CC BY
10Russian Journal of Nonlinear Dynamics, 2022, vol. 18, no. 2, pp. 309-328. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd220211
MATHEMATICAL PROBLEMS OF NONLINEARITY
MSC 2010: 34A08, 35A22, 26A33
Homotopy Analysis Method and Time-fractional NLSE with Double Cosine, Morse, and New Hyperbolic
Potential Traps
U.Ghosh, T. Das, S. Sarkar
A brief outline of the derivation of the time-fractional nonlinear Schrödinger equation (NLSE) is furnished. The homotopy analysis method (HAM) is applied to study time-fractional NLSE with three separate trapping potential models that we believe have not been investigated yet. The first potential is a double cosine potential [V(x) = V1 cos x + V2 cos2x], the second one is the Morse potential [V(x) = V1 e-2ßx + V2e-ßx], and a hyperbolic potential [V(x) = = V0 tanh(x) sech(x)] is taken as the third model. The fractional derivatives and integrals are described in the Caputo and Riemann Liouville sense, respectively. The solutions are given in the form of convergent series with easily computable components. A physical analysis with graphical representations explicitly reveals that HAM is effective and convenient for solving nonlinear differential equations of fractional order.
Keywords: time fractional nonlinear Schrödinger equation (NLSE), homotopy analysis method (HAM), Caputo derivative, Riemann - Liouville fractional integral operator, trapping potential
1. Introduction
Nature and almost its all phenomena follow nonlinear rules [1]. That means from the outset the equation that describes them is simply nonlinear [2]. In general, nonlinear differential
Received January 27, 2021 Accepted February 14, 2022
Uttam Ghosh [email protected] Susmita Sarkar [email protected]
University of Calcutta, Department of Applied Mathematics 92 A. P. C. Road, Kolkata, India
Tapas Das
Kodalia Prasanna Banga High School (H.S.) South 24 Parganas 700146, India
equations are very hard to tackle. This is the only reason why we choose to study any physical system modeled by the linear differential equations. To give an example, the famous undamped pendulum equation is nonlinear, but it is studied in linear form imposing certain approximations and restrictions.
The theory of linear differential equation has been studied extensively for the past 200 years and is a fairly complete and well-rounded body of knowledge [3]. That's why researchers are aiming their focus on different nonlinear differential equations. Finding the exact or approximate solutions is the top priority for many nonlinear problems. A great deal of effort has also been made to find robust and stable numerical and analytical methods for solving nonlinear problems. The renowned techniques include the homotopy perturbation method, the variational and topological method, the generalized quasi-linearization concept, the multiscale finite element method, and the analog equation method. For more on this the reader is referred to [4-12].
Along with this development, the theory of derivatives of fractional (noninteger) orders stimulates a considerable amount of interest in almost all areas of applied science and engineering [13, 14]. Many research outcomes suggest that to know a physical system from the deep, fractional mode of the differential equation is a better choice than the usual integer-order differential equation [15]. Fractional derivatives and integrals have a nonlocal point property and it is an excellent tool for the description of memory and hereditary properties of various materials and processes. Maybe this is the reason why there are several different or somehow interrelated fractional derivative definitions and fractional integral rules. This paper is concerned with Caputo's fractional derivatives and Riemann-Liouville integral rules.
So the recent trend of studies, that is, the nonlinearity of a problem and a fractional differential equation has created a new subject, generally called fractional nonlinear differential (or partial) equations (FNLDEs or FNLPDEs) [16-31]. This subject is attracting many researchers, and one can easily find numerous papers that affirm the fact that interest is increasing at an exponential rate. Still, there is an uncountable number of problems waiting for proper investigation. This is the prime responsible factor that has motivated us to arrange this present study. In this paper, we choose the time-fractional nonlinear Schrodinger equation (fNLSE) for three unexplored trapping potentials, we believe, which will surely mark an impact on practical applications.
The nonlinear Schrodinger equation (NLSE) appears as a model representing the propagation of optical pulses in the single-mode fibers [32]. In a normal study of undergraduate optics, we generally assume that, when a light beam propagates through a material, the properties of the material are not affected by the light itself. However, if the intensity of the light is large enough (in the case of laser), the properties of the medium (such as refractive index, etc.) are affected and the study of the propagation of a light beam becomes more complicated with nonlinearity as the principle of superposition does not remain valid. This is the domain of nonlinear optics where many new effects are observed like self-focusing and second harmonic generation.
Various versions of NLSE have been studied in other subjects like semiconductor physics, plasma physics, Bose-Einstein condensation, solid mechanics to name a few [33, 34]. In our case, we have chosen three separate trapping potentials viz double cosine potential, Morse potential, and a hyperbolic potential for investigation. The entire study is based on the analytical tool homotopy analysis method (HAM).
The paper is organized as follows: Section 2 will start with basic definitions, notations, and properties of fractional calculus. Section 3 is concerned with a brief theoretical approach to time-fractional NLSE. Then the homotopy analysis method (HAM) is outlined in Section 4. Section 5 discusses the implementation of the method in actual problems. Results and discussions are presented in Section 6 and, finally, the conclusion comes in Section 7.
2. Preliminaries of fractional calculus
2.1. The Caputo derivative
The Caputo derivative [35] of fractional order a > 0 of the function f (t), with t > 0, is
realized as %D"f(t) = ^f (t) and explicitly defined as
f(n) (t )
dr, n — 1 < a < n E N,
CD?f (t)=Pn - a) J (t - r)a+1- ' ' (2.1)
dn
—f(t), a = ne N.
dtn
The main advantage of the Caputo derivative is that the initial conditions for fractional differential equations with the Caputo derivatives take the same form as for integer-order differential equations, i.e., they contain the limit values of integer-order derivatives of unknown functions at the lower terminal t = a. In this paper "fractional derivative" always refers to the Caputo derivative and Df(-) will be used as its short notation. Keeping all the definition intact as it is, we declare D?{■) = §,{■) ^ §,{■).
2.2. The Riemann —Liouville derivative
The left and right Riemann-Liouville fractional derivatives are defined as
t
1 /dx n
1 f m —-
J
t
(2.2) (2.3)
where n E N, n — 1 ^ a < n.
If a acts as an integer, then aD?f{t) = f(t), tD«f(t) = f(t).
2.3. Fractional integral
The Riemann-Liouville fractional integral operator [36] of order a > 0, of a function f (t) E E C^, n > —1, is defined as
f t 1
J(t — r)a-1dr, t > 0,
aD-a f (t) = Jaf (t) = { r(a) J ' (2.4)
J
. J0f (t) = f (t).
Many properties of Ja can be found in [37, 38]. We mention only those which will be used in this paper:
jjf f (t) = Ja+f f (t), jajf f (t) = Jf Jaf (t),
JtatJ = + Ja+1, where a, f3 > 0 and 7 > -1. t r(1+ y + a)
t
1
a
Two more important properties of fractional calculus are useful here:
C Da J?f (t) = f (t), (2.5)
n-1 tk
JtaDfm = f(t) - ' where n-l<a<ne N. (2.6)
k=0 '
3. A brief theoretical approach to time-fractional NLSE
In this section, we will try to frame the time-fractional nonlinear Schrödinger equation. Before the development of the fractional nonlinear Schrödinger equation, the fractional Schrödinger equation was discovered by N.Laskin [39]. He argued and showed that the space-fractional Schrödinger equation follows all the quantum mechanics rules crafted on the Levy-path. On the other hand, the time-fractional Schrödinger equation violates most of the quantum mechanics rules. Like the linear Schrödinger equation, the nonlinear Schrödinger equation is a field equation. Since we are not dealing with quantum mechanics, taking a time-fractional nonlinear Schrödinger equation has no burden at all. Time-fractional field equations like time-fractional NLSE may help to study the time delay effect, the roughness of a signal with respect to time. In general, At refers infinitesimal change in time and ordinary differentiation is realizable as only for At ^ 0. But if we take (At)a, where 0 < a < 1, as an element, then (At)a > At. So taking a fractional time will provide an extra time frame to study the homogeneity and roughness of a system.
The derivation of the time-fractional nonlinear Schrödinger equation is not an easy task and it is not unique at all. Since it is a field equation it should be extracted from the fractional Euler - Lagrange equation through a suitable Lagrangian. Suitable does not mean that it is physical, as a Lagrangian is always a nonphysical entity. We take the fractional Euler - Lagrange equation as
I
Here q and L are the generalized coordinate and the Lagrangian function, respectively. The above equation is free from constraints. To know more about this form of the fractional Euler -Lagrange equation, the reader should refer to [40, 41]. We assume a suitable Lagrangian (but it may not be physical) in the form
L = \vitDtv + aD?v) + vuxx + ClV(x)vu + c2(uv)m, (3.2)
where v is the complex conjugate of u and c1, c2 are constants. V(x) acts as a potential function. Now setting q = v, we have
dL
— = C-iV(x)u + c2in(uv)rn~1u + uxx, dL \ ( dL \ i
u,
dtDtv) \daDfv J 2
Inserting these in Eq. (3.1), we have
iDau + uxx + c1V (x)u + c2m(uv)m-1 u = 0. (3.3)
We can set Dfu = + aD")u [42]. Now m is a free constant. Taking m = 3, Eq. (3.3) can
be written as
iDau + uxx + c1V (x)u + 3c2\u\2 u = 0, (3.4)
which is the time-fractional nonlinear Schrodinger equation that will be taken in Section 5. It should be mentioned here that this type of model equation has been used earlier [43, 44], but we have provided a basic theoretical proof before using it.
The nonlinear Schrodinger equation (NLSE), also known as the Gross-Pitaevskii equation, deals with a many-particle quantum system. The NLSE with a potential function has practical application when magnetic potential traps are used to trap neutral particles with magnetic moments. Bose-Einstein condensate (BEC) via the nonlinear Schrodinger equation is one of such topics. In the case of optical fiber, a high-power laser beam can also change the refractive index of a medium, and as a result, the light beam can face focusing and defocusing phenomena. In such a situation, the external perturbation can play an important role in the study [45]. Basically, NLSE provides a soliton solution when dispersion and nonlinear effect balance each other. The classical form
= --^xx^ +
gives bright soliton (focusing) when k < 0 and for k > 0 it delivers dark soliton (defocusing). This equation is oversimplified and can't deal with the external perturbation. In the literature, it is a common practice to add a term like ±V(x)^ (potential trap) on the right side to deal with the perturbed condition. Equation (3.4) is a perfect example of the situation. Looking at (3.4), we can say that, to have a perfect soliton, c1 V(x)u + 3c2\u\2u must balance the dispersive term uxx. Several earlier studies had been done on the topic [46] to check the fact how initial pulse acts on the perturbed or trapped condition [47]. In our work we are taking a few model potentials that have not been investigated yet, to study the pulse behavior while going through the trapped system. The constants c1 and c2 are also important to make the effective nonlinear term c1V(x)u + 3c2\u\2u balance or off-balance with the dispersive term. Our model, as well as the technique of solution, can be used for other potentials also and we believe that is the merit of our study.
4. Homotopy analysis method (HAM)-An outline
The HAM [48] was introduced by Liao in 1992. To discuss the basic concept we start with the following nonlinear fractional differential equation:
NopDau(x, t)] = 0, (4.1)
where Nop is a nonlinear operator, u(x, t) is an unknown function to be determined and x, t denote independent variables. By means of the HAM one first constructs the zero-order deformation equation
(1 — p)C[j>(x, t, p) — u0(x, t)] = H(x, t)Nop[Dau(x, t)], (4.2)
where p E [0, 1] is the embedding parameter, h = 0 is an auxiliary parameter, H(x, t) denotes a nonzero auxiliary function, u0(x, t) is an initial guess, and L is an auxiliary linear operator which has the property L(C) = 0, where C is a constant. Clearly, when p = 0 and p = 1, the following holds:
0(x, t, 0) = u0(x, t), 0(x, t, 1) = u(x, t). (4.3)
As p increases from 0 to 1, the solution 0(x, t, p) varies from the initial guess u0(x, t) to the actual solution u(x, t).
Liao expanded 0(x, t, p) in Taylor series with respect to the embedding parameter p as follows:
0(x, t, p) = Uo(x, t) + ^ um(x, t)pm,
m=1
where
um(x, t)
1 dmfi(x, t, p)
m!
dpn
(4.4)
(4.5)
p=0
Assume that the auxiliary linear operator L, the initial guess u0(x, t), the auxiliary parameter h, and the auxiliary function H(x,t) are selected such that the series (4.4) is convergent at p = 1, then
0(x, t, 1) = u(x, t) = u0(x, t) + ^ um(x, t),
(4.6)
m=1
which must be one of the solutions of the original nonlinear fractional differential equation given by Eq. (4.1).
Let us define the vector
um = {uo(x, t), ui(x, t), u2(x, t), ..., um(x, t)}. (4.7)
Differentiating (4.2) m times with respect to p, then setting p = 0 and finally dividing by m!, we have the mth-order deformation equation
C\um(x, fy Xmum-1 ] = (x, tyRm(um— 1),
where
and
Rm(um_ i)
1
d
m-1
(m — 1)! I dp
Xm
■N
m-1 op
J2Um(x, t)pm
m=0
p=0
0, m < 1,
1, m > 1.
(4.8)
(4.9)
(4.10)
Note that the higher-order deformation equation (4.8) is governed by the same linear operator L. The term Rm(um-l) can be expressed simply by u0(x, t), ul(x, t), u2(x, t), ..., um(x, t). It should be remembered that the Caputo differentiation is a linear operation and in every higherorder deformation, we have to do Caputo fractional differentiation. So we can solve the higherorder deformation equations one after the other and reach the solution manually at least up to a few terms of Eq. (4.6). Alternatively one can use mathematical computing software like Matlab or Mathematica or Maple to do the job.
It is to be mentioned here that, as the nonlinear problems are harder to solve than linear problems, there are a few limited methods to tackle them [49]. One parallel method like the homotopy analysis method (HAM) is the homotopy perturbation method (HPM) [50]. The HAM is more powerful and effective than the HPM. The reasons are as follows:
• The convergence of the solution for HAM depends on the four factors (I) Initial guess of the solution (II) Auxiliary operators (III) Auxiliary function H(x, t) and (IV) Auxiliary parameter h.
• In HPM the entire solution depends on (I) and (II) only.
That's why HPM can provide a convergence solution if the initial guess is good, but no such restriction is there for HAM. Even if someone does not choose a good initial guess, HAM delivers a fairly good convergent solution depending on the analysis of the parameter h. It makes HPM a special case of HAM [51]. On the other hand, Adomian's decomposition method (ADM) is similar to HPM [52]. In that sense, the chosen method of this paper is a robust technique to tackle nonlinear fractional differential equations.
5. Implementation of HAM in selected problems
In this study, we consider the following time-fractional NLS equation:
iDfu + Au + F{\uf, u) = 0 (0 < a < 1) and i = V^l. (5.1)
Originally this equation with a = 1 appears in nonlinear optical fibers, planar wave guides, and in the case of Bose -Einstein condensates confined to highly anisotropic cigar-shaped traps. The function F(\u\2, u) is a real function. The fractional model equation, i.e., (5.1) has been taken because it provides a deeper insight into the nonlinear effect concerning the time scale or, more precisely, the fractional time scale. Here u(x, t) is a propagating wave function and in general Au = uxx + uyy + uzz.
There are various types of choices available for F(\u\2, u), but in the case of our study we will concentrate on the form related to Kerr type nonlinearity, i.e.,
F(\u\2, u) = a\u\2u + bV(x)u. (5.2)
The term V(x) is called the trapping potential, and a, b are simple constants. The one-dimensional situation requires Au = uxx. Therefore, the main equation takes the form
iDau + uxx + a\u\2 u + bV (x)u = 0. (5.3)
This equation has been derived in Section 3 where c1 = b and 3c2 = a.
By means of HAM, we choose the linear operator L and the nonlinear operator Nop that act as
da
m = (5.4)
Nop = [D— i^xx — iaW2^ — ibV (x)0], (5.5)
with the property L(C) = 0, where C is constant and 0 = 0(x, t, p). The zeroth order deformation equation reads
(1 — p)L[0(x, t, p) — u0(x, t)] = H(x, t)Nop[0(x, t, p)]. (5.6)
It is clear that for p = 0 and p = 1 equation (5.6) provides
0(x, t, 0) = u0(x, t), 0(x, t, 1) = u(x, t). (5.7)
Therefore, the mth-order deformation equation remains the same as given by Eq. (4.8) where
Rm(um-1) = Dtum-1 — i(um-1 )xx — iaRNh\m — ibV (x)um-1, (5.8)
where the nonlinear third term is
RNL,\m =
d
\m— 1
(m - 1)! dpm-1
m— 1
UkPk
k=0
m1
J2Uk Pk
k=0
(5.9)
p=0
Now, depending on the trapping potential V(x) and the initial choice of u(x, 0) = u0 = f (x), one can study numerous problems. To make this paper self-contained, we wish to construct a formula-based mathematics to tackle different problems of the same type. The initial condition f (x) may be complex, so we will use f(x) to denote the complex conjugate of f(x). Let us derive a few specific terms in detail. In the case of m = 1 Eq. (5.9) can be written as
I go 0 2 0 2
rnl\i = TnrK'oP I uoP = \uo\ uo = uquQuQ.
0! dp0
For m = 2 we have
1 d
RNL\2 =
(2 - 1)! dp
J2Uk Pk
k=0
Ukpk
k=0
p=0
d_
dp
d
\u0 + u1 p\2 (u0 + u1p)
p=0
= [k + uip)(u0 + UiP)(Uo + UlP)}
p=0
Using the partial derivative product rule, it is easy to obtain
rnl\-2 = [2m0«o«i +uluo\-
In the same way,
R-nl I3 = [uo('2uoU2 +u1('2Tqu0
Now there is a great freedom to choose the auxiliary linear operator L, the nonzero auxiliary parameter h, and the auxiliary function H(x, t). We can assume that all of them are properly chosen so that the solution 0(x, t, p) exists for 0 ^ p ^ 1. Under the condition H(x, t) = 1 from Eq. (4.8) we have
um = Xmum-1 + hJt Rm(um-1), (5.10)
where it is clear that J f L-1.
Selecting u0 = f (x) and using (4.10), (5.8) and the rules of Section 2, we have
u1 = X1u0 + hja R1(u0) = hjaR1(u0) = hja iDtu0 - i(u0 )xx - iaRNL \ 1 - ibV (x)u0]
= hJ?
d2
-''■g-lfW ~ ibV(x)f(x) - ia\f(x)\2f(x)
= ihg(x)
t°
r(1 + a)'
(5.11)
In the above derivation we have used the rule Df u0 = 0 in the Caputo sense as u0 is independent of t and
g(x) = fxx (x) + bV (x)f (x) + a\f (x)\2 f (x), (5.12)
where g(x) is the complex conjugate g(x). Similarly, using (4.10), (5.8) and (2.6), it is easy to derive
u2 = X1u1 + hJf R^u), (5.13)
1
2
or
u2 = ui + hJf [D?ui — )xx — ibV(x)ul — iaRNL\2]
= —ih(1 + h)g(x)
where
±2a
r(1 + a)
— h2 gi(x)
r(1 + 2a):
(5.14)
0*0 = + '2af{x,)f{x)g{x) - g{x)f'2{x). (5.15)
We have used the rule (2.6), which gives the fact Jf [Dfu1 ] = u1. The other symbols are used for
o o 2
the meaning (-)T = and (-)xx = ^r(-)- Finally, the approximate solution is realized from
the equation u = ^ ue for a few terms. In the next subsections, we will address time-fractional e=o
NLSE for three separate trapping potentials, namely, double cosine, Morse, and a hyperbolic type. To the best of our knowledge, these have not been investigated to date.
5.1. Double cosine potential
Double Cosine potential
h-curve for double Cosine potential solution
Real paxt of u(x, t) Imaginary part of u(x, t) Absolute value of u(x, t)
(b)
Fig. 1. (a) Nature of the double cosine potential curve from Eq. (5.16). (b) The h-curves for the values x = 0.3, t = 0.1, a = 0.8, a = b =1, V1 = 1, V2 = 2 plotted from Eq. (5.17)
The double cosine potential [53] that we have taken is
V(x) = V1 cos x + V2 cos2x. (5.16)
This potential has various versions. Basically, the types of these potentials (Fig. 1a) are used to study the Bloch state solution of quantum states of a single particle. Selecting the initial condition f (x) = sin x, the approximate solution reads
u(x, t) = sin x + u1(x, t) + u2(x, t) + ••• , (5.17)
where u1, u2 come from (5.11)-(5.14) and hence
g(x) = (bV2 — 1) sin x + sin 2x + (a — 2bV2) sin3 x, g1(x) = gxx(x) + [(bVi cos x + bV2 cos 2x) + a sin2 x] g(x).
(5.18)
(5.19)
a
t
The approximate solution obtained, i.e., Eq. (5.17), heavily depends on the auxiliary parameter h. In [54], Abbasbandy et al. have proved that the value of h, from the so-called h-curve, is the main single factor that can control obtaining the convergence series solution. In general, the h-curve is drawn by plotting any achieved approximate solution of a particular problem with h, where h acts as a variable keeping all used variables as a fixed parameter. Liao [55] proposed that the valid region of an h-curve is the part that is horizontal with the h axis. We can choose any suitable value of h from the horizontal part of the curve to construct the converging approximate solution. This is how the auxiliary parameter h provides the unique facility to ensure the convergence of the solution and that's why the homotopy analysis method (HAM) is superior to other analytical methods. Here Figure 1 shows the h-curves for the double cosine potential case. The first four terms are used to draw the h curves. We have taken the real part, the imaginary part of the solutions, and the absolute solution. The graph shows that the appropriate value of h lies in the range -2 < h < 1. Since h = 0 is impossible to employ the homotopy analysis method, we have selected the value h = -0.5. Using this value of h, here are the surface plots of the approximate solution (5.17). Figure 2, Figure 3, and Figure 4 are the real, imaginary, and absolute plots, respectively.
16.5 16 15.5 15 14.5 14 1
490.5 490 489.5i 489 488.5 488] 1
Fig. 2. Real part of the solution (5.17) where the parameter values are a = b =1, V1 = 1, V2 =2, h = -0.5
5.2. Morse potential
The Morse potential [56] in reduced form is written as
V (x) = Vle-2fix + V2 e-l3x, (5.20)
where f is a parameter which controls the width of the potential. The smaller f means the larger well. In general the Morse potential (Fig. 5a) is a model for a vibrating diatomic molecule. It is better than the simple harmonic model because it includes the effect of bond breaking,
Fractional parameter a = 0.3
Fractional parameter a = 0.5
0.5 x^q 0 -10_i5
Fractional parameter a = 0.8
0 -10~5 Fractional parameter a = 1.0
1680 1679.5 1679 1678.5 1678 1677.5, 1
0 -10
0 -10~5
Fractional parameter a = 0.3
Fractional parameter a — 0.5
-0.3727
-0.3727
-0.3728
-0.3728
-0.3729
-0.3729 1
0 -10
Fractional parameter a = 0.8
-0.8425 -0.8425 -0.8425 -0.8425 -0.8425 -0.8425J 1
0 -10
-0.531 -0.531 -0.531 -0.531 -0.531 -0.531 1
0 -10"5 Fractional parameter a = 1.0
0 -10"
Fig. 3. Imaginary part of the solution (5.17) where the parameter values are a = b = 1, V1 = 1, V2 =2,
h = 0.5
Fractional parameter a = 0.3
Fractional parameter a = 0.5
0 -10"
Fractional parameter a = 0.8
0 -10"
0 -10~5 Fractional parameter a= 1.0
1680 1679.5 1679 1678.5 1678 1677.5 1
0 -10
Fig. 4. Absolute solution (\u\) of (5.17) where the parameter values are a = b =1, V1 = 1, V2 =2,
h = 0.5
anharmonic type bond stretching factors in the prospect of the theoretical study. We start with the initial condition f (x) = eix and H(x, t) = 1. The approximate solution takes the form
u(x, t) = eix + u1 (x, t) + u2(x, t) +----, (5.21)
Morse potential
h-curve for Morse potential solution
Fig. 5. (a) Nature of the Morse potential curve from Eq. (5.20). (b) The h-curves for the values x t = 0.1, a = 0.8, a = b =1, V1 = 1, V2 = 2, f = 0.05 plotted from Eq. (5.21)
Fractional parameter a = 0.3
0.7,
°'5 cr-io-5
Fractional parameter a = 0.8
Fractional parameter a = 0.5
397 L
396.5 396 395.5 395 394.5
0 -10
Fractional parameter a = 1.0
2175.5 2175 2174.5 2174 2173.5 2173
0 -10"
0 -10
Fig. 6. Real part of the solution (5.21) where the parameter values are a = b =1, V1 = 1, V2 = 2, h = -0.5, / = 0.05
where
g(x) = s(x)eix, (5.22)
s(x) = a + bV(x) - 1, (5.23)
g1(x) = [sxx(x) + 2iSx(x) + s2(x) e*x. (5.24)
The h curves for this case have been shown in Fig. 5. The three curves are due to real(u), imaginary(u), and absolute(u). Here we also find that the value h, from the horizontal part of
Fractional parameter a = 0.3
Fractional parameter a = 0.5
Fractional parameter a = 0.8
0.5
0.5
Fractional parameter a =1.0
0-10"
Fig. 7. Imaginary part of the solution (5.21) where the parameter values are a = b = 1, V1 h = -0.5, 3 = 0.05
1, V,
Fractional parameter a = 0.3
Fractional parameter a = 0.5
0 -10"
Fractional parameter a = 0.8
0 -10"
0 -10"
Fig. 8. Absolute solution (|u|) of (5.21) where the parameter values are a = b =1, V1 = 1, V2 =2, h = -0.5, f = 0.05
the graphs, lies in the range h £ (-2, 1). We choose again the value h = -0.5. The surface plots for real(u), imaginary(u), and absolute(u) are shown in Fig. 6, Fig. 7, and Fig. 8, respectively.
5.3. Hyperbolic potential
Here we introduce a new type of potential function
V(x) = V0 tanh(x) sech(x). (5.25)
This potential (Fig. 9a) has only one maximum and one minimum irrespective of the value of V0. If the sign of the V0 is changed, the maximum becomes a minimum and the minimum becomes a maximum. Here the approximate solution reads
u(x, t) = sechx + u1(x, t) + u2(x, t) + ••• , (5.26)
where the initial condition has been taken: f (x) = sech x. Here
g(x) = sech x + (a — 2) sech3 x + bV0 tanh x sech2 x, (5.27)
g1 (x) = gxx(x) + [bV(x) + a sech2 x] g(x). (5.28)
Now Fig. 9 shows the h curves for this case. The three curves are due to real(u), imaginary(u), and absolute(u) extracted from Eq. (5.26). The horizontal part of the graphs suggests that h e G (—2, 1). We choose again the value h = —0.5. Using this value, the surface plots for real(u), imaginary(u), and absolute(u) are shown in Fig. 10, Fig. 11, and Fig. 12, respectively.
h-curve for hyperbolic
(a) (b)
Fig. 9. (a) Nature of the hyperbolic potential curve from Eq. (5.25). (b) The h-curves for the values x = = 0.5, t = 0.1, a = 0.8, a = b =1, V0 = 1 plotted from Eq. (5.26)
6. Results and discussions
In this section, we will try to analyze the solutions and surface plots that we have achieved in the previous section. If we look at the three h curves depicted in Figs. 1, 5 and 9, we find that the h curves for real and absolute solutions show the same type of evolution and all the imaginary parts show different trends of evolution. This means that the solutions or the surface plots for the three potential cases will show the same trend. If we look at the surface plots in
Fractional parameter a = 0.3
Fractional parameter a = 0.5
(T-IO-5 Fractional parameter a = 0.8
1805 1804.5 1804 1803.5
0 -10
246.2 246 245.8 245.6 245.4 245.2 245 1
0 -10
Fractional parameter a= 1.0
6190 6189.8 6189.6 6189.4 6189.2 6189 6188.8 1
0 -10-5
Fig. 10. Real part of the solution (5.26) where the parameter values are a = b =1, V0 = 1, h = —0.5
Fractional parameter a = 0.3
Fractional parameter a = 0.5
0.0733 0.0733 0.0733 0.0733 0.0733 0.0733 1
0-10"5 Fractional parameter a = 0.8
0.1044
0.1044
0.1044
0.1044
0.1044 1
0 -10
Fractional parameter a = 1.0
0 -10"
0 -10"
Fig. 11. Imaginary part of the solution (5.26) where the parameter values are a = b = 1, V0 = 1, h = —0.5
Figs. 2, 3, 6, 7, 10 and 11, we find that the real and absolute solutions show the same type of behavior. On the other hand, all the imaginary solutions, extracted from Eqs. (5.17), (5.21), (5.26), show different behavior as compared to their real and absolute solutions. This shows that our solutions are coherent with the evolution h curve.
Fractional parameter a = 0.3
Fractional parameter a = 0.5
0 -10"
Fractional parameter a = 0.8
246.2 246 245.8 245.6 245.4 245.2 245i 1
0 -10
Fractional parameter a= 1.0
1805 1804.5 1804 1803.5
6190 6189.8 6189.6 6189.4 6189.2 6189 6188.8 1
0 -10
0 -10
Fig. 12. The absolute solution (|u|) of (5.26) where the parameter values are a = b =1, V0 = 1, h = -0.5
The theoretical reason for taking time-fractional NLSE is given in Section 3. We know that in the case of nonlinear PDE, like NLSE, setting a large time scale kills the effect of dispersion. The presence of the fractional parameter a in all the solutions also supports this fact. Here a provides an extra tool to study the disappearance of the dispersion effect in nonlinear optics.
If the surface plots are examined carefully, we find that, as a goes from a lower value to a = 1, the real and the absolute solutions retain their shape, but the height or the Z-plane value is increasing. In the case of double cosine and Morse potential, this means that a clear self-de-focusing effect is going on. In other words, we can say that the nonlinear media may have a negative refractive index. The fractional parameter a is acting as a dialing parameter that controls the dispersion or the self-defocusing with time. In practical applications, the self-focusing phenomena of a light beam in optical fibers are problematic as they produce pulse distortion, noise, and catastrophic damage of optical components. In this sense, finding a defocusing effect is quite encouraging. We also feel that an initial condition is a key to studying the process via the homotopy analysis method. It is possible to form a soliton in the case of double cosine potential and Morse potential trap, but we need further research on this topic.
In a real situation, the soliton formation process may have a broad range of initial pulse shapes. It is also true that keeping a soliton shape constant throughout the progress is also a delicate matter and the effect of perturbation is equally important. In our last potential, i.e., V0 tanh(x)sech(;c), we have achieved a bright soliton solution. Here the initial pulse was soliton itself. It is clear that the initial pulse remains soliton despite facing a perturbation in the form of a potential function. This signifies that the dispersion effect and the nonlinearity are perfectly balanced. The fractional parameter a has a role here also. As a increases from a lower value to a = 1, the height or the Z-value is increasing, keeping the shape fixed. So the solution is stable. Controlling the a values, it is possible to study the temporal and roughness effect of the soliton in the complex time domain. That's the advantage of using fractional differential equations.
7. Conclusion
In this paper, the homotopy analysis method (HAM) is applied to study the time-fractional nonlinear Schrodinger equation with three separate trapping potential models, namely, the double cosine potential, the Morse potential, and a hyperbolic potential, respectively. The standard definitions of Caputo and Riemann - Liouville type fractional calculus rules have been applied to compose the entire study. A brief derivation of the time-fractional Schrodinger equation has been done with the fractional Euler - Lagrange equation. We have presented several figures for the display of the obtained approximate solutions and corresponding h-curves. It is seen that all the solutions heavily depend on the auxiliary parameter h. Controlling h, it is possible to achieve a convergence series solution. In that sense, the homotopy analysis method (HAM) is far more superior to other analytical tools for studying nonlinear differential or partial differential equations with initial boundary conditions. Finally, we discussed the effects of the fractional parameter on the obtained solutions.
Two-dimensional treatment of the present problem (Au = and V = V(x, y)) can
provide more information, we will try that in the near future. We would also like to see the same problem for the quintic nonlinear Schrodinger equation where Eq. (5.2) takes F(|u|2, u) = = alul4u + bV(x)u. Trapping potential for the nonlinear Schrodinger equation is an emerging subject and we hope that our work will generate a considerable amount of research interest in this field.
Conflict of interest
The authors declare that they have no conflict of interest.
Acknowledgments
We would like to express our sincere thanks to the referees for their positive suggestions that helped to improve this paper greatly.
References
[1] Thompson, J. M. T. and Stewart, H. B., Nonlinear Dynamics and Chaos, 2nd ed., Chichester: Wiley, 2002.
[2] Struble, R.A., Nonlinear Differential Equations, Mineola, N.Y.: Dover, 2018.
[3] Simmons, G.F., Differential Equations with Applications and Historical Notes, 3rd ed., Boca Raton, Fla.: CRC, 2017.
[4] Mohyud-Din, S. T., Noor, M. A., and Noor, K.I., Some Relatively New Techniques for Nonlinear Problems, Math. Probl. Eng., 2009, vol. 2009, Art. ID 234849, 25 pp.
[5] He, J.-H., Homotopy Perturbation Technique, Comput. Methods Appl. Mech. Engrg., 1999, vol. 178, nos. 3-4, pp. 257-262.
[6] Nirenberg, L., Variational and Topological Methods in Nonlinear Problems, Bull. Amer. Math. Soc. (N. S.), 1981, vol. 4, no. 3, pp. 267-302.
[7] Lakshmikantham, V. and Vatsala, A. S., Generalized Quasilinearization for Nonlinear Problems, Math. Appl., vol. 440, Dordrecht: Kluwer, 1998.
[8] Efendiev, Y., Hou, T., and Ginting, V., Multiscale Finite Element Methods for Nonlinear Problems and Their Applications, Commun. Math. Sci, 2004, vol. 2, no. 4, pp. 553-589.
[9] Katsikadelis, J. T., The Analog Equation Method: A Boundary-Only Integral Equation Method for Nonlinear Static and Dynamic Problems in General Bodies, Theoret. Appl. Mech., 2002, vol. 27, pp. 13-38.
[10] Podlubny, I., Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, New York: Acad. Press, 1998.
[11] Applications of Fractional Calculus in Physics, R.Hilfer (Ed.), River Edge, N.J.: World Sci., 2000.
[12] New Trends in Nanotechnology and Fractional Calculus Applications: Selected Papers from the Internat. Workshop on New Trends in Science and Technology (NTST'08) and the Internat. Workshop on Fractional Differentiation and Its Applications (FDA'08) Ankara, Turkey (Nov, 2008), D.Baleanu, Z. B. Guveng, J. A. Tenreiro Machado (Eds.), New York: Springer, 2010.
[13] Dalir, M. and Bashour, M., Applications of Fractional Calculus, Appl. Math. Sci. (Ruse), 2010, vol. 4, nos. 21-24, pp. 1021-1032.
[14] Yang, A.-M., Zhang, Y.-Zh., Cattani, C., Xie, G.-N., Rashidi, M.M., Zhou, Y.-J., and Yang, X.-J., Application of Local Fractional Series Expansion Method to Solve Klein-Gordon Equations on Cantor Sets, Abstr. Appl. Anal., 2014, Art. ID 372741, 6 pp.
[15] Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, including Papers from the Minisymposium on Fractional Derivatives and Their Applications (EN0C'2005, Eindhoven, Aug 2005), and the 2nd Symposium on Fractional Derivatives and Their Applications (ASME-DETC'2005, Long Beach, Calif., Sept 2005), J.Sabatier, O.P. Agrawal, J. A. Tenreiro Machado (Eds.), Dordrecht: Springer, 2007.
[16] Abdel-Salam, E. A.-B. and Gumma, E. A. E., Analytical Solution of Nonlinear Space-Time Fractional Differential Equations Using the Improved Fractional Riccati Expansion Method, Ain Shams Eng. J., 2015, vol. 6, no. 2, pp. 613-620.
[17] Singh, J., Kumar, D., and Sushila, D., Homotopy Perturbation Sumudu Transform Method for Nonlinear Equations, Adv. Theor. Appl. Mech., 2011, vol. 4, no. 4, pp. 165-175.
[18] Guo, Sh., Mei, L., Li, Y., and Sun, Y., The Improved Fractional Sub-Equation Method and Its Applications to the Space-Time Fractional Differential Equations in Fluid Mechanics, Phys. Lett. A, 2012, vol. 376, no. 4, pp. 407-411.
[19] Saadatmandi, A. and Dehghan, M., A New Operational Matrix for Solving Fractional-Order Differential Equations, Comput. Math. Appl., 2010, vol. 59, no. 3, pp. 1326-1336.
[20] Safari, M., Ganji, D. D., and Moslemi, M., Application of He's Variational Iteration Method and Ado-mian's Decomposition Method to the Fractional KdV-Burgers-Kuramoto Equation, Comput. Math. Appl, 2009, vol. 58, nos. 11-12, pp. 2091-2097.
[21] Saha Ray, S. and Bera, R. K., An Approximate Solution of a Nonlinear Fractional Differential Equation by Adomian Decomposition Method, Appl. Math. Comput., 2005, vol. 167, no. 1, pp. 561571.
[22] Wu, G.-Ch. and Lee, E. W. M., Fractional Variational Iteration Method and Its Application, Phys. Lett. A, 2010, vol. 374, no. 25, pp. 2506-2509.
[23] Wang, H., Wang, K., and Sircar, T., A Direct O(N log2 N) Finite Difference Method for Fractional Diffusion Equations, J. Comput. Phys., 2010, vol. 229, no. 21, pp. 8095-8104.
[24] Wu, J.-L., A Wavelet Operational Method for Solving Fractional Partial Differential Equations Numerically, Appl. Math. Comput., 2009, vol. 214, no. 1, pp. 31-40.
[25] Bekir, A., Guner, O., and Cevikel, A. C., Fractional Complex Transform and Exp-Function Methods for Fractional Differential Equations, Abstr. Appl. Anal., 2013, Art. ID 426462, 8 pp.
[26] Yan, Y., Pal, K., and Ford, N. J., Higher Order Numerical Methods for Solving Fractional Differential Equations, BIT Numer. Math., 2014, vol. 54, no. 2, pp. 555-584.
[27] Shah, A., Khan, R. A., Khan, A., Khan, H., and Gomez-Aguilar, J.F., Investigation of a System of Nonlinear Fractional Order Hybrid Differential Equations under Usual Boundary Conditions for Existence of Solution, Math. Methods Appl. Sci., 2021, vol. 44, no. 2, pp. 1628-1638.
[28] Aydogan, S.M., Gomez-Aguilar, J. F., Baleanu, D., Rezapour, Sh., and Samei, M.E., Approximate Endpoint Solutions for a Class of Fractional ^-Differential Inclusions by Computational Results, Fractals, 2020, vol. 28, no. 8, 2040029, 18 pp.
[29] Ali, G. and Gomez-Aguilar, J. F., Approximation of Partial Integro Differential Equations with a Weakly Singular Kernel Using Local Meshless Method, Alex. Eng. J., 2020, vol. 59, no. 4, pp. 2091-2100.
[30] Abdeljawad, Th., Atangana, A., Gomez-Aguilar, J. F., and Jarad, F., On a More General Fractional Integration by Parts Formulae and Applications, Phys. A, 2019, vol. 536, 122494, 17 pp.
[31] Khan, A., Khan, H., Gomez-Aguilar, J.F., and Abdeljawad, Th., Existence and Hyers-Ulam Stability for a Nonlinear Singular Fractional Differential Equations with Mittag-Leffler Kernel, Chaos Solitons Fractals, 2019, vol. 127, pp. 422-427.
[32] Sulem, C. and Sulem, P.-L., The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse, Appl. Math. Sci., vol. 139, New York: Springer, 2007.
[33] Gippius, N., Tikhodeev, S., Kulakovskii, V., Krizhanovskii, D., and Tartakovskii, A., Nonlinear Dynamics of Polariton Scattering in Semiconductor Microcavity: Bistability vs. Stimulated Scattering, Europhys. Lett., 2004, vol. 67, no. 6, pp. 997-1003.
[34] Lü, X., Zhu, H.-W., Yao, Zh.-Zh., Meng, X.-H., Zhang, Ch., Zhang, Ch.-Y., and Tian, B., Multi-soliton Solutions in Terms of Double Wronskian Determinant for a Generalized Variable-Coefficient Nonlinear Schrödinger Equation from Plasma Physics, Arterial Mechanics, Fluid Dynamics and Optical Communications, Ann. Physics, 2008, vol. 323, no. 8, pp. 1947-1955.
[35] Caputo, M. and Fabrizio, M., On the Singular Kernels for Fractional Derivatives: Some Applications to Partial Differential Equations, Progr. Fract. Differ. Appl., 2021, vol. 7, no. 2, pp. 79-82.
[36] Heymans, N. and Podlubny, I., Physical Interpretation of Initial Conditions for Fractional Differential Equations with Riemann - Liouville Fractional Derivatives, Rheol. Acta, 2006, vol. 45, pp. 765-771.
[37] Biswas, S., Ghosh, U., Sarkar, S., and Das, S., Approximate Solution of Space-Time Fractional KdV Equation and Coupled KdV Equations, J. Phys. Soc. Jpn, 2020, vol. 89, 014002, 10 pp.
[38] Pakhira, R., Ghosh, U., and Sarkar, S., Study of Memory Effect in an Inventory Model with Linear Demand and Shortage, I. J. Math. Sci. Comput., 2019, vol. 5, no. 2, pp. 54-70.
[39] Laskin, N., Fractional Quantum Mechanics and Levy Path Integrals, Phys. Lett. A, 2000, vol. 268, nos. 4-6, pp. 298-305.
[40] Agrawal, O.P., Formulation of Euler - Lagrange Equations for Fractional Variational Problems, J. Math. Anal. Appl., 2002, vol. 272, no. 1, pp. 368-379.
[41] Hasan, E.H., Fractional Variational Problems of Euler - Lagrange Equations with Holonomic Constrained Systems, Appl. Phys. Res., 2016, vol. 8, no. 3, pp. 60-65.
[42] Klimek, M., Lagrangean and Hamiltonian Fractional Sequential Mechanics, Czechoslov. J. Phys., 2002, vol. 52, no. 11, pp. 1247-1253.
[43] Antar, N. and Pamuk, N., Exact Solutions of Two-Dimensional Nonlinear Schrödinger Equations with External Potentials, Appl. Math. Comput., 2013, vol. 2, no. 6, pp. 152-158.
[44] Khan, N. A., Jamil, M., and Ara, A., Approximate Solutions to Time-Fractional Schrödinger Equation via Homotopy Analysis Method, Int. Sch. Res. Notices, 2012, vol. 2012, Art. ID 197068, 11 pp.
[45] Devi, A. and De, A. K., Theoretical Investigation on Nonlinear Optical Effects in Laser Trapping of Dielectric Nanoparticles with Ultrafast Pulsed Excitation, Opt. Express, 2016, vol. 24, no. 19, pp. 21485-21496.
[46] Stanislavova, M. and Stefanov, A. G., Ground States for the Nonlinear Schrödinger Equation under a General Trapping Potential, J. Evol. Equ., 2021, vol. 21, no. 1, pp. 671-697.
[47] Sakaguchi, H. and Tamura, M., Scattering and Trapping of Nonlinear Schrödinger Solitons in External Potentials, J. Phys. Soc. Japan, 2004, vol. 73, no. 3, pp. 503-506.
[48] Liao, S., Beyond Perturbation: Introduction to the Homotopy Analysis Method, Boca Raton, Fla.: Chapman & Hall/CRC, 2003.
[49] Fermi, E., Pasta, J., and Ulam, S., Studies of Nonlinear Problems: Los Alamos Report LA-1940, 1955, 22 pp.
[50] He, J.-H. and El-Dib, Y. O., Homotopy Perturbation Method for Fangzhu Oscillator, J. Math. Chem., 2020, vol. 58, no. 10, pp. 2245-2253.
[51] Liao, S., Comparison between the Homotopy Analysis Method and Homotopy Perturbation Method, Appl. Math. Comput, 2005, vol. 169, no. 2, pp. 1186-1194.
[52] Ellahi, R., Abbasbandy, S., Hayat, T., and Zeeshan, A., On Comparison of Series and Numerical Solutions for Second Painlevé Equation, Numer. Methods Partial Differential Equations, 2010, vol. 26, no. 5, pp. 1070-1078.
[53] Hai, K., Hai, W., and Chen, Q., Exact Manipulations to Bloch States of a Particle in a Double Cosine Potential, Phys. Lett. A, 2007, vol. 367, no. 6, pp. 445-449.
[54] Abbasbandy, S. and Jalili, M., Determination of Optimal Convergence-Control Parameter Value in Homotopy Analysis Method, Numer. Algorithms, 2013, vol. 64, no. 4, pp. 593-605.
[55] Liao, S., Homotopy Analysis Method in Nonlinear Differential Equations, Berlin: Springer, 2012.
[56] Nasarenko, A. Ya., Spiridonov, V. P., Butayev, B. S., and Zasorin, E. Z., Second-Order Perturbation Approach to Anharmonic Analysis of Molecules by Electron Diffraction: 3. Morse-Like Potential Function of SnCl2, SnBr2, SnI2, Ga2O, and Tl2O, J. Mol. Struct., 1985, vol. 119, nos. 3-4, pp. 263270.
