2}\right)+F^{(iv)}\right\\}$ $\displaystyle+2$
$\displaystyle\left\\{\underline{-\frac{2}{3}\tau^{-5/3}AF}+\tau^{-2/3}A_{\tau}F+\underline{\tau^{-2/3}AF^{\prime}\left(-\frac{1}{3}\frac{\zeta_{0}}{\tau}\right)}+\tau^{-2/3}F^{\prime}\frac{\varphi_{\tau}}{\tau^{1/3}}\right\\}$
(175) $\displaystyle+$
$\displaystyle\underline{\tau^{-5/3}AF+6\,A\,\tau^{-5/3}FF^{\prime}+\tau^{-5/3}AF^{\prime\prime\prime}}+\beta\tau^{-2}\partial_{\zeta}^{-1}\left[\tau^{-1/3}A_{\Theta\Theta}F+\tau^{-2/3}\varphi_{\Theta\Theta}F^{\prime}\right]=0.$

Multiplying (171) evaluated at $\zeta=\zeta_{0}$ by $A\,\tau^{-5/3}$
eliminates the terms underlined in (175) simplifying the latter to

$\displaystyle\varphi\,\tau^{-2}\left[-\frac{4}{3}F^{\prime}-\frac{2}{3}\left(F^{\prime}+\zeta_{0}F^{\prime\prime}\right)+F^{\prime}+3\left(FF^{\prime\prime}+F^{\prime
2}\right)+F^{(iv)}\right]+$ (176) $\displaystyle
2\mathop{\tau^{-2/3}\frac{\varphi_{\tau}}{\tau^{1/3}}F^{\prime}}_{\mathcal{O}(\epsilon^{2a+c})}+2\mathop{\tau^{-2/3}A_{\tau}F}_{\mathcal{O}(\epsilon^{1+\frac{5}{3}a})}+\mathop{3\,A\,\tau^{-5/3}FF^{\prime}}_{\mathcal{O}(\epsilon^{1+\frac{5}{3}a})}+\beta\tau^{-2}\partial_{\zeta}^{-1}\left[\mathop{\tau^{-1/3}A_{\Theta\Theta}F}_{\mathcal{O}(\epsilon^{1+\frac{7}{3}a+2b})}+\mathop{\tau^{-2/3}\varphi_{\Theta\Theta}F^{\prime}}_{\mathcal{O}(\epsilon^{\frac{8}{3}a+c+2b})}\right],$

where the expression in the first brackets also vanish because equation (171)
differentiated once and evaluated at $\zeta=\zeta_{0}$ yields the same
expression. As a result, we are left with five terms in (176) having, in
general, four different exponents in the respective orders of $\epsilon$: 1
$2a+c$, 2 $1+\frac{5}{3}a$, 3 $1+\frac{7}{3}a+2b$, 4 $\frac{8}{3}a+c+2b$.
Consideration of all possible matching combinations leaves reasonable only two
options:

  1. 1.

$\leavevmode\hbox to10.08pt{\vbox
to10.08pt{\pgfpicture\makeatletter\hbox{\hskip 5.03984pt\lower-5.03984pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.83984pt}{0.0pt}\pgfsys@curveto{4.83984pt}{2.673pt}{2.673pt}{4.83984pt}{0.0pt}{4.83984pt}\pgfsys@curveto{-2.673pt}{4.83984pt}{-4.83984pt}{2.673pt}{-4.83984pt}{0.0pt}\pgfsys@curveto{-4.83984pt}{-2.673pt}{-2.673pt}{-4.83984pt}{0.0pt}{-4.83984pt}\pgfsys@curveto{2.673pt}{-4.83984pt}{4.83984pt}{-2.673pt}{4.83984pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.25pt}{-1.6111pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\hbox{{1}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=\leavevmode\hbox
to10.08pt{\vbox to10.08pt{\pgfpicture\makeatletter\hbox{\hskip
5.03984pt\lower-5.03984pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.83984pt}{0.0pt}\pgfsys@curveto{4.83984pt}{2.673pt}{2.673pt}{4.83984pt}{0.0pt}{4.83984pt}\pgfsys@curveto{-2.673pt}{4.83984pt}{-4.83984pt}{2.673pt}{-4.83984pt}{0.0pt}\pgfsys@curveto{-4.83984pt}{-2.673pt}{-2.673pt}{-4.83984pt}{0.0pt}{-4.83984pt}\pgfsys@curveto{2.673pt}{-4.83984pt}{4.83984pt}{-2.673pt}{4.83984pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.25pt}{-1.6111pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\hbox{{2}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ yields
$c=1-\frac{1}{3}a$, in which case $\leavevmode\hbox to10.08pt{\vbox
to10.08pt{\pgfpicture\makeatletter\hbox{\hskip 5.03984pt\lower-5.03984pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.83984pt}{0.0pt}\pgfsys@curveto{4.83984pt}{2.673pt}{2.673pt}{4.83984pt}{0.0pt}{4.83984pt}\pgfsys@curveto{-2.673pt}{4.83984pt}{-4.83984pt}{2.673pt}{-4.83984pt}{0.0pt}\pgfsys@curveto{-4.83984pt}{-2.673pt}{-2.673pt}{-4.83984pt}{0.0pt}{-4.83984pt}\pgfsys@curveto{2.673pt}{-4.83984pt}{4.83984pt}{-2.673pt}{4.83984pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.25pt}{-1.6111pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\hbox{{3}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=\leavevmode\hbox
to10.08pt{\vbox to10.08pt{\pgfpicture\makeatletter\hbox{\hskip
5.03984pt\lower-5.03984pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.83984pt}{0.0pt}\pgfsys@curveto{4.83984pt}{2.673pt}{2.673pt}{4.83984pt}{0.0pt}{4.83984pt}\pgfsys@curveto{-2.673pt}{4.83984pt}{-4.83984pt}{2.673pt}{-4.83984pt}{0.0pt}\pgfsys@curveto{-4.83984pt}{-2.673pt}{-2.673pt}{-4.83984pt}{0.0pt}{-4.83984pt}\pgfsys@curveto{2.673pt}{-4.83984pt}{4.83984pt}{-2.673pt}{4.83984pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.25pt}{-1.6111pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\hbox{{4}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$. In this case, there
is a possibility of a slow developing long-wave instability.

  2. 2.

$\leavevmode\hbox to10.08pt{\vbox
to10.08pt{\pgfpicture\makeatletter\hbox{\hskip 5.03984pt\lower-5.03984pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.83984pt}{0.0pt}\pgfsys@curveto{4.83984pt}{2.673pt}{2.673pt}{4.83984pt}{0.0pt}{4.83984pt}\pgfsys@curveto{-2.673pt}{4.83984pt}{-4.83984pt}{2.673pt}{-4.83984pt}{0.0pt}\pgfsys@curveto{-4.83984pt}{-2.673pt}{-2.673pt}{-4.83984pt}{0.0pt}{-4.83984pt}\pgfsys@curveto{2.673pt}{-4.83984pt}{4.83984pt}{-2.673pt}{4.83984pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.25pt}{-1.6111pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\hbox{{2}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=\leavevmode\hbox
to10.08pt{\vbox to10.08pt{\pgfpicture\makeatletter\hbox{\hskip
5.03984pt\lower-5.03984pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.83984pt}{0.0pt}\pgfsys@curveto{4.83984pt}{2.673pt}{2.673pt}{4.83984pt}{0.0pt}{4.83984pt}\pgfsys@curveto{-2.673pt}{4.83984pt}{-4.83984pt}{2.673pt}{-4.83984pt}{0.0pt}\pgfsys@curveto{-4.83984pt}{-2.673pt}{-2.673pt}{-4.83984pt}{0.0pt}{-4.83984pt}\pgfsys@curveto{2.673pt}{-4.83984pt}{4.83984pt}{-2.673pt}{4.83984pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.25pt}{-1.6111pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\hbox{{4}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ yielding $c=1-a-2b$,
while $\leavevmode\hbox to10.08pt{\vbox
to10.08pt{\pgfpicture\makeatletter\hbox{\hskip 5.03984pt\lower-5.03984pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.83984pt}{0.0pt}\pgfsys@curveto{4.83984pt}{2.673pt}{2.673pt}{4.83984pt}{0.0pt}{4.83984pt}\pgfsys@curveto{-2.673pt}{4.83984pt}{-4.83984pt}{2.673pt}{-4.83984pt}{0.0pt}\pgfsys@curveto{-4.83984pt}{-2.673pt}{-2.673pt}{-4.83984pt}{0.0pt}{-4.83984pt}\pgfsys@curveto{2.673pt}{-4.83984pt}{4.83984pt}{-2.673pt}{4.83984pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.25pt}{-1.6111pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\hbox{{1}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=\leavevmode\hbox
to10.08pt{\vbox to10.08pt{\pgfpicture\makeatletter\hbox{\hskip
5.03984pt\lower-5.03984pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.83984pt}{0.0pt}\pgfsys@curveto{4.83984pt}{2.673pt}{2.673pt}{4.83984pt}{0.0pt}{4.83984pt}\pgfsys@curveto{-2.673pt}{4.83984pt}{-4.83984pt}{2.673pt}{-4.83984pt}{0.0pt}\pgfsys@curveto{-4.83984pt}{-2.673pt}{-2.673pt}{-4.83984pt}{0.0pt}{-4.83984pt}\pgfsys@curveto{2.673pt}{-4.83984pt}{4.83984pt}{-2.673pt}{4.83984pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.25pt}{-1.6111pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\hbox{{3}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ produces
$c=1+\frac{1}{3}a+2b$. Altogether, this leads to $b=-\frac{1}{3}a$ and
therefore $\leavevmode\hbox to10.08pt{\vbox
to10.08pt{\pgfpicture\makeatletter\hbox{\hskip 5.03984pt\lower-5.03984pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.83984pt}{0.0pt}\pgfsys@curveto{4.83984pt}{2.673pt}{2.673pt}{4.83984pt}{0.0pt}{4.83984pt}\pgfsys@curveto{-2.673pt}{4.83984pt}{-4.83984pt}{2.673pt}{-4.83984pt}{0.0pt}\pgfsys@curveto{-4.83984pt}{-2.673pt}{-2.673pt}{-4.83984pt}{0.0pt}{-4.83984pt}\pgfsys@curveto{2.673pt}{-4.83984pt}{4.83984pt}{-2.673pt}{4.83984pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.25pt}{-1.6111pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\hbox{{1}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=\leavevmode\hbox
to10.08pt{\vbox to10.08pt{\pgfpicture\makeatletter\hbox{\hskip
5.03984pt\lower-5.03984pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.83984pt}{0.0pt}\pgfsys@curveto{4.83984pt}{2.673pt}{2.673pt}{4.83984pt}{0.0pt}{4.83984pt}\pgfsys@curveto{-2.673pt}{4.83984pt}{-4.83984pt}{2.673pt}{-4.83984pt}{0.0pt}\pgfsys@curveto{-4.83984pt}{-2.673pt}{-2.673pt}{-4.83984pt}{0.0pt}{-4.83984pt}\pgfsys@curveto{2.673pt}{-4.83984pt}{4.83984pt}{-2.673pt}{4.83984pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.25pt}{-1.6111pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\hbox{{2}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=\leavevmode\hbox
to10.08pt{\vbox to10.08pt{\pgfpicture\makeatletter\hbox{\hskip
5.03984pt\lower-5.03984pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
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instability is fast $(a<0)$, but still long-wave $(b>0)$.

Given that the most interesting and physically relevant case is the second
one, i.e. if instability develops at short times then it will dominate the
subsequent dynamics and the case (i) becomes irrelevant, let us proceed with
its analysis:

$\displaystyle
2\,A_{\tau}F+2\frac{\varphi_{\tau}}{\tau^{1/3}}F^{\prime}+3\,A\,\tau^{-1}F\,F^{\prime}+\beta\,\tau^{-5/3}A_{\Theta\Theta}\int^{\zeta_{0}}{F\,\mathrm{d}\widehat{\zeta}}+\beta\,\tau^{-2}\varphi_{\Theta\Theta}F=0.$
(177)

The challenge of applying the Kadomtsev & Petviashvili (1970) type analysis to
(177) consists, in particular, in the lesser degree of localization of the
soliton (139,165) compared to the plane 1D case (6) as we saw in §3.2. The
goal, however, is still the same – to decompose (177) in functionally
independent parts, which would lead to an amplitude equation for the
perturbation.

Differentiating (177) w.r.t. $\zeta_{0}$,

$\displaystyle\left(2\,A_{\tau}+\beta\,\tau^{-2}\varphi_{\Theta\Theta}\right)F^{\prime}+2\frac{\varphi_{\tau}}{\tau^{1/3}}F^{\prime\prime}+3\,A\,\tau^{-1}\left(F\,F^{\prime}\right)^{\prime}+\beta\,\tau^{-5/3}A_{\Theta\Theta}F=0,$
(178)

multiplying by $F^{\prime}$ and integrating w.r.t. $\zeta_{0}$, for example
for $a>0$ from $\zeta_{0}$ to $\infty$ as dictated by the asymptotic behavior
(145), in the limit $\zeta_{0}=-\infty$ we get at the leading order

$\displaystyle 2\,A_{\tau}+\beta\,\tau^{-2}\varphi_{\Theta\Theta}=0.$ (179)

In arriving at (179) we took into account that
$\int_{\zeta_{0}}^{\infty}{F^{\prime}\left(FF^{\prime}\right)^{\prime}\,\mathrm{d}\zeta}=\frac{1}{2}\int_{\zeta_{0}}^{\infty}{F^{\prime
3}\,\mathrm{d}\zeta}$ and in the limit $\zeta_{0}\rightarrow-\infty$ the
following integrals simplify
$\int_{\zeta_{0}}^{\infty}{F^{\prime}F^{\prime\prime}\,\mathrm{d}\zeta}=\frac{1}{2}\left.F^{\prime
2}\right|_{\zeta_{0}}\sim\mathrm{const}$,
$\int_{\zeta_{0}}^{\infty}{FF^{\prime}\,\mathrm{d}\zeta}=\frac{1}{2}\left.F^{2}\right|_{\zeta_{0}}=0$,
as well as

$\displaystyle\lim_{\zeta_{0}\rightarrow-\infty}{\frac{\int_{\zeta_{0}}^{\infty}{F^{\prime
3}\,\mathrm{d}\zeta}}{\int_{\zeta_{0}}^{\infty}{F^{\prime
2}\,\mathrm{d}\zeta}}}=0,$ (180)

since in the limit $\zeta_{0}\rightarrow-\infty$ the integral
$\int_{\zeta_{0}}^{\infty}{F^{\prime 2}\,\mathrm{d}\zeta}$ diverges as
$\sim\zeta_{0}$, while the integral $\int_{\zeta_{0}}^{\infty}{F^{\prime
3}\,\mathrm{d}\zeta}$ grows slower than $\zeta_{0}$ due to cancellation of
integrals of a fast oscillating function for $\zeta_{0}\rightarrow-\infty$ –
the property also known as the Riemann-Lebesgue lemma in the case of Fourier
analysis. Hence, compared to the approach of Kadomtsev & Petviashvili (1970),
we used the different rate of divergence of the corresponding integrals (180).
Note that integration in the case $a<0$ would have to be from $-\infty$ to
$\zeta_{0}$ with the limit taken as $\zeta_{0}\rightarrow+\infty$ due to the
asymptotic behavior of the soliton reversed compared to (145).

Similarly, multiplying (178) by $F^{\nu}$ with $\nu>3$ and integrating w.r.t.
$\zeta_{0}$, for example for $a>0$ from $\zeta_{0}$ to $\infty$, leads to

$\displaystyle
2\frac{\varphi_{\tau}}{\tau^{1/3}}+3\,A\,\tau^{-1}\mathcal{I}_{32}+\beta\,\tau^{-5/3}A_{\Theta\Theta}\mathcal{I}_{12}=0,$
(181)

where $\mathcal{I}_{32}=\mathcal{I}_{3}/\mathcal{I}_{2}>0$ and
$\mathcal{I}_{12}=\mathcal{I}_{1}/\mathcal{I}_{2}<0$ with the corresponding
finite integrals $\mathcal{I}_{1-3}$ defined as follows

$\displaystyle\mathcal{I}_{1}$
$\displaystyle=\int_{\zeta_{0}}^{\infty}{F^{\nu+1}\,\mathrm{d}\zeta}\
\text{converges for}\ \nu>1,$ (182a) $\displaystyle\mathcal{I}_{2}$
$\displaystyle=\int_{\zeta_{0}}^{\infty}{F^{\nu}F^{\prime\prime}\,\mathrm{d}\zeta}\
\text{converges for}\ \nu>3,$ (182b) $\displaystyle\mathcal{I}_{3}$
$\displaystyle=\int_{\zeta_{0}}^{\infty}{F^{\nu}\left(FF^{\prime}\right)^{\prime}\,\mathrm{d}\zeta}=\left.F^{\nu+1}F^{\prime}\right|_{\zeta_{0}}^{\infty}-\nu\int_{\zeta_{0}}^{\infty}{F^{\nu}F^{\prime
2}\,\mathrm{d}\zeta}\ \text{converges for}\ \nu>2;$ (182c)

note that in the last integral
$\lim_{\zeta_{0}\rightarrow-\infty}{\left.F^{\nu+1}F^{\prime}\right|_{\zeta_{0}}^{\infty}}=0$.
In the deduction of (181) we also took into account that
$\int_{-\infty}^{\infty}{F^{\nu}F^{\prime}\,\mathrm{d}\zeta}=0$ as well as
$F(\zeta_{0})\sim(-\zeta_{0})^{-1/2}$ for $\zeta_{0}\rightarrow-\infty$ as per
(145).

[] []

Figure 6: Behavior of two independent solutions (a,b) to (185) corresponding
to the short-time asymptotics of (184); insets show the oscillatory behavior
near the origin.

As a result, the perturbation evolution is determined by the system (179,181),
which after the Fourier transform in the transverse direction becomes:

$\displaystyle 2\,\widehat{A}_{\tau}-\beta\,k^{2}\,\tau^{-2}\widehat{\varphi}$
$\displaystyle=0,$ (183a) $\displaystyle
2\frac{\widehat{\varphi}_{\tau}}{\tau^{1/3}}+3\,\widehat{A}\,\tau^{-1}\mathcal{I}_{32}-\beta\,k^{2}\,\tau^{-5/3}\widehat{A}\,\mathcal{I}_{12}$
$\displaystyle=0,$ (183b)

and can be reduced to a single equation after elimination of
$\widehat{\varphi}$ and substitution $\widehat{A}=\tau^{-1}\widehat{\eta}$:

$\displaystyle\widehat{\eta}_{\tau\tau}+\frac{\beta
k^{2}}{4}\tau^{-10/3}\left(3\,\mathcal{I}_{32}\,\tau^{3/2}-\beta
k^{2}\mathcal{I}_{12}\right)\widehat{\eta}=0.$ (184)

The first observation to make about equation (184) is that the transverse
wavenumber $k$ can be scaled out by $\tau\rightarrow k^{3}\widetilde{\tau}$
and hence no critical wavenumber exists, also in agreement with the
conclusions of §3.3. While equation (184) corresponds to the short time
instability, i.e. case (ii), within this asymptotic approximation we can
consider the short- and long-time behavior of (184) in the proper multiple-
scale sense. Clearly, for short times it is the second term in the brackets of
(184), which is dominant, thus leading to

$\displaystyle\widehat{\eta}_{\widetilde{\tau}\widetilde{\tau}}+\widetilde{\tau}^{-10/3}\widehat{\eta}=0,$
(185)

after the straightforward scaling out of the constant with the help of
redefining the time variable and taking into account that
$\mathcal{I}_{12}<0$. The solution of (185) is a linear combination of two
independent modes

$\displaystyle\widehat{\eta}(\widetilde{\tau})=C_{1}\widetilde{\tau}^{1/2}J_{3/4}{\left(\frac{3}{2}\widetilde{\tau}^{-2/3}\right)}+C_{2}\widetilde{\tau}^{1/2}J_{-3/4}{\left(\frac{3}{2}\widetilde{\tau}^{-2/3}\right)},$
(186)

which are shown in figure 6 – the first is approaching a constant plateau,
while the second one grows linearly in time. The long-time asymptotics of
(184) is dictated by the first term in the brackets, which after scaling out
the numerical coefficient produces

$\displaystyle\widehat{\eta}_{\widetilde{\tau}\widetilde{\tau}}\pm\widetilde{\tau}^{-8/3}\widehat{\eta}=0,$
(187)

where the plus sign corresponds to $\beta>0$ and negative to $\beta<0$. The
solutions of (187) read for $\beta>0$:

$\displaystyle\widehat{\eta}(\widetilde{\tau})$
$\displaystyle=C_{1}\widetilde{\tau}\left[-\varrho\cos{\varrho}+\sin{\varrho}\right]=C_{2}\widetilde{\tau}\left[\cos{\varrho}+\varrho\sin{\varrho}\right],$
(188)

where $\varrho=3\,\widetilde{\tau}^{-1/3}$, and are illustrated in figure 7.
In the case $\beta<0$, the solution becomes

$\displaystyle\widehat{\eta}(\widetilde{\tau})$
$\displaystyle=C_{1}\widetilde{\tau}\left[\varrho\cosh{\varrho}-\sinh{\varrho}\right]=C_{2}\widetilde{\tau}\left[\cosh{\varrho}-\varrho\sinh{\varrho}\right]$
(189)

and is illustrated in figure 7. In both cases, one of the solutions approaches
a non-zero constant, while the other one grows linearly (the one in figure 7
is shown on logarithmic scale). Thus, taking into account the transformation
$\widehat{A}=\tau^{-1}\widehat{\eta}$ connecting $\widehat{A}$ and
$\widehat{\eta}$, we conclude that initial perturbations, measured relative to
the unit amplitude of the self-similar solution as per (166), are able to grow
from infinitesimal values and approach some finite value, so that nonlinear
effects start playing a role – this behavior is atypical for linear stability
problems, usually exhibiting either exponential growth or decay, and more
characteristic to nonlinear behavior predicated earlier in §3.3 based on the
properties of equation (157). Therefore, it is the short-time behavior
governed by (184) which dictates the transverse stability properties of ncKdV,
and makes the appearance of transverse instability possible. This situation is
not unusual for stability problems involving time-dependent base states such
as in the Rayleigh-Plateau instability of a growing cylindrical liquid blob
(Krechetnikov, 2017).

[] []

Figure 7: Behavior of two independent solutions to (187) corresponding to (a)
$\beta>0$ and (b) $\beta<0$.

##  4 Conclusions

With the goal to study stability of axisymmetric solitary waves, in the
present work we deduced a proper envelope equation for solitary waves on deep
water, which proves to include an inverse-square potential and hence be of
Gross-Pitaevskii type (24,28); in the shallow water limit we rederived a ncKdV
equation (133) by including surface tension effects and under asymptotic
assumptions different from what was known before. In the former case, our
derivation is set apart from previous studies which postulated that the
corresponding NLS for axisymmetric case has the Laplace operator unchanged –
our analysis (§§2.1,2.2) demonstrates that the covariance principle does not
apply to envelope equations despite their “universal” character. Given the
novelty of the deduced GP equation for deep water waves, we studied its
general properties – conservation laws (§2.3), Hamiltonian structure (§2.6),
finite-time-singularity (§2.3) – as well as axisymmetric base states along
with geometric and mechanistic interpretations of their varieties (§2.4,
Appendix A).

The stability analysis in the deep water case was performed with the help of
both spectral (in the limit of long wavelengths, cf. §2.5) and Hamiltonian
(for general wavelengths, cf. §2.6) methods, which complement each other. The
challenge of the spectral stability problem (81) was its singular nature
dictated by the particularities of the base states (§2.4), which nevertheless
enable analytical approaches. We revealed the crucial differences in stability
characteristics between cylindrical and plane solitons: namely, there is a
threshold in the Weber number $We_{c}=\frac{1}{2}$ above which instability
appears in the deep water case as opposed to the nearly plane NLS (2), the 1D
plane solitons of which are always unstable to transverse perturbations
(Zakharov & Rubenchik, 1974) regardless of the value of $We$. Thus, surface
tension must be sufficiently strong to induce a transverse instability of a
cylindrical soliton on the deep water444A qualitative interpretation one may
offer is that in the case of a plane soliton surface tension breaks it similar
to a Rayleigh-Plateau instability of a rectilinear liquid column, which takes
place for any magnitude of surface tension as long as it is non-zero, while in
the case of a cylindrical soliton the Rayleigh-Plateau instability competes
with stabilizing effect of the transverse curvature in the plane of the
soliton propagation as well as with the time-dependence of the base state..

In the shallow water case, we performed an analysis (§3.4) in the spirit of
Kadomtsev & Petviashvili (1970) extending it not only to cylindrical geometry
but also to self-similar solitons (139), with the resulting linear amplitude
equation (184), which governs perturbation evolution, being highly-
nonautonomous and exhibiting transient growth of transverse perturbations
regardless of the value of the Weber number in contrast to its plane
counterpart, where there is a non-zero critical Weber number555Obviously, the
effect of the solid bottom plays a stabilizing role in the case of plane
solitons and thus requires strong enough surface tension to induce a
transverse instability, i.e. to get into a Rayleigh-Plateau regime. On the
other hand, in the case of a cylindrical soliton the effect of the base
soliton time-dependence overpowers any other effects thus leading to transient
growth of perturbations, which should trigger nonlinear effects before the
subsequent linear dynamics would lead to a decay of the perturbation.. For
long times the stability picture is consistent with the intuition that the
dynamics should approach that of npKdV (§3.3). Also, for general wavenumbers,
from the reduction to a 2D eigenvalue problem (157) in the self-similar plane,
we made an unexpected conclusion that the transverse perturbations must have
an irregular structure in the azimuthal $\theta$-direction, cf. §3.3.
Numerical study of (157), however, represents a challenge for future efforts.

While here we explored only the basic properties of the GP and ncKdV
solutions, one might expect that similar to the standard (near planar)
versions of these equations, the behavior of their solutions is very rich (Cai
et al., 2002). Including higher-order terms (Dysthe, 1979) or generalization
onto finite depth (Hasimoto & Ono, 1972) of the GP equation may offer further
insights in the axisymmetric water waves, same as establishing a relation
between GP and ncKdV similar to that between NLS and KdV (Boyd & Chen, 2001)
as well as considering the near-critical values of the Weber number
$We\rightarrow We_{c}=\frac{1}{3}$ in the ncKdV equation, which should bring
up fifth-order derivatives (Green, 1983; Hunter & Scheurle, 1988). Also, in
the derivation of ncKdV equation with surface tension (§3.1) we neglected the
resonance between the linear (carrier) wave speed $c_{0}$ and the linear phase
speed $\omega(k)/k\approx g^{1/2}(1+We\,k^{2}h^{2})^{1/2}h$, which exhibits
itself in the far-field (Boyd, 1998; Grimshaw & Iooss, 2003; Grimshaw, 2005)
and occurs because the graph $\omega(k)/k$ is not monotonic when
$0<We<\frac{1}{3}$; for $We>\frac{1}{3}$ the graph of $\omega(k)/k$ is
monotonic and hence the derivation of ncKdV does not require corrections.

Both types of envelope equations – on deep and shallow water – could be
amenable to the inverse scattering transform methods, which may serve as yet
another method for studying transverse stability as it was done by Zakharov
(1975) for the KdV solitons. While some solutions of the cKdV (135) were
constructed with inverse scattering transform, cf. Calogero & Degasperis
(1978); Johnson (1979); Freeman (1980), the feasibility of the inverse
scattering for the NLS with an inverse-square potential has not been fully
explored yet, cf. Murphy (2019) and references therein, even though NLS with
potentials could be suitable to inverse scattering transform analysis and
represents an active area of research, cf. Sasaki (2008); Fajun & Li (2019).

## Acknowledgements

his work was partially supported by the National Science Foundation (NSF)
CAREER award under Grant No. 1054267 and the Natural Sciences and Engineering
Research Council of Canada (NSERC) under Grant No. 04374. Declaration of
interests: the author reports no conflict of interest.

##  Appendix A Mechanical analogy

[] [] []

Figure 8: Isolines of canonical Hamiltonians (a)
$\dot{\xi}^{2}+\frac{1}{\xi^{2}}=1$, (b) $\dot{\xi}^{2}-\frac{1}{\xi^{2}}=1$,
and (c) $-\dot{\xi}^{2}+\frac{1}{\xi^{2}}=1$.

Equation (28) admits a mechanistic interpretation by looking for solutions in
the form $\psi_{0}=\mathrm{H}^{1/2}\,e^{\mathrm{i}S(R)}$, which leads to the
Hamiltonian

$\displaystyle-\frac{\lambda_{\infty}}{\chi_{\infty}}p^{2}+\frac{\lambda_{\infty}^{\prime}}{\chi_{\infty}}\frac{1}{R^{2}}\equiv\mathrm{H},$
(190)

where $p=S_{R}$ is the momentum, while the factors
$\frac{\lambda_{\infty}}{\chi_{\infty}}$ and
$\frac{\lambda_{\infty}^{\prime}}{\chi_{\infty}}$ may assume all possible sign
combinations as per (29). Of course, motion of a particle in a central field
such as our inverse-square potential would also bring in conservation of
angular momentum, which in turn leads to an “effective” $\sim R^{-2}$
potential (Arnold, 1989) as well; hence the potential in (190) can be
considered as a total one. Most importantly, if both central and effective
potentials are $\sim R^{-2}$, then there is no local minimum of the total
potential and hence no non-zero lower bound for the particle trajectory. In
arriving at the Hamiltonian (190) we omitted the terms
$S_{RR}+\frac{1}{R}S_{R}$ as in the WKB method, which is to be verified a
posteriori. Without loss, we may scale out the factors in (190) and thus
consider canonical Hamiltonians as in figure 8. Clearly, all trajectories are
unbounded. This can also be seen by introducing an “effective” time $t$ to
arrive at the corresponding Hamitonian systems:
$\ddot{\xi}=\pm\frac{4}{\xi^{3}}$, integration of which indeed gives diverging
trajectories as per the phase portraits in figure 9; these trajectories
correspond to solutions of (28) oscillatory at infinity. Hence
$S_{R}\approx\left(-\frac{\chi_{\infty}}{\lambda_{\infty}}\mathrm{H}\right)^{1/2}\left(1-\frac{\lambda_{\infty}^{\prime}}{\chi_{\infty}}\frac{1}{2\mathrm{H}R^{2}}+\ldots\right)$
for large $R$ and thus neglecting the omitted terms $S_{RR}+\frac{1}{R}S_{R}$
in (190) is justified.

[] []

Figure 9: Phase plots of (a) $\ddot{\xi}=-\frac{4}{\xi^{3}}$ and (b)
$\ddot{\xi}=\frac{4}{\xi^{3}}$.

##  Appendix B Variance

Finally, let us derive an equation for the variance
$\mathcal{V}(\tau)=\int{R^{2}|\psi|^{2}\,\mathrm{d}\nu}$. Taking the first
derivative, we find

$\mathrm{i}\frac{\mathrm{d}\mathcal{V}}{\mathrm{d}\tau}=\mathrm{i}\int{R^{2}\left(\overline{\psi}\psi_{\tau}+\psi\overline{\psi}_{\tau}\right)\,\mathrm{d}\nu}=-\lambda_{\infty}\int{R^{2}\left(\overline{\psi}\Delta_{R}\psi-\psi\Delta_{R}\overline{\psi}\right)\,\mathrm{d}\nu}\\\
-\mu_{\infty}\int{\frac{1}{R^{2}}\left(\overline{\psi}\psi_{\theta\theta}-\psi\overline{\psi}_{\theta\theta}\right)\,\mathrm{d}\nu},$
(191)

which follows from multiplying (28) by $\overline{\psi}$, i.e. equation (42),
and subtracting the complex conjugate of (28) multiplied by $\psi$:

$\displaystyle-\mathrm{i}\,\psi\,\overline{\psi}_{\tau}+\lambda_{\infty}\psi\,\Delta_{R}\overline{\psi}+\frac{\lambda_{\infty}^{\prime}}{R^{2}}|\psi|^{2}+\frac{\mu_{\infty}}{R^{2}}\psi\overline{\psi}_{\theta\theta}-\chi_{\infty}\,|\psi|^{4}=0,$
(192)

results in

$\displaystyle\mathrm{i}\,\partial_{\tau}|\psi|^{2}=-\lambda_{\infty}\left(\overline{\psi}\Delta_{R}\psi-\psi\Delta_{R}\overline{\psi}\right)-\frac{\mu_{\infty}}{R^{2}}\left(\overline{\psi}\psi_{\theta\theta}-\psi\overline{\psi}_{\theta\theta}\right).$
(193)

The first integral on the rhs of (191) is calculated via integration by parts
and use of BCs (41):

$\displaystyle\int_{0}^{\infty}{R^{2}\left[\overline{\psi}\left(\psi_{RR}+\frac{1}{R}\psi_{R}\right)-\psi\left(\overline{\psi}_{RR}+\frac{1}{R}\overline{\psi}_{R}\right)\right]R\,\mathrm{d}R}$
(194) $\displaystyle=$ $\displaystyle
R^{3}\left[\psi_{R}\overline{\psi}-\overline{\psi}_{R}\psi\right]_{0}^{\infty}-\int_{0}^{\infty}{\left[\psi_{R}\left(R^{3}\overline{\psi}\right)_{R}-\overline{\psi}_{R}\left(R^{3}\psi\right)_{R}\right]\mathrm{d}R}+\int_{0}^{\infty}{R^{2}\left(\overline{\psi}\psi_{R}-\psi\overline{\psi}_{R}\right)\,\mathrm{d}R}$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}{3R^{2}\left(\overline{\psi}_{R}\psi-\psi_{R}\overline{\psi}\right)\,\mathrm{d}R}+\int_{0}^{\infty}{R^{2}\left(\overline{\psi}\psi_{R}-\psi\overline{\psi}_{R}\right)\,\mathrm{d}R}=2\int_{0}^{\infty}{R^{2}\left(\overline{\psi}_{R}\psi-\psi_{R}\overline{\psi}\right)\,\mathrm{d}R}.$

The second integral on the rhs of (191) is calculated also via integration by
parts and use of periodicity in $\theta$:

$\displaystyle\begin{split}\int_{0}^{2\pi}{\left(\overline{\psi}\psi_{\theta\theta}-\psi\overline{\psi}_{\theta\theta}\right)\,\mathrm{d}\theta}=\left[\overline{\psi}\psi_{\theta}-\psi\overline{\psi}_{\theta}\right]_{0}^{2\pi}-\int_{0}^{2\pi}{\left(\overline{\psi}_{\theta}\psi_{\theta}-\psi_{\theta}\overline{\psi}_{\theta}\right)\,\mathrm{d}\theta}=0.\end{split}$
(195)

Therefore, the first derivative of the variance (191) becomes:

$\displaystyle\mathrm{i}\frac{\mathrm{d}\mathcal{V}}{\mathrm{d}\tau}=-2\lambda_{\infty}\int{R\left(\overline{\psi}_{R}\psi-\psi_{R}\overline{\psi}\right)\,\mathrm{d}\nu}.$
(196)

Differentiating $\mathcal{V}$ second time, we obtain

$\displaystyle-\frac{\mathrm{i}}{2\lambda_{\infty}}\frac{\mathrm{d}^{2}\mathcal{V}}{\mathrm{d}\tau^{2}}$
$\displaystyle=\int{R\left(\overline{\psi}_{R}\psi_{\tau}-\psi_{R}\overline{\psi}_{\tau}+\overline{\psi}_{R\tau}\psi-\psi_{R\tau}\overline{\psi}\right)\,\mathrm{d}\nu}$
$\displaystyle=\int_{0}^{2\pi}{\mathrm{d}\theta}\int_{0}^{\infty}{R^{2}\left(\overline{\psi}_{R}\psi_{\tau}-\psi_{R}\overline{\psi}_{\tau}\right)\,\mathrm{d}R}+\int_{0}^{2\pi}{R^{2}\left[\overline{\psi}_{\tau}\psi-\psi_{\tau}\overline{\psi}\right]_{R=0}^{\infty}\,\mathrm{d}\theta}$
$\displaystyle-\int_{0}^{2\pi}{\mathrm{d}\theta}\int_{0}^{\infty}{\left[\overline{\psi}_{\tau}\left(R^{2}\psi\right)_{R}-\psi_{\tau}\left(R^{2}\overline{\psi}\right)_{R}\right]\,\mathrm{d}R}$
$\displaystyle=2\int_{0}^{2\pi}{\mathrm{d}\theta}\int_{0}^{\infty}{R^{2}\left(\overline{\psi}_{R}\psi_{\tau}-\psi_{R}\overline{\psi}_{\tau}\right)\,\mathrm{d}R}-2\int_{0}^{2\pi}{\mathrm{d}\theta}\int_{0}^{\infty}{R\left(\overline{\psi}_{\tau}\psi-\psi_{\tau}\overline{\psi}\right)\,\mathrm{d}R}$
$\displaystyle=2\int{R\left(\overline{\psi}_{R}\psi_{\tau}-\psi_{R}\overline{\psi}_{\tau}\right)\,\mathrm{d}\nu}-2\int{\left(\overline{\psi}_{\tau}\psi-\psi_{\tau}\overline{\psi}\right)\,\mathrm{d}\nu}.$
(197)

The integrand in the second integral on the rhs of (B) is computed by adding
(42) to (192):

$\displaystyle\mathrm{i}\left(\overline{\psi}_{\tau}\psi-\psi_{\tau}\overline{\psi}\right)=\lambda_{\infty}\left(\psi\,\Delta_{R}\overline{\psi}+\overline{\psi}\,\Delta_{R}\psi\right)+\frac{2\lambda_{\infty}^{\prime}}{R^{2}}|\psi|^{2}+\frac{\mu_{\infty}}{R^{2}}\left(\psi\overline{\psi}_{\theta\theta}+\overline{\psi}\psi_{\theta\theta}\right)-2\chi_{\infty}\,|\psi|^{4}.$

To compute the corresponding integral, we note that
$\psi\,\Delta_{R}\overline{\psi}+\overline{\psi}\,\Delta_{R}\psi=2\left(\psi^{r}\,\Delta_{R}\psi^{r}+\psi^{i}\,\Delta_{R}\psi^{i}\right)$
and

$\displaystyle\begin{split}\int_{0}^{\infty}{\phi\Delta_{R}\phi\,R\,\mathrm{d}R}&=\left.\phi_{R}\phi
R\right|_{0}^{\infty}-\int_{0}^{\infty}{\phi_{R}\left(\phi\,R\right)_{R}\,\mathrm{d}R}+\int_{\mathbb{R}}{\phi\phi_{R}\,\mathrm{d}R}\\\
&=\left.\phi_{R}\phi
R\right|_{0}^{\infty}-\int_{0}^{\infty}{R\,\phi_{R}^{2}\,\mathrm{d}R},\end{split}$
(198)

so that applying the latter result to $\phi=\psi^{r},\psi^{i}$ and using BCs
(41) we get

$\displaystyle\int_{0}^{\infty}{\left(\psi\,\Delta_{R}\overline{\psi}+\overline{\psi}\,\Delta_{R}\psi\right)\,R\,\mathrm{d}R}=-\int_{0}^{\infty}{|\psi_{R}|^{2}\,R\,\mathrm{d}R}.$
(199)

Similarly, since
$\psi\,\overline{\psi}_{\theta\theta}+\overline{\psi}\,\psi_{\theta\theta}=2\left(\psi^{r}\,\psi^{r}_{\theta\theta}+\psi^{i}\,\psi^{i}_{\theta\theta}\right)$
and

$\displaystyle\int_{0}^{2\pi}{\frac{\phi\phi_{\theta\theta}}{R}\,\mathrm{d}\theta}=\left.\frac{\phi\phi_{\theta}}{R}\right|_{0}^{2\pi}-\int_{0}^{2\pi}{\frac{\phi_{\theta}^{2}}{R}\,\mathrm{d}\theta}=-\int_{0}^{2\pi}{\frac{\phi_{\theta}^{2}}{R}\,\mathrm{d}\theta},$
(200)

altogether the second integral on the rhs of (B) becomes

$\mathrm{i}\int{\left(\overline{\psi}_{\tau}\psi-\psi_{\tau}\overline{\psi}\right)\,\mathrm{d}\nu}=-2\lambda_{\infty}\int{|\psi_{R}|^{2}\,\mathrm{d}\nu}+2\lambda_{\infty}^{\prime}\int{\frac{|\psi|^{2}}{R^{2}}\,\mathrm{d}\nu}\\\
-2\mu_{\infty}\int{\frac{|\psi_{\theta}|^{2}}{R^{2}}\,\mathrm{d}\nu}-2\chi_{\infty}\int{|\psi|^{4}\,\mathrm{d}\nu}.$
(201)

As for the first integral on the rhs of (B), it is computed by multiplying
(28) with $\overline{\psi}_{R}$ and subtracting the complex conjugate of (28)
multiplied by $\psi_{R}$, which results in

$\mathrm{i}\left(\overline{\psi}_{R}\psi_{\tau}-\psi_{R}\overline{\psi}_{\tau}\right)+\lambda_{\infty}\mathop{\left(\overline{\psi}_{R}\,\Delta_{R}\psi+\psi_{R}\,\Delta_{R}\overline{\psi}\right)}_{\text{does
not
contribute}}+\left[\frac{\lambda_{\infty}^{\prime}}{R^{2}}-\chi_{\infty}|\psi|^{2}\right]\left(\overline{\psi}_{R}\psi+\psi_{R}\,\overline{\psi}\right)\\\
+\frac{\mu_{\infty}}{R^{2}}\mathop{\left(\overline{\psi}_{R}\,\psi_{\theta\theta}+\psi_{R}\,\overline{\psi}_{\theta\theta}\right)}_{\text{does
not contribute}}=0,$ (202)

where
$\overline{\psi}_{R}\psi+\psi_{R}\,\overline{\psi}=\frac{\mathrm{d}}{\mathrm{d}R}|\psi|^{2}$,
$\overline{\psi}_{R}\,\Delta_{R}\psi+\psi_{R}\,\Delta_{R}\overline{\psi}=2\left(\psi^{r}_{R}\,\Delta_{R}\psi^{r}+\psi^{i}_{R}\,\Delta_{R}\psi^{i}\right)$
and
$\overline{\psi}_{R}\,\psi_{\theta\theta}+\psi_{R}\,\overline{\psi}_{\theta\theta}=2\left(\psi^{r}_{R}\,\psi^{r}_{\theta\theta}+\psi^{i}_{R}\,\psi^{i}_{\theta\theta}\right)$;
also, given that

$\int_{0}^{\infty}{R^{2}\,\phi_{R}\Delta_{R}\phi\,\mathrm{d}R}=\int_{0}^{\infty}{R^{2}\,\phi_{R}\frac{1}{R}\frac{\mathrm{d}}{\mathrm{d}R}\left(R\phi_{R}\right)\,\mathrm{d}R}\\\
=\frac{1}{2}\int_{0}^{\infty}{\frac{\mathrm{d}}{\mathrm{d}R}\left(R\phi_{R}\right)^{2}\,\mathrm{d}R}=\frac{1}{2}\left.\left(R\phi_{R}\right)^{2}\right|_{0}^{\infty}=0,$
(203)

by applying this result to $\phi=\psi^{r},\psi^{i}$ with BCs (41) we find

$\displaystyle\int_{0}^{\infty}{R^{2}\left(\overline{\psi}_{R}\,\Delta_{R}\psi+\psi_{R}\,\Delta_{R}\overline{\psi}\right)\,\mathrm{d}R}=0.$
(204)

In the same fashion,

$\int_{0}^{2\pi}\int_{0}^{\infty}{\phi_{R}\phi_{\theta\theta}\,\mathrm{d}\theta\mathrm{d}R}=\int_{0}^{\infty}{\left[\left.\phi_{R}\phi_{\theta}\right|_{0}^{2\pi}-\int_{0}^{2\pi}{\phi_{\theta}\phi_{R\theta}\,\mathrm{d}\theta}\right]\,\mathrm{d}R}\\\
=-\frac{1}{2}\int_{0}^{2\pi}{\int_{0}^{\infty}{\frac{\mathrm{d}}{\mathrm{d}R}\left(\phi_{\theta}^{2}\right)\,\mathrm{d}R}\,\mathrm{d}\theta}=-\frac{1}{2}\int_{0}^{2\pi}{\left.\phi_{\theta}^{2}\right|_{R=0}^{\infty}\,\mathrm{d}\theta}=0,$
(205)

so that the first integral on the rhs of (B) becomes

$\mathrm{i}\int_{0}^{\infty}\int_{0}^{2\pi}{R^{2}\left(\overline{\psi}_{R}\psi_{\tau}-\psi_{R}\overline{\psi}_{\tau}\right)\,\mathrm{d}R}=-\lambda_{\infty}^{\prime}\int_{0}^{2\pi}{\left.|\psi|^{2}\right|_{R=0}^{\infty}\,\mathrm{d}\theta}+\chi_{\infty}\int{R\,|\psi|^{2}\frac{\mathrm{d}}{\mathrm{d}R}|\psi|^{2}\,\mathrm{d}\nu}\\\
=2\pi\lambda_{\infty}^{\prime}|\psi(\tau,0)|^{2}+\chi_{\infty}\int{R\,|\psi|^{2}\frac{\mathrm{d}}{\mathrm{d}R}|\psi|^{2}\,\mathrm{d}\nu},$
(206)

where in the first term on the rhs in the above expression we used the
equality

$\displaystyle\int_{0}^{\infty}{R^{2}\left[\frac{1}{R^{2}}\right]\frac{\mathrm{d}}{\mathrm{d}R}|\psi|^{2}\,\mathrm{d}R}$
$\displaystyle=\int_{0}^{\infty}{\frac{\mathrm{d}}{\mathrm{d}R}|\psi|^{2}\,\mathrm{d}R}=\left.|\psi|^{2}\right|_{0}^{\infty},$
(207)

and in the last term

$\int_{0}^{\infty}{R^{2}|\psi|^{2}\frac{\mathrm{d}}{\mathrm{d}R}|\psi|^{2}\,\mathrm{d}R}=\frac{1}{2}\int_{0}^{\infty}{R^{2}\frac{\mathrm{d}}{\mathrm{d}R}\left(|\psi|^{2}\right)^{2}\,\mathrm{d}R}\\\
=\frac{1}{2}\left[\left.R^{2}|\psi|^{4}\right|_{0}^{\infty}-2\int_{0}^{\infty}{R\,|\psi|^{4}\,\mathrm{d}R}\right].$
(208)

As a result, the second derivative of the variance (B)

$\frac{1}{4\lambda_{\infty}}\frac{\mathrm{d}^{2}\mathcal{V}}{\mathrm{d}\tau^{2}}=\mathrm{i}\int_{0}^{2\pi}{\mathrm{d}\theta}\int_{0}^{\infty}{R^{2}\left(\overline{\psi}_{R}\psi_{\tau}-\psi_{R}\overline{\psi}_{\tau}\right)\,\mathrm{d}R}\\\
-\mathrm{i}\int_{0}^{2\pi}{\mathrm{d}\theta}\int_{0}^{\infty}{R\left(\overline{\psi}_{\tau}\psi-\psi_{\tau}\overline{\psi}\right)\,\mathrm{d}R}$
(209)

becomes

$\frac{1}{4\lambda_{\infty}}\frac{\mathrm{d}^{2}\mathcal{V}}{\mathrm{d}\tau^{2}}=2\pi\lambda_{\infty}^{\prime}|\psi(\tau,0)|^{2}-\chi_{\infty}\int{|\psi|^{4}\,\mathrm{d}\nu}+2\lambda_{\infty}\int{|\psi_{R}|^{2}\,\mathrm{d}\nu}\\\
-2\lambda_{\infty}^{\prime}\int{\frac{|\psi|^{2}}{R^{2}}\,\mathrm{d}\nu}+2\mu_{\infty}\int{\frac{|\psi_{\theta}|^{2}}{R^{2}}\,\mathrm{d}\nu}+2\chi_{\infty}\int{|\psi|^{4}\,\mathrm{d}\nu}.$
(210)

Using the expression (52) for the Hamiltonian, equation (209) can be rewritten
in the form (53).

##  Appendix C Asymptotics of ground states at infinity

To find next-order corrections, we introduce a phase $\varphi(x)$ such that
$|\varphi(x)|\ll x$, so that plugging $y(x)=C\cos{\left[x+\varphi(x)\right]}$
into (62) produces

$-\cos{\left[x+\varphi(x)\right]}\,(1+\varphi^{\prime})^{2}-\sin{\left[x+\varphi(x)\right]}\varphi^{\prime\prime}+\cos{\left[x+\varphi(x)\right]}\\\
\frac{d}{x^{2}}\cos{\left[x+\varphi(x)\right]}+\frac{C^{2}}{x}\cos^{3}{\left[x+\varphi(x)\right]}=0.$
(211)

After cancelling out the leading-order solution
$\cos{\left[x+\varphi(x)\right]}$, the next order terms are

$\displaystyle-2\varphi^{\prime}(x)+\frac{C^{2}}{x}\cos^{2}{\left[x+\varphi(x)\right]}=0\
\Rightarrow\
\frac{\mathrm{d}\varphi}{\mathrm{d}x}=C^{2}\frac{1+\cos{2\left[x+\varphi(x)\right]}}{4x}.$
(212)

Since we are studying the asymptotics at infinity, it is convenient to
integrate the latter equation from $x$ to some large, but finite $x_{\infty}$,
which yields

$\displaystyle\varphi(x_{\infty})-\varphi(x)=\frac{C^{2}}{4}\left(\ln{x_{\infty}}-\ln{x}\right)+\frac{C^{2}}{4}\left[\operatorname{Ci}(2x_{\infty})-\operatorname{Ci}(2x)\right],$
(213)

where the last term follows from

$\displaystyle\int_{x}^{x_{\infty}}{\frac{\cos{2\left[x+\varphi(x)\right]}}{x}\mathrm{d}x}=\int_{x}^{x_{\infty}}{\frac{\cos{2\widetilde{x}}}{\widetilde{x}-\varphi(x)}\frac{\mathrm{d}\widetilde{x}}{1+\varphi^{\prime}(x)}}\mathop{\Longrightarrow}_{x\gg
1}\int_{x}^{x_{\infty}}{\frac{\cos{2\widetilde{x}}}{\widetilde{x}}\mathrm{d}\widetilde{x}}$
(214) $\displaystyle=$
$\displaystyle\int_{2x}^{2x_{\infty}}{\frac{\cos{2\widetilde{x}}}{2\widetilde{x}}\mathrm{d}(2\widetilde{x})}=\int_{2x}^{\infty}{\frac{\cos{2\widetilde{x}}}{2\widetilde{x}}\mathrm{d}(2\widetilde{x})}-\int_{2x_{\infty}}^{\infty}{\frac{\cos{2\widetilde{x}}}{2\widetilde{x}}\mathrm{d}(2\widetilde{x})}\equiv-\operatorname{Ci}{(2x)}+\operatorname{Ci}{(2x_{\infty})}.$

Letting
$\varphi(x_{\infty})=\frac{C^{2}}{4}\ln{x_{\infty}}+\frac{C^{2}}{4}\operatorname{Ci}{(2x_{\infty})}$,
we find the phase

$\displaystyle\varphi(x)=\frac{C^{2}}{4}\ln{x}+\frac{C^{2}}{4}\operatorname{Ci}{(2x)}=\frac{C^{2}}{4}\ln{x}+\mathcal{O}\left(\frac{1}{x}\right),$
(215)

where we took into account the asymptotics of the cosine integral for large
argument $z$:

$\displaystyle\operatorname{Ci}{z}=\frac{1}{z}\sin{z}-\frac{1}{z^{2}}\cos{z}+\ldots.$
(216)

Altogether, the solution $y(x)$ of (62) for large $x$ reads

$\displaystyle
y(x)=C\cos{\left[x+\frac{C^{2}}{4}\ln{x}+\mathcal{O}\left(\frac{1}{x}\right)\right]},$
(217)

which is equation (71). To find the next-order correction we may add a new
phase function $\psi(x)$ to (71), i.e.
$y(x)=\cos{\left[x+\frac{C^{2}}{4}\ln{x}+\psi(x)\right]}$; straightforward
calculations yield

$\displaystyle\psi(x)=\frac{C^{2}}{4}\operatorname{Ci}{(2x)}\approx\frac{C^{2}}{4}\left[\frac{1}{2x}\sin{2x}-\frac{1}{4x^{2}}\cos{2x}\right]+\ldots.$
(218)

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# Enhancing Conformal Prediction Using E-Test Statistics

A.A. Balinsky  Department of Mathematics, Cardiff University, UK  A.D.
Balinsky  Department of Computing, Imperial College London, UK

###### Abstract

Conformal Prediction (CP) serves as a robust framework that quantifies
uncertainty in predictions made by Machine Learning (ML) models. Unlike
traditional point predictors, CP generates statistically valid prediction
regions, also known as prediction intervals, based on the assumption of data
exchangeability. Typically, the construction of conformal predictions hinges
on p-values. This paper, however, ventures down an alternative path,
harnessing the power of e-test statistics to augment the efficacy of conformal
predictions by introducing a BB-predictor (bounded from the below predictor).

##  1 Introduction

We will assume that the reader is familiar with the basics of CP, see [FZV23,
AB23] for recent reviews and introductions.

The first key benefit of CP is that it is distribution-free: CP doesn’t
require assumptions about the underlying data distribution, making it
adaptable to diverse scenarios. As such, CP is usually based on rank-based
statistics and permutations. The second key benefit is that CP can be built
under the exchangeability assumption, which is weaker than the i.i.d.
assumption. Recall that the exchangeability of $L_{1},\ldots,L_{n+1}$ means
that their joint distribution is unchanged under permutations:

$(L_{1},\ldots,L_{n+1})\stackrel{{\scriptstyle\text{d}}}{{=}}(L_{\sigma(1)},\ldots,L_{\sigma(n+1)})$

for all permutations $\sigma$.

The next Lemma is the main mathematical tool of standard CP theory and is
based on the idea of the p-value from hypothesis testing in statistics:

######  Lemma 1.

Suppose $L_{1},\ldots,L_{n+1}$ are exchangeable random variables. Set

$U=\\#\\{i=1,\ldots,n+1:\ L_{i}\geq L_{n+1}\\},$

i.e., the number of $L$ that are at least as large as the last one. Then

$P\left\\{\frac{U}{n+1}>\epsilon\right\\}\geq 1-\epsilon$

for all $\epsilon\in[0,1]$

In this article, we would like to develop another approach to CP by using the
theory of hypothesis testing based on e-test statistics, [GDHK19]. The idea
behind e-test statistics is very simple and is a straightforward application
of Markov’s inequality: if $E$ is non-negative random variables with the
expectation $\mathbf{E}(E)\leq 1$, then

$P(E\geq 1/\alpha)\leq\alpha,$

for any positive $\alpha$. If, for example, $\alpha=0.05$, then the event
$E\geq 20$ will have probability less than 5%. Though elementary, Markov’s
inequality in combination with Cramer-Chernoff’s method turns out to be very
powerful and surprisingly sharp [Mas07].

In the case of an unknown probability distribution, evaluating the mean value
of a random variable becomes challenging, especially if we lack an upper bound
estimate for this variable. Such an estimate is necessary to normalize the
random variable before applying Markov’s inequality. Naturally, as often
occurs in Machine Learning, we can approximate the mean value empirically if
we have a sufficient number of samples. However, we cannot provide a guarantee
similar to Lemma 1. The following section will present our main theoretical
result (Theorem 1) on addressing this issue in the context of exchangeable
random variables.

##  2 Main Result

The main theoretical result of this article is the following theorem

######  Theorem 1.

Suppose $L_{1},\ldots,L_{n+1}$ are exchangeable non-negative random variables.
Set

$F=\frac{L_{n+1}}{\left(\sum\limits_{j=1}^{n+1}L_{j}\right)/(n+1)}.$

Then the expectation $\mathbf{E}(F)=1$ and

$P\\{F\geq 1/\alpha\\}\leq\alpha,$

for any positive $\alpha$.

###### Proof.

Let us introduce the following random variables

$F_{i}=\frac{L_{i}}{\left(\sum\limits_{j=1}^{n+1}L_{j}\right)/(n+1)},\ \ \ \
i=1,2,\ldots,n+1.$

Due to the exchangeability of $L_{1},\ldots,L_{n+1}$, all random variables
$F_{i},\ i=1,\ldots,n+1$ are identically distributed, so they all have the
same expectation $\mathbf{E}(F_{i})$.

$\mathbf{E}(F_{1}+F_{2}+\ldots+F_{n+1})=(n+1)\times\mathbf{E}(F_{n+1})=(n+1)\times\mathbf{E}(F).$

From definition of $F_{i}$ we can see that $F_{1}+F_{2}+\ldots+F_{n+1}=n+1$.
Thus, $(n+1)\times\mathbf{E}(F)=n+1$, and $\mathbf{E}(F)=1$. Markov’s
inequality implies that

$P\\{F\geq 1/\alpha\\}\leq\alpha$

for any positive $\alpha$. ∎

Now, $F\geq 1/\alpha$ is equivalent to

$(n+1)L_{n+1}\geq\frac{1}{\alpha}(L_{1}+\ldots L_{n}+L_{n+1}),$

$(\alpha(n+1)-1)L_{n+1}\geq L_{1}+\ldots+L_{n},$

$\left(\alpha+\frac{\alpha-1}{n}\right)L_{n+1}\geq\frac{L_{1}+\ldots+L_{n}}{n}).$

######  Corollary 1.1.

Suppose $L_{1},\ldots,L_{n+1}$ are exchangeable non-negative random variables.
Then for any positive $\alpha$

$P\left\\{\frac{L_{n+1}}{\frac{L_{1}+\ldots+L_{n}}{n}}\geq\frac{1}{\alpha}\left(\frac{1}{1+\frac{1-1/\alpha}{n}}\right)\right\\}\leq\alpha.$
(1)

So, we can now introduce our new BB-predictor (bounded from the below
predictor)

$P\left\\{L_{n+1}\geq\frac{1}{\alpha}\left(\frac{1}{1+\frac{1-1/\alpha}{n}}\right)\times{\frac{L_{1}+\ldots+L_{n}}{n}}\right\\}\leq\alpha.$
(2)

Up until now, our presentation has been largely theoretical. We have yet to
provide examples of interesting exchangeable random variables. In the next
section, we elucidate how machine learning is a source of such examples. We
also delve into the renowned MNIST dataset to observe how CP performs in this
scenario.

##  3 Experiments and Results

In Conformal Prediction, a non-conformity measure is a function that assigns a
score to a data sample, reflecting its degree of conformity with other data
samples. The pursuit of quantifying similarity and conformity is of utmost
importance in Machine Learning (ML). To this end, we utilize ML techniques to
construct a robust measure of similarity. Following this, we use a calibration
set to enable instance-based modelling. The amalgamation of these two
paradigms—model-based and instance-based—results in Inductive Conformal
Prediction. This versatile framework provides statistically valid prediction
regions.

Let’s examine the case of supervised machine learning through the lens of
hypothesis testing theory. Our null hypothesis posits that an observation
$z=(x,y)=(sample,label)$ originates from an unknown probability distribution
of correct data. To reject this null hypothesis (i.e., to declare the label
incorrect), we require a statistic that increases for incorrect labels and
decreases for correct ones. This requirement closely mirrors the concept of a
loss function in Machine Learning (ML). Such a loss function can be derived
from ML algorithms that learn through optimization. Let’s delve deeper into
this with some specific examples.

######  Definition.

Let $Z$ be a set of possible observations. A non-conformity measure $A$ is a
sequence of equivariant maps $A^{(n+1)}$ from $Z^{n+1}$ to $\mathbf{R}^{n+1}$
with $n=0,1,2,\ldots$.

Given a non-conformity measure $A$ and a sequence of observations
$z_{1},z_{2},\ldots,z_{n+1}$, the output will be a sequence of non-conformity
scores $\alpha_{1},\alpha_{2},\ldots,\alpha_{n+1}$. Intuition: A small non-
conformity score means that the sample is similar to other samples, i.e. from
the same distribution.

In plain English, the definition above tell us, that

$\alpha_{i}=A_{n+1}(z_{i};z_{1},\ldots,z_{i-1},z_{i+1},\ldots,z_{n+1}),$

where $A_{n+1}:Z^{n+1}\rightarrow\mathbf{R}$ and is invariant under
permutation of the last $n$ variables.

Suppose we equip $Z^{n+1}$ with permutation invariant probability measures. In
this case, we can utilize the sequence of non-conformity scores
$\alpha_{1},\alpha_{2},\ldots,\alpha_{n+1}$ as exchangeable random variables,
as per Lemma 1 and Theorem 1. It is intuitively expected that $\alpha$ will be
small for data of interest.

Let’s consider the simplest possible scenario where the function $A_{n+1}$
depends solely on the first argument, that is,
$\alpha_{i}=LossFunction(z_{i})$.

For our numerical experiments, we will employ the MNIST (Modified National
Institute of Standards and Technology) database of handwritten digits. This
database is a popular choice for training and testing various image processing
systems and machine learning models. It comprises 60,000 training images and
10,000 testing images. Each image is a grayscale representation of a
handwritten digit, sized at $28\times 28$ pixels. Let’s examine some examples:

Figure 1:  The first nine images from the MNIST database of handwritten
digits.

We will train a very simple TensorFlow Keras model Figure 2

Figure 2:  Our model.

with SparseCategoricalCrossentropy loss function and ’adam’ optimizer. For
reproducibility we use $tf.random.set\\_seed(2024)$.

Upon training the model with the training images and labels over $25$ epochs,
we achieved an accuracy of $99.22$% on the training data and $98.17$% on the
test data. Thus, the model appears to be quite effective. Let’s examine some
instances where the predictions were incorrect: Figure 3

Figure 3:  Some images that the model predicts the wrong labels.

Once the model is trained and the loss function is established, we can shift
our focus away from the training data and concentrate solely on the test data.
We partition the test data randomly into two subsets: the CalibrationSet,
which comprises 50% of the test data, and the ConformalPredictionTestSet,
which also makes up 50% of the test data. For the CalibrationSet, a plot of
the sorted values of the loss function is depicted in Figure 4.

Figure 4:  Values of the loss function on the CalibrationSet.

###  3.1 Inductive Conformal Prediction with e-test statistics

With $\alpha=0.05$, $n=5000$, the BB-predictor (2) tells us that for an
(image, label) with the image from from the ConformalPredictionTestSet.

$P\\{LossFunction\geq 1.5002\\}<0.05.$

Applying this criterion to the ConformalPredictionTestSet, we obtain that with
probability more than 95%

  * •

Examples without labels: 0

  * •

Examples with one label: 4944

  * •

Examples with more than one label: 56

Let us demonstrate some examples with multiple labels Figure 5.

Figure 5:  Some images that have multiple labels prediction with their labels.

###  3.2 Inductive Conformal Prediction with p-value statistics

Given $\epsilon=0.05$, $n=5000$, Lemma 1 instructs us to compute the loss of
the element at index 4750 in the ordered list of losses for images from the
CalibrationSet. This value is found to be 0.0247. When we apply this to the
ConformalPredictionTestSet, we find that with a probability exceeding 95

  * •

Examples without labels: 238

  * •

Examples with one label: 4762

  * •

Examples with more than one label: 0

Here are some examples without labels as depicted in Figure 6:

Figure 6:  Some images that do not have labels.

Indeed, it’s evident that the standard Inductive Conformal Prediction
employing p-value statistics and our variant of Inductive Conformal Prediction
using BB-predictor (2) yield distinct outcomes. Furthermore, we can introduce
a monotonic function to the exchangeable random variables. While this alters
the mean value, it leaves the rank-based statistics unchanged.

##  4 Conclusion

This article embarks on an exploration of Conformal Prediction (CP), a robust
framework specifically designed to quantify the uncertainty inherent in
machine learning predictions. Notably, CP is capable of generating
statistically valid prediction regions, eliminating the need for assumptions
about the data distribution.

The authors propose an enhancement to CP through the incorporation of e-test
statistics and introducing a new BB-predictor. These statistics leverage
Markov’s inequality on exchangeable positive random variables, presenting a
fresh perspective on the problem.

The crux of the theoretical results is the demonstration that for exchangeable
non-negative random variables, a certain ratio consistently holds an
expectation of 1. Moreover, the probability of this ratio can be effectively
constrained using Markov’s inequality.

We underscore the importance of non-conformity measures and their crucial role
in supervised learning. The MNIST dataset is employed as the experimental
example, further reinforcing the concepts discussed.

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