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Modulation Instability, Dark and Singular Soliton for Weakly Nonlocal Schrodinger Equation

Ali Danladi1,∗, Alhaji Tahir2, Hadi Rezazadeh3

1Department of Mathematics, Federal University, Dutse, Nigeria
alidanladi14@gmail.com, ali.danladi@fud.edu.ng

2Department of Mathematics, Faculty of Physical Sciences, Modibbo Adama University, Yola, Nigeria
atahir@mau.edu.ng

3Faculty of Engineering Technology, Amol University of Special Modern Technologies, Amol, Iran
h.rezazadeh@ausmt.ac.ir

∗Correspondence: alidanladi14@gmail.com, ali.danladi@fud.edu.ng

Abstract. The optical soliton solution of nonlinear complex models holds significant importance innonlinear optics and communication systems. Considering nonlinear complex models, often describedby equations like the nonlinear Schrodinger equation (NLSE), plays a crucial role in defining thebalance between dispersive and nonlinear effects, enabling the formation and maintenance of solitonsover long distances. This stability is crucial for signal integrity in optical communication systems.The investigation of optical soliton solutions from nonlinear complex models is sometimes compli-cated. With this in mind, we employed an effective method of extended Tanh function with a Riccattidifferential equation to retrieve the dark, singular and periodic wave solutions for the weakly nonlocalnonlinear Schrodinger equation with parabolic law. The obtained solutions were verified by back-substitution in the original equations, with the aid of a Mathematica to affirm the robustness of thechosen approach. Respective 2D and 3D graphs for some of the obtained results was portrayed bychoosing suitable values of the parameters that were involved. An analysis of instability that resultsin the modulation of the steady-state as a result of co-action between the nonlinear and dispersiveeffects was performed on the proposed model where the condition for stable wave under small per-turbation was obtained and presented. The gain spectrum plot for the modulation instability wasportrayed.
1. IntroductionPartial Differential Equations (PDEs) serve as powerful mathematical tools for modeling and under-standing diverse phenomena in science and engineering. These equations involve multiple variablesand their partial derivatives, making them well-suited for describing complex physical processessuch as heat conduction, fluid dynamics, and electromagnetic fields. The investigation of PDEsencompasses various aspects, including analytical and numerical methods for solving these equa-tions. Analytically, researchers explore techniques like separation of variables, integral transforms,
Received: 3 Mar 2025.
Key words and phrases. Nonlocal Schrodinger equation with PL; Extended tanh approach; Modulation Instability.1

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Eur. J. Math. Anal. 10.28924/ada/ma.5.14 2and series solutions to obtain exact solutions or gain insights into the behavior of solutions. Onthe other hand, numerical methods, such as finite difference, finite element, and spectral methods,provide computational tools to approximate solutions for PDEs in cases where analytical solu-tions are challenging or impossible to obtain. Additionally, the study of stability, existence, anduniqueness of solutions, as well as the development of new solution methods, continues to be avibrant area of research within the broader field of partial differential equations. Understandingand harnessing the mathematical intricacies of PDEs play a crucial role in advancing our compre-hension of the natural world and technological applications. Numerous scholars have focused theirefforts on determining exact solutions for NPDEs, with these equations finding applications acrossdiverse scientific and technological domains, including but not limited to mathematical physics,fluid dynamics, optical fibers, and economics [10, 14, 27].In recent years, various methodologies have been developed for determining precise solutions toNPDEs. These approaches include the Lie symmetry method [16, 18, 20, 26, 30], the Kudryashovmethod [21, 22, 33], the sine-Gordon expansion method [6], the invariant subspace method [13, 28],the Sardar subequation method [3, 11], and others [1, 5, 8, 9, 29]. These techniques contributeto the exploration of exact solutions for a wide range of PDEs. The NLSE is a complex PDEthat plays a crucial role in various fields of physics and engineering, describing the evolution ofcomplex wave packets. It arises in different contexts, including nonlinear optics, plasma physics,fluid dynamics, and condensed matter physics. The NLSE and its variants have significant impor-tance in understanding and predicting the behavior of wave-like phenomena in diverse real-worldapplications.In nonlinear optics, the NLSE governs the propagation of intense laser beams through nonlinearmedia. This equation describes the interactions among optical waves, considering effects suchas self-focusing, self-phase modulation, and optical solitons. Optical solitons, which are stable,localized wave packets that can maintain their shape during propagation, and applications in long-distance communication systems. The NLSE helps optimize and control these phenomena in thedesign of optical communication systems and laser technologies.The NLSE also appears in the study of ultra-cold atomic gases, particularly in the context ofBose-Einstein condensates. In this scenario, the NLSE describes the dynamics of the macroscopicwave function of the condensate. Understanding the NLSE for BECs is crucial for investigatingphenomena such as matter-wave solitons and vortices, which have applications in precision mea-surements and quantum information processing.The NLSE arises in the study of Langmuir waves in plasmas, where it describes the evolution of theelectron plasma wave. Nonlinear effects become significant in high-intensity laser-plasma inter-actions and can lead to the generation of harmonics and other phenomena. This has applicationsin areas such as controlled nuclear fusion research and the development of high-power particle

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Eur. J. Math. Anal. 10.28924/ada/ma.5.14 3accelerators.NLSE is fundamental in understanding the behavior of optical pulses in fiber optic communica-tion systems. Fiber optic channels exhibit nonlinear effects such as self-phase modulation andcross-phase modulation, which can distort transmitted signals. The NLSE is essential for modelingand mitigating these effects, ensuring the reliability and efficiency of long-distance communicationnetworks.Variants of the NLSE are used to model the propagation of water waves in oceans and otherbodies of water. Nonlinear effects, such as wave steepening and wave breaking, can be describedusing these equations. Understanding these phenomena is crucial for predicting and mitigating theimpact of tsunamis, storm surges, and other oceanic events.NLSE variants are employed in the study of biological systems, such as modeling the propagationof nerve impulses. The NLSE can describe the nonlinear dynamics of excitable media, providinginsights into the behavior of electrical signals in biological tissues.Moreover, the NLSE [4, 12, 23], which is an essential fully-integrated nonlinear dispersive par-tial differential equation (PDE), has found extensive application in elucidating diverse phenomenalike deep water waves, rogue waves, plasmas, and nonlinear optics, including atomic physics. TheNLSE serves as a comprehensive description of nonlinear dispersive processes. Zhou et al. [34]considered the weakly nonlocal NLSE having PL nonlinearity with external potential as
iΦt + λ1Φxx +

(
λ2|Φ|2xx + λ3|Φ|2 + λ4|Φ|4

)
Φ = 0. (1)

The study of Schrodinger equations with nonlinearity is an important area of research in math-ematical physics. In recent years, the weakly nonlocal Schrodinger equation with parabolic lawnonlinearity has gained significant attention due to its numerous applications in various fields suchas quantum mechanics, nonlinear optics, and fluid dynamics.In this paper, we present Modulation instability analysis and the novel exact solutions for theweakly nonlocal Schrodinger equation with parabolic law nonlinearity, derived using the methodextended tanh function. The solutions we obtain, which include dark soliton, bright soliton, andtraveling wave solutions, have not been previously reported in the literature and demonstrate thecomplex dynamics of the considered equation. Dark optical soliton describes the soli- tary waveswith lower intensity than the background, bright optical soliton describes the solitary waves whosepeak intensity is larger than the background and the sin- gular optical soliton is a solitary wavewith discontinuous derivatives; examples of such solitary waves include compactions, which havefinite (compact) support, and peakons, whose peaks have a discontinuous first derivative [35, 36].Dark soli- ton propagates without changing its shape, but it is not made by a normal pulse; rather,it is a lack of energy in a continuous-time beam. The intensity is constant, but for a short timeduring which it jumps to zero and back again, thus generating a "dark pulse"’. Those solitons can

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Eur. J. Math. Anal. 10.28924/ada/ma.5.14 4actually be generated introducing short dark pulses in much longer standard pulses. Dark soli-tons are more difficult to handle than standard solitons, but they have shown to be more stableand robust to losses. Bright optical soliton causes a temporary increase in an associated waveamplitude [52]. The combined dark-bright optical soliton carries the combine features of the darkand bright optical solitons. Moreover, the modulation instability analysis for the weakly nonlocalNLSE having PL nonlinearity with external potential will also be presented.Our findings have significant implications for the study of nonlinear phenomena in physical sys-tems. The ability to obtain exact solutions to such complex equations is crucial in understandingthe underlying physics and designing new experiments. Additionally, with the aid of mathemat-ica, our results are verified by back-substitution in the original equations, affrming the robustnessof our approach. The suggested is not only direct and simple but also suitable for constructingnew results, paving the way for future applications on NLSEs with dual power law and perturbedNLSEs with Kerr law. This is particularly useful for identifying solitons in photorefractive andpolymer materials. The proposed technique holds potential for further applications in natural sci-ence models, aiding in the investigation of other mathematical challenges and characterizing thebehavior of nonlinear models.The remaining sections of this manuscript are organized as follows: In Section 2, we give thedescription of the approach.In section 3, we apply the method the governing equation. Insection4, we give the graphical representation of some of the obtained results. In section 5, we presentthe analysis of modulation instability for the model and its physical description. In section 6, weprovide the conclusion of the study.
2. Description of the approachThe method of extended tanh is often employed in obtaining soliton solutions due to is effectivenessin capturing the inherent characteristics of solitons such as their amplitude, width, and velocity.The following are some of the rationale for using the method: Flexibility, asymptotic behavior, Easeof manipulation accuracy, existence of exact solutions.In order to present the modified extended tanh approach, we need to consider the nonlinear partialdifferential equation of the form:

ρ (U,Ux , Uxx , UtUx , ...) = 0; (2)
In transforming (2); we use the wave transformations:

U(x, t) = U(ξ); ξ = (x − µt) (3)
where µ is a nonzero constant. Substitution of the transformation (3) into Eq. (2), it reduces to anordinary differential equation of the polynomial form:

Ξ
(
U(ξ), U ′(ξ), U ′′(Ξ), U ′′′(Ξ), ...

)
= 0. (4)

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Eur. J. Math. Anal. 10.28924/ada/ma.5.14 5Furthermore, the solution is considered to be a finite series of the form:
u(ξ) = a0 +

n=N∑
n=1

anψ
n(ξ) +

n=N∑
n=1

bn
ψn

(ξ) (5)
where, a0, an, bn, n = 1, 2, 3, ...N are constants which to be computed; and N is a positive integerwhich is to be determined by balancing the highest order derivative and with the highest nonlinearterms in the equation. Also, ψ(ξ) satisfies the Riccati’s differential equation:

ψ′(ξ) = ψ2(ξ) + b (6)
where b is a constant. Furthermore, the Riccati differential equation in (6) has solutions of theform:
• For Hyperbolic solution, if b < 0, then

Φ(ξ) = −
√
−b Tanh

√
−b ξ, Φ(ξ) = −

√
−b Coth

√
−b ξ

• For rational solution, if b = 0, then 1
ξ

• For Periodic solution, if b > 0, then
Φ(ξ) =

√
b Tan

√
b ξ, Φ(ξ) =

√
b Cot −

√
−b ξNow, after we substitute Eq. (5) and its derivatives together with the Riccati Eq. (6), into Eq.(4), it gives a polynomial in Φ(ξ) . Collecting the coefficients of the same power of Φ(ξ) in thepolynomial and setting each of them to zero, we shall get a set of algebraic equations. Solvingthese system of algebraic equation with the aid of symbolic computation using Mathematica to getthe values of a0, an, bn, (n = 1, 2, 3, ...) and b. Finally, substituting these values into Eq. (5) fromwhich we obtain the solution of Eq. (4).

3. ApplicationIn this section, the application of the modified extended tanh expansion method with Riccati differ-ential equation [49] to our equation (1) shall be presented.Consider the transformation:
φ(x, t) = Φ(ξ)e i$; ξ = x − νt and $ = ωx − r t + δ (7)

Substituting equation (7) into equation (1), gives the following nonlinear ODE:
2λ2Φ2Φ′′ + λ1Φ′′ + λ4Φ5 + λ3Φ3 + Φ

(
2λ2Φ′2 − λ1ω

2 + r
)

= 0. (8)
From the real part and

(v − 2λ1ω) Φ′ = 0. (9)
From the imaginary part. Thus, we have the constraint condition as

v = 2λ1ω. (10)

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Eur. J. Math. Anal. 10.28924/ada/ma.5.14 6Balancing between Φ2Φ′′ and Φ5 , that is:
2N + N + 2 = 5N =⇒ N = 1. (11)

With equation (11), our equation (5) takes the form:
φ(Ω) = a1Φ(Ω) + a0 +

b1

Φ(Ω)
(12)

Substituting Eq. (12), its first and second derivative along with equation (6) into equation (8), weget a polynomial in powers of Φ(Ω). Summing the coefficients of Φ(Ω) with the same power andequating each summation to zero, gives a set of algebraic equations. Solving these set of algebraicequations, yields the following cases of solutions:
case oneWhen a0 = 0; a1 = − i

√
6
√
λ2√

λ4
; b1 = 0; r → 1

16

(
λ1

(
2λ3
λ2

+ 16ω2
)
− λ4λ

2
1

λ2
2

+
3λ2

3
λ4

)
;

w = λ1λ4−3λ2λ3

24λ2
2

, we obtain the following solutions to equation (3):If W < 0, then we have
Φ1(ξ) = ±

i
√

3λ2λ3 − λ1λ4 tanh
(√

3λ2λ3−λ1λ4Ω

2
√

6λ2

)
2
√
λ2

√
λ4

e i(ωx−r t+δ). (13)
Φ2(ξ) = ±

i
√

3λ2λ3 − λ1λ4 coth
(√

3λ2λ3−λ1λ4Ω

2
√

6λ2

)
2
√
λ2

√
λ4

e i(ωx−r t+δ). (14)If W > 0, then we have
Φ3(ξ) = ∓−

i
√
λ1λ4 − 3λ2λ3 tan

(√
λ1λ4−3λ2λ3Ω

2
√

6λ2

)
2
√
λ2

√
λ4

e i(ωx−r t+δ). (15)
Φ4(ξ) = ±

i
√
λ1λ4 − 3λ2λ3 cot

(√
λ1λ4−3λ2λ3Ω

2
√

6λ2

)
2
√
λ2

√
λ4

e i(ωx−r t+δ). (16)
case twoWhen a0 = 0; a1 = i

√
6
√
λ2√

λ4
; b1 = i(3λ2λ3−λ1λ4)

8
√

6λ
3/2
2

√
λ4

; r =
λ1(3λ2(4λ2ω

2+λ3)−λ1λ4)
12λ2

2
;

w = 3λ2λ3−λ1λ4

48λ2
2

; we obtain the following solutions to equation (1):If W < 0, then we have

Φ5(ξ) = ±
i
√
λ1λ4 − 3λ2λ3csch(√ λ1λ4

3
−λ2λ3Ω

2λ2

)
√

2
√
λ2

√
λ4

e i(ωx−r t+δ). (17)
If W > 0, then we have

Φ6(ξ) = ∓
i
√

3λ2λ3 − λ1λ4 csc

(√
λ2λ3−

λ1λ4
3

Ω

2λ2

)
√

2
√
λ2

√
λ4

e i(ωx−r t+δ). (18)

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Eur. J. Math. Anal. 10.28924/ada/ma.5.14 7

4. Graphical Representations of ResultsIn this section, the 2D and 3D graphical representation for some of the obtained results for theweakly nonlocal NLSE having PL nonlinearity, which is given by equations (1) shall be presentedby choosing different values of parameters that are involved.

Figure 1. 3D and 2D Abs plot for Φ1(ξ).
In figure (1) above, we have the surface profile of 2D and 3D absolute plot represen-tation of equation (13) ( that is, Φ1(ξ)) by taking the following parameter values (δ =

−1.5; ) (λ4 = 1.5; ) (λ1 = 0.5; ) (λ3 = 1.5; ) (λ2 = 1.15; ) (ω = 1.5; ). The range of values of xfor which the graphs were plotted is [-10,10]. The 2D graphs was plotted by taking values of t at
0, 2, 4, .

Figure 2. 3D and 2D Abs plot for Φ3(ξ).
In figure (2) below, we have the surface profile of 2D and 3D absolute plot represen-tation of equation (15) ( that is, Φ3(ξ)) by taking the following parameter values (δ =

−1.5; ) (λ4 = 1.5; ) (λ1 = 0.5; ) (λ3 = 1.5; ) (λ2 = 1.15; ) (ω = 1.5; ). The range of values of xfor which the graphs were plotted is [-10,10]. The 2D graphs was plotted by taking values of t at

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Eur. J. Math. Anal. 10.28924/ada/ma.5.14 8

Figure 3. 3D and 2D IM plot for Φ3(ξ).
the origin.
In figure (3) above, we have the surface profile of 2D and 3D imaginary plot represen-tation of equation (15) ( that is, Φ3(ξ)) by taking the following parameter values (δ =

−1.5; ) (λ4 = 1.5; ) (λ1 = 0.5; ) (λ3 = 1.5; ) (λ2 = 1.15; ) (ω = 1.5; ). The range of values of xfor which the graphs were plotted is [-10,10]. The 2D graphs was plotted by taking values of t atthe origin.

Figure 4. 3D and 2D Abs plot for Φ5(ξ).
In figure (4) above, we have the surface profile of 2D and 3D absolute plot represen-tation of equation (17) ( that is, Φ5(ξ)) by taking the following parameter values (δ =

−1.5; ) (λ4 = 1.5; ) (λ1 = 0.5; ) (λ3 = 1.5; ) (λ2 = 1.15; ) (ω = 1.5; ). The range of values of xfor which the graphs were plotted is [-10,10]. The 2D graphs was plotted by taking values of t atthe origin.In figure (5) above, we have the surface profile of 2D and 3D real plot representa-tion of equation (17) ( that is, Φ5(ξ)) by taking the following parameter values (δ =

−1.5; ) (λ4 = 1.5; ) (λ1 = 0.5; ) (λ3 = 1.5; ) (λ2 = 1.15; ) (ω = 1.5; ). The range of values of x

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Eur. J. Math. Anal. 10.28924/ada/ma.5.14 9

Figure 5. 3D and 2D real plot for Φ5(ξ).
for which the graphs were plotted is [-10,10]. The 2D graphs was plotted by taking values of t atthe origin.

Figure 6. 3D and 2D real plot for Φ6(ξ).
In figure (6) above, we have the surface profile of 2D and 3D real plot representa-tion of equation (18) ( that is, Φ6(ξ)) by taking the following parameter values (δ =

−1.5; ) (λ4 = 1.5; ) (λ1 = 0.5; ) (λ3 = 1.5; ) (λ2 = 1.15; ) (ω = 1.5; ). The range of values of xfor which the graphs were plotted is [-10,10]. The 2D graphs was plotted by taking values of t atthe origin.
5. Modulation Instability (MI) AnalysisVarious nonlinear phenomena display an instability that results in the modulation of the steady-state as a result of co-action between the nonlinear and dispersive effects [37]. In this section, wederive modulation instability for the weakly nonlocal Schrodinger equation with parabolic law.
Theorem 1: Suppose that equation (1) characterizes a wave system featuring a non-trivial disper-sion relation, alongside nonlinear components. Given that φ(x, t) represents a solution to equation,it is possible, under appropriate circumstances, for a range of parameters to exist where MI takesplace. Over time, the amplitude and configuration of the wave undergo significant alterations dueto instability, resulting in a rapid growth of small perturbations.

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Eur. J. Math. Anal. 10.28924/ada/ma.5.14 10

Proof : To explore the MI of equation (1), let take the initial assumption that equation (1) issubjected to a small perturbation in the following manner:
φ(x, t) =

(√
m + Φ(x, t)

)
e imx ; (19)

where m is the normalized optical power,φ(x, t) shows real valued amplitude of perturbation withrelatively dispersion. Substituting equation (19) into equation (1) and linearizing, gives
iΦt + (λ1 + λ2m) Φxx + iλ1m (Φx + Φ∗x) +

(
(λ4 − λ1)m2 + λ3m

)
(Φ + Φ∗) = 0, (20)

where Φ∗ denotes complex conjugate.Assume the solutions of equation (20) to be of the form
Φ(x, t) = A1e

(i(x$−ξt)) + A2e
(−i(x$−ξt)) (21)

Φ(x, t)∗ = A1e
(−i(x$−ξt)) + A2e

(i(x$−ξt)) (22)Where $ and ξ represent normalize wave number and frequency of perturbation respectively.Substituting (21) and (22) into (20) and collect the coefficients of e(−i(x$−ξt)), and

e(i(x$−ξt)), and solve the determinant of the resulting matrix of coefficients we obtain the followingdispersion relation:
−λ2

1$
4 − 2λ1λ2m

3$2 + 2λ2λ4m
3$2 − λ2

2m
2$4 − 2λ2

1m
2$2 + 2λ2λ3m

2$2

2λ1λ4m
2$2 − 2λ1mξ$ − 2λ1λ2m$

4 + 2λ1λ3m$
2 + ξ2 = 0 (23)

Solving the dispersion relation(23) for ξ, we obtain:
ξ = λ1m$ ∓

(
λ2

1$
4 + 2λ1λ2m

3$2 + λ2
2m

2$4 + 3λ2
1m

2$2 + 2λ1λ2m$
4

−2λ2λ4m
3$2 − 2λ2λ3m

2$2 − 2λ1λ4m
2$2 − 2λ1λ3m$

2

)
1
2 . (24)

In a situation whereby(
λ2

1$
4 + 2λ1λ2m

3$2 + λ2
2m

2$4 + 3λ2
1m

2$2 + 2λ1λ2m$
4

−2λ2λ4m
3$2 − 2λ2λ3m

2$2 − 2λ1λ4m
2$2 − 2λ1λ3m$

2

)
> 0 ,the wave number is real for

any real value of m and the steady-state is stable against small perturbations. However, in contraryto the above condition, the steady-state solution turns to be unstable, that is, the wave number
becomes imaginary, when ( λ2

1$
4 + 2λ1λ2m

3$2 + λ2
2m

2$4 + 3λ2
1m

2$2 + 2λ1λ2m$
4

−2λ2λ4m
3$2 − 2λ2λ3m

2$2 − 2λ1λ4m
2$2 − 2λ1λ3m$

2

)
< 0

and the perturbation grows exponentially. Under this condition, the growth rate of modulationstability gain spectrum G(m) may be given as
G(r) = 2Im(ξ)=

2Im

(
λ1m$ ∓

(
λ2

1$
4 + 2λ1λ2m

3$2 + λ2
2m

2$4 + 3λ2
1m

2$2 + 2λ1λ2m$
4

−2λ2λ4m
3$2 − 2λ2λ3m

2$2 − 2λ1λ4m
2$2 − 2λ1λ3m$

2

))
. (25)

The figure below shows the gained dispersion relation to investigate the steady-state stabilityby taking the following parameter values λ1 → 2, λ2 → 2.4, λ3 → 1.5, λ4 → 2.3 under distinct

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Eur. J. Math. Anal. 10.28924/ada/ma.5.14 11

Figure 7. Gain spectrum of MI under different values of m
values of m.

6. ConclusionsVariants of the NLSE are used to model the propagation of water waves in oceans and otherbodies of water. Nonlinear effects, such as wave steepening and wave breaking, can be describedusing these equations. Understanding these phenomena is crucial for predicting and mitigatingthe impact of tsunamis, storm surges, and other oceanic events. Furthermore, NLSE variant areemployed in the study of biological systems, such as modeling the propagation of nerve impulses.The NLSE can describe the nonlinear dynamics of excitable media, providing insights into the be-havior of electrical signals in biological tissues. Also, NLSE which are essential in fully-integratednonlinear dispersive partial differential equation (PDE), have found extensive application in elu-cidating diverse phenomena like deep water waves, rogue waves, plasmas, and nonlinear optics,including atomic physics. The NLSE serves as a comprehensive description of nonlinear dispersiveprocesses. With all of these in mind, we found the courage and motivation in this study to providethe distinct types of exact soliton solutions for the weakly nonlocal Schrodinger equation. We ob-tain dark and periodic singular soliton solutions via the reliable approach of the modified extendedtanh function method. The obtained solutions were verified by back-substitution in the originalequations, with the aid of a Mathematica to affirm the robustness of the chosen approach. The ob-tained results were portrayed using two-dimensional and three-dimensional graphs. Additionally,

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Eur. J. Math. Anal. 10.28924/ada/ma.5.14 12modulation instability was performed to study the stationary state of the governing model. Resultsare helpful in the progress of the concerned system. The gained results will be of high importancein the interaction of quantum-mechanical fluctuations, granular matters, and other fields of weaklynonlocal Schrodinger with parabolic law applications. The achieved results are also useful in var-ious naturally occurring phenomena, industries, geophysics, civil engineering, pharmaceutical, andmany others. It is suggested that the method used is also useful for the other nonlinear models ofdifferent fields of science and engineering.
References

[1] M.J. Ablowitz, Nonlinear Dispersive Waves: Asymptotic Analysis and Solitons, Cambridge Univ. Press, 47 (2011).[2] A.R.Z.U. Akbulut, M. Mirzazadeh, M.S. Hashemi, K. Hosseini, S. Salahshour, C. Park, Triki?Biswas Model: ItsSymmetry Reduction, Nucci’s Reduction and Conservation Laws, Int. J. Mod. Phys. B 37 (2023), 2350063.[3] L. Akinyemi, U. Akpan, P. Veeresha, H. Rezazadeh, M. Inc, Computational Techniques to Study the Dynamics ofGeneralized Unstable Nonlinear Schrodinger Equation, J. Ocean Eng. Sci. (2022).[4] L. Akinyemi, M. ?enol, M. Mirzazadeh, M. Eslami, Optical Solitons for Weakly Nonlocal Schrodinger Equation WithParabolic Law Nonlinearity and External Potential, Optik 230 (2021), 166281.[5] G. Akram, H.U. Rehman, Numerical Solution of Eighth Order Boundary Value Problems in Reproducing KernelSpace, Numer. Algorithms 62 (2013), 527–540.[6] K.K. Ali, A.M. Wazwaz, M.S. Osman, Optical Soliton Solutions to the Generalized Nonautonomous NonlinearSchrodinger Equations in Optical Fibers via the Sine-Gordon Expansion Method, Optik 208 (2020), 164132.[7] M.O. Al-Amr, H. Rezazadeh, K.K. Ali, A. Korkmazki, N1-Soliton Solution for Schrodinger Equation With CompetingWeakly Nonlocal and Parabolic Law Nonlinearities, Commun. Theor. Phys. 72 (2020), 065503.[8] T. An, N.H.M. Shahen, S.N. Ananna, M.F. Hossain, T. Muazu, Exact and Explicit Travelling-Wave Solutions to theFamily of New 3D Fractional WBBM Equations in Mathematical Physics, Results Phys. 19 (2020), 103517.[9] S.N. Ananna, P.P. Gharami, T. An, M. Asaduzzaman, The Improved Modified Extended Tanh-Function Method toDevelop the Exact Travelling Wave Solutions of a Family of 3D Fractional WBBM Equations, Results Phys. 41(2022), 105969.[10] M.I. Asjad, N. Ullah, H.U. Rehman, D. Baleanu, Optical Solitons for Conformable Space-Time Fractional Non-LinearModel, J. Math. Comput. Sci. 27 (2022), 1–9.[11] H.M. Baskonus, M.S. Osman, H.U. Rehman, M. Ramzan, M. Tahir, S. Ashraf, On Pulse Propagation of Soliton WaveSolutions Related to the Perturbed Chen?Lee?Liu Equation in an Optical Fiber, Opt. Quantum Electron. 53 (2021),1–17.[12] M.D. Feit, J.A. Fleck Jr., A. Steiger, Solution of the Schrodinger Equation by a Spectral Method, J. Comput. Phys.47 (1982), 412–433.[13] R.K. Gazizov, A.A. Kasatkin, Construction of Exact Solutions for Fractional Order Differential Equations by InvariantSubspace Method, Comput. Math. Appl. 66 (2013), 576–584.[14] V. Gokulakrishnan, R. Srinivasan, Impulsive Effects on Stabilization of Stochastic Nonlinear Reaction-DiffusionSystems With Time Delays and Boundary Feedback Control, J. Math. Comput. Sci. 28 (2023), 350–362.[15] M.S. Hashemi, A Novel Approach to Find Exact Solutions of Fractional Evolution Equations With Non-SingularKernel Derivative, Chaos Solitons Fractals 152 (2021), 111367.[16] M.S. Hashemi, D. Baleanu, Lie Symmetry Analysis of Fractional Differential Equations, Chapman and Hall/CRC,New York, 2020.

https://doi.org/10.28924/ada/ma.5.14


Eur. J. Math. Anal. 10.28924/ada/ma.5.14 13

[17] M.S. Hashemi, M. Inc, B. Kilic, A. Akcol, On Solitons and Invariant Solutions of the Magneto-Electro-ElasticCircular Rod, Waves Random Complex Media 26 (2016), 259–271.[18] M.S. Hashemi, M.C. Nucci, S. Abbasbandy, Group Analysis of the Modified Generalized Vakhnenko Equation,Commun. Nonlinear Sci. Numer. Simul. 18 (2013), 867–877.[19] K. Hosseini, S. Salahshour, M. Mirzazadeh, Bright and Dark Solitons of a Weakly Nonlocal Schrodinger EquationInvolving the Parabolic Law Nonlinearity, Optik 227 (2021), 166042.[20] L. Gagnon, P. Winternitz, Lie Symmetries of a Generalised Nonlinear Schrodinger Equation: I. The SymmetryGroup and Its Subgroups, J. Phys. A: Math. Gen. 21 (1988), 1493.[21] M.A. Iqbal, Y. Wang, M.M. Miah, M.S. Osman, Study on Date Jimbo Kashiwara Miwa Equation With ConformableDerivative Dependent on Time Parameter To and the Exact Dynamic Wave Solutions, Fractal Fract. 6 (2021), 4.[22] N.A. Kudryashov, Highly Dispersive Solitary Wave Solutions of Perturbed Nonlinear Schrodinger Equations, Appl.Math. Comput. 371 (2020), 124972.[23] W. Liu, Y. Zhang, Z. Luan, Q. Zhou, M. Mirzazadeh, M. Ekici, A. Biswas, Dromion-Like Soliton Interactions forNonlinear Schrodinger Equation With Variable Coefficients in Inhomogeneous Optical Fibers, Nonlinear Dyn. 96(2019), 729–736.[24] M. Mirzazadeh, A. Sharif, M.S. Hashemi, A. Akcol, S.M. El Din, Optical Solitons With an Extended (3+1)-Dimensional Nonlinear Conformable Schrodinger Equation Including Cubic-Quintic Nonlinearity, Results Phys.49 (2023), 106521.[25] M.C. Nucci, P.G.L. Leach, The Determination of Nonlocal Symmetries by the Technique of Reduction of Order, J.Math. Anal. Appl. 251 (2000), 871–884.[26] M.S. Osman, D. Baleanu, A.R. Adem, K. Hosseini, M. Mirzazadeh, M. Eslami, Double-Wave Solutions and LieSymmetry Analysis to the (2+1)-Dimensional Coupled Burgers Equations, Chin. J. Phys. 63 (2020), 122–129.[27] M.N. Rasoulizadeh, M.J. Ebadi, Z. Avazzadeh, O. Nikan, An Efficient Local Meshless Method for the Equal WidthEquation in Fluid Mechanics, Eng. Anal. Bound. Elem. 131 (2021), 258–268.[28] H.U. Rehman, I. Iqbal, S. Subhi Aiadi, N. Mlaiki, M.S. Saleem, Soliton Solutions of Klein Fock Gordon EquationUsing Sardar Subequation Method, Mathematics 10 (2022), 3377.[29] H.U. Rehman, N. Ullah, M.A. Imran, Highly Dispersive Optical Solitons Using Kudryashov’s Method, Optik 199(2019), 163349.[30] R. Sahadevan, P. Prakash, On Lie Symmetry Analysis and Invariant Subspace Methods of Coupled Time FractionalPartial Differential Equations, Chaos Solitons Fractals 104 (2017), 107–120.[31] F.L. Xia, F. Jarad, M.S. Hashemi, M.B. Riaz, A Reduction Technique to Solve the Generalized Nonlinear DispersivemK (m, n) Equation With New Local Derivative, Results Phys. 38 (2022), 105512.[32] Y. Xie, L. Li, Y. Kang, New Solitons and Conditional Stability to the High Dispersive Nonlinear SchrodingerEquation With Parabolic Law Nonlinearity, Nonlinear Dyn. 103 (2021), 1011–1021.[33] A. Zafar, M. Raheel, M. Asif, K. Hosseini, M. Mirzazadeh, L. Akinyemi, Some Novel Integration Techniques toExplore the Conformable M-Fractional Schrodinger?Hirota Equation, J. Ocean Eng. Sci. 7 (2022), 337–344.[34] Q. Zhou, D. Yao, S. Ding, Y. Zhang, F. Chen, X. Liu, Spatial Optical Solitons in Fifth Order and Seventh OrderWeakly Nonlocal Nonlinear Media, Optik 124 (2013), 5683–5686.[35] M. Daniel, L. Kavitha, R. Amuda, Soliton Spin Excitations in an Anisotropic Heisenberg Ferromagnet With Octupole-Dipole Interaction, Phys. Rev. B 59 (1999), 13774.[36] M. Daniel, Magnetization Reversal Through Soliton Flip in a Biquadratic Ferromagnet With Varying ExchangeInteractions, Phys. Rev. B 66 (2002), 184433.[37] G.P. Agrawal, Nonlinear Fiber Optics, Academic Press, New York, 2013.

https://doi.org/10.28924/ada/ma.5.14

	References

