# Comparison of different proximity potentials for asymmetric colliding nuclei

Ishwar Dutt  Rajeev K. Puri<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of
Physics, Panjab University, Chandigarh 160 014, India

###### Abstract

Using the different versions of phenomenological proximity potential as well
as other parametrizations within the proximity concept, we perform a detailed
comparative study of fusion barriers for asymmetric colliding nuclei with
asymmetry parameter as high as 0.23. In all, 12 different proximity potentials
are robust against the experimental data of 60 reactions. Our detailed study
reveals that the surface energy coefficient as well as radius of the colliding
nuclei depend significantly on the asymmetry parameter. All models are able to
explain the fusion barrier heights within $\pm 10\%$ on the average. The
potentials due to Bass 80, AW 95, and Denisov DP explain nicely the fusion
cross sections at above- as well as below-barrier energies.

###### pacs:

24.10.-i, 25.70.Jj, 25.70.-z.

##  I Introduction

The fusion of colliding nuclei with neutron -rich/ -deficient content and at
the extreme of isospin plane has attracted a large number of studies in recent
years canto06 ; Aguilera95 ; Silva97 ; Aguilera90 ; Cavallera90 ; Vega90 ;
Sonz98 ; Vinod96 ; Trotta01 ; Stefanini06 ; stefanini08 . This renewed
interest is due to the availability of radioactive-ion beams that can produce
nuclei at the extreme of isospin canto06 ; Stefanini06 ; stefanini08 .
Further, this field has also been enriched with several new phenomena that put
a stringent test on theoretical models derived to study the fusion phenomenon
in heavy-ion reactions.

As is evident from the literature, no experiment can extract information about
the fusion barriers directly. All experiments measure the fusion differential
cross sections canto06 ; Aguilera95 ; Silva97  and then with the help of
theoretical model, one extracts the fusion barriers. Theoretical models are
very helpful in understanding the nuclear interactions at a microscopic level.
A vast number of theoretical models and potentials have become available in
recent years that can explain one or the other features of fusion dynamics id1
; rkp1 ; rkp2 ; blocki77 ; wr94 ; ms2000 ; gr09 ; bass73 ; bass77 ; cw76 ;
aw95 ; ngo80 ; deni02 ; ngo75 ; deni07 .

In the galaxy of different theoretical models, proximity potential blocki77
enjoys very popular status. This phenomenological potential is a benchmark and
backbone for all microscopic/macroscopic fusion models. It is almost mandatory
to compare the potential and parametrize it within the proximity concept for
wider acceptability. In recent years, several refinements and modifications
have been proposed over original proximity potential wr94 ; ms2000 . Further
with the passage of time, different versions of the same model are also
available id1 . Many of these modifications are based on the isospin effects
either through the surface energy coefficients or via nuclear radius. It would
be of interest to test these potentials in the isospin plane and to see how
these different potentials will perform when asymmetry in the neutron/proton
content is very large.

Recently, we carried out a detailed comparative systematic study of different
fusion models for symmetric colliding nuclei id1 . Here we plan to extend this
study for those colliding nuclei that have larger neutron/proton content. In
this study, we shall compare as many as 12 proximity potentials with different
versions. This will include four versions of proximity potential, three
versions of potential due to Bass and Winther each and the latest potential
due to Ngô and a modified version of the Denisov potential. Section II, deals
with formalism in detail, Sec. III contains the results, and a summary is
presented in Sec. IV.

##  II  Formalism

In this section, we present the details of various proximity potentials used
for the calculation of fusion barriers. When two surfaces approach each other
within a distance of 2 - 3 fm, additional force due to the proximity of the
surface is labeled as proximity potential. Various versions of these
potentials take care of different aspects including the isospin dependence. In
the following, we discuss each of them in detail.

###  II.1 $\rm Proximity~{}1977~{}(Prox~{}77)$

The basis of proximity potential is the theorem that states that _“the force
between two gently curved surfaces in close proximity is proportional to the
interaction potential per unit area between the two flat surfaces”_. According
to the original version of proximity potential 1977 blocki77 , the interaction
potential $V_{N}(r)$ between two surfaces can be written as:

$V_{N}(r)=4\pi\gamma
b\overline{R}\Phi\left(\frac{{r}-C_{1}-C_{2}}{b}\right)~{}~{}\rm MeV.$ (1)

In this, the mean curvature radius, $\overline{R}$ has the form

$\overline{R}=\frac{C_{1}C_{2}}{C_{1}+C_{2}},$ (2)

quite similar to the one used for reduced mass. Here

$C_{i}=R_{i}\left[1-\left(\frac{b}{R_{i}}\right)^{2}+\cdots\cdots\right],$ (3)

${\rm R_{i}}$, the effective sharp radius, reads as

$R_{i}=1.28A^{1/3}_{i}-0.76+0.8A^{-1/3}_{i}{~{}\rm
fm}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(i=1,2).$ (4)

In Eq. (1), $\Phi(\xi=\frac{{r}-C_{1}-C_{2}}{b})$ is a universal function that
depends on the separation between the surfaces of two colliding nuclei only.
As we see, both these factors do not depend on the isospin content. However,
$\gamma$, the surface energy coefficient, depends on the neutron/proton excess
as

$\gamma=\gamma_{0}\left[1-k_{s}\left(\frac{N-Z}{N+Z}\right)^{2}\right],$ (5)

where N and Z are the total number of neutrons and protons. In the present
version, $\gamma_{0}$ and $k_{s}$ were taken to be $0.9517~{}\rm MeV/fm^{2}$
and $1.7826$, respectively. Note that for the symmetric colliding pair i.e. (N
= Z), $\gamma=\gamma_{0}=0.9517~{}\rm MeV/fm^{2}$. If the
$\left(\frac{N-Z}{N+Z}\right)$ ratio is $0.5$, $\gamma$ reduces to
$0.5276~{}\rm MeV/fm^{2}$. Defining asymmetry parameter
$A_{s}=\left[\frac{N_{1}+N_{2}-(Z_{1}+Z_{2})}{N_{1}+N_{2}+(Z_{1}+Z_{2})}\right]$,
one notices drastic reduction in the magnitude of the potential with asymmetry
of the colliding pair. Interestingly, most of the modified proximity type
potentials use different values of the parameter $\gamma$ wr94 ; ms2000 .

The universal function $\Phi\left(\xi\right)$ was parameterized with the
following form:

$\Phi\left(\xi\right)=\left\\{\begin{array}[]{l
r}-\frac{1}{2}\left(\xi-2.54\right)^{2}-0.0852\left(\xi-2.54\right)^{3},\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{
for $\xi\leq 1.2511$ },\\\ -3.437\exp\left(-\xi/0.75\right),\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{
for $\xi\geq 1.2511$ }.\end{array}\right.$ (6)

The surface width $b$ has been evaluated close to unity. Using the above form,
one can calculate the nuclear part of the interaction potential ${V_{N}(r)}$.
This model is referred to as Prox 77 and the corresponding potential as
$V_{N}^{Prox~{}77}(r)$.

###  II.2 $\rm Proximity~{}1988~{}(Prox~{}88)$

Later on, using the more refined mass formula due to Möller and Nix mn81 , the
value of coefficients $\gamma_{0}$ and $k_{s}$ were modified yielding the
values = 1.2496 $\rm MeV/fm^{2}$ and 2.3, respectively. Reisdorf wr94  labeled
this modified version as “Proximity 1988”. Note that this set of coefficients
give stronger attraction compared to the above set. Even a more recent
compilation by Möller and Nix mn95  yields similar results. We labeled this
potential as Prox 88.

###  II.3 $\rm Proximity~{}2000~{}(Prox~{}00)$

Recently, Myers and Świa̧tecki ms2000  modified Eq. (1) by using up-to-date
knowledge of nuclear radii and surface tension coefficients using their
droplet model concept. The prime aim behind this attempt was to remove
descripency of the order of $4\%$ reported between the results of Prox 77 and
experimental data ms2000 . Using the droplet model ms80 , matter radius
$C_{i}$ was calculated as

$C_{i}=c_{i}+\frac{N_{i}}{A_{i}}t_{i}~{}~{}~{}~{}(i=1,2),$ (7)

where $c_{i}$ denotes the half-density radii of the charge distribution and
$t_{i}$ is the neutron skin of the nucleus. To calculate $c_{i}$, these
authors ms2000  used two-parameter Fermi function values given in Ref. dv87
and the remaining cases were handled with the help of parametrization of
charge distribution described below. The nuclear charge radius (denoted as
$R_{00}$ in Ref. bn94 ), is given by the relation:

$R_{00i}=\sqrt{\frac{5}{3}}\left<r^{2}\right>^{1/2}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle=1.240A_{i}^{1/3}\left\\{1+\frac{1.646}{A_{i}}-0.191\left(\frac{A_{i}-2Z_{i}}{A_{i}}\right)\right\\}{~{}\rm
fm}$ $\displaystyle(i=1,2),$ (8)

where $<r^{2}>$ represents the mean-square nuclear charge radius. According to
Ref. bn94 , Eq. (8) was valid for the even-even nuclei with $8\leq Z<38$ only.
For nuclei with $Z\geq 38$, the above equation was modified by Pomorski _et
al_. bn94  as;

$\displaystyle
R_{00i}=1.256A_{i}^{1/3}\left\\{1-0.202\left(\frac{A_{i}-2Z_{i}}{A_{i}}\right)\right\\}{~{}\rm
fm}$ $\displaystyle~{}~{}~{}~{}~{}~{}(i=1,2).$ (9)

These expressions give good estimate of the measured mean square nuclear
charge radius $<r^{2}>$. In the present model, authors used only Eq. (8). The
half-density radius, $c_{i}$, was obtained from the relation:

$c_{i}=R_{00i}\left(1-\frac{7}{2}\frac{b^{2}}{R_{00i}^{2}}-\frac{49}{8}\frac{b^{4}}{R_{00i}^{4}}+\cdots\cdots\right)~{}~{}~{}~{}~{}~{}~{}(i=1,2).$
(10)

Using the droplet model ms80 , neutron skin $t_{i}$ reads as;

$t_{i}=\frac{3}{2}r_{0}\left[\frac{JI_{i}-\frac{1}{12}c_{1}Z_{i}A^{-1/3}_{i}}{Q+\frac{9}{4}JA^{-1/3}_{i}}\right](i=1,2).$
(11)

Here $r_{0}$ is $1.14$ fm, the value of nuclear symmetric energy coefficient
$J=32.65$ MeV and $c_{1}=3e^{2}/5r_{0}=0.757895$ MeV. The neutron skin
stiffness coefficient Q was taken to be 35.4 MeV. The nuclear surface energy
coefficient $\gamma$ in terms of neutron skin was given as;

$\gamma=\frac{1}{4\pi
r^{2}_{0}}\left[18.63{\rm(MeV)}-Q\frac{\left(t^{2}_{1}+t^{2}_{2}\right)}{2r^{2}_{0}}\right],$
(12)

where $t_{1}$ and $t_{2}$ were calculated using Eq. (11). The universal
function $\Phi(\xi)$ was reported as;

$\Phi\left(\xi\right)=\left\\{\begin{array}[]{ll}-0.1353+\sum\limits_{n=0}^{5}\left[c_{n}/\left(n+1\right)\right]\left(2.5-\xi\right)^{n+1},\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{
for \quad$0<\xi\leq 2.5$},\\\
-0.09551\exp\left[\left(2.75-\xi\right)/0.7176\right],\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{
for $\quad\xi\geq 2.5$}.\end{array}\right.$ (13)

The values of different constants $c_{n}$ were: $c_{0}=-0.1886$,
$c_{1}=-0.2628$, $c_{2}=-0.15216$, $c_{3}=-0.04562$, $c_{4}=0.069136$ and
$c_{5}=-0.011454$. For $\xi>2.74$, the above exponential expression is the
exact representation of the Thomas-Fermi extension of the proximity potential.
This potential is labeled Prox 00.

###  II.4 $\rm Modified~{}Proximity~{}2000~{}(Prox~{}00DP)$

Recently, Royer and Rousseau gr09  modified Eq. (8) with slightly different
constants as;

$\displaystyle
R_{00i}=1.2332A^{1/3}_{i}\left[1+\frac{2.348443}{A_{i}}\right.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle\left.-0.151541\left(\frac{A_{i}-2Z_{i}}{A_{i}}\right)\right]{~{}\rm
fm}~{}~{}(i=1,2).$ (14)

It is obtained by analyzing as many as 2027 masses with N, Z $\geq$ 8 and a
mass uncertainty $\leq$ 150 keV. Further, the accuracy of the above formula is
mainly improved by adding the Coulomb diffuseness correction or the charge
exchange correction to the mass formulas gr09 . We implement this radius in
the proximity 2000 version instead of the form given in the proximity 2000.
This new version of the proximity potential is labeled Prox 00DP id1 .

###  II.5 $\rm Bass~{}1973~{}(Bass~{}73)$

This model is based on the assumption of liquid drop model bass73 . Here
change in the surface energy of two fragments due to their mutual separation
is represented by exponential factor. By multiply with geometrical arguments,
one can obtained the nuclear part of the interaction potential as

$V_{N}(r)^{Bass~{}73}=-\frac{d}{R_{12}}a_{s}A_{1}^{1/3}A_{2}^{1/3}exp(-\frac{r-R_{12}}{d}){~{}\rm
MeV},$ (15)

with ${R_{12}}=r_{0}(A_{1}^{1/3}+A_{2}^{1/3}),~{}d=1.35~{}{\rm fm}$ and
$a_{s}=17.0~{}{\rm MeV}$. The cut-off distance $R_{12}$ is chosen to yield
saturation density in the overlap region and $r_{o}=1.07~{}{\rm fm}$
corresponding half of the maximum density for individual nucleus. We labeled
this potential Bass 73.

###  II.6 $\rm Bass~{}1977~{}(Bass~{}77)$

In this model, nucleus-nucleus potential is derived from the information based
on the experimental fusion cross sections by using the liquid drop model and
general geometrical arguments. The nuclear part of the potential (for
spherical nuclei with frozen densities) can be written as bass77

$\displaystyle
V_{N}\left(r\right)^{Bass~{}77}=-4\pi\gamma\frac{R_{1}R_{2}}{R_{1}+R_{2}}f\left(r-R_{1}-R_{2}\right)$
$\displaystyle\qquad\qquad=-\frac{R_{1}R_{2}}{R_{1}+R_{2}}\Phi\left(r-R_{1}-R_{2}\right)~{}{\rm
MeV},$ (16)

with

$\frac{df}{ds}=-1,\qquad\qquad{\rm for}\qquad\qquad s=0.$ (17)

Note that $f\left(s=r-R_{1}-R_{2}\right)$ and
$\Phi\left(s=r-R_{1}-R_{2}\right)$ are the universal functions. Here radius
$R_{i}$ is written as

$R_{i}=1.16A^{1/3}_{i}-1.39A^{-1/3}_{i}~{}{\rm fm}~{}~{}~{}~{}(i=1,2).$ (18)

The form of the universal function $\Phi\left(s\right)$ reads as

$\Phi\left(s\right)=\left[A\exp\left(\frac{s}{d_{1}}\right)+B\exp\left(\frac{s}{d_{2}}\right)\right]^{-1},$
(19)

with $A=0.0300$ MeV-1fm, $B=0.0061$ MeV-1fm, $d_{1}=3.30$ fm and $d_{2}=0.65$
fm. Note that where $b=1$, $\xi$ and s turn out to be the same quantities.
This model was very successful in explaining the barrier heights, positions,
and cross sections over a wide range of incident energies and masses of
colliding nuclei. We labeled this potential Bass 77.

###  II.7 $\rm Bass~{}1980~{}(Bass~{}80)$

The above potential form was further improved by Bass wr94 . Here
$\Phi\left(s=r-R_{1}-R_{2}\right)$ is now given as:

$\Phi\left(s\right)=\left[0.033\exp\left(\frac{s}{3.5}\right)+0.007\exp\left(\frac{s}{0.65}\right)\right]^{-1},$
(20)

with central radius, $R_{i}$ as

$R_{i}=R_{s}\left(1-\frac{0.98}{R_{s}^{2}}\right)~{}~{}~{}~{}(i=1,2),$ (21)

where $R_{s}$ is same as given by Eq. (4). We labeled this potential as Bass
80.

###  II.8 $\rm Christensen~{}and~{}Winther~{}1976~{}(CW~{}76)$

Christensen and Winther cw76  derived the nucleus-nucleus interaction
potential by analyzing the heavy-ion elastic-scattering data, based on the
semiclassical arguments and the recognition that optical-model analysis of
elastic scattering determines the real part of the interaction potential only
in the vicinity of a characteristic distance. The nuclear part of the
empirical potential due to Christensen and Winther is written as

$V_{N}^{CW~{}76}\left(r\right)=-50\frac{R_{1}R_{2}}{R_{1}+R_{2}}\Phi\left(r-R_{1}-R_{2}\right)~{}{\rm
MeV}.$ (22)

This form of the geometrical factor is similar to that of $\rm Bass~{}77$ with
different radius parameters

$R_{i}=1.233A^{1/3}_{i}-0.978A^{-1/3}_{i}~{}{\rm fm}~{}~{}~{}~{}(i=1,2).$ (23)

The universal function $\Phi(s=r-R_{1}-R_{2}$ ) has the following form

$\Phi\left(s\right)=\exp\left(-\frac{r-R_{1}-R_{2}}{0.63}\right).$ (24)

This model was tested for more than 60 reactions and we labeled it CW 76.

###  II.9 $\rm Broglia~{}and~{}Winther~{}1991~{}(BW~{}91)$

A refined version of the above potential was derived by Broglia and Winther
wr94 , by taking Woods-Saxon parametrization with subsidiary condition of
being compatible with the value of the maximum nuclear force predicted by the
proximity potential Prox 77. This refined potential resulted in

$V_{N}^{BW~{}91}(r)=-\frac{V_{0}}{1+\exp\left(\frac{r-R_{0}}{0.63}\right)}~{}{\rm
MeV};$ (25) ${\rm
with},~{}V_{0}=16\pi\frac{R_{1}R_{2}}{R_{1}+R_{2}}{\gamma}{a},$ (26)

here $a=0.63$ fm and

$R_{0}=R_{1}+R_{2}+0.29.$ (27)

Here radius $R_{i}$ has the form

$R_{i}=1.233A^{1/3}_{i}-0.98A^{-1/3}_{i}~{}{\rm fm}~{}~{}~{}~{}(i=1,2).$ (28)

The form of the surface energy coefficient $\gamma$ is quite similar to the
one used in Prox 77 with slight difference

$\gamma=\gamma_{o}\left[1-k_{s}\left(\frac{N_{p}-Z_{p}}{A_{p}}\right)\left(\frac{N_{t}-Z_{t}}{A_{t}}\right)\right],$
(29)

where ${~{}\rm\gamma_{0}}$ = 0.95 $~{}\rm~{}MeV/fm^{2}$ and $k_{s}=1.8$. Note
that the second term used in this potential gives different results when the
projectile is symmetric ($N=Z$) and the target is asymmetric ($N>Z$). This
form will also give different results for larger mass asymmetry $\eta_{A}$.
Note that the radius used in this potential has same form like that of Bass
with different constants. We labeled this potential as BW 91.

###  II.10 $\rm Aage~{}Winther~{}(AW~{}95)$

Winther adjusted the parameters of the above potential through an extensive
comparison with experimental data for heavy-ion elastic scattering. This
refined adjustment to slight different values of “$a$” and $R_{i}$ as aw95

${a}=\left[\frac{1}{1.17(1+0.53(A_{1}^{-1/3}+A_{2}^{-1/3}))}\right]~{}\rm fm,$
(30)

and

$R_{i}=1.20A^{1/3}_{i}-0.09~{}\rm fm~{}~{}~{}~{}(i=1,2).$ (31)

Here, $R_{0}=R_{1}+R_{2}$ only. We labeled this potential as AW 95.

###  II.11 $\rm Ng$ô$~{}1980~{}(\rm Ng$ô $80)$

In earlier attempts, based on the microscopic picture of a nucleus and on the
idea of energy density formalism, the potential from Ngô and collaborators
enjoy special status ngo75 . In this model, calculations of the ion-ion
potential are performed within the framework of energy density formalism due
to Bruckener _et al_., using a sudden approximation bk68 . The need of
Hartree-Fock densities as input in this model limited its scope. This not only
made calculations tedious, but it also hindered its application to heavier
nuclei. The above-stated parametrization was improved by H. Ngô and Ch. Ngô
ngo80 , by using a Fermi-density distribution for nuclear densities as

$\rho_{n,p}(r)=\frac{\rho_{n,p}(0)}{1+\exp\left[(r-C_{n,p})/0.55\right]}~{},$
(32)

where $C$ represents the central radius of the distribution and is defined in
Prox 77 (see Eq. (3) with b = 1 fm). Here $\rho_{n,p}(0)$ is given by

$\rho_{n}(0)=\frac{3}{4\pi}\frac{N}{A}\frac{1}{r^{3}_{0_{n}}};~{}~{}~{}~{}~{}~{}~{}\rho_{p}(0)=\frac{3}{4\pi}\frac{Z}{A}\frac{1}{r^{3}_{0_{p}}}~{}.$
(33)

Ngô parameterized the nucleus-nucleus interaction potential in the spirit of
proximity concept. The interaction potential can be divided into the
geometrical factor and a universal function. The nuclear part of the
parameterized potential is written as ngo80 ;

$V_{N}^{Ngo~{}80}\left(r\right)=\overline{R}\Phi\left(r-C_{1}-C_{2}\right)~{}\rm
MeV,$ (34)

where $\overline{R}$ is defined by Eq. (2). Now the nuclear radius $R_{i}$
reads as:

$R_{i}=\frac{NR_{n_{i}}+ZR_{p_{i}}}{A_{i}}~{}~{}~{}~{}~{}(i=1,2).$ (35)

The equivalent sharp radius for protons and neutrons are given as;

$R_{p_{i}}=r_{0_{pi}}A^{1/3}_{i};~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}R_{n_{i}}=r_{0_{ni}}A^{1/3}_{i},$
(36)

with

$r_{0_{pi}}=1.128~{}{\rm fm};~{}r_{0_{ni}}=1.1375+1.875\times
10^{-4}A_{i}~{}\rm fm.$ (37)

The above different radius formulas for the neutrons and protons take isotopic
dependence into account. The universal function $\Phi(s=r-C_{1}-C_{2})$ (in
$\rm MeV/fm$) is noted by

$\Phi\left(s\right)=\left\\{\begin{array}[]{l
r}-33+5.4\left(s-s_{0}\right)^{2},&\mbox{ for $s<s_{0}$ },\\\
-33\exp\left[-\frac{1}{5}\left(s-s_{0}\right)^{2}\right],&\mbox{ for $s\geq
s_{0}$ },\end{array}\right.$ (38)

and $s_{0}=-1.6$ fm. We labeled this potential as Ngô 80.

###  II.12 New Denisov Potential (Denisov DP)

Denisov deni02  performed numerical calculations and parametrized the
potential based on 7140 pair within semi-microscopic approximation. In total,
119 spherical or near spherical nuclei along the $\beta$-stability line from
16O to 212Po were taken. The potential is evaluated for any nucleus-nucleus
combinations at 15 distances between ions around the touching point. By using
this database, a simple analytical expression for the nuclear part of the
interaction potential VN(r) between two spherical nuclei is presented as;

$\displaystyle
V_{N}\left(r\right)=-1.989843\frac{R_{1}R_{2}}{R_{1}+R_{2}}\Phi\left(r-R_{1}-R_{2}-2.65\right)$
$\displaystyle\times\left[1+0.003525139\left(\frac{A_{1}}{A_{2}}+\frac{A_{2}}{A_{1}}\right)^{3/2}\right.$
$\displaystyle\left.-0.4113263\left(I_{1}+I_{2}\right)\right],$ (39)

with

$I_{i}=\frac{N_{i}-Z_{i}}{A_{i}}~{}~{}~{}~{}(i=1,2).$ (40)

The effective nuclear radius $R_{i}$ is given as;

$\displaystyle
R_{i}=R_{ip}\left(1-\frac{3.413817}{R^{2}_{ip}}\right)+~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle 1.284589\left(I_{i}-\frac{0.4A_{i}}{A_{i}+200}\right)(i=1,2),$
(41)

where, proton radius $R_{ip}$ is given by Eq. (8) and
$\Phi\left(s=r-R_{1}-R_{2}-2.65\right)$ is given by the following complex
form:

$\Phi(s)=\left\\{\begin{array}[]{l}1-s/0.7881663+1.229218s^{2}-0.2234277s^{3}\\\
-0.1038769s^{4}\\\
-\frac{R_{1}R_{2}}{R_{1}+R_{2}}\left(0.1844935s^{2}+0.07570101s^{3}\right)\\\
+\left(I_{1}+I_{2}\right)\left(0.04470645s^{2}+0.03346870s^{3}\right),\\\
\qquad\qquad\qquad\qquad{\rm for}\quad\quad-5.65\leq s\leq 0,\\\
\left[1-s^{2}[0.05410106\frac{R_{1}R_{2}}{R_{1}+R_{2}}\exp(-\frac{s}{1.760580})\right.\\\
\left.-0.5395420(I_{1}+I_{2})\exp(-\frac{s}{2.424408})]\right]\\\
\times\exp(-\frac{s}{0.7881663}),\\\ \qquad\qquad\qquad\qquad\qquad\qquad{\rm
for}\qquad s\geq 0.\end{array}\right.$ (42)

Here $A_{i}$, $N_{i}$, $Z_{i}$, $R_{i}$, and $R_{ip}$ are, respectively, the
mass number, the number of neutrons, the number of protons, the effective
nuclear radius, and the proton radius of the target and projectile. The above
form of the universal function not only depends on the separation distance s,
but also has complex dependence on the mass as well as on the relative neutron
excess content. The above parametrization is derived for different
combinations of nuclei between 16O and 212Po.

As stated in the subsection II.4, a new radius formula has become available
recently gr09 . We here extend the above potential due to Denisov to include
this radius in its parametrization. This modified new version of the potential
is labeled as Denisov DP id1 . Note that this new implementation was reported
to yield very close agreement (within 1%) with experimental data for symmetric
colliding pairs id1 .

If one looks on the different versions of potentials (Bass 73, Bass77, Bass
80, and CW 76), one notices that although the form of the radius is different,
it is still isospin independent. Further, the corresponding universal
functions are also isospin independent. The newer versions of Winther (BW 91
and AW 95) have incorporated a $\gamma$ similar to the one used in the Prox 77
potential with a slightly different form. Here isospin content is calculated
separately for the target/ projectile. The latest version of Ngô (Ngô 80) has
some isospin dependence in the radius parameter. In most of the above
mentioned potentials, modifications are made either through the surface energy
coefficients or via nuclear radii. Both of these technical parameters can have
sizable effects on the outcome of a reaction id2 .

Using the above sets of models, the nuclear part of the interaction potential
is calculated.

By adding the Coulomb potential to a nuclear part, one can compute the total
potential $V_{T}(r)$ for spherical colliding pairs as

$\displaystyle V_{T}(r)=V_{N}(r)+V_{C}(r),$ (43)
$\displaystyle=V_{N}(r)+\frac{Z_{1}Z_{2}e^{2}}{r}.$ (44)

Since the fusion happens at a distance larger than the touching configuration
of colliding pair, the above form of the Coulomb potential is justified. One
can extract the barrier height $V^{theor}_{B}$ and barrier position
$R^{theor}_{B}$ using the following conditions

$\frac{dV_{T}(r)}{dr}|_{r=R^{theor}_{B}}=0;~{}~{}{\rm{and}}~{}~{}\frac{d^{2}V_{T}(r)}{dr^{2}}|_{r=R^{theor}_{B}}\leq
0.$ (45)

The knowledge of the shape of the potential as well as barrier position and
height, allows one to calculate the fusion cross section at a microscopic
level. To study the fusion cross sections, we shall use the model given by
Wong wg72 . In this formalism, the cross section for complete fusion is given
by

$\sigma_{fus}=\frac{\pi}{k^{2}}\sum_{l=0}^{l_{max}}\left(2l+1\right)T_{l}\left(E_{cm}\right),$
(46)

where $k=\sqrt{\frac{2\mu E}{\hbar^{2}}}$ and here $\mu$ is the reduced mass.
The center-of-mass energy is denoted by $E_{cm}$. In the above formula,
$\l_{max}$ corresponds to the largest partial wave for which a pocket still
exists in the interaction potential and T${}_{\l}\left(E_{cm}\right)$ is the
energy-dependent barrier penetration factor and is given by,

$T_{\l}\left(E_{cm}\right)=\left\\{1+\exp\left[\frac{2\pi}{\hbar\omega_{\l}}\left(V^{theor}_{B_{\l}}-E_{cm}\right)\right]\right\\}^{-1},$
(47)

where $\hbar\omega_{l}$ is the curvature of the inverted parabola. If we
assume that the barrier position and width are independent of $\l$, the fusion
cross section reduces to

$\displaystyle\sigma_{fus}(mb)=\frac{10R^{theor^{2}}_{B}\hbar\omega_{0}}{2E_{cm}}\times~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle\ln\left\\{1+\exp\left[\frac{2\pi}{\hbar\omega_{0}}\left(E_{cm}-V^{theor}_{B}\right)\right]\right\\}.$
(48)

For Ecm$>>$V${}^{theor}_{B}$, the above formula reduces to well-known sharp
cut-off formula

$\sigma_{fus}(mb)=10\pi
R^{theor^{2}}_{B}\left(1-\frac{V^{theor}_{B}}{E_{cm}}\right),$ (49)

whereas for Ecm$<<$V${}^{theor}_{B}$, the above formula reduces to

$\sigma_{fus}(mb)=\frac{10R^{theor^{2}}_{B}\hbar\omega_{0}}{2E_{cm}}\exp\left[\frac{2\pi}{\hbar\omega_{0}}\left(E_{cm}-V^{theor}_{B}\right)\right].$
(50)

We used Eq. (48) to calculate the fusion cross sections.

From the above brief discussion, it is clear that the main stress is made on
the surface energy coefficients $\gamma$ and nuclear radii to incorporate the
isospin dependence in the nuclear potential. Definitely, the response of the
isospin dependent potentials will be different for asymmetric nuclei compared
to symmetric nuclei. At intermediate energies, a strong effect was reported
for the asymmetric reactions as well as for the mass dependence of the
reaction rkp .

##  III Results and Discussions

The present study is conducted using a variety of the above-mentioned
potentials. In total, 60 asymmetric reactions with compound mass between 29
and 294 (that have been experimentally explored) are taken for the present
study. All nuclei considered here are assumed to be spherical in nature;
however, deformation as well as orientation of the nuclei also affect the
fusion barriers deni07 . For uniform comparison of different models, we
consider all colliding nuclei to be spherical. The lightest reaction taken is
that of 12C + 17O, whereas heaviest one is of 86Kr + 208Pb. The asymmetry
$A_{s}$ of the colliding nuclei varies between 0.02 and 0.23. The other form
of the asymmetry used in the literature is the mass asymmetry $\eta_{A}$ rkp1
; rkp2 . In the present analysis, $\eta_{A}$ varies between 0.0 and 0.97. Note
that the non zero value of $A_{s}$ will involve complex interplay of the
isospin degree of freedom which has strong role at intermediate energies as
well. The variation of $\eta$ alters the physical outcome of a reaction with
$\eta\approx 0.0$ leading to high dense matter and maximum collision volume
whereas a larger value of $\eta\approx 1.0$ will not be able to compress the
matter to higher density rkp .

As stated above, the isospin dependence of the different potentials enters via
surface energy coefficient $\gamma$. In Fig. 1, we display the variation of
$\gamma$ (in $\rm MeV~{}fm^{-2}$) with asymmetry parameter $A_{s}$. Here we
compare three versions of the surface energy coefficient $\gamma$ used in Prox
77, Prox 88, and Prox 00 potentials along with the relation suggested in AW 95
potential. For the present analysis, the mass of the reacting partner is kept
fixed equal to $A_{1}=A_{2}=40$. The $A_{s}$ was increased by increasing the
neutrons and decreasing the protons. For example, ${}^{40}_{20}Ca_{20}$ \+
${}^{40}_{20}Ca_{20}$ has $A_{s}$ = 0.0. For $A_{s}$ = 0.2, we chose the
reaction of ${}^{40}_{16}S_{24}$ \+ ${}^{40}_{16}S_{24}$ whereas for $A_{s}$ =
0.4, the reaction was ${}^{40}_{12}Mg_{28}$ \+ ${}^{40}_{12}Mg_{28}$. In all
cases, the mass of the reacting partner is kept fixed, whereas the ratio
$A_{s}$ is varied by converting the proton into neutrons. At the end of this
series, we have the reaction of ${}^{40}_{10}Ne_{30}$ \+ ${}^{40}_{10}Ne_{30}$
having $A_{s}$ = 0.5. From the figure, we see that the surface energy
coefficient $\gamma$ used in the latest proximity potentials Prox 00/ Prox
00DP as well as in original version Prox 77 is less sensitive toward the
asymmetry and isospin dependence, whereas the one used in the Prox 88
potential has a stronger dependence on the asymmetry of the reacting nuclei.
The coefficient $\gamma$ of AW 95 yields same results like Prox 77. Since
nuclear potential $V_{N}(r)$ depends directly on $\gamma$, one can conclude
that the potentials calculated within Prox 88 and Prox 77 will be far less
attractive for larger asymmetries compared to the one generated using Prox 00.
When colliding nuclei are symmetric (N = Z; $A_{s}$ = 0.0), such dependence
does not play a role. In many studies rkp1 , one finds that neutron excess,
leads to more attraction. In these studies, the total mass of the colliding
pair is not fixed and as a result, this dependence is more of mass dependence
than of isospin dependence.

In Fig. 2, we display the dependence of different nuclear radii on the
asymmetry parameter $A_{s}$. As noted above, this parameter also plays
significant role in nuclear potential and finally in the barrier calculations.
We show the dependence of different forms of nuclear radii used in various
potentials on the asymmetry parameter. We see that, the radius used in the
Prox 77 (also in Prox 88) as well as in Bass versions (i.e., Bass 73, Bass 77,
and Bass 80), and in all versions from Winther (CW 76, BW 91, and AW 95) are
independent of the asymmetry content, whereas, the one used in the Prox 00,
Prox 00DP (and Denisov DP), and Ngô 80 versions depends on the asymmetry
content of the colliding pairs.

From Figs. 1 and 2, we see that both these parameters can lead to significant
change in the nuclear potential and ultimately in the fusion barriers even if
the universal function $\Phi$(s) is kept the same.

In Fig. 3, we display the nuclear part of the interaction potential $V_{N}(r)$
at a distance of $C_{1}+C_{2}+1$ fm for the same sets of the reactions as
depicted in Figs. 1 and 2. In addition, a series of heavier reacting partners
with mass $A_{1}=A_{2}=80$ is also taken. We display four versions of
proximity potential, three versions from Bass and Winther and one each of the
latest versions of Ngô and Denisov each. We see a systematic decrease in the
attractive strength of the potentials with asymmetry content $A_{s}$. The
decrease is stronger for the Prox 88 version compared to Prox 77, Prox 00, and
Prox 00DP. The Bass 73, Bass 77, and Bass 80 versions of the potential are
independent of the asymmetry content. One also notices a very weak dependence
in the Ngô 80 potential. Two of the three versions of Winther potential have
significant dependence on the asymmetry of the reaction. The Winther 1976
potential, however, does not show such dependence due to the absence of
$\gamma$ term in the potential. The Denisov DP potential also shows a linear
decrease in the strength of the potential with asymmetry content. These
variations are stronger for heavier colliding nuclei. This figure shows true
isospin dependence of the nuclear potential as the mass of the colliding
nuclei is kept fixed. All those potentials that do not depend on the asymmetry
parameter $A_{s}$ will not show any change in the structure.

We now shift from the systematic study to the study involving real nuclei. As
stated above, here 60 reactions with $A_{s}$ between 0.02 and 0.23 and
$\eta_{A}$ between 0.0 and 0.97 are taken. For all these reactions,
experimental fusion barriers are known Aguilera95 ; Silva97 ; Aguilera90 ;
Cavallera90 ; Vega90 ; Sonz98 ; Vinod96 ; Trotta01 ; Stefanini06 ; stefanini08
; Padron02 ; tighe93 ; vaz81 ; beck03 ; rath09 ; kolata98 ; Morsad90 ;
trotta2000 ; gomes91 ; newton04 ; Liu05 ; Kolata04 ; Prasad96 ; Sinha01 ;
Szanto90 ; Stefanini02 ; quirz01 ; Stefanini2000 ; Baby2000 ; capurro02 ;
Vandana01 ; Stelson90 ; Mitsuoka07 . In Fig. 4, we display the fusion barrier
heights $V_{B}$ and barrier positions $R_{B}$ versus experimental values for
the above mentioned reactions involving 12 different potentials. For the
clarity of the figure, only 60 asymmetric reactions studied experimentally and
covering the whole range of the mass and asymmetry are displayed. We see no
clear difference with fusion barrier heights and positions. The fusion barrier
heights can be reproduced within $\pm 10\%$ in all cases on the average. Due
to the large uncertainty in the fusion barrier positions, no definite trend
and conclusion can be drawn as is observed for the symmetric colliding nuclei
id1 . To further understand the role of isospin content, we display, in Fig.
5, the percentage difference of the fusion barrier heights $\Delta V_{B}(\%)$
defined as

$\Delta V_{B}~{}(\%)=\frac{V_{B}^{theor}-V_{B}^{expt}}{V_{B}^{expt}}\times
100,$ (51)

verses asymmetry parameter $A_{s}$. In some cases, only the latest versions of
the potential are shown. Interestingly, we see that Prox 77 and Ngô 80 fail to
reproduce the barrier heights satisfactorily, whereas Prox 88, Bass 80, AW 95,
Prox 00DP, and Denisov DP do a far better job compared to other potentials. We
do not see any systematic deviation/improvement in the fusion barrier heights
with the asymmetry of the colliding nuclei. We see that the potentials Prox
88, Bass 80, AW 95, and Denisov DP can reproduce the empirical barrier heights
within $\pm 5\%$ (see the shaded regions in Fig. 5), whereas others need $\pm
10\%$ to produce the same result.

The comparison of the fusion barrier positions outcome is shown in Fig. 6. We
see that due to large uncertainty in the measurements of fusion barrier
positions, a large deviation is seen and all the models are able to reproduce
the results within $\pm 10\%$. The precise values of the fusion barrier
heights $V_{B}$ (in MeV) and positions $R_{B}$ (in fm) are shown in the Tables
1 and 2, for 60 asymmetric colliding nuclei involving significant variations
of asymmetry $A_{s}$ as well as mass asymmetry $\eta_{A}$. The experimental
(or empirical) barriers displayed in Tables 1 and 2 and in Figs. 4 - 6 are
obtained by fitting the cross sections in the approach, when shapes of both
colliding nuclei are spherical. A large number of experimental data are
available for different reactions; however, we restrict ourselves to the
latest one only.

In Figs. 7 and 8, we display the fusion cross-sections $\sigma_{fus}$ (in mb)
as a function of center-of-mass energy $E_{cm}$ for the reactions of 48Ca +
96Zr Stefanini06 , 28Si + 92Zr newton01 , 12C + 92Zr newton01 , 16O + 208Pb
Morton01  (in Fig. 7) and 16O + 50Ti Neto90 , 16O + 112Sn Vandana01 , 16O +
116Sn Vandana01 , and 16O + 120Sn Baby2000  (in Fig. 8). Here the latest
versions of proximity parametrizations along with original proximity potential
and its modifications are shown for clarity. As we see, Bass 80, Denisov DP,
and AW 95 do a better job for all the systems, whereas Prox 77 and Ngô 80 fail
to come closer to the experimental data. The above results are in agreement
with the one obtained for symmetric colliding nuclei id1 .

##  IV Summary

We performed a systematic study of the role of isospin dependence on fusion
barriers by employing as many as 12 different proximity-based potentials. Some
of the potentials have isospin dependence via the surface energy coefficient
as well as via nuclear radius. We noted that the nuclear part of the potential
becomes more shallow with asymmetry of the reaction. On the other hand, a
detailed comparison of different potentials does not show any preference for
the isospin-dependent potential. Our comparison for 60 reactions reveals that
all models can explain the fusion barrier heights within $\pm 10\%$. The
potentials from Prox 88, Bass 80, AW 95, and Denisov DP perform better than
others. The fusion cross sections are nicely explained by Bass 80, AW 95, and
Denisov DP potentials at below as well as above barrier energies.

This work was supported by a research grant from the Department of Atomic
Energy, Government of India.

## References

  * (1) L. F. Canto, P. R. S. Gomes, R. Donangelo, and M. S. Hussein, Phys. Rep. 424, 1 (2006).
  * (2) E. F. Aguilera _et al._ , Phys. Rev. C 52, 3103 (1995).
  * (3) C. P. Silva _et al._ , Phys. Rev. C 55, 3155 (1997).
  * (4) E. F. Aguilera _et al._ , Phys. Rev. C 41, 910 (1990).
  * (5) S. Cavallaro _et al._ , Nucl. Phys. A513, 174 (1990).
  * (6) J. J. Vega _et al._ , Phys. Rev. C 42, 947 (1990).
  * (7) A. A. Sonzogni _et al._ , Phys. Rev. C 57, 722 (1998).
  * (8) A. M. Vinodkumar _et al._ , Phys. Rev. C 53, 803 (1996).
  * (9) M. Trotta _et al._ , Phys. Rev. C 65, 011601(R) (2001).
  * (10) A. M. Stefanini _et al._ , Phys. Rev. C 73, 034606 (2006).
  * (11) A. M. Stefanini _et al._ , Phys. Rev. C 78, 044607 (2008).
  * (12) I. Dutt and R. K. Puri, Phys. Rev. C 81, 044615 (2010).
  * (13) J. Blocki, J. Randrup, W. J. Świa̧tecki and C. F. Tsang, Ann. Phys. (NY) 105, 427 (1977).
  * (14) W. Reisdorf, J. Phys. G: Nucl. Part. Phys. 20, 1297 (1994).
  * (15) W. D. Myers and W. J. Świa̧tecki, Phys. Rev. C 62, 044610 (2000).
  * (16) R. Bass, Phys. Lett. B47, 139 (1973); Nucl. Phys. A231, 45 (1974).
  * (17) R. Bass, Phys. Rev. Lett. 39, 265 (1977).
  * (18) P. R. Christensen and A. Winther, Phys. Lett. B65, 19 (1976).
  * (19) A. Winther, Nucl. Phys. A594, 203 (1995).
  * (20) V. Y. Denisov, Phys. Lett. B 526, 315 (2002).
  * (21) H. Ngô and C. Ngô, Nucl. Phys. A348, 140 (1980).
  * (22) G. Royer and R. Rousseau, Eur. Phys. J. A 42, 541 (2009).
  * (23) R. K. Puri _et al._ , Eur. Phys. J. A 23, 429 (2005); R. Arora _et al._ , _ibid._ 8, 103 (2000); R. K. Puri _et al._ , _ibid._ 3, 277 (1998); R. K. Puri _et al._ , Phys. Rev. C 43, 315 (1991); R. K. Puri _et al._ , Phys. Rev. C 45, 1837 (1992); N. K. Dhiman _et al._ , Acta. Phys. Pol. B 38, 2133 (2007); N. K. Dhiman _et al._ , _ibid._ 37, 1855 (2006).
  * (24) R. K. Gupta _et al._ , Phys. Rev. C 47, 561 (1993); R. K. Gupta _et al._ , J. Phys. G: Nucl. Part. Phys. 18, 1533 (1992); S. S. Malik _et al._ , Pramana J. Phys. 32, 419 (1989); R. K. Puri _et al._ , Europhys. Lett. 9, 767 (1989); R. K. Puri _et al._ , J. Phys. G: Nucl. Part. Phys. 18, 903 (1992).
  * (25) C. Ngô, B. Tamain, M. Beiner, R. J. Lombard, D. Mas, and H. H. Deubler, Nucl. Phys. A252, 237 (1975).
  * (26) V. Y. Denisov and N. A. Pilipenko, Phys. Rev. C 76, 014602 (2007); A. S. Umar and V. E. Oberacker, Phys. Rev. C 77, 064605 (2008); M. Ismail, W. M. Seif, and M. M. Botros Nucl. Phys. A828, 333 (2009).
  * (27) P. Möller and J. R. Nix, Nucl. Phys. A361, 117 (1981).
  * (28) P. Möller, J. R. Nix, W. D. Myers, and W. J. Świa̧tecki, At. Data Nucl. Data Tables 59, 185 (1995).
  * (29) W. D. Myers and W. J. Światecki, Ann. Phys. 55, 395 (1969); Nucl. Phys. A336, 267 (1980).
  * (30) C. W. de Jager, H. de Vries, and C. de Vries, At. Data Nucl. Data Tables 14, 479 (1974); H. de Vries, C. W. de Jager, and C. de Vries, ibid. 36 495 (1987).
  * (31) B. Nerlo-Pomorska and K. Pomorski, Z. Phys. A 348, 169 (1994).
  * (32) K. A. Brueckner, J. R. Buchler, and M. Kelly, Phys. Rev. 173, 944 (1968).
  * (33) I. Dutt and R. K. Puri, Phys. Rev. C 81, 047601 (2010).
  * (34) C. Y. Wong, Phys. Lett. B42, 186 (1972); Phys. Rev. Lett. 31, 766 (1973).
  * (35) R. K. Puri _et al._ , Nucl. Phys. A 575, 733 (1994); R. K. Puri _et al._ , Phys. Rev. C 54, R28 (1996); S. Kumar _et al._ , _ibid._ 58, 3494 (1998); J. Singh _et al._ , _ibid._ 62, 044617 (2000); R. K. Puri _et al._ , J. Comput. Phys. 162, 245 (2000); S. Kumar _et al._ , Phys. Rev. C 78, 064602 (2008); Y. K. Vermani _et al._ , J. Phys. G: Nucl. Part. Phys. 37, 015105 (2010); Y. K. Vermani _et al._ , Phys. Rev. C 79, 064613 (2009); Y. K. Vermani _et al._ , J. Phys. G: Nucl. Part. Phys. 36, 105103 (2009); Y. K. Vermani _et al._ , Europhys. Lett. 85, 62001 (2009); A. Sood _et al._ , Phys. Rev. C 79, 064618 (2009).
  * (36) I. Padron _et al._ , Phys. Rev. C 66, 044608 (2002).
  * (37) R. J. Tighe _et al._ , Phys. Rev. C 47, 2699 (1993).
  * (38) C. Beck _et al._ , Phys. Rev. C 67, 054602 (2003).
  * (39) L. C. Vaz, J. M. Alexander, and G. R. Satchler, Phys. Rep. 69, 373 (1981).
  * (40) J. J. Kolata _et al._ , Phys. Rev. Lett. 81, 4580 (1998).
  * (41) A. Morsad _et al._ , Phys. Rev. C 41, 988 (1990).
  * (42) M. Trotta _et al._ , Phys. Rev. Lett. 84, 2342 (2000).
  * (43) P. K. Rath _et al._ , Phys. Rev. C 79, 051601(R) (2009).
  * (44) P. R. S. Gomes _et al._ , Nucl. Phys. A534, 429 (1991).
  * (45) J. O. Newton _et al._ , Phys. Rev. C 70, 024605 (2004).
  * (46) Z. H. Liu _et al._ , Eur. Phys. J. A 26, 73 (2005).
  * (47) J. J. Kolata _et al._ , Phys. Rev. C 69, 047601 (2004).
  * (48) N. V. S. V. Prasad _et al._ , Nucl. Phys. A603, 176 (1996).
  * (49) V. Tripathi _et al._ , Phys. Rev. C 65, 014614 (2001).
  * (50) S. Sinha _et al._ , Phys. Rev. C 64, 024607 (2001).
  * (51) E. M. Szanto _et al._ , Phys. Rev. C 41, 2164 (1990).
  * (52) A. M. Stefanini _et al._ , Phys. Rev. C 65, 034609 (2002).
  * (53) E. M. Martínez-Quiroz _et al._ , Phys. Rev. C 63, 054611 (2001).
  * (54) A. M. Stefanini _et al._ , Phys. Rev. C 62, 014601 (2000).
  * (55) L. T. Baby _et al._ , Phys. Rev. C 62, 014603 (2000).
  * (56) O. A. Capurro _et al._ , Phys. Rev. C 65, 064617 (2002).
  * (57) P. H. Stelson _et al._ , Phys. Rev. C 41, 1584 (1990).
  * (58) S. Mitsuoka _et al._ , Phys. Rev. Lett. 99, 182701 (2007).
  * (59) J. O. Newton, C. R. Morton, M. Dasgupta, J. R. Leigh, J. C. Mein, D. J. Hinde, H. Timmers, and K. Hagino, Phys. Rev. C 64, 064608 (2001).
  * (60) C. R. Morton, A. C. Berriman, M. Dasgupta, D. J. Hinde, J. O. Newton, K. Hagino, and I. J. Thompson, Phys. Rev. C 60, 044608 (1999).
  * (61) R. L. Neto _et al._ , Nucl. Phys. A512, 333 (1990)

Figure 1: (Color online) The variation of the surface energy coefficient
$\gamma$ $(\rm MeV~{}fm^{-2})$ with asymmetry parameter $A_{s}$. We display
the results using $\gamma$ from Prox 77, Prox 88, Prox 00, and AW 95 for
masses of reacting partner $A_{1}$ = $A_{2}$ = 40 units.  Figure 2: (Color
online) Same as Fig. 1, but for various radii used in the literature. Figure
3: (Color online) The strength of the nuclear potential $\rm V_{N}~{}(MeV)$
calculated at a distance equal to $C_{1}+C_{2}+1$ fm as a function of
asymmetry parameter $A_{s}$ for the reacting partners having masses $A_{1}$ =
$A_{2}$ = 40 and $A_{1}$ = $A_{2}$ = 80 units. Here $C_{i}$ denotes the
central radius id1 . The dotted lines denote the value of the potential at
$A_{s}$ = 0.0 (for $A_{1}$ = $A_{2}$ = 40 only) using proximity potentials.
Figure 4: The theoretical fusion barrier heights $V_{B}$ (MeV) and positions
$R_{B}$ (fm) are displayed as a function of experimentally extracted values.
The shaded area represents the region within which all 12 proximity potentials
are able to reproduce experimental data. Figure 5: The percentage difference
$\Delta V_{B}(\%)$ of theoretical fusion barrier heights over experimental one
as a function of asymmetry parameter $A_{s}$. Here only 60 reactions covering
the whole mass and asymmetry range are taken. The shaded area is marked only
for those potentials where the deviation is within $\pm 5\%$.  Figure 6: Same
as Fig. 5, but for percentage difference $\Delta R_{B}(\%)$. Figure 7: (Color
online) The fusion cross sections $\sigma_{fus}$ (mb) as a function of center-
of-mass energy $E_{c.m.}~{}\rm(MeV)$. For the clarity, only latest versions of
different proximity potentials are shown. The experimental data are taken from
Stefanini 2006 Stefanini06 , Newton 2001 newton01 , and Morton 1999 Morton01 .
Figure 8: (Color online) Same as Fig. 7, but for different systems explained
in the text. The experimental data are taken from Neto 1990 Neto90 , Tripathi
2001 Vandana01 , and Baby 2000 Baby2000 .

Table 1: The fusion barrier heights VB (in MeV) and positions RB (in fm) using different proximity potentials for 60 asymmetric systems. The corresponding experimental values are also listed.  Reaction | Prox 77 | Prox 88 | Prox 00 | Prox 00DP | Empirical |
---|---|---|---|---|---|---
| VB | RB | VB | RB | VB | RB | VB | RB | VB | RB | $Ref.$
7Li + 27Al | 6.52 | 7.78 | 6.34 | 8.03 | 6.80 | 7.45 | 6.34 | 8.08 | 7.38 | 7.36 |  Padron02
12C + 17O | 8.22 | 7.56 | 7.98 | 7.81 | 8.46 | 7.39 | 7.93 | 7.92 | 8.20 | 7.76 |  tighe93
11B + 27Al | 10.68 | 7.94 | 10.39 | 8.19 | 11.09 | 7.64 | 10.62 | 8.05 | 11.20 | 7.69 |  Padron02
6Li + 59Co | 12.64 | 8.41 | 12.31 | 8.66 | 12.58 | 8.49 | 11.78 | 9.14 | 12.00 | 7.60 |  beck03
4He + 164Dy | 17.71 | 9.90 | 17.36 | 10.15 | 17.36 | 10.20 | 16.01 | 11.09 | 17.14 | 10.32 |  vaz81
4He + 209Bi | 21.30 | 10.44 | 20.89 | 10.64 | 20.63 | 10.81 | 19.20 | 11.70 | 20.98 | 10.04 |
|  |  |  |  |  |  |  |  |  $\pm$0.05 |  $\pm$0.01 |  kolata98
26Mg + 30Si | 25.61 | 8.64 | 24.97 | 8.89 | 25.05 | 8.86 | 24.71 | 9.01 | 24.80 | 9.05 |  Morsad90
6He + 238U | 22.06 | 11.22 | 21.69 | 11.42 | 22.56 | 10.97 | 21.21 | 11.74 | 20.28 | 12.50 |  trotta2000
6Li + 144Sm | 25.26 | 9.80 | 24.72 | 10.05 | 25.18 | 9.85 | 23.69 | 10.53 | 24.65 | 10.20 |  rath09
14N + 59Co | 28.19 | 8.83 | 27.50 | 9.08 | 28.13 | 8.87 | 27.37 | 9.16 | 26.13 | 9.60 |  gomes91
7Li + 159Tb | 25.50 | 10.20 | 25.00 | 10.45 | 26.76 | 10.15 | 24.32 | 10.77 | 23.81 | 11.03 |  vaz81
24Mg + 35Cl | 31.18 | 8.60 | 30.39 | 8.85 | 30.36 | 8.90 | 30.04 | 8.98 | 30.70 | 8.84 |  Cavallera90
16O + 58Ni | 33.32 | 8.85 | 32.51 | 9.10 | 33.52 | 8.82 | 32.72 | 9.09 | 31.67 | 9.30 |  newton04
18O + 64Ni | 32.08 | 9.25 | 31.35 | 9.50 | 32.32 | 9.20 | 31.58 | 9.42 | 32.50 | 9.04 |  Silva97
12C + 92Zr | 33.88 | 9.38 | 33.12 | 9.63 | 33.98 | 9.37 | 32.78 | 9.79 | 32.31 | 9.68 |  newton04
6Li + 208Pb | 31.17 | 10.57 | 30.59 | 10.77 | 31.11 | 10.60 | 29.49 | 11.25 | 30.10 | 11.00 |  Liu05
16O + 72Ge | 36.79 | 9.22 | 35.94 | 9.42 | 36.80 | 9.23 | 35.96 | 9.45 | 35.40 | 9.70 |  Aguilera95
36S + 48Ca | 44.63 | 9.51 | 43.65 | 9.76 | 44.67 | 9.55 | 43.70 | 9.78 | 43.30 |  |  stefanini08
10Be + 209Bi | 40.50 | 11.02 | 39.78 | 11.22 | 40.59 | 10.99 | 39.11 | 11.44 | 37.60 | 13.50 |  Kolata04
19F + 93Nb | 50.34 | 9.74 | 49.24 | 9.99 | 49.27 | 10.02 | 49.27 | 10.02 | 46.60 | 9.20 |
|  |  |  |  |  |  |  |  |  $\pm$0.10 |  $\pm$0.10 |  Prasad96
12C + 152Sm | 48.37 | 10.28 | 47.41 | 10.48 | 48.98 | 10.17 | 47.60 | 10.49 | 46.39 | 10.77 |  vaz81
16O + 116Sn | 53.56 | 9.94 | 52.43 | 10.19 | 53.48 | 10.01 | 52.35 | 10.23 | 50.94 | 10.36 |  Vandana01
18O + 124Sn | 51.99 | 10.27 | 50.97 | 10.52 | 51.89 | 10.33 | 50.81 | 10.55 | 49.30 | 10.98 |  Sinha01
48Ca + 48Ca | 53.96 | 9.89 | 52.84 | 10.09 | 53.93 | 9.89 | 52.86 | 10.11 | 51.70 | 10.38 |  Trotta01
27Al + 70Ge | 57.62 | 9.59 | 56.34 | 9.84 | 57.74 | 9.58 | 57.74 | 9.58 | 55.10 | 10.20 |  Aguilera90
40Ca + 48Ti | 61.67 | 9.46 | 60.27 | 9.71 | 60.71 | 9.64 | 60.71 | 9.64 | 58.17 | 9.97 |
|  |  |  |  |  |  |  |  |  $\pm$0.62 |  $\pm$0.07 |  Sonz98
35Cl + 54Fe | 62.04 | 9.46 | 60.62 | 9.71 | 60.85 | 9.66 | 60.27 | 9.79 | 58.59 | 10.14 |  Szanto90
37Cl + 64Ni | 64.41 | 9.82 | 63.03 | 10.07 | 64.02 | 9.91 | 63.37 | 10.05 | 60.60 | 10.59 |  Vega90
46Ti + 46Ti | 67.15 | 9.56 | 65.64 | 9.81 | 66.34 | 9.70 | 65.38 | 9.87 | 63.30 | 10.27 |  Stefanini02
12C + 204Pb | 60.73 | 10.84 | 59.61 | 11.09 | 60.96 | 10.85 | 59.08 | 11.22 | 57.55 | 11.34 |  newton04
16O + 144Sm | 64.16 | 10.31 | 62.86 | 10.56 | 64.01 | 10.38 | 62.47 | 10.63 | 61.03 | 10.85 |  newton04
40Ar + 58Ni | 68.84 | 9.72 | 67.33 | 9.97 | 67.93 | 9.92 | 67.93 | 9.92 | 66.32 | 10.16 |  vaz81
37Cl + 73Ge | 72.43 | 10.00 | 70.91 | 10.25 | 71.88 | 10.11 | 70.74 | 10.30 | 69.20 | 10.60 |  quirz01
28Si + 92Zr | 74.52 | 10.00 | 72.95 | 10.25 | 72.72 | 10.30 | 72.35 | 10.34 | 70.93 | 10.19 |  newton04
16O + 186W | 73.09 | 10.86 | 71.74 | 11.06 | 71.39 | 11.18 | 70.03 | 11.40 | 68.87 | 11.12 |  newton04
48Ti + 58Ni | 82.70 | 9.89 | 80.91 | 10.14 | 81.34 | 10.13 | 81.34 | 10.13 | 78.80 | 9.80 |
|  |  |  |  |  |  |  |  |  $\pm$0.30 |  $\pm$0.30 |  Vinod96
32S + 89Y | 82.52 | 10.06 | 80.78 | 10.31 | 81.38 | 10.23 | 80.62 | 10.36 | 77.77 | 10.30 |  newton04
36S + 90Zr | 82.99 | 10.30 | 81.30 | 10.55 | 82.35 | 10.41 | 81.10 | 10.60 | 79.00 | 10.64 |  Stefanini2000
16O + 208Pb | 79.38 | 11.09 | 77.96 | 11.29 | 79.30 | 11.13 | 77.78 | 11.35 | 74.90 | 11.76 |  Liu05
35Cl + 92Zr | 88.58 | 10.25 | 86.75 | 10.50 | 87.64 | 10.39 | 86.41 | 10.56 | 82.94 | 10.20 |  newton04
28Si + 120Sn | 89.43 | 10.49 | 87.65 | 10.69 | 88.12 | 10.65 | 88.12 | 10.65 | 85.89 | 11.04 |  Baby2000
19F + 197Au | 85.70 | 11.15 | 84.16 | 11.35 | 85.33 | 11.20 | 85.33 | 11.20 | 81.61 | 11.32 |  newton04
16O + 238U | 86.86 | 11.39 | 85.37 | 11.59 | 87.46 | 11.30 | 85.81 | 11.56 | 80.81 | 11.45 |  newton04
35Cl + 106Pd | 99.86 | 10.48 | 97.85 | 10.68 | 98.75 | 10.62 | 97.45 | 10.74 | 94.30 | 11.27 |  capurro02
58Ni + 60Ni | 102.83 | 10.16 | 100.67 | 10.41 | 102.07 | 10.26 | 102.07 | 10.26 | 96.00 | 10.26 |  newton04
32S + 116Sn | 101.78 | 10.49 | 99.75 | 10.74 | 100.65 | 10.64 | 99.73 | 10.76 | 97.36 | 10.80 |  Vandana01
40Ca + 90Zr | 103.60 | 10.30 | 101.46 | 10.55 | 102.57 | 10.43 | 102.10 | 10.48 | 96.88 | 10.53 |  newton04
48Ca + 96Zr | 99.33 | 10.80 | 97.46 | 11.00 | 98.73 | 10.90 | 97.28 | 11.04 | 95.90 | 11.21 |  Stefanini06
28Si + 144Sm | 108.00 | 10.78 | 105.90 | 10.98 | 105.40 | 11.04 | 105.03 | 11.13 | 103.89 | 10.93 |  newton04
50Ti + 93Nb | 112.74 | 10.71 | 110.54 | 10.96 | 111.25 | 10.87 | 110.38 | 10.99 | 106.90 |  |  Stelson90
40Ca + 124Sn | 123.11 | 10.90 | 120.78 | 11.10 | 121.55 | 11.01 | 121.55 | 11.01 | 112.93 | 10.08 |  newton04
28Si + 208Pb | 133.90 | 11.56 | 131.59 | 11.76 | 131.10 | 11.79 | 131.10 | 11.79 | 128.07 | 11.45 |  newton04

TABLE 1:-(continued).
Reaction Prox 77 Prox 88 Prox 00 Prox 00DP Empirical VB RB VB RB VB RB VB RB
VB RB $Ref.$ 40Ar + 165Ho 141.27 11.49 138.78 11.69 138.61 11.71 138.61 11.71
141.38 11.48 vaz81  32S + 232Th 163.08 11.92 160.39 12.12 162.32 11.94 160.97
12.02 155.73 11.18 newton04  40Ca + 192Os 174.70 11.71 171.71 11.96 173.90
11.74 173.07 11.79 168.07 11.05 newton04  48Ti + 208Pb 200.34 12.18 197.08
12.38 197.08 12.34 197.08 12.34 190.10 Mitsuoka07  56Fe + 208Pb 233.61 12.33
229.84 12.58 229.74 12.45 229.74 12.45 223.00 Mitsuoka07  64Ni + 208Pb 247.56
12.56 243.66 12.76 245.68 12.53 245.68 12.53 236.00 Mitsuoka07  70Zn + 208Pb
262.60 12.71 258.53 12.91 259.01 12.76 259.01 12.76 250.60 Mitsuoka07  86Kr +
208Pb 308.05 12.99 303.40 13.24 306.16 12.92 304.56 12.98 299.20 Mitsuoka07

Table 2: Fusion barrier heights VB (in MeV) and positions RB (in fm) are displayed using other different proximity potentials for 60 asymmetric systems. The limited numbers of reactions in certain cases are due to the restriction posed in different potentials. Reaction | Bass 80 | Ngo 8̂0 | AW 95 | Denisov DP
---|---|---|---|---
| VB | RB | VB | RB | VB | RB | VB | RB
7Li + 27Al | 6.20 | 8.35 | - | - | 6.31 | 8.27 | - | -
12C + 17O | 7.79 | 8.13 | - | - | 7.89 | 8.10 | - | -
11B + 27Al | 10.13 | 8.50 | - | - | 10.24 | 8.49 | - | -
6Li + 59Co | 12.00 | 8.97 | - | - | 12.14 | 8.97 | - | -
4He + 164Dy | 16.87 | 10.51 | - | - | 17.12 | 10.44 | - | -
4He + 209Bi | 20.30 | 11.00 | - | - | 20.62 | 10.95 | - | -
26Mg + 30Si | 24.33 | 9.20 | 25.65 | 8.76 | 24.42 | 9.20 | 23.84 | 9.29
6He + 238U | 21.10 | 11.83 | - | - | 21.60 | 11.59 | - | -
6Li + 144Sm | 24.08 | 10.36 | - | - | 24.34 | 10.34 | - | -
14N + 59Co | 26.79 | 9.40 | - | - | 26.90 | 9.43 | - | -
7Li + 159Tb | 24.33 | 10.76 | - | - | 24.67 | 10.72 | - | -
24Mg + 35Cl | 29.61 | 9.16 | 31.19 | 8.72 | 29.67 | 9.21 | 29.21 | 9.23
16O + 58Ni | 31.69 | 9.41 | 33.42 | 8.94 | 31.78 | 9.44 | 31.14 | 9.50
18O + 64Ni | 30.53 | 9.81 | 32.18 | 9.33 | 30.70 | 9.76 | 29.91 | 9.93
12C + 92Zr | 32.26 | 9.94 | - | - | 32.43 | 9.93 | - | -
6Li + 208Pb | 29.72 | 11.14 | - | - | 30.08 | 11.11 | - | -
16O + 72Ge | 35.02 | 9.73 | 36.92 | 9.29 | 35.14 | 9.79 | 34.46 | 9.83
36S + 48Ca | 42.48 | 10.07 | 44.69 | 9.59 | 42.69 | 10.04 | 42.11 | 10.09
10Be + 209Bi | 38.70 | 11.59 | - | - | 39.29 | 11.48 | - | -
19F + 93Nb | 48.01 | 10.25 | 50.57 | 9.78 | 48.24 | 10.26 | 47.56 | 10.32
12C + 152Sm | 46.13 | 10.79 | - | - | 46.45 | 10.82 | - | -
16O + 116Sn | 51.11 | 10.45 | 53.85 | 9.97 | 51.36 | 10.50 | 50.61 | 10.55
18O + 124Sn | 49.57 | 10.83 | 52.18 | 10.34 | 49.98 | 10.80 | 49.04 | 10.93
48Ca + 48Ca | 51.39 | 10.40 | 54.06 | 9.94 | 51.74 | 10.39 | 51.13 | 10.42
27Al + 70Ge | 54.97 | 10.11 | 57.86 | 9.60 | 55.13 | 10.12 | 54.77 | 10.09
40Ca + 48Ti | 58.83 | 9.97 | 61.90 | 9.47 | 58.91 | 9.99 | 58.76 | 9.93
35Cl + 54Fe | 59.18 | 9.92 | 62.28 | 9.47 | 59.28 | 9.98 | 59.11 | 9.92
37Cl + 64Ni | 61.47 | 10.33 | 64.67 | 9.87 | 61.71 | 10.37 | 61.37 | 10.33
46Ti + 46Ti | 64.10 | 10.07 | 67.45 | 9.56 | 64.21 | 10.07 | 64.09 | 10.02
12C + 204Pb | 58.04 | 11.40 | 57.86 | 9.60 | 58.53 | 11.38 | 55.13 | 10.12
16O + 144Sm | 61.34 | 10.82 | 64.59 | 10.31 | 61.68 | 10.83 | 60.85 | 10.88
40Ar + 58Ni | 65.75 | 10.23 | 69.19 | 9.71 | 65.91 | 10.22 | 65.71 | 10.20
37Cl + 73Ge | 69.19 | 10.51 | 72.81 | 9.98 | 69.48 | 10.48 | 69.21 | 10.49
28Si + 92Zr | 71.21 | 10.51 | 74.96 | 9.97 | 71.44 | 10.53 | 71.32 | 10.45
16O + 186W | 69.86 | 11.37 | 73.44 | 10.85 | 70.36 | 11.34 | 69.47 | 11.45
48Ti + 58Ni | 79.08 | 10.35 | 83.24 | 9.87 | 79.28 | 10.43 | 79.28 | 10.35
32S + 89Y | 78.91 | 10.52 | 83.07 | 10.03 | 79.15 | 10.59 | 79.18 | 10.50
36S + 90Zr | 79.40 | 10.76 | 83.56 | 10.26 | 79.82 | 10.76 | 79.54 | 10.73
16O + 208Pb | 75.92 | 11.60 | 79.76 | 11.07 | 76.52 | 11.60 | 75.55 | 11.66
35Cl + 92Zr | 84.76 | 10.71 | 89.22 | 10.16 | 85.09 | 10.71 | 85.11 | 10.65
28Si + 120Sn | 85.56 | 10.95 | 90.04 | 10.39 | 85.94 | 10.93 | 85.98 | 10.86
19F + 197Au | 82.04 | 11.66 | 86.19 | 11.07 | 82.76 | 11.60 | 81.92 | 11.67
16O + 238U | 83.12 | 11.90 | 87.20 | 11.37 | 83.85 | 11.88 | 82.77 | 11.98
35Cl + 106Pd | 95.67 | 10.89 | 100.71 | 10.38 | 96.09 | 10.92 | 96.24 | 10.81
58Ni + 60Ni | 98.53 | 10.56 | 103.77 | 10.02 | 98.80 | 10.61 | 99.09 | 10.53
32S + 116Sn | 97.53 | 10.95 | 102.68 | 10.39 | 97.93 | 10.98 | 98.18 | 10.85
40Ca + 90Zr | 99.28 | 10.71 | 104.55 | 10.15 | 99.58 | 10.75 | 99.93 | 10.66
48Ca + 96Zr | 95.05 | 11.26 | 100.03 | 10.74 | 95.87 | 11.23 | 95.61 | 11.19
28Si + 144Sm | 103.60 | 11.19 | 109.04 | 10.61 | 104.12 | 11.20 | 104.31 | 11.12
50Ti + 93Nb | 108.10 | 11.17 | 113.83 | 10.59 | 108.77 | 11.13 | 108.87 | 11.06
40Ca + 124Sn | 118.07 | 11.31 | 124.29 | 10.73 | 118.66 | 11.31 | 119.36 | 11.14
28Si + 208Pb | 128.56 | 11.97 | 135.03 | 11.37 | 129.53 | 11.92 | 130.06 | 11.79
40Ar + 165Ho | 135.70 | 11.90 | 142.75 | 11.29 | 136.97 | 11.86 | 137.38 | 11.73
32S + 232Th | 156.86 | 12.28 | 164.65 | 11.67 | 158.17 | 12.25 | - | -
40Ca + 192Os | 168.22 | 12.07 | 176.93 | 11.45 | 169.44 | 12.05 | 171.15 | 11.82
48Ti + 208Pb | 193.15 | 12.49 | 203.09 | 11.86 | 195.26 | 12.44 | 196.99 | 12.15
56Fe + 208Pb | 225.72 | 12.59 | 237.53 | 11.89 | 228.16 | 12.52 | 230.95 | 12.18
64Ni + 208Pb | 239.28 | 12.81 | 251.83 | 12.10 | 242.40 | 12.68 | 245.24 | 12.31
70Zn + 208Pb | 253.99 | 12.92 | 267.37 | 12.20 | 257.54 | 12.78 | 260.75 | 12.37
86Kr + 208Pb | 298.65 | 13.15 | - | - | 303.25 | 12.98 | 308.13 | 12.32


# Exactly solvable quadratic differential equation systems through generalized
inversion

Ádám Bácsi<EMAIL_ADDRESS>Department of Mathematics and Computational
Sciences, Széchenyi István University
H-9026 Győr, Egyetem tér 1, Hungary  MTA-BME Lendület Topology and Correlation
Research Group, Budapest University of Technology and Economics,
1521 Budapest, Hungary  Albert Tihamér Kocsis  Department of Mathematics and
Computational Sciences, Széchenyi István University
H-9026 Győr, Egyetem tér 1, Hungary

###### Abstract

We study the autonomous systems of quadratic differential equations of the
form
$\dot{x}_{i}(t)=\mathbf{x}(t)^{T}\mathbf{A}_{i}\mathbf{x}(t)+\mathbf{v}_{i}^{T}\mathbf{x}(t)$
with $\mathbf{x}(t)=(x_{1}(t),x_{2}(t),\dots,x_{i}(t),\dots)$ which, in
general, cannot be solved exactly. In the present paper, we present a subclass
of analytically solvable quadratic systems, whose solution is realized through
a multi-dimensional generalization of the inversion which transforms a
quadratic system into a linear system. We provide a constructive algorithm
which, on one hand, decides whether the system of differential equations is
analytically solvable with the inversion transformation and, on the other
hand, provides the solution. The presented results apply for arbitrary, finite
number of variables.

###### keywords:

Quadratic differential equation system , Analytical solution , Generalized
inversion

##  1 Introduction

Linear, ordinary, autonomous systems of differential equations are celebrated
for their analytical integrability which makes them excellent tools in almost
all fields of science. This solvability breaks down when entering the realm of
non-linear ordinary differential equation systems where no general solution
method exists. In most cases, non-linear systems are handled with
sophisticated numerical methods such as Runge-Kutta schemes[1].

Despite of the great accuracy of numerical methods, it is also desirable to
find analytical techniques to solve non-linear systems. Beside the inherent
beauty of exact solutions, they can also play the role of benchmark for
numerical methods. In this work, we focus on first-order, quadratic,
autonomous differential equation systems (QDEs) which emerge in various
research fields including models of population dynamics[2], fluid dynamics[3],
control systems [4] and even quantum dynamics [5]. Classification of two-
variable quadratic systems have been studied extensively [6, 7, 8, 9].

There exists no general method to find analytical solution of QDEs. Except for
the special case, when the number of variables is just one. By denoting this
variable with $x(t)$, the quadratic differential equation is written as

$\displaystyle\dot{x}=ax^{2}+vx$ (1)

with $a\neq 0$ and $v$ being constants. An additional constant term on the
right-handside could be eliminated by shifting $x(t)$. By introducing
$y(t)=x(t)^{-1}$, the equation is transformed to the linear equation of

$\displaystyle\dot{y}=-a-vy$ (2)

and, hence, the quadratic differential equation (1) becomes exactly solvable.

In this paper, we investigate a class of QDEs with more than one variable
which can be solved exactly by transforming the QDE to a linear differential
equation system (LDE). The transformation is realized through a multi-variable
extension of $y(t)=x(t)^{-1}$, similar to the generalized spherical inversions
presented in Refs. [10, 11].

##  2 Generalized inversion transformation

Let us consider some differentiable functions $x_{1}(t)$, $x_{2}(t)$, …,
$x_{n}(t)$ whose dynamics is governed by the QDE

$\displaystyle\dot{x}_{i}=\mathbf{x}^{T}\mathbf{A}_{i}\mathbf{x}+\mathbf{v}_{i}^{T}\mathbf{x}$
(3)

where $\mathbf{A}_{i}$ are $n\times n$ matrices which are chosen to be
symmetric and $\mathbf{v}_{i}^{T}$ are row vectors of $n$ components. The
entries of $\mathbf{A}_{i}$ and $\mathbf{v}_{i}$ are independent from $t$ and
are assumed to be real without loss of generality111In case of complex
entries, the differential equations can be split up to their real and
imaginary parts.. Our goal is to determine if the variables $x_{i}(t)$ can be
transformed with a multivariable variant of the inversion in such a way that
the QDE (3) is transformed to an LDE. The multivariable inversion is defined
as

$\displaystyle
y_{i}(t)=\frac{x_{i}(t)}{\mathbf{x}(t)^{T}\mathbf{B}\mathbf{x}(t)}$ (4)

with some symmetric matrix $\mathbf{B}$ which does not depend on time. Note
that we use the same $\mathbf{B}$ matrix for all $i$. This ensures that the
transformation is easily inverted by

$\displaystyle
x_{i}(t)=\frac{y_{i}(t)}{\mathbf{y}(t)^{T}\mathbf{B}\mathbf{y}(t)}\,.$ (5)

The transformations (4) and (5) will henceforth be referred to as
$B$-transformation and inverse $B$-transformation, respectively. The variables
$y_{i}(t)$ are expected to obey the linear differential equations

$\displaystyle\dot{y_{i}}=\mathbf{m}_{i}^{T}\mathbf{y}+w_{i}\,.$ (6)

with time-independent row vectors $\mathbf{m}_{i}^{T}$ and the scalars
$w_{i}$. Since Eq. (6) is exactly solvable for arbitrary initial conditions,
the QDE can also be solved by applying the inverse $B$-transformation (5). The
goal of the present paper is to provide a procedure based on which one can
decide whether a given QDE, defined through $\mathbf{A}_{i}$ and
$\mathbf{v}_{i}$, can be $B$-transformed to an LDE.

To start our study, let us calculate the time derivative of Eq. (5)

$\displaystyle\dot{x}_{i}=\frac{\dot{y}_{i}}{\mathbf{y}^{T}\mathbf{B}\mathbf{y}}-\frac{y_{i}}{\left(\mathbf{y}^{T}\mathbf{B}\mathbf{y}\right)^{2}}\big{(}\dot{\mathbf{y}}^{T}\mathbf{B}\mathbf{y}+\mathbf{y}^{T}\mathbf{B}\dot{\mathbf{y}}\big{)}\,.$
(7)

Substituting the LDE differential equations (6) and then transforming all
$y_{i}$ variables back to $x_{i}$ by means of Eq. (4), we obtain

$\displaystyle\dot{x}_{i}=\mathbf{m}_{i}^{T}\mathbf{x}+w_{i}\left(\mathbf{x}^{T}\mathbf{B}\mathbf{x}\right)-\mathbf{x}^{T}\mathbf{B}\mathbf{w}x_{i}-x_{i}\mathbf{w}^{T}\mathbf{B}\mathbf{x}-x_{i}\frac{\mathbf{x}^{T}\left(\mathbf{M}^{T}\mathbf{B}+\mathbf{B}\mathbf{M}\right)\mathbf{x}}{\mathbf{x}^{T}\mathbf{B}\mathbf{x}}$
(8)

with the matrix $\mathbf{M}$ built from the row vectors $\mathbf{m}_{i}^{T}$
and the vector $\mathbf{w}$ consisting of the constants $w_{i}$.

By comparing Eq. (8) with the original QDE (3), one can note that Eq. (8) is
not necessarily a quadratic differential equation. The last term has an
$\mathbf{x}$-dependent denominator and, hence, is neither quadratic nor linear
in general. This term scales with $\mathbf{x}$ as a linear term but becomes an
actual linear term only if

$\displaystyle\mathbf{M}^{T}\mathbf{B}+\mathbf{B}\mathbf{M}=-\lambda\mathbf{B}$
(9)

holds with some scalar $-\lambda$. The minus sign is chosen for later
convenience. The condition (9) means that $\mathbf{B}$ must be the
eigenmatrix[12] of $\mathbf{M}$ with the eigenvalue of $-\lambda$. In this
case, the differential equations (8) become

$\displaystyle\dot{x}_{i}=\left(\mathbf{m}_{i}^{T}+\lambda\mathbf{e}_{i}^{T}\right)\mathbf{x}+\mathbf{x}^{T}\left(w_{i}\mathbf{B}-\mathbf{B}\left(\mathbf{w}\circ\mathbf{e}_{i}^{T}\right)-\left(\mathbf{e}_{i}\circ\mathbf{w}^{T}\right)\mathbf{B}\right)\mathbf{x}$
(10)

where $\circ$ denotes dyadic product and $\mathbf{e}_{i}^{T}$ is the unit row
vector with $1$ in the $i$th entry and zero otherwise. The equation is already
a quadratic differential equation containing quadratic and linear terms in
$\mathbf{x}$.

Comparing the linear terms of Eqs. (10) and (3), we find

$\displaystyle\mathbf{V}=\mathbf{M}+\lambda\mathbf{I}$ (11)

where $\mathbf{V}$ is constructed from the row vectors $\mathbf{v}_{i}^{T}$
and $\mathbf{I}$ is the $n\times n$ identity matrix. Substituting (11) into
the condition (9), we obtain

$\displaystyle\mathbf{V}^{T}\mathbf{B}+\mathbf{B}\mathbf{V}=\lambda\mathbf{B}$
(12)

indicating that the matrix $\mathbf{B}$ must be a symmetric eigenmatrix of
$\mathbf{V}$ as well.

Beside the linear terms, the quadratic terms of Eq. (10) and (3) must also
equal. Since both $\mathbf{A}_{i}$ and the kernel in the quadratic terms in
Eq. (10) are symmetric, they must equal

$\displaystyle\mathbf{A}_{i}=w_{i}\mathbf{B}-\mathbf{B}\left(\mathbf{w}\circ\mathbf{e}_{i}^{T}\right)-\left(\mathbf{e}_{i}\circ\mathbf{w}^{T}\right)\mathbf{B}$
(13)

for all $i$. Note that the second and third terms are matrices full of zeros
except for the $i$th row and $i$th column, respectively. Therefore, omitting
the $i$th row and column in the equation leads to the proportionality
condition

$\displaystyle\tilde{\mathbf{A}}_{i}^{i}=w_{i}\tilde{\mathbf{B}}^{i}$ (14)

where $\tilde{\mathbf{A}}_{i}^{i}$ ($\tilde{\mathbf{B}}^{i}$) is the
$(n-1)\times(n-1)$ matrix which is obtained from $\mathbf{A}_{i}$
($\mathbf{B}$) by skipping the $i$th row and column. For the $i$th column of
Eq. (13), we have

$\displaystyle\mathbf{a}_{i}^{i}\equiv\mathbf{A}_{i}\mathbf{e}_{i}=w_{i}\mathbf{B}\mathbf{e}_{i}-\mathbf{B}\mathbf{w}-\left(\mathbf{e}_{i}^{T}\mathbf{B}\mathbf{w}\right)\mathbf{e}_{i}\,.$
(15)

To summarize, the QDE as given in Eq. (3) can be transformed with a
$B$-transformation to an LDE if we find a symmetric matrix $\mathbf{B}$
fulfilling Eq. (12) with some constant $\lambda$ and, at the same time,
obeying Eqs. (14) and (15) with some constants $w_{i}$. The linear terms of
the LDE can be calculated by expressing $\mathbf{V}$ from Eq. (11).

##  3 Algorithm

Let us provide a detailed description of the procedure based on the findings
of the previous section. The starting point of the algorithm is Eq. (3)
determined by the matrices $\mathbf{A}_{i}$ and vectors $\mathbf{v}_{i}$.

  * 1.

As a first step, one has to build up the matrix $\mathbf{V}$ from the row
vectors $\mathbf{v}_{i}^{T}$ and find all symmetric eigenmatrices based on
$\mathbf{V}^{T}\mathbf{B}+\mathbf{B}\mathbf{V}=\lambda\mathbf{B}$. Note that
the eigenmatrices are definite only up to an overall multiplication factor
similarly to the case of usual eigenvectors.

To obtain the eigenmatrices, one may calculate the vector-eigenvalues $s_{j}$
of $\mathbf{V}^{T}$ and the corresponding right-handside eigenvectors
$\mathbf{r}_{j}$, i.e., $\mathbf{V}^{T}\mathbf{r}_{j}=s_{j}\mathbf{r}_{j}$.
The sum of the geometric multiplicity of all eigenvalues is denoted by $N$.
The matrix-eigenvalues of $\mathbf{V}$ are given by $s_{1}+s_{1}$,
$s_{1}+s_{2}$, …, $s_{1}+s_{n}$, $s_{2}+s_{2}$, $s_{2}+s_{3}$, …,$s_{N}+s_{N}$
and the eigenmatrix corresponding to $s_{j}+s_{m}$ is
$\mathbf{P}_{jm}=\left(\mathbf{r}_{j}\circ\mathbf{r}_{m}^{T}+\mathbf{r}_{m}\circ\mathbf{r}_{j}^{T}\right)/2$.
Note that the number of symmetric eigenmatrices is $N(N+1)/2$ which is utmost
$n(n+1)/2$ when $n$ eigenvectors are found.

In case of degenerate matrix eigenvalues $\lambda$, all linear combinations
within the subspace are potential $\mathbf{B}$ matrices. For an example, see
Sec. 5.

  * 2.

For each potential eigenmatrix $\mathbf{B}$, the proportionality condition
(14) must be checked for all $i$ with some constants $w_{i}$, which may also
be zero.

If Eq. (14) cannot be fulfilled with any potential $\mathbf{B}$ matrix, then
the QDE is not solvable by $B$-transforming to an LDE. If Eq. (14) is
satisfied with a $\mathbf{B}$ matrix and some $w_{i}$ constants, one may
proceed to Step 3.

  * 3.

For the potential pairs of $\mathbf{B}$ and $\mathbf{w}$ surviving Step 2, one
has to check Eq. (15) for all $i$. If it holds true for all $i$, then the QDE
(3) can be transformed to an LDE of the form of Eq. (6) where the constants
$w_{i}$ are the coefficients of proportionality found in Eq. (14) in Step 2
and the vectors $\mathbf{m}_{i}^{T}$ are the row vectors of
$\mathbf{M}=\mathbf{V}-\lambda\mathbf{I}$ with $\lambda$ the eigenvalue
corresponding to the eigenmatrix $\mathbf{B}$.

  * 4.

The solution of the QDE can be obtained by solving first the linear equation
(6) as

$\displaystyle\mathbf{y}(t)=e^{\mathbf{M}t}\mathbf{y}_{0}+\mathbf{y}_{p}(t)$
(16)

where
$\mathbf{y}_{0}=\mathbf{x}_{0}/(\mathbf{x}_{0}^{T}\mathbf{B}\mathbf{x}_{0})$
is the initial condition with $\mathbf{x}_{0}=\mathbf{x}(t=0)$ and
$\mathbf{y}_{p}(t)=\left(e^{\mathbf{M}t}-\mathbf{I}\right)\mathbf{M}^{-1}\mathbf{w}$
if $\mathbf{M}$ is invertible. If $\mathbf{M}$ is not invertible, the function
$\mathbf{y}_{p}(t)$ has linear time-dependence in the nullspace of
$\mathbf{M}$. Using the inverse $B$ transformation, the solution of the QDE is
obtained as

$\displaystyle\mathbf{x}(t)=\frac{e^{\mathbf{M}t}\mathbf{x}_{0}+\left(\mathbf{x}_{0}^{T}\mathbf{B}\mathbf{x}_{0}\right)\mathbf{y}_{p}(t)}{e^{-\lambda
t}+2\mathbf{y}_{p}(t)^{T}\mathbf{B}e^{\mathbf{M}t}\mathbf{x}_{0}+\left(\mathbf{x}_{0}^{T}\mathbf{B}\mathbf{x}_{0}\right)\left(\mathbf{y}_{p}(t)^{T}\mathbf{B}\mathbf{y}_{p}(t)\right)}\,.$
(17)

Before presenting examples, let us make a remark on the region where the
$B$-transformation is undefined, i.e., where
$\mathbf{x}^{T}\mathbf{B}\mathbf{x}=0$. This region can be any quadratic
hypersurface depending on the specific structure of the $\mathbf{B}$ matrix.
The region also involves the nullspace of $\mathbf{B}$, i.e., the points where
$\mathbf{B}\mathbf{x}=0$. In Sec. 4, the hypersurface contains two straight
lines while in Sec. 5, it consists of two cone-shaped surfaces. This raises
the question how the system behaves if the initial condition $\mathbf{x}_{0}$
is an element of the region, i.e.
$\mathbf{x}_{0}^{T}\mathbf{B}\mathbf{x}_{0}=0$.

First, one can prove that if the initial condition of the QDE is on the
hypersurface, the dynamics is constrained to the hypersurface for the whole
time-evolution. To justify this, we define
$b(t)=\mathbf{x}(t)^{T}\mathbf{B}\mathbf{x}(t)$ whose dynamics is derived from
the quadratic equations (3) as
$\dot{b}=b\left(\lambda-2\mathbf{w}^{T}\mathbf{B}\mathbf{x}\right)$. If the
initial condition is on the hypersurface, i.e., $b(0)=0$, then the
differential equation solves to $b(t)=0$.

Second, it can also be shown that inside the hypersurface, the quadratic
differential equation is analytically solvable and the solution is obtained
simply by taking $\mathbf{x}_{0}^{T}\mathbf{B}\mathbf{x}_{0}=0$ in Eq. (17)
which reads as

$\displaystyle\mathbf{x}(t)=\frac{e^{\mathbf{M}t}\mathbf{x}_{0}}{e^{-\lambda
t}+2\mathbf{y}_{p}(t)^{T}\mathbf{B}e^{\mathbf{M}t}\mathbf{x}_{0}}\,.$ (18)

Note that (18) does not obviously solve the quadratic differential equation
because it was derived from Eq. (17) which was computed through the $B$
transformation. However, by substituting (18) directly into (3) and taking
advantage of $\mathbf{x}_{0}^{T}\mathbf{B}\mathbf{x}_{0}=0$ and the properties
of $\mathbf{B}$, one can prove that the QDE is solved indeed by (18). Hence,
if a QDE is exactly solvable by a $B$-transformation, the integrability is
preserved also in the region where the generalized inversion is undefined.

##  4 Example, two-dimensional quadratic system

In this section, we consider the quadratic equations of

$\displaystyle\dot{x}_{1}$ $\displaystyle=-x_{1}^{2}+4x_{1}x_{2}-x_{1}+2x_{2}$
$\displaystyle\dot{x}_{2}$ $\displaystyle=x_{1}^{2}+2x_{2}^{2}+x_{1}$ (19)

from which we read out

$\displaystyle\mathbf{A}_{1}=\left[\begin{array}[]{cc}-1&2\\\
2&0\end{array}\right]\qquad\mathbf{A}_{2}=\left[\begin{array}[]{cc}1&0\\\
0&2\end{array}\right]\qquad\mathbf{v}_{1}^{T}=\left[\begin{array}[]{cc}-1&2\end{array}\right]\qquad\mathbf{v}_{2}^{T}=\left[\begin{array}[]{cc}1&0\end{array}\right]\,.$
(26)

The first step is to obtain the eigenmatrices of $\mathbf{V}$. The right-
handside eigenvectors of $\mathbf{V}^{T}$ are given by

$\displaystyle\mathbf{r}_{1}=\left[\begin{array}[]{c}2\\\
-2\end{array}\right]\qquad\mathbf{r}_{2}=\left[\begin{array}[]{c}1\\\
2\end{array}\right]$ (31)

with the eigenvalues of $s_{1}=-2$ and $s_{2}=1$. The potential $\mathbf{B}$
matrices are given by

$\displaystyle\mathbf{P}_{11}$
$\displaystyle=\mathbf{r}_{1}\circ\mathbf{r}_{1}^{T}=\left[\begin{array}[]{cc}4&-4\\\
-4&4\end{array}\right]\qquad$ $\displaystyle\lambda_{11}=-4$
$\displaystyle\mathbf{P}_{22}$
$\displaystyle=\mathbf{r}_{2}\circ\mathbf{r}_{2}^{T}=\left[\begin{array}[]{cc}1&2\\\
2&4\end{array}\right]\qquad$ $\displaystyle\lambda_{22}=2$
$\displaystyle\mathbf{P}_{12}$
$\displaystyle=\frac{1}{2}\left(\mathbf{r}_{1}\circ\mathbf{r}_{2}^{T}+\mathbf{r}_{2}\circ\mathbf{r}_{1}^{T}\right)=\left[\begin{array}[]{cc}2&1\\\
1&-4\end{array}\right]\qquad$ $\displaystyle\lambda_{12}=-1$

which are non-degenerate eigenmatrices.

The second step is to check the proportionality condition of Eq. (14) and
determine the coefficients $w_{i}$.

Proportionality check, Eq. (14) |  $\tilde{A}_{1}^{1}=[0]$ |  $\tilde{A}_{2}^{2}=[1]$
---|---|---
$\mathbf{B}=\mathbf{P}_{11}$ |  $\tilde{\mathbf{B}}^{1}=[1]\rightarrow w_{1}=0$ |  $\tilde{\mathbf{B}}^{2}=[4]\rightarrow w_{2}=\frac{1}{4}$
$\mathbf{B}=\mathbf{P}_{22}$ |  $\tilde{\mathbf{B}}^{1}=[4]\rightarrow w_{1}=0$ |  $\tilde{\mathbf{B}}^{2}=[1]\rightarrow w_{2}=1$
$\mathbf{B}=\mathbf{P}_{12}$ |  $\tilde{\mathbf{B}}^{1}=[-4]\rightarrow w_{1}=0$ |  $\tilde{\mathbf{B}}^{2}=[2]\rightarrow w_{2}=\frac{1}{2}$

The table shows that all eigenmatrices obey the proportionality condition
(14). Note that for a two-dimensional QDE, the proportionality check always
simplifies to comparison of $1\times 1$ matrices, which is in most cases
trivially fulfilled. For higher dimensions, however, (14) might mean a much
stricter condition. For an example, see Sec. 5.

In the example, we continue with the third step by checking Eq. (15) for each
$i$. In the table below, $rhs$ stands for the right-handside of Eq. (15)
evaluated with $\mathbf{B}$ and $\mathbf{w}$.

$\mathbf{a}_{i}^{i}$ check, Eq. (15) |  $\mathbf{a}_{1}^{1}=\left[\begin{array}[]{c}-1\\\ 2\end{array}\right]$ |  $\mathbf{a}_{2}^{2}=\left[\begin{array}[]{c}0\\\ 2\end{array}\right]$
---|---|---
$\mathbf{B}=\mathbf{P}_{11}$ |  $rhs=\left[\begin{array}[]{c}2\\\ -1\end{array}\right]\quad{\color[rgb]{1,0,0}\times}$ |  $rhs=\left[\begin{array}[]{c}0\\\ -1\end{array}\right]\quad{\color[rgb]{1,0,0}\times}$
$\mathbf{B}=\mathbf{P}_{22}$ |  $rhs=\left[\begin{array}[]{c}-4\\\ -4\end{array}\right]\quad{\color[rgb]{1,0,0}\times}$ |  $rhs=\left[\begin{array}[]{c}0\\\ -4\end{array}\right]\quad{\color[rgb]{1,0,0}\times}$
$\mathbf{B}=\mathbf{P}_{12}$ |  $rhs=\left[\begin{array}[]{c}-1\\\ 2\end{array}\right]\quad{\color[rgb]{0,1,0}\checkmark}$ |  $rhs=\left[\begin{array}[]{c}0\\\ 2\end{array}\right]\quad{\color[rgb]{0,1,0}\checkmark}$

The investigation shows that $\mathbf{a}_{1}^{1}$ and $\mathbf{a}_{2}^{2}$ are
reproduced only by $\mathbf{P}_{12}$.

The result of the algorithm is that the QDE can be transformed to an LDE by
$\mathbf{B}=\mathbf{P}_{12}$. The linear system is obtained as

$\displaystyle\mathbf{M}=\mathbf{V}-\lambda_{12}\mathbf{I}=\left[\begin{array}[]{cc}0&2\\\
1&1\end{array}\right]\qquad\mathbf{w}=\left[\begin{array}[]{c}0\\\
1/2\end{array}\right]\qquad\qquad\begin{array}[]{rcl}\dot{y}_{1}&=&2y_{2}\\\
\dot{y}_{2}&=&y_{1}+y_{2}+\frac{1}{2}\end{array}$ (38)

which can be solved analytically for arbitrary initial conditions.

Figure 1: Phase portrait of the quadratic system, Eq. (19). The green dots
indicate the fix points of the system. The dashed lines comprise the region
where the $B$ transformation is undefined.

The analytical solution of the quadratic system can be given based on Eq.
(17). The phase portrait is shown in Fig. 1. The two dashed lines comprise the
region where the $B$ transformation is undefined, i.e., where
$\mathbf{x}^{T}\mathbf{B}\mathbf{x}=(\mathbf{x}^{T}\mathbf{r}_{1})\cdot(\mathbf{x}^{T}\mathbf{r}_{2})=0$.

##  5 Three-dimensional example

In this section, the algorithm is demonstrated in an example with three
variables $x_{1}(t)$, $x_{2}(t)$ and $x_{3}(t)$. The quadratic differential
equations are given by

$\displaystyle\begin{array}[]{ccll}\dot{x}_{1}&=&x_{1}^{2}+7x_{2}^{2}-4x_{1}x_{2}&+5x_{1}\\\
&&&\\\ \dot{x}_{2}&=&-2x_{2}^{2}+x_{1}x_{2}+2x_{1}x_{3}-7x_{2}x_{3}&+2x_{2}\\\
&&&\\\ \dot{x}_{3}&=&-x_{2}^{2}-7x_{3}^{2}-4x_{2}x_{3}&-x_{3}\end{array}$ (44)

from which we can read out

$\displaystyle\mathbf{A}_{1}=\left[\begin{array}[]{ccc}1&-2&0\\\ -2&7&0\\\
0&0&0\end{array}\right]\qquad\mathbf{A}_{2}=\left[\begin{array}[]{ccc}0&\frac{1}{2}&1\\\
\frac{1}{2}&-2&-\frac{7}{2}\\\
1&-\frac{7}{2}&0\end{array}\right]\qquad\mathbf{A}_{3}=\left[\begin{array}[]{ccc}0&0&0\\\
0&-1&-2\\\ 0&-2&-7\end{array}\right]$ (54)

and

$\displaystyle\mathbf{V}=\left[\begin{array}[]{ccc}5&0&0\\\ 0&2&0\\\
0&0&-1\end{array}\right]\,.$ (58)

The eigenvectors of $\mathbf{V}^{T}$ are simply obtained as
$\mathbf{r}_{i}=\mathbf{e}_{i}$ with the eigenvalues of $s_{1}=5$, $s_{2}=2$
and $s_{3}=-1$. Hence, the eigenmatrices of $\mathbf{V}$ matrices are as
follows.

$\displaystyle\mathbf{P}_{11}$
$\displaystyle=\left[\begin{array}[]{ccc}1&0&0\\\ 0&0&0\\\
0&0&0\end{array}\right]\qquad$ $\displaystyle\lambda_{11}=10$
$\displaystyle\mathbf{P}_{22}$
$\displaystyle=\left[\begin{array}[]{ccc}0&0&0\\\ 0&1&0\\\
0&0&0\end{array}\right]\qquad$ $\displaystyle\lambda_{22}=4$
$\displaystyle\mathbf{P}_{33}$
$\displaystyle=\left[\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\
0&0&1\end{array}\right]\qquad$ $\displaystyle\lambda_{33}=-2$
$\displaystyle\mathbf{P}_{12}$
$\displaystyle=\left[\begin{array}[]{ccc}0&\frac{1}{2}&0\\\ \frac{1}{2}&0&0\\\
0&0&0\end{array}\right]\qquad$ $\displaystyle\lambda_{12}=7$
$\displaystyle\mathbf{P}_{13}$
$\displaystyle=\left[\begin{array}[]{ccc}0&0&\frac{1}{2}\\\ 0&0&0\\\
\frac{1}{2}&0&0\end{array}\right]\qquad$ $\displaystyle\lambda_{13}=4$
$\displaystyle\mathbf{P}_{23}$
$\displaystyle=\left[\begin{array}[]{ccc}0&0&0\\\ 0&0&\frac{1}{2}\\\
0&\frac{1}{2}&0\end{array}\right]\qquad$ $\displaystyle\lambda_{23}=1$

Note that the eigenmatrices $\mathbf{P}_{22}$ and $\mathbf{P}_{13}$ are
degenerate. Therefore, their linear combination

$\displaystyle\mathbf{P}_{4}=\mathbf{P}_{22}+b\mathbf{P}_{13}=\left[\begin{array}[]{ccc}0&0&\frac{b}{2}\\\
0&1&0\\\ \frac{b}{2}&0&0\end{array}\right]$ (62)

is also a potential $\mathbf{B}$ matrix with an arbitrary value of $b$.

We continue the algorithm with Step 2, the proportionality check.

Prop. check, Eq. (14) |  $\tilde{A}_{1}^{1}=\left[\begin{array}[]{cc}7&0\\\ 0&0\end{array}\right]$ |  $\tilde{A}_{2}^{2}=\left[\begin{array}[]{cc}0&1\\\ 1&0\end{array}\right]$ |  $\tilde{A}_{3}^{3}=\left[\begin{array}[]{cc}0&0\\\ 0&-1\end{array}\right]$
---|---|---|---
$\mathbf{B}=\mathbf{P}_{11}$ |  $\tilde{\mathbf{B}}^{1}=\left[\begin{array}[]{cc}0&0\\\ 0&0\end{array}\right]\rightarrow w_{1}=0$ |  $\tilde{\mathbf{B}}^{2}=\left[\begin{array}[]{cc}1&0\\\ 0&0\end{array}\right]\qquad{\color[rgb]{1,0,0}\times}$ |  –
$\mathbf{B}=\mathbf{P}_{33}$ |  $\tilde{\mathbf{B}}^{1}=\left[\begin{array}[]{cc}0&0\\\ 0&1\end{array}\right]\qquad{\color[rgb]{1,0,0}\times}$ |  – |  –
$\mathbf{B}=\mathbf{P}_{12}$ |  $\tilde{\mathbf{B}}^{1}=\left[\begin{array}[]{cc}0&0\\\ 0&0\end{array}\right]\rightarrow w_{1}=0$ |  $\tilde{\mathbf{B}}^{2}=\left[\begin{array}[]{cc}0&0\\\ 0&0\end{array}\right]\rightarrow w_{2}=0$ |  $\tilde{\mathbf{B}}^{3}=\left[\begin{array}[]{cc}0&\frac{1}{2}\\\ \frac{1}{2}&0\end{array}\right]\qquad{\color[rgb]{1,0,0}\times}$
$\mathbf{B}=\mathbf{P}_{23}$ |  $\tilde{\mathbf{B}}^{1}=\left[\begin{array}[]{cc}0&\frac{1}{2}\\\ \frac{1}{2}&0\end{array}\right]\qquad{\color[rgb]{1,0,0}\times}$ |  – |  –
$\mathbf{B}=\mathbf{P}_{4}$ |  $\tilde{\mathbf{B}}^{1}=\left[\begin{array}[]{cc}1&0\\\ 0&0\end{array}\right]\rightarrow w_{1}=7$ |  $\tilde{\mathbf{B}}^{2}=\left[\begin{array}[]{cc}0&\frac{b}{2}\\\ \frac{b}{2}&0\end{array}\right]\rightarrow w_{2}=\frac{2}{b}$ |  $\tilde{\mathbf{B}}^{3}=\left[\begin{array}[]{cc}0&0\\\ 0&1\end{array}\right]\rightarrow w_{3}=-1$

Note that only $\mathbf{P}_{4}$ obeys the proportionality check but no
restriction on $b$ has been obtained. Hence, in Step 3, we only investigate
$\mathbf{P}_{4}$.

$\mathbf{a}_{i}^{i}$ check, Eq. (15) |  $\tilde{a}_{1}^{1}=\left[\begin{array}[]{c}1\\\ -2\\\ 0\end{array}\right]$ |  $\tilde{a}_{2}^{2}=\left[\begin{array}[]{c}\frac{1}{2}\\\ -2\\\ -\frac{7}{2}\end{array}\right]$ |  $\tilde{a}_{3}^{3}=\left[\begin{array}[]{c}0\\\ -2\\\ -7\end{array}\right]$
---|---|---|---
$\mathbf{B}=\mathbf{P}_{4}$ |  $rhs=\left[\begin{array}[]{c}b\\\ -\frac{2}{b}\\\ 0\end{array}\right]{\color[rgb]{0,1,0}\checkmark}\quad\mbox{if $b=1$}$ |  $rhs=\left[\begin{array}[]{c}\frac{b}{2}\\\ -\frac{2}{b}\\\ -\frac{7b}{2}\end{array}\right]{\color[rgb]{0,1,0}\checkmark}\,\mbox{if $b=1$}$ |  $rhs=\left[\begin{array}[]{c}0\\\ -\frac{2}{b}\\\ -7b\end{array}\right]{\color[rgb]{0,1,0}\checkmark}\,\mbox{if $b=1$}$

It has been found that the quadratic system transformed to a linear system
with the matrix

$\displaystyle\mathbf{B}=\left[\begin{array}[]{ccc}0&0&\frac{1}{2}\\\ 0&1&0\\\
\frac{1}{2}&0&0\end{array}\right]$ (66)

and the resulting linear differential equations are given as

$\displaystyle\begin{array}[]{rcl}\dot{y}_{1}&=&y_{1}+7\\\
\dot{y}_{2}&=&-2y_{2}+2\\\ \dot{y}_{3}&=&-5y_{3}-1\end{array}$ (70)

which are exactly solvable for arbitrary initial conditions. The analytical
solution can be given based on Eq. (17).

Note that the $B$ transformation is undefined in the region where
$\mathbf{x}^{T}\mathbf{B}\mathbf{x}=x_{1}x_{3}+x_{2}^{2}=0$. This quadratic
equation determines two cone-shaped surfaces touching each other at the origin
as shown in Fig. 2.

Figure 2: The hypersurface determined by
$\mathbf{x}^{T}\mathbf{B}\mathbf{x}=0$ consists of two cone-shaped surfaces.
The hypersurface contains the two special directions determined by
$\bm{\ell}_{1}$ (blue) and $\bm{\ell}_{3}$ (red).

##  6 Conclusion

We studied the system of quadratic differential equations of the form of Eq.
(3). Although these differential equation systems cannot be solved in general,
we have presented a procedure by means of which some quadratic systems can be
solved analytically. The solution is performed through the $B$-transformation
of $\mathbf{y}=\mathbf{x}/(\mathbf{x}^{T}\mathbf{B}\mathbf{x})$ which is a
multi-dimensional generalization of the inversion $y=x^{-1}$.

In Sec. 3, we have described the algorithm which allows one to decide whether
a quadratic system can be transformed with a $B$-transformation to a linear
differential equation system. If so, the algorithm also yields the linear
system which is exactly solvable. The algorithm works for arbitrary number of
variables.

## References

  * [1] A. Iserles, A First Course in the Numerical Analysis of Differential Equations, Cambridge University Press, 1996.
  * [2] N. Champagnat, P.-E. Jabin, G. Raoul, Convergence to equilibrium in competitive Lotka–Volterra and chemostat systems, Comptes Rendus Mathematique 348 (23-24) (2010) 1267–1272.
  * [3] W. Coppel, A new class of quadratic systems, Journal of Differential Equations 92 (1991) 360–372.  doi:10.1016/0022-0396(91)90054-D.
  * [4] D. Elliott, Bilinear control systems: Matrices in action, 1st Edition, Applied Mathematical Sciences, Springer, 2009.
  * [5] A. Bácsi, C. P. Moca, B. Dóra, Dissipation-induced Luttinger liquid correlations in a one-dimensional Fermi gas, Phys. Rev. Lett. 124 (2020) 136401\.  doi:10.1103/PhysRevLett.124.136401.
  * [6] T. Date, Classification and analysis of two-dimensional real homogeneous quadratic differential equation systems, Journal of Differential Equations 32 (3) (1979) 311 – 334.  doi:10.1016/0022-0396(79)90037-8.
  * [7] D. Schlomiuk, N. Vulpe, Geometry of quadratic differential systems in the neighborhood of infinity, Journal of Differential Equations 215 (2) (2005) 357 – 400.  doi:https://doi.org/10.1016/j.jde.2004.11.001.
URL http://www.sciencedirect.com/science/article/pii/S0022039604004437

  * [8] J. Chavarriga, B. García, J. Llibre, J. S. P. [del Río], J. A. Rodríguez, Polynomial first integrals of quadratic vector fields, Journal of Differential Equations 230 (2) (2006) 393 – 421.  doi:https://doi.org/10.1016/j.jde.2006.07.022.
URL http://www.sciencedirect.com/science/article/pii/S0022039606002750

  * [9] J. Reyn, Phase portraits of planar quadratic systems, Springer, 2007.
  * [10] J. L. Ramírez, G. N. Rubiano, A geometrical construction of inverse points with respect to an ellipse, International Journal of Mathematical Education in Science and Technology 45 (8) (2014) 1254–1259.  doi:10.1080/0020739X.2014.914255.
  * [11] J. L. Ramírez, G. N. Rubiano, A generalization of the spherical inversion, International Journal of Mathematical Education in Science and Technology 48 (1) (2017) 132–149.  doi:10.1080/0020739X.2016.1201598.
  * [12] J. e Feng, J. Lam, Y. Wei, Spectral properties of sums of certain Kronecker products, Linear Algebra and its Applications 431 (9) (2009) 1691 – 1701.  doi:10.1016/j.laa.2009.06.004.