.91 | 80.02 | 84.06 | 84.20 | 85.36 | 85.41 | 95.69 | 96.00
$3$ | 76.98 | 77.06 | 80.83 | 81.19 | 86.66 | 87.04 | 92.35 | 91.89 | 97.90 | 97.78
$4$ | 71.12 | 71.48 | 77.79 | 78.47 | 87.33 | 88.33 | 94.47 | 95.33 | 97.61 | 98.44
$5$ | 82.70 | 80.55 | 83.52 | 83.93 | 87.90 | 88.91 | 94.31 | 95.32 | 99.04 | 99.19
Table 2: Means and standard deviations of accuracy (%) of heads in MH-UI on
CIFAR-$10$ and SVHN datasets.

Head | Head$1$ | Head$2$ | Head$3$ | Head$4$ | Head$5$ | Head$6$ | Head$7$ | Head$8$ | Head$9$
---|---|---|---|---|---|---|---|---|---
CIFAR-$10$ | $67.6\pm 1.1$ | $71.6\pm 1.3$ | $81.6\pm 1.1$ | $84.4\pm 1.2$ | $88.3\pm 0.8$ | $89.6\pm 0.5$ | $92.0\pm 0.4$ | $93.5\pm 0.4$ | $93.2\pm 0.6$
SHVN | $65.8\pm 3.0$ | $69.6\pm 2.7$ | $79.8\pm 2.3$ | $85.1\pm 1.5$ | $88.6\pm 1.2$ | $91.3\pm 1.1$ | $92.5\pm 0.7$ | $94.6\pm 0.3$ | $95.2\pm 0.3$

Here, we match the first- and second-order moments between $\boldsymbol{y}$
and $Dir(\boldsymbol{\alpha})$ according to the moment matching method in [18]
by

$\displaystyle\begin{cases}\text{E}\left[\boldsymbol{y}\right]=\frac{\boldsymbol{\alpha}}{\alpha_{0}}\\\
\text{Var}\left[\boldsymbol{y}\right]=\frac{\boldsymbol{\alpha}\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}(\alpha_{0}-\boldsymbol{\alpha})}{\alpha_{0}^{2}(\alpha_{0}+1)}\end{cases},$
(3)

where $\alpha_{0}=\sum_{n=1}^{N}\alpha_{n}$ and
$\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}$
is Hadamard product. The expectation and variance of $\boldsymbol{y}$ are
ddenoted by $\text{E}\left[\boldsymbol{y}\right]$ and
$\text{Var}\left[\boldsymbol{y}\right]$, respectively, and are defined as

$\displaystyle\begin{cases}\text{E}\left[\boldsymbol{y}\right]=\frac{1}{N}\sum_{n=1}^{N}\boldsymbol{y}_{n}\\\
\text{Var}\left[\boldsymbol{y}\right]=\frac{1}{N-1}\sum_{n=1}^{N}(\boldsymbol{y}_{n}-\text{E}\left[\boldsymbol{y}\right])^{2}\end{cases}.$
(4)

According to (3), we can obtain the value of $\boldsymbol{\alpha}$. The
uncertainty of an input image can be evaluated by max probability
(Max.P.$=\max_{n}m_{n}$) and entropy (Ent.$=-\sum_{n=1}^{N}m_{n}\ln(m_{n})$),
and $m_{n}=\frac{\alpha_{n}-1}{\alpha_{0}-N}$. When Max.P is larger or Ent. is
smaller than the threshold value, we consider the image is attacked.

The general adversarial attack method is to confuse the final predictions,
which accumulate errors from shallower layers and work with those of other
heads difficultly. Such attack samples have large cognitive uncertainty. Since
each head represents different depth of the DNN, their features are distinct.
Therefore, some heads will become less confident when encountering the attack
samples and the possibility of error recognition of the final predictions will
be increased. When taking into account them of other heads, which performs
similarly to an ensemble model, the aggregated predictions can be corrected.
The change brought by the increase of cognitive uncertainty to the head output
can be reflected on $\boldsymbol{\alpha}$ and then amplified, which allows us
to evaluate $\boldsymbol{\alpha}$ to determine whether the input image is
attacked.

Table 3: Means and standard deviations of accuracy (%) of each method on CIFAR-$10$ and SVHN datasets. Method | MC dropout | DPN | RKL | DS-UI | MH-UI (ours)
---|---|---|---|---|---
CIFAR-$10$ | $93.22\pm 0.67$ | $93.55\pm 0.34$ | $93.67\pm 0.58$ | $92.89\pm 0.77$ | $93.11\pm 0.61$
SVHN | $93.60\pm 0.14$ | $93.29\pm 0.23$ | $93.32\pm 0.24$ | $93.50\pm 0.34$ | $94.12\pm 0.11$

---

(a) Max.P. on CIFAR-$10$

(b) Ent. on CIFAR-$10$

(c) Max.P. on SVHN

(d) Ent. on SVHN

Fig. 2: Performance on CIFAR-$10$ and SVHN datasets. $\epsilon$ is the step
size in the FGSM and selected in the set $\\{\frac{n}{20}\\}_{n=1}^{20}$. The
dashed and solid lines in each subfigure present the MU-UI and the referred
methods, respectively.

---

(a) Max.P. with ResNet backbone

(b) Ent. with ResNet backbone

Fig. 3: Performance on ResNet$18$ backbone with CIFAR-$10$ dataset. $\epsilon$
is the step size in the FGSM and selected in the set
$\\{\frac{n}{20}\\}_{n=1}^{20}$. $8$ heads were used to connect all basiclock
in ResNet$18$. The dashed and solid lines in each subfigure present the MH-UI
and the referred methods, respectively.

##  3 EXPERIMENTAL RESULTS AND DISCUSSIONS

###  3.1 Implementation Details

In our experiments, we used GoogLeNet [17] and ResNet [19] as the backbone and
followed the experimental settings in [10, 12, 13]. We connected $9$ heads
(excluding the final classifier) after each inception output for the GoogLeNet
model and $7$ heads for the ResNet model. In the training procedure, we first
trained the backbones. we applied Adam optimizer by following [12] with $100$
epochs. We used $1$-cycle learning rate scheme, where we set initial learning
rate as $7.5\times 10^{-4}$ and cycle length as $70$ epochs for each dataset.
All other referred methods were undertaken in the same training way. Then, we
trained the heads in sequence. For the heads, the learning rate was fixed as
$0.001$ with $50$ epoch for each dataset. We compared the MH-UI with MC
dropout [20], DPN [12], RKL [13], and DS-UI [10] for adversarial attack
detection in CIFAR-$10$ [21] and SVHN [22] datasets.

We introduced max probability and entropy of the estimated Dirichlet
distribution as metrics for uncertainty measurement and adopted the area under
receiver operating characteristic curve (AUROC) [12, 13, 10]. The larger the
values of AUROC, the better the performance. For all the methods, we conducted
five runs and reported the mean of AUROC.

###  3.2 Ablation Study

We conducted ablation study on the CIFAR-$10$ dataset under adversarial attack
detection to discuss the head combination configurations, as well as the
combination of head effect. We set five different combinations of heads in our
experiment. Table 1 shows the setting details and means of AUROC results. It
can be observed that the model with lager number of heads used in the MH-UI
performed better in most of instances. Thus, we choose the combination of all
the head $\left\\{1,2,3,4,5,6,7,8,9,10\right\\}$ in the following experiment,
which performed best in the ablation study.

###  3.3 Adversarial Attack Detection

Table 2 illustrates the accuracies on both CIFAR-$10$ and SVHN datasets of
different heads in the proposed MH-UI and the referred methods. Different
heads perform diversely and gradually better among the increasing depth, while
the final accuracies of the last head in Table 3 achieve comparable
performance compared with the referred methods. The classification accuracies
in Table 3 is for convergence evaluation of model training in each method
under clean image setting. And all the methods can converge well and obtain
high accuracies. Our method can achieve competitive results and does not
reduce the accuracy of the original framework.

Figure 2(a) and (b) illustrate the performance of each method in the
adversarial attack detection for CIFAR-$10$ dataset. We used FGSM attack
method in experiment and set $\epsilon$ in the FGSM in the set
$\left\\{\frac{n}{20}\right\\}^{20}_{n=1}$. Comparing to different methods,
the MH-UI converges to lager value when $\epsilon$ grows larger
($\epsilon>0.2$). And as $\epsilon$ is small, the MH-UI can perform similar to
the best one (RKL).

Figure 2(c) and (d) illustrate experimental result of each method in the
attacked sample detection on SVHN dataset. As the SVHN dataset needs to
classify the digital features relatively simple, all methods perform better
than those on CIFAR-$10$. The MH-UI work better in two evaluation criteria.
And when $\boldsymbol{\epsilon}$ approaches to $1$, the AUROC of detecting
attack can converge to $98\%$ in both Max.P. and Ent.

In summary, compared with other UI methods, the above results show that our
MH-UI method is effective on detecting adversarial attack in different
datasets.

###  3.4 Generalized Applicability

Furthermore, our method can be applied to other models. We applied ResNet$18$
[19] as backbone. Different from the head used in GoogLeNet, we used a three-
layer FC net with $1024$ hidden units for each hidden layer in each one. For
other comparing methods, we also built them with ResNet$18$ as backbone.
Figure 3 illustrates each method’s experimental result with ResNet backbone.
Our method also works well in ResNet. When the samples are under heavy attack
($\boldsymbol{\epsilon}$ >$0.2$) the AUROC of Max.P. and Ent. can converge to
$99\%$. These result shows that our method can also be used in other models
backbone such as ResNet.

##  4 CONCLUSION

In this paper, we have proposed a multi-head uncertainty inference (MH-UI)
framework for detecting adversarial attack examples. We adopted a multi-head
architecture with multiple prediction heads to obtain predictions from
different depths in the DNNs and introduced shallow information for the UI.
Using independent heads at different depths, the normalized predictions are
assumed to follow the same Dirichlet distribution and the distribution
parameter of it is estimated by moment matching. The proposed MH-UI is
flexible with different head numbers and backbone structures. Experimental
results show that the proposed MH-UI framework can outperform all the referred
UI methods in the adversarial attack detection task with different settings.

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