
This work investigates the generalized nonlinear Schrödinger equation (NLSE), which imitates the wave transmission along optical fibers. This model incorporates a quintuple power-law of non-linearity and nonlinear chromatic dispersion. To demonstrate the significance and motivation for this work, a review of the prior research is presented in the literature. Three integration strategies are applied during the study process in order to produce a variety of novel solutions. These techniques include the modified exp-function approach, the general projective Riccati method (GPRM), and the new generalized method. The extracted solutions include bright solitons, singular solitons, dark solitons, and trigonometric solutions.
Citation: Islam Samir, Hamdy M. Ahmed, Wafaa Rabie, W. Abbas, Ola Mostafa. Construction optical solitons of generalized nonlinear Schrödinger equation with quintuple power-law nonlinearity using Exp-function, projective Riccati, and new generalized methods[J]. AIMS Mathematics, 2025, 10(2): 3392-3407. doi: 10.3934/math.2025157
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This work investigates the generalized nonlinear Schrödinger equation (NLSE), which imitates the wave transmission along optical fibers. This model incorporates a quintuple power-law of non-linearity and nonlinear chromatic dispersion. To demonstrate the significance and motivation for this work, a review of the prior research is presented in the literature. Three integration strategies are applied during the study process in order to produce a variety of novel solutions. These techniques include the modified exp-function approach, the general projective Riccati method (GPRM), and the new generalized method. The extracted solutions include bright solitons, singular solitons, dark solitons, and trigonometric solutions.
The field of telecommunications engineering has benefited greatly from optical soliton dynamics in recent decades [1,2,3]. There are two types of telecommunications engineering: Wireless communications, which transmit data over long distances using radio waves rather than wires, and wired communications, which use subterranean communications cables. Fiber optics elucidates nonlinear responses of incident light characteristics, including phase and polarization [4]. Numerous optical phenomena are produced by these nonlinear interactions. To enhance the performance of telecommunications engineering, many novel ideas have been proposed. Propagation dynamics of solitons is one of them. These consist of the nonlinear refractive index and chromatic dispersion (CD) of the optical fiber. The nonlinear Schrödinger model is widely applied to mimic many physical phenomena in different fields such as plasma physics, condensed matter, nonlinear optics, fluid dynamics, solid-state physics, biochemistry, and many more [5,6,7,8,9]. The soliton theory is essential to many models, including those in fiber optics and other fields. These models such as the Gerdjikov–Ivanov equation [10], Radhakrishnan—Kundu-–Lakshmanan equation [11], Chen–Lee–Liu equation [12,13], complex Ginzburg–Landau model [14,15], Gilson–Pickering equation [16,17] and Lakshmanan–Porsezian–Daniel model [18]. One of the basic models for nonlinear waves is NLSE, which describes nonlinear phenomena in optical fibers and is also used to describe spatial solitons [19]. It is necessary to solve nonlinear partial differential equations (NLPDEs) analytically in order to understand these nonlinear phenomena [20,21,22]. Many academics are interested in the extraction of solitons in NLPDEs. There are efficient techniques that have been put forth to get exact solutions to NLPDEs including extended hyperbolic function technique [23], the Sardar sub-equation method [24], the extended Tanh–Coth scheme [25], the Jacobi elliptic function method [26] and many more.
This work examines generalized NLSE with nonlinear chromatic dispersion and quintuple power law non-linearity. This model is as follows: [27]:
iψt+α(|ψ|pψ)xx+(k1|ψ|2m+k2|ψ|2m+n+k3|ψ|2m+2n+k4|ψ|2m+n+r+k5|ψ|2m+2n+r+(|ψ|r)xx)ψ=0, | (1.1) |
where ψx,t represents the wave profile. α is the coefficient of the nonlinear CD. k1,k2,k3,k4, and k5 are the coefficients of nonlinear refractive index. p≥0 and r,m, and n are positive constants. When p=0, it becomes the case of linear chromatic dispersion in the exceptional case. The possible values of the parameters remain unknown, though. Benjamin–Feir stability analysis could be used to aid with this issue. This project is distinct and should be handled later. This work focuses on soliton solutions with non-negative parameters. Optical solitons have become a popular topic in fibre optics during the past decade due to their potential to block pulse transmission over intercontinental distance transmission [28,29]. Pre-existing and current publications serve as models for studying the genesis and existence of optical solitons. In order to obtain stationary soliton solutions, the proposed model was examined using the G′/G expansion approach [30]. In addition, the modified extended mapping approach is implemented to derive dark and singular solitons [27].
Three integration strategies are used in this study to find novel and different solutions for Eq (1.1). These techniques including the modified exp-function method, the general projective Riccati method (GPRM), and new generalized method. With the aid of these techniques, different solutions can be offered, including trigonometric solutions, singular solitons, dark solitons, and bright solitons.
This paper's structure will be as follows. Section 2 provides a brief introduction to the proposed techniques. Analytical solutions for Eq (1.1) are then provided in Section 3 using the proposed techniques. Section 4 presents both 2D and 3D graphs to show the properties of the propagating waves. In the last section, the manuscript is concluded.
This section contains concise summaries of the modified exp-function approach [31], GPRM [32] and new generalized method [4]
Let us consider NLPDE as follows:
F(ψ,ψt,ψx,ψxx,...)=0, | (2.1) |
where F is polynomial in ψ(x,t).
Suppose ψ(x,t)=ψ(ξ), ξ=k(x−ct). Then, Eq (2.1) is transformed to a nonlinear ordinary differential equation (NLODE) as follows:
Q(ψ,ψ′,ψ′′,ψ′′′,...)=0. | (2.2) |
Step 1. The solution to the generated NLODE is considered as follows:
ψ(ξ)=N∑j=−Naj(exp(−ϕ(ξ)))j, | (2.3) |
where aj are constants to be determined and ϕ meets the following auxiliary equation
ϕ′(ξ)=−√λ1+λ2(exp(−ϕ(ξ)))2,λ1,λ2∈R. | (2.4) |
Step 2. Applying the balance rule in Eq (2.2), N can be raised.
Step 3. A set of nonlinear equations is raised by putting (2.3) and (2.4) into (2.2), then gathering and equating all the terms of (exp(−ϕ(ξ)))j to zero. The extracted system can be handled by Mathematica or Maple software tools to obtain all unknowns.
Step 4. Finally, we obtain the new exact solutions for NLPDE by combining Eq (2.3) with the general solutions of Eq (2.4).
Step 5. To obtain more exact solutions, the steps from (1) to (4) will be repeated again by using the following auxiliary equation:
ϕ′(ξ)=−λ1exp(ϕ(ξ))−λ2exp(−ϕ(ξ)),λ1,λ2∈R. | (2.5) |
The general solutions of Eq (2.4) are:
Ψ(ξ)=−ln(−√λ1λ2csch[√λ1ξ]),λ1>0,λ2>0, | (2.6) |
Ψ(ξ)=−ln(√−λ1λ2sec[√−λ1ξ]),λ1<0,λ2>0, | (2.7) |
Ψ(ξ)=−ln(√λ1−λ2sech[√λ1ξ]),λ1>0,λ2<0, | (2.8) |
Ψ(ξ)=−ln(√−λ1λ2csc[√−λ1ξ]),λ1<0,λ2>0, | (2.9) |
while the general solutions of Eq (2.5) are
Ψ(ξ)=−ln(√λ1λ2tan[√λ1λ2ξ]),λ1λ2>0, | (2.10) |
Ψ(ξ)=−ln(−√λ1λ2cot[√λ1λ2ξ]),λ1λ2>0, | (2.11) |
Ψ(ξ)=−ln(√λ1−λ2tanh[√−λ1λ2ξ]),λ1λ2<0, | (2.12) |
Ψ(ξ)=−ln(√λ1−λ2coth[√−λ1λ2ξ]),λ1λ2<0. | (2.13) |
Step 1. Assuming that Eq (2.2) has a solution of the form
ψ(ξ)=s0+N∑i=1ϕi−1(ξ)(siϕ(ξ)+βiφ(ξ)), | (2.14) |
where the functions ϕ(ξ) and φ(ξ) fulfill the next equation:
ϕ′(ξ)=ϵϕ(ξ)φ(ξ),φ′(ξ)=σ+ϵφ2(ξ)−δϕ(ξ), | (2.15) |
where
φ2(ξ)=−ϵ(σ−2δϕ(ξ)+δ2+τσϕ2(ξ)). | (2.16) |
The following are the general solutions to Eq (2.15):
Set (1): ϵ=τ=−1:
ϕ(ξ)=σsech[√σξ]1+δsech[√σξ],φ(ξ)=√σtanh[√σξ]1+δsech[√σξ], | (2.17) |
Set (2): ϵ=−1, τ=1:
ϕ(ξ)=σcsch[√σξ]1+δcsch[√σξ],φ(ξ)=√σcoth[√σξ]1+δcsch[√σξ], | (2.18) |
Set (3): ϵ=1, τ=−1:
ϕ(ξ)=σsec[√σξ]1+δsec[√σξ],φ(ξ)=√σtan[√σξ]1+δsec[√σξ], | (2.19) |
Set (4): ϵ=τ=1:
ϕ(ξ)=σcsc[√σξ]1+δcsc[√σξ],φ(ξ)=−√σcot[√σξ]1+δcsc[√σξ]. | (2.20) |
Step 2. Along with (2.15) and (2.16), (2.14) is substituted into (2.2). A polynomial of φ(ξ) and ϕ(ξ) is thus obtained. A collection of nonlinear equations is generated via equating terms of the identical power to zero. The extracted system can be handled by Mathematica or Maple software tools to assess all unknowns. Consequently, we obtain exact solutions for Eq (2.1).
Step 1. The solution of the obtained NLODE is considered as follows:
ψ(ξ)=α0+N∑i=1αi+βiϕ′(ξ)iϕ(ξ)i, | (2.21) |
and ϕ(ξ) fulfills the next equation:
ϕ′(ξ)=√−σ+ϕ(ξ)2, | (2.22) |
and
ϕ(n)(ξ)=ϕ(ξ),n≥2andnis even,ϕ(n)(ξ)=ϕ′(ξ),n≥2andnis odd. | (2.23) |
Equation (2.22) has the following solution:
Φ(ξ)=aeξ+σ4aeξ, | (2.24) |
where σ and a are constants.
Step 2. Substituting (2.21) into (2.2) along with (2.22) and its derivatives (2.23). This substitution yields a polynomial of the form 1Φ(ξ)(Φ′(ξ)Φ(ξ)). Once all of the terms in this polynomial are combined and their values are set to zero, an overdetermined system of algebraic equations is produced. All of the unknowns in this system can be found by solving it with Mathematica. Consequently, we have analytical solutions for Eq (2.1).
Our goal is to achieve analytic solutions in the following form for Eq (1.1):
ψ(x,t)=V(kx)ei(wt+θ), | (3.1) |
where θ represents the phase constant and w is the wave number. k is a constant. Inserting Eq (3.1) into Eq (1.1) yields
αk2(p+1)Vp+1V′′+αk2p(p+1)VpV′2+k4V2m+n+r+2+k5V2m+2n+r+2+k3V2m+2n+2+k2V2m+n+2+k1V2m+2+k2rVr+1V′′+k2(r−1)rVrV′2−wV2=0. | (3.2) |
Taking p=2m+r and using the transformation
V=Q1n, | (3.3) |
Eq (3.2) becomes
n2Q2(−w+k1Q2mn+k2Q2m+nn+k3Q2m+2nn+k4Q2m+n+rn+k5Q2m+2n+rn)+k2r(r−1)QrnQ′2+αk2(r+2m)(r+2m+1)Q′2Q2m+rn+k2rQrn(nQQ′′+(1−n)Q′2)αk2(2m+r+1)Q2m+rn(nQQ′′+(1−n)Q′2)=0. | (3.4) |
Setting n=r=2m, Eq (3.4) becomes
2αmk2(1+4m)QQ′′+αk2(1+2m)(1+4m)Q′2+4k5m2Q4+4m2(k3+k4)Q3+4k2m2Q2+4k1m2Q+4k2m2Q′′−4m2w=0. | (3.5) |
Balancing QQ″ with Q4 in Eq (3.5) results in N=1.
By applying the modified exponential method, the following form is the solution to Eq (3.5)
Q(x)=a0+a1exp(−ϕ(x))+a−1exp(ϕ(x)). | (3.6) |
Applying the steps discussed in Section 2.1, we get the next results:
a0=−αk3−αk4−24αk3m2−24αk4m2+8k5m2−10αk3m−10αk4m4αk5(20m2+9m+1),a−1=0,w=a0k2(a20λ2(αa0(8m2+6m+1)−8m2)+a21λ1(αa0(8m2+6m+1)−4m2))4a21m2,λ2=−4a21k5m2αk2(24m2+10m+1),λ1=2a21k1m2−2a20k2λ2(αa0(3m+1)(4m+1)−6m2)a21k2(αa0(3m+1)(4m+1)−2m2). |
Afterwards, we have
ψ(x,t)={a0−√−4a20k5(αa0(12m2+7m+1)−6m2)+αk1(24m2+10m+1)2k5(αa0(12m2+7m+1)−2m2)×csch(√2x√m2(4a20k5(αa0(12m2+7m+1)−6m2)+αk1(24m2+10m+1))α(24m2+10m+1)(αa0(12m2+7m+1)−2m2))}1/nei(wt+θ), | (3.7) |
ψ(x,t)={a0+√4a20k5(αa0(12m2+7m+1)−6m2)+αk1(24m2+10m+1)2k5(αa0(12m2+7m+1)−2m2)×sech(√2x√m2(4a20k5(αa0(12m2+7m+1)−6m2)+αk1(24m2+10m+1))α(24m2+10m+1)(αa0(12m2+7m+1)−2m2))}1/nei(wt+θ), | (3.8) |
ψ(x,t)={a0+√4a20k5(αa0(12m2+7m+1)−6m2)+αk1(24m2+10m+1)2k5(αa0(12m2+7m+1)−2m2)×csc(√2x√−m2(4a20k5(αa0(12m2+7m+1)−6m2)+αk1(24m2+10m+1))α(24m2+10m+1)(αa0(12m2+7m+1)−2m2))}1/nei(wt+θ), | (3.9) |
ψ(x,t)={a0+√4a20k5(αa0(12m2+7m+1)−6m2)+αk1(24m2+10m+1)2k5(αa0(12m2+7m+1)−2m2)×sec(√2x√−m2(4a20k5(αa0(12m2+7m+1)−6m2)+αk1(24m2+10m+1))α(24m2+10m+1)(αa0(12m2+7m+1)−2m2))}1/nei(wt+θ). | (3.10) |
A singular solitary is provided by Eq (3.7), while a bright solitary is provided by Eq (3.8). Equations (3.9) and (3.10) represent singular periodic solutions.
Substituting by the auxiliary equation of (2.5) instead of Eq (2.4), we obtain the next results for Eq (1.1).
Result (1):
a0=a−1=0,λ1=−k12k2λ2,a1=√2kλ2√−k3−k4,w=αa21k2λ21(8m2+6m+1)4m2,k5=−αk2λ22(24m2+10m+1)4a21m2,k2=−αk2λ1λ2(4m+1)22m2. |
The following solutions are then provided
ψ(x,t)={√k1tanh(√k12x)√−k3−k4}1/nei(wt+θ),k1>0,k3+k4<0, | (3.11) |
ψ(x,t)={√k1coth(√k12x)√−k3−k4}1/nei(wt+θ),k1>0,k3+k4<0, | (3.12) |
ψ(x,t)={√−k1tan(√−k12x)√−k3−k4}1/nei(wt+θ),k1<0,k3+k4<0, | (3.13) |
ψ(x,t)={−√−k1cot(√−k12x)√−k3−k4}1/nei(wt+θ),k1<0,k3+k4<0. | (3.14) |
The dark soliton is provided by Eq (3.11), whereas the singular soliton is provided by Eq (3.12). Equations (3.13) and (3.14) represent singular periodic solutions.
Result (2):
a0=0,a−1=−a1λ1λ2,λ1=−k18k2λ2,a1=√2kλ2√−k3−k4,w=4αa21k2λ21(8m2+6m+1)m2,k5=−αk2λ22(24m2+10m+1)4a21m2,k2=−2αk2λ1λ2(4m+1)2m2. |
Afterwards, we have
ψ(x,t)={√k1tanh(√k1x2√2)(coth2(√k1x2√2)+1)2√−k3−k4}1/nei(wt+θ),k1>0,k3+k4<0, | (3.15) |
ψ(x,t)={√−k1csc(√−k1x√2)√−k3−k4}1/nei(wt+θ),k1<0,k3+k4<0. | (3.16) |
A singular solitary is provided by Eq (3.15) while a singular periodic solution is provided by Eq (3.16).
By applying GPRM, solution of Eq (3.5) is represented as:
Q(x)=s0+s1ϕ(x)+β1φ(x). | (3.17) |
Applying Step-2 of the proposed method, we get the following results:
Case (1). When τ=−1 and ϵ=−1
s0=0,w=αβ61(δ2−1)2k2(8m2+6m+1)16m2s41,β1=k√2√−(k3+k4),s1=√α√1−δ2k2(4m+1)√16k2k3m2+16k2k4m2,σ=2k1k2,k5=−αk2(24m2+10m+1)16β21m2. |
Afterwards, we have
ψ(x,t)={√α√δ2−1k1(4m+1)sech(√2√k1x)+2√k1√k2mtanh(√2√k1x)2√k2√−k3−k4(δmsech(√2√k1x)+m)}1/nei(wt+θ),k1>0,k3+k4<0,k2>0,α(δ2−1)>0. | (3.18) |
Case (2). When τ=1 and ϵ=−1
s0=0,w=αβ61(δ2+1)2k2(8m2+6m+1)16m2s41,s1=√α√−δ2−1k2(4m+1)√16k2k3m2+16k2k4m2,β1=k√2√−k3−k4,σ=2k1k2,k5=−αk2(24m2+10m+1)16β21m2. |
Afterwards, we have
ψ(x,t)={√α√δ2+1k1(4m+1)csch(√2√k1x)+2√k1√k2mcoth(√2√k1x)2√k2√−k3−k4(δmcsch(√2√k1x)+m)}1/nei(wt+θ),k1>0,k3+k4<0,k2>0,α>0. | (3.19) |
Case (3). When τ=−1 and ϵ=1
s0=0,w=αβ61(δ2−1)2k2(8m2+6m+1)16m2s41,s1=√α√1−δ2k2(4m+1)√16k2k3m2+16k2k4m2,β1=k√2√−k3−k4,σ=−2k1k2,k5=−αk2(24m2+10m+1)16β21m2. |
Afterwards, we have
ψ(x,t)={√α√δ2−1k1(−4m−1)+2√−k1√k2msin(√2√−k1x)2√k2√−k3−k4m(δ+cos(√2√−k1x))}1/nei(wt+θ),k1<0,k3+k4<0,k2>0,α(δ2−1)>0. | (3.20) |
Case (4). When τ=1 and ϵ=1
s0=0,w=αβ61(δ2+1)2k2(8m2+6m+1)16m2s41,s1=√α√−δ2−1k2(4m+1)√16k2k3m2+16k2k4m2,β1=k√2√−k3−k4,σ=−2k1k2,k5=−αk2(24m2+10m+1)16β21m2. |
Afterwards, we have
ψ(x,t)={−√α√δ2+1k1(4m+1)−2√−k1√k2mcos(√2√−k1x)2√k2√−k3−k4m(δ+sin(√2√−k1x))}1/nei(wt+θ),k1<0,k3+k4<0,k2>0,α>0. | (3.21) |
By applying the new generalized method, the following form is the solution to Eq (3.5)
Q(x)=α0+β1ϕ′(x)+α1ϕ(x), | (3.22) |
Applying Step-2 of the proposed method, We obtain the findings listed below:
Result (1):
α0=m2(2k2σ−α21k3−α21k4)αk2(20m2+9m+1)σ,β1=0,α1=√αk√24m2+10m+1√σ2√k5m,w=α0k2(α20σ(8m2−αα0(8m2+6m+1))+α21(αα0(8m2+6m+1)−4m2))4α21m2,k2=k2(−24α0m2σ+6αα20(4m+1)2σ−αα21(4m+1)2)4α21m2,k1=k2(−2αα30(12m2+7m+1)σ+12α20m2σ+αα21α0(12m2+7m+1)−2α21m2)2α21m2. |
Then, we have the following solutions
ψ(x,t)={14(8a√αk√24m2+10m+1√σex√k5(4a2me2x+mσ)−(k3+k4)(6m+1)k5(5m+1)+8m2α+20αm2+9αm)}1/nei(wt+θ). | (3.23) |
When σ=±4a2, Eq (3.23) gives a bright soliton solution
ψ(x,t)={14(2√αk√(4m+1)(6m+1)sech(x)√k5m−(k3+k4)(6m+1)k5(5m+1)+8m2α+20αm2+9αm)}1/nei(wt+θ), | (3.24) |
and a singular soliton solution
ψ(x,t)={14(2√−αk√(4m+1)(6m+1)csch(x)√k5m−(k3+k4)(6m+1)k5(5m+1)+8m2α+20αm2+9αm)}1/nei(wt+θ). | (3.25) |
Result (2):
α0=m2(2k2+β21k3+β21k4)αk2(20m2+9m+1),α1=0,β1=√−αk√24m2+10m+12√k5m,w=k2(β21−α20)(αβ21(8m2+6m+1)+8α0m2−αα20(8m2+6m+1))4β21m2,k2=k2(12α0m2+αβ21(4m+1)2−3αα20(4m+1)2)2β21m2,k1=k2(−αα0β21(12m2+7m+1)+αα30(12m2+7m+1)−6α20m2+2β21m2)β21m2. |
Afterwards, we have
ψ(x,t)={√−αk√24m2+10m+1(aex−σe−x4a)2√k5m(σe−x4a+aex)+2m2α(20m2+9m+1)}1/nei(wt+θ). | (3.26) |
When σ=±4a2, Eq (3.26) gives a dark soliton solution
ψ(x,t)={14(2√−αk√(4m+1)(6m+1)tanh(x)√k5m−(k3+k4)(6m+1)k5(5m+1)+8m2α+20αm2+9αm)}1/nei(wt+θ), | (3.27) |
and a singular soliton solution
ψ(x,t)={14(2√−αk√(4m+1)(6m+1)coth(x)√k5m−(k3+k4)(6m+1)k5(5m+1)+8m2α+20αm2+9αm)}1/nei(wt+θ). | (3.28) |
The nature of some extracted solutions is demonstrated by the presentation of graphic simulations. A bright solitary solution of Eq (3.24) with m=n=1,k=k3=k4=α=−2,k5=−0.36 is shown in Figure 1. A dark solitary solution of Eq (3.27) with n=m=1,k=k3=α=−2,k4=2,k5=0.56 is graphically illustrated in Figure 2. The extracted solutions demonstrate a certain balance between dispersion and nonlinearity for the investigated model. This results in solitons that have the advantage of being highly stable, as they can travel very long distances while still maintaining their velocity and shape. Figure 3 depicts the graphical illustration of a singular solitary solution of Eq (3.28) with n=m=1,k=k3=α=−2,k4=1.75,k5=2. This solution represents a rare phenomenon in nonlinear physics, characterized by a point of singularity or divergence in intensity. It captures the abrupt change at the point, offering insight into the interplay of nonlinearity and dispersion in forming exotic solitary waves. Figure 4 depicts the graphical illustration of a singular periodic solution of Eq (3.16) with k1=k3=k4=−2. This solution displays a periodically repeating wave with a point of singularity, showing valuable insights into the behavior of nonlinear systems with recurring singularities.
The generalized NLSE, which simulates wave transmission via optical fibers, was examined in this work. This model incorporates a quintuple power-law of non-linearity and nonlinear chromatic dispersion. Three integration techniques are implemented to conduct this study. These techniques are the modified exp-function method, the general projective Riccati method, and the new generalized method. Different solutions are derived for the studied model, including singular solitons, bright solitons, dark solitons, and singular periodic solutions. These findings may be useful for optical communication and for a deeper comprehension of many phenomena that arise in various physical systems that are governed by the current model.
Islam Samir: Formal analysis, Software; Hamdy M. Ahmed: Validation, Methodology; Wafaa B. Rabie: Resources, Writing–review & editing; W. Abbas: Writing–review & editing, Investigation; Ola Mostafa: Software, Investigation. All authors have read and approved the final version of the manuscript for publication.
The authors declare that they have not used Artificial Intelligence tools in the creation of this article.
All authors declare no conflicts of interest in this paper.
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