# Interpolating supersymmetric pair of Fokker-Planck equations

Choon-Lin Ho  Department of Physics, Tamkang University, Tamsui 25137, Taiwan

###### Abstract

We consider Fokker-Planck equations that interpolate a pair of
supersymmetrically related Fokker-Planck equations with constant coefficients.
Based on the interesting property of shape-invariance, various one-parameter
interpolations of the solutions of the supersymmetric pair of Fokker-Planck
systems can be directly constructed.

##  I Introduction

One of the basic tools widely employed to study the effecst of fluctuations in
macroscopic systems is the Fokker-Planck equation (FPE) R ; F . Owing to its
broad applicability, it is of great interest to obtain solutions of the FPE
for various physical situations.

However, as for any equation in science, it is generally not easy to find
analytic solutions of FPEs. In most cases, one can only solve the equation
approximately, or numerically. Nevertheless, methods of solving the FEP have
been developed in the past. These include a change of variables, eigenfunction
expansion, variational approach, perturbation expansion, Green’s function,
moment method, path integral, the continued-fraction method, etc. R ; F . Lie
symmetry methods SS  and similarity method Ho1 ; Ho2 ; Ho3  have also been
considered in solving and classifying the FPE.

Needless to say, any method to obtain exactly solvable FPE is always welcome.
Some recent work on various aspects of solution of the FPE can be found in CCR
; DM ; KSF ; LM ; Ho4 ; WH ; OK .

It is an interesting fact that the one-dimensional FPE with constant drift and
diffusion coefficients can be transformed into a corresponding Schrödinger
equation. As such, any method that solves a Schrödinger equation exactly can
be carried over to solving the FPE. An important such method is the Darboux
transformation (the time-independent version is more commonly known as the
supersymmetric (SUSY) method in physics literature) susy . This method has
been employed to enlarge the number of solvable FPEs BB ; Ros ; SMJ ; SR ; IN
; Ho5 . Recently, we have further considered FPEs which are related by Darboux
transformation through their corresponding Schrödinger equations. Under
appropriate conditions, we have studied how an exactly solvable FPE can be
obtained by the SUSY transformation from the solutions of a known FPE. These
two FPEs we shall call the SUSY FPE partners Ho5 .

In this note, we would like to present a simple way to generate exactly
solvable FPE with solutions that interpolate those of two SUSY partners of
FPEs. They form a one-parameter family of deformed FPEs between the pair of
SUSY FPEs.

##  II Fokker-Planck and Schrödinger equations

The FPE is

$\frac{\partial}{\partial
t}P(x,t;\mathbf{a}_{0})=\left[-\frac{\partial}{\partial
x}D(x;\mathbf{a}_{0})+\frac{\partial^{2}}{\partial
x^{2}}\right]\,P(x,t;\mathbf{a}_{0}).$ (1)

The function $P(x,t;\mathbf{a}_{0})$ describes the distribution of particles
in a system. $\mathbf{a}_{0}$ is a set of parameters characterizing the
system. $D(x;\mathbf{a}_{0})$ is the drift coefficient that represents the
external force acting on the particle. The constant coefficient of the second
derivative term in (1), which represents diffusion effect, is set to unity
without loss of generality. The drift coefficient can be given by a drift
potential $W(x;\mathbf{a}_{0})$ as
$D(x;\mathbf{a}_{0})=-2W^{\prime}(x;\mathbf{a}_{0})$, where the prime denotes
the derivative with respect to $x$. The function $W(x;\mathbf{a}_{0})$ is
called the prepotential because, as will be shown below, it determines the
potential of a Schrödinger equation related to the FPE. As we will encounter
various functions involving parameters other than $\mathbf{a}_{0}$, we find it
convenient to use subscript/superscript to indicate the parameters involved,
so we write $P_{0}(x,t)\equiv P(x,t;\mathbf{a}_{0})$, $D_{0}(x)\equiv
D(x;\mathbf{a}_{0})$ and $W_{0}(x)\equiv W(x;\mathbf{a}_{0})$.

By setting

$P_{0}(x,t)=e^{-W_{0}(x)}\Psi(x,t),~{}~{}\Psi(x,t)=e^{-\lambda(\mathbf{a}_{0})\,t}\phi(x;\mathbf{a}_{0}),$
(2)

we can recast (1) into

$H^{(0)}\phi=\lambda^{(0)}\phi^{(0)},$ (3)

where

$H^{(0)}\equiv-\frac{\partial^{2}}{\partial x^{2}}+W_{0}^{\prime
2}-W_{0}^{\prime\prime},$ (4)

$\lambda^{(0)}\equiv\lambda(\mathbf{a}_{0})$ and
$\phi^{(0)}(x)\equiv\phi(x;\mathbf{a}_{0})$. In these cases we use superscript
to indicate the parameter involved. The function $\phi^{(0)}$ satisfies the
stationary Schrödinger equation with the Hamiltonian $H^{(0)}$ and eigenvalue
$\lambda^{(0)}$ (for clarity of presentation, we will not indicate the
independent variables and parameters if no confusion arises). It is clear that
$W_{0}$ defines the potential of the Schrödinger equation and the zero-mode
$\phi^{(0)}_{0}\equiv\exp(-W_{0})$ : $H^{(0)}\phi^{(0)}_{0}=0$.

So every FPE (1) with constant diffusion can be associated with a Schrödinger
equation (3). As such every solution of the Schrödinger equation gives a
solution of the FPE. Suppose all the normalized eigenfunctions
$\phi^{(0)}_{n}\equiv\phi_{n}(x;\mathbf{a}_{0})$ ($n=0,1,2,\ldots$) of
$H^{(0)}$ with eigenvalues
$\lambda^{(0)}_{n}\equiv\lambda_{n}(\mathbf{a}_{0})$ ($\lambda^{(0)}_{0}=0$)
are solved, then by (2) the general solution of the FPE is

$\displaystyle P_{0}(x,t)$ $\displaystyle=$
$\displaystyle\phi^{(0)}_{0}(x)\,\Psi^{(0)}(x,t),$ (5)
$\displaystyle\Psi^{(0)}(x,t)$ $\displaystyle=$
$\displaystyle\sum_{n}c_{n}\phi^{(0)}_{n}(x)\exp(-\lambda^{(0)}_{n}\,t).$

The constant coefficients $c_{n}$ ($n=0,1,\ldots$) are determined from the
initial profile $P_{0}(x,0)=\phi^{(0)}_{0}(x)\sum_{n}c_{n}\phi^{(0)}_{n}(x)$
by

$\displaystyle
c_{n}=\int_{-\infty}^{\infty}\phi^{(0)}_{n}(x)\left(\phi^{(0)-1}_{0}(x)P_{0}(x,0)\right)dx.$
(6)

The stationary distribution as $t\to\infty$ is
$P_{0}(x)=\phi_{0}^{2}=\exp(-2W_{0})$. If $P_{0}(x,t)$ is normalized, i.e.,
$\int\,P_{0}(x,t)dx=1$, then $c_{0}=1$.

##  III Supersymmetric pair of FPEs

The connection between FPE and the Schrödinger equation allows one to generate
a Fokker-Planck system from a known one by using the supersymmetric structure
of the Schrödinger equation Ho4 . Note that $H^{(0)}$ is factorizable,
$H^{(0)}=A^{\dagger}\,A$, where $A=\partial_{x}+W_{0}^{\prime}$ and
$A^{\dagger}=-\partial_{x}+W_{0}^{\prime}$. The supersymmetric partner
(Darboux transformed) Hamiltonian is given by

$\widetilde{H}=AA^{\dagger}=-\frac{\partial^{2}}{\partial x^{2}}+W_{0}^{\prime
2}+W_{0}^{\prime\prime}.$ (7)

The eigenvalues and the normalized eigenfunctions of the corresponding
Schrödinger equation

$\widetilde{H}\tilde{\phi}_{n}=\tilde{\lambda}_{n}\tilde{\phi}_{n},~{}~{}n=0,1,2,\ldots$
(8)

are related to those of (3) by

${\tilde{\lambda}}_{n}=\lambda^{(0)}_{n+1},\ \ \ \ \
{\tilde{\phi}}_{n}=\frac{1}{\sqrt{\lambda^{(0)}_{n+1}}}A\,\phi^{(0)}_{n+1},~{}~{}~{}n=0,1,2,\ldots.$
(9)

$\widetilde{H}$ has the same spectrum as that of $H^{(0)}$ except
$\lambda^{(0)}_{0}=0$ since $A_{0}\phi_{0}=0$. The ground state of the SUSY
partner has the same energy $\lambda^{(0)}_{1}$ as the first excited state of
the original system.

The SUSY Hamiltonian $\widetilde{H}$ does not allows us to associate it with a
FPE because of the “plus” sign between the two prepotential terms in (7).
However, this is made possible if the potentials in the two Hamiltonians are
shape-invariant. Shape invariance means that the two potentials are similar in
shape and differ only in the parameters appearing in them susy . It turns out
to be a sufficient condition for exact-solvability, and it is amazing that
most of the known one-dimensional exactly solvable quantal systems possess
this property. Mathematically shape invariance means the condition

$W^{\prime}(x;\mathbf{a}_{0})^{2}+W^{\prime\prime}(x;\mathbf{a}_{0})=W^{\prime}(x;\mathbf{a}_{1})^{2}-W^{\prime\prime}(x;\mathbf{a}_{1})+R(\mathbf{a}_{0}).$
(10)

Here $\mathbf{a}_{1}$ is a function of $\mathbf{a}_{0}$ and
$R(\mathbf{a}_{0})$ is an $x$-independent function of $\mathbf{a}_{0}$. It
turns out that for all the well-known SUSY one-dimensional solvable
potentials, $\mathbf{a}_{1}$ differs from $\mathbf{a}_{0}$ only by constant
shifts, i.e., $\mathbf{a}_{1}=\mathbf{a}_{0}+\boldsymbol{\delta}$. The
condition (10) then gives the eigenvalues as susy

$\lambda_{n}^{(0)}=\lambda_{n}(\mathbf{a}_{0})=\sum_{k=0}^{n-1}R(\mathbf{a}_{0}+k\boldsymbol{\delta}).$
(11)

The presence of shape invariance permits us to associate a FPE with the
Hamiltonian

$H^{(1)}\equiv W_{1}^{\prime 2}-W_{1}^{\prime\prime},$ (12)

where $W_{1}(x)\equiv W_{0}(x;\mathbf{a}_{1})$. It is obvious that the
eigenvalues $\lambda_{n}^{(1)}$ and the normalized eigenfunctions
$\phi_{n}^{(1)}(x)$ of $H^{(1)}$ are given by

$\lambda^{(1)}_{n}=\lambda_{n}(\mathbf{a}_{1}),\ \ \
\phi_{n}^{(1)}=\frac{\alpha}{\sqrt{\lambda^{(0)}_{n+1}}}A\,\phi^{(0)}_{n+1},~{}~{}~{}n=0,1,2,\ldots,~{}~{}\alpha=\pm
1.$ (13)

The first relation in (13) is obtained as follows. From (9) and (11) we have

$\displaystyle\lambda^{(1)}_{n}$ $\displaystyle=$
$\displaystyle\tilde{\lambda}_{n}-R(\mathbf{a}_{0})$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{n}R(\mathbf{a}_{0}+k\boldsymbol{\delta})$
$\displaystyle=$
$\displaystyle\sum_{k=0}^{n-1}R(\mathbf{a}_{1}+k\boldsymbol{\delta})=\lambda_{n}(\mathbf{a}_{1}).$

The FPE corresponding to $H^{(1)}$ has drift coefficient
$D^{(1)}(x)=-2W_{1}^{\prime}$. The two FPEs with drift coefficients related by
$\mathbf{a}_{0}$ and $\mathbf{a}_{1}$ are called the supersymmetric pair FPEs.
The phase $\alpha=\pm 1$ depends on the SUSY systems.

The general solution of the partner FPE is

$\displaystyle P_{1}(x,t)$ $\displaystyle=$ $\displaystyle
e^{-W_{1}}\,\Psi^{(1)}(x,t),$ (15) $\displaystyle\Psi^{(1)}(x,t)$
$\displaystyle=$
$\displaystyle\sum_{n}d_{n}\phi^{(1)}_{n}\exp(-\lambda^{(1)}_{n}t).$

But in order for $P_{1}(x,t)$ to be SUSY-related to the solution $P_{0}(x,t)$
(5) of the original FPE, we require $\Psi^{(1)}(x,t)\sim A\Psi^{(0)}(x,t)$,
i.e., we want $\Psi^{(1)}(x,t)$ to be the SUSY transform of $\Psi^{(0)}(x,t)$.
Now we have

$\displaystyle A\Psi^{(0)}(x,t)$ $\displaystyle=$ $\displaystyle
A_{0}\left(\phi^{(0)-1}_{0}P_{0}(x,t)\right)=\sum_{n=0}^{\infty}\,c_{n}\left(A\phi^{(0)}_{n}\right)\exp(-\lambda^{(0)}_{n}t)$
(16) $\displaystyle=$ $\displaystyle
e^{-R(\mathbf{a}_{0})t}\sum_{n=0}^{\infty}\,c_{n+1}\alpha\sqrt{\lambda^{(0)}_{n+1}}\phi^{(1)}_{n}\exp(-\lambda^{(1)}_{n}t).$

Hence we can take

$\displaystyle d_{n}$ $\displaystyle=$
$\displaystyle\alpha\sqrt{\lambda^{(0)}_{n+1}}c_{n+1},$ (17)
$\displaystyle\Psi^{(1)}(x,t)$ $\displaystyle=$ $\displaystyle
e^{R(\mathbf{a}_{0})t}A\Psi^{(0)}(x,t).$

This gives the SUSY partner solution of $P_{0}(x,t)$

$\displaystyle P_{1}(x,t)$ $\displaystyle=$ $\displaystyle
e^{-(W_{1}-R(\mathbf{a}_{0})t)}\left(\partial_{x}+W_{0}^{\prime}\right)\left(e^{W_{0}}\,P_{0}(x,t)\right)$
(18) $\displaystyle=$
$\displaystyle\phi^{(1)}_{0}\sum_{n=0}^{\infty}\,\alpha\sqrt{\lambda^{(0)}_{n+1}}c_{n+1}\phi^{(1)}_{n}\exp(-\lambda^{(1)}_{n}t).$

##  IV Interpolating SUSY pair of FPEs

Using the above relation between a pair of SUSY FPEs with shape invariance, we
can generate a one-parameter family of FPEs that interpolates between the two
FPE SUSY pair.

Take a parameter $s\in[0,1]$. Suppose we can construct a set of $s$-dependent
parameters $\mathbf{a}_{s}$ such that

$\mathbf{a}_{s}=\mathbf{a}_{0}{\rm\
for}~{}~{}s=0;~{}~{}~{}\mathbf{a}_{s}=\mathbf{a}_{1}{\rm\ for}~{}~{}s=1.$ (19)

Then we can associate with each $s$ a prepotential $W_{s}(x)\equiv
W(x;\mathbf{a}_{s})$ so that $W_{s}(x)=W_{0}(x)$ for $s=0$, and
$W_{s}(x)=W_{1}(x)$ for $s=1$. The FPE interpolating the SUSY pair of FPEs is
defined by the drift coefficient $D_{s}(x)=-2W_{s}^{\prime}(x)$. Let us call
this FPE the $s$-defomed FPE of the SUSY pair of FPEs.

Now comes a simple observation that allows us to construct solutions of the
$s$-deformed FPE that interpolates those of the two SUSY-related FPEs defined
by $W_{0}$ and $W_{1}$. It is clear that the Schrödinger equation associated
with the $s$-deformed FPE has eigenfunctions
$\phi^{(s)}_{n}(x)\equiv\phi_{n}(x;\mathbf{a}_{s})$ with eigenvalues
$\lambda_{n}^{(s)}\equiv\lambda_{n}(\mathbf{a}_{s})$. Then by (III), we have

$\displaystyle\phi^{(s)}_{n}(x$ $\displaystyle)$
$\displaystyle=\phi^{(0)}_{n}(x),~{}\lambda^{(s)}_{n}(x)=\lambda^{(0)}_{n}(x)~{}~{}{\rm
when}~{}~{}s=0,$ (20) $\displaystyle\phi^{(s)}_{n}(x$ $\displaystyle)$
$\displaystyle=\phi^{(1)}_{n}(x),~{}\lambda^{(s)}_{n}(x)=\lambda^{(1)}_{n}(x)~{}~{}{\rm
when}~{}~{}s=1.$

The normalized solution interpolating the solutions of the SUSY pair is then
given by

$P_{s}(x,t)=N\phi^{(s)}_{0}\sum_{n=0}^{\infty}\,\left[(1-s)c_{n}+s\alpha\sqrt{\lambda^{(0)}_{n+1}}c_{n+1}\right]\phi^{(s)}_{n}\exp(-\lambda^{(s)}_{n}t),$
(21)

where the noramlizing constant is

$N=\left[(1-s)c_{0}+s\alpha\sqrt{\lambda^{(0)}_{n+1}}c_{1}\right]^{-1}.$ (22)

The above construction is valid for any $\mathbf{a}_{s}$ as a function of $s$
satisfying (19). There could be many ways to construct such $\mathbf{a}_{s}$.
For illustration purpose, in the next section, we present two examples with
the simplest choice of $\mathbf{a}_{s}$. As mentioned before, for all the
well-known one-dimensional exactly solvable shape-invariant quantum systems
possessing, the relevant parameters of the two SUSY partner potentials are
simply related by constant shifts, i.e.
$\mathbf{a}_{1}=\mathbf{a}_{0}+\boldsymbol{\delta}$ where
$\boldsymbol{\delta}$ is a set of constants. Thus the simplest choice of
$\mathbf{a}_{s}$ is

$\mathbf{a}_{s}=\mathbf{a}_{0}+s\boldsymbol{\delta}.$ (23)

As shown in [21], this choice appears naturally in most of the one-dimensional
solvable quantum models, if one interpolates the SUSY pair of the Hamiltonians
by their transformed versions, i.e.,

$(1-s)\left[e^{W_{0}}H_{0}e^{-W_{0}}\right]+s\left[e^{W_{1}}H_{1}e^{-W_{1}}\right].$
(24)

This requirement, which involves the Hamiltonians $H_{0}$ and $H_{1}$, is more
restrictive than (19), which only involves the parameters $\mathbf{a}_{0}$ and
$\mathbf{a}_{1}$. The construction presented here is valid for general
$\mathbf{a}_{2}$.

##  V Examples

As illustration, we construct the one-parameter deformed families of two
examples of SUSY FPE pairs, with the dirft potential related to the radial
oscillator and the Morse potential.

### The radial oscillator

The prepotential for the radial oscillator potential is
$W_{0}(x;\mathbf{a}_{0})=\omega x^{2}/4-(\ell+1)\ln x$, where $x\in[0,\infty)$
and $\omega,\ell>0$. In this case $\mathbf{a}_{0}=(\omega,\ell)$ and
$\boldsymbol{\delta}=(0,1)$, i.e., $\mathbf{a}_{1}=(\omega,\ell+1)$ susy . As
only the parameter $\ell$ is shifted in the SUSY transformation, we shall use
its values to label the two SUSY pair.

The wavefunctions are

$\phi^{(\ell)}_{n}(x)\equiv\phi_{n}(x;\mathbf{a}_{0})=N_{n\ell}y^{\frac{\ell+1}{2}}e^{-\frac{y}{2}}L_{n}^{\ell+\frac{1}{2}}(y),~{}~{}y\equiv\frac{1}{2}\omega
x^{2},$ (25)

The normalization constant is

$N_{n\ell}=(2\omega)^{\frac{1}{4}}\left[\left(\begin{array}[]{c}n+\ell+\frac{1}{2}\\\
n\end{array}\right)\Gamma(\ell+\frac{3}{2})\right]^{-1},$ (26)

where the two factors in the square-bracket are the binomial coefficient and
the Gamma function, respectively. Eigenvalues are given by
$\lambda_{n}^{(0)}=2n\omega,~{}~{}n=0,1,2,\ldots$.

It is easy to check that

$A\phi^{(\ell)}_{n+1}(x)=-\sqrt{\lambda^{(\ell)}_{n+1}}\phi^{(\ell+1)}_{n}(x),$
(27)

so the factor $\alpha$ in (13) is $\alpha=-1$ in this case.

The choice (23) leads to the interpolating prepotential
$W_{s}(x)=W(x;\mathbf{a}_{s})$ with $\mathbf{a}_{s}=(\omega,\ell+s)$.

Fig. 1 gives the plots of the radial oscillator drift potential $W_{s}(x)$ and
several normalized $P_{s}(x,t)$ for different $s$ with $\omega=\ell=1$. For
the graphs of $P_{s}(x,t)$, we have scaled down the prepotential $W_{0}(x)$ 3
times for a better visual comparison with $P_{s}(x,t)$. The coefficients
$c_{n}$ used in (21) are $c_{0}=5,c_{1}=c_{2}=1$ and $c_{n}=0(n\geq 3)$.

### The Morse potential

In this case the drift potential is $W_{0}(x;\mathbf{a}_{0})=\alpha x+\beta
e^{-x},~{}~{}\alpha,\beta>0,~{}~{}\mathbf{a}_{0}=(\alpha,\beta)$. Under SUSY
transformation, $\mathbf{a}_{1}=(\alpha-1,\beta)$ susy . Similar to the
precious example, here we shall use the only changing parameter $\alpha$ to
label the two SUSY pair.

Wavefunctions are given by:

$\phi^{(\alpha)}_{n}(x)\equiv\phi_{n}(x;\mathbf{a}_{0})=N_{n\alpha}y^{\alpha-n}e^{-\frac{y}{2}}L_{n}^{2(\alpha-n)}(y),~{}~{}y\equiv
2\beta e^{-x},$ (28)

with normalization constant

$N_{n\alpha}=\left[\frac{2(\alpha-n)\Gamma(n+1)}{\Gamma(2\alpha-n+1)}\right]^{\frac{1}{2}}.$
(29)

and the corresponding eigenvalues
$\lambda_{n}^{(0)}=\alpha^{2}-(\alpha-n)^{2},~{}~{}n=0,1,2,\ldots,{\rm\
largest\ integer}<\alpha$. It is easy to check that the factor $\alpha$ in
(13) is $\alpha=1$ in this case. The choice (23) gives
$\mathbf{a}_{s}=(\alpha-s,\beta)$.

In Fig. 2 we plot the Morse drift potential $W_{s}(x)$ and $P_{s}(x,t)$ for
$\alpha=5,\beta=1$, $c_{0}=3,c_{1}=2,c_{2}=1$ and $c_{n}=0(n\geq 3)$. Again
for easy visual comparison, we have plotted $(W_{0})+3)/4$ instead of $W_{0}$
in the graphs of $P_{s}(x,t)$. .

From both Fig. 1 and 2, it is seen that the function $P_{s}(x,t)$ is confined
within the potential well $W_{s}(x)$. For small times, $P_{s}(x,t)$ could look
very different for different values of $s$. But for large times, they become
quite similar. This is expected as $t\to\infty$, only the $n=0$ term in
$P_{s}(x,t)$ remains as $\lambda_{0}^{(s)}=0$, i.e.,
$P_{s}(x,t)\sim\phi_{0}^{(s)2}$, which are similar in shape.

##  VI Summary

We have presented a simple construction of one-parameter deformed FPEs with
solutions that interpolate a pair of SUSY FPEs. The construction is very
general: it only requires the condition of shape-invariance, which amazingly
is satisfied by most of the well-known exactly solvable one-dimensional
quantum systems, and the interpolating condition of the parameters related by
the shape-invariance, namely, Eq. (19).

We have demonstrated the procedure by two examples using the simplest
interpolation rule, the linear shift (23). Other deformations are possible, as
long as the rule (19) is satisfied. For instance, following [21], one can
consider simply interpolating the two SUSYHamiltonians,

$(1-s)H_{0}+sH_{1}$ (30)

instead of the gauge-transformed Hamiltonians as in (24). In this case the
parameters $\mathbf{a}_{s}$ are mostly nonlinear in $s$ OPS . In this case the
parameters $\mathbf{a}_{s}$ are mostly nonlinear in $s$ OPS . For example, in
the radial oscillator case, the parameter $\ell$ is deformed to
$[\sqrt{4(\ell+1)(\ell+2s)}-1]/2$. Nonetheless, our construction is
applicable.

## Acknowledgments

The work is supported in part by the Ministry of Science and Technology (MoST)
of the Republic of China under Grant NSTC 112-2112-M-032-007.

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Figure 1: The first plot presents graphs of $W_{s}(x,t)$ of the radial
oscillator. The next three plots show the scaled radial oscillator drift
potential $W_{0}(x)/3$ (large dashed line) and the normalized $P_{s}(x,t)$
versus $x$ with $\omega=\ell=1$, $c_{0}=5,c_{1}=c_{2}=1$ and $c_{n}=0(n\geq
3)$, time $t=0.1$, $0.03$, $0.5$, $1.0$. The deformed parameter $s$ are $s=0$
(solid), $0.3$ (dashed), $0,7$ (dotdashed), and $1.0$ (dotted).

Figure 2: The first plot presents the graphs of $W_{s}(x,t)$ of the Morse
potential. The next three plots show the scaled Morse drift potential
$(W_{0}(x)+3)/4$ (large dashed line) and $P_{s}(x,t)$ versus $x$ for
$\alpha=5,\beta=1$, $c_{0}=3,c_{1}=2,c_{2}=1$ and $c_{n}=0(n\geq 3)$ and time
$t=0.01$, $0.05$, $1.0$, $5.0$. The deformed parameter $s$ are $s=0$ (solid),
$0.3$ (dashed), $0,7$ (dotdashed), and $1.0$ (dotted).