
Citation: Shimao Fan, Michael Herty, Benjamin Seibold. Comparative model accuracy of a data-fitted generalized Aw-Rascle-Zhang model[J]. Networks and Heterogeneous Media, 2014, 9(2): 239-268. doi: 10.3934/nhm.2014.9.239
[1] | 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. AIMS Mathematics, 2025, 10(2): 3392-3407. doi: 10.3934/math.2025157 |
[2] | Elsayed M. E. Zayed, Mona El-Shater, Khaled A. E. Alurrfi, Ahmed H. Arnous, Nehad Ali Shah, Jae Dong Chung . Dispersive optical soliton solutions with the concatenation model incorporating quintic order dispersion using three distinct schemes. AIMS Mathematics, 2024, 9(4): 8961-8980. doi: 10.3934/math.2024437 |
[3] | Muhammad Bilal, Javed Iqbal, Ikram Ullah, Aditi Sharma, Hasim Khan, Sunil Kumar Sharma . Novel optical soliton solutions for the generalized integrable (2+1)- dimensional nonlinear Schrödinger system with conformable derivative. AIMS Mathematics, 2025, 10(5): 10943-10975. doi: 10.3934/math.2025497 |
[4] | Mahmoud Soliman, Hamdy M. Ahmed, Niveen Badra, M. Elsaid Ramadan, Islam Samir, Soliman Alkhatib . Influence of the $ \beta $-fractional derivative on optical soliton solutions of the pure-quartic nonlinear Schrödinger equation with weak nonlocality. AIMS Mathematics, 2025, 10(3): 7489-7508. doi: 10.3934/math.2025344 |
[5] | Yazid Alhojilan, Islam Samir . Investigating stochastic solutions for fourth order dispersive NLSE with quantic nonlinearity. AIMS Mathematics, 2023, 8(7): 15201-15213. doi: 10.3934/math.2023776 |
[6] | Dumitru Baleanu, Kamyar Hosseini, Soheil Salahshour, Khadijeh Sadri, Mohammad Mirzazadeh, Choonkil Park, Ali Ahmadian . The (2+1)-dimensional hyperbolic nonlinear Schrödinger equation and its optical solitons. AIMS Mathematics, 2021, 6(9): 9568-9581. doi: 10.3934/math.2021556 |
[7] | Jinfang Li, Chunjiang Wang, Li Zhang, Jian Zhang . Multi-solitons in the model of an inhomogeneous optical fiber. AIMS Mathematics, 2024, 9(12): 35645-35654. doi: 10.3934/math.20241691 |
[8] | Ninghe Yang . Exact wave patterns and chaotic dynamical behaviors of the extended (3+1)-dimensional NLSE. AIMS Mathematics, 2024, 9(11): 31274-31294. doi: 10.3934/math.20241508 |
[9] | Abeer S. Khalifa, Hamdy M. Ahmed, Niveen M. Badra, Wafaa B. Rabie, Farah M. Al-Askar, Wael W. Mohammed . New soliton wave structure and modulation instability analysis for nonlinear Schrödinger equation with cubic, quintic, septic, and nonic nonlinearities. AIMS Mathematics, 2024, 9(9): 26166-26181. doi: 10.3934/math.20241278 |
[10] | Nafissa T. Trouba, Huiying Xu, Mohamed E. M. Alngar, Reham M. A. Shohib, Haitham A. Mahmoud, Xinzhong Zhu . Soliton solutions and stability analysis of the stochastic nonlinear reaction-diffusion equation with multiplicative white noise in soliton dynamics and optical physics. AIMS Mathematics, 2025, 10(1): 1859-1881. doi: 10.3934/math.2025086 |
The nonlinear Schrödinger equations (NLSEs) are extensively used to describe numerous crucial phenomena and dynamic processes in various fields such as fluid dynamics, plasma, chemistry, biology, optical fibers[1,2,3,4], nuclear physics, stochastic mechanics, biomolecule dynamics, dynamics of accelerators, and Bose-Einstein condensates [5,6,7,8,9,10]. The last few decades have seen significant advancements in the field of nonlinear optics [11,12,13]. These equations also appear in other forms of nonlinearities, such as cubic-quintic (CQ), cubic-quintic-septic (CQS), power-law, logarithmic nonlinearities, and various other forms. There is a growing interest in studying soliton pulses that can propagate without changing their shape in optical fibers [14,15].
Therefore, obtaining the exact soliton solutions of these NLSEs can help us to understand these phenomena better. In recent years, several effective approaches have been developed to construct accurate solutions of these equations. For example, the simple equation method [16], Kudryashov's method [17], the Jacobi elliptic function method [18], the inverse scattering method [19], the extended trial function method [20], the tanh method [21], the F-expansion method [22], generalized extended tanh-function method, the sine–cosine method [23], the generalized Riccati equation method [24], the new $ \phi^6 $-model expansion method [25], Hirota bilinear method [26], the exp-function method [27], the Darboux transformation [28], the auxiliary equation method [29], the Binary Bell polynomials [30], the extended hyperbolic function method [31], the homogeneous balance Method [32], the $ (\frac{G'}{G}) $-expansion method [33,34], the exponential rational function method [35], the Bäcklund transformation [36], the homotopy perturbation method [37], the modified Kudryashov method, the sine-Gordon expansion approach [38], the Riccati-Bernoulli sub-ODE method [39], the modified extended direct algebraic method [40], the truncated Painlevé expansion method [41], the $ (G'/G, 1/G) $-expansion method [42], the tan$ (\phi/2) $-expansion method [43,44], the soliton ansatz method [45,46], and so on.
Researches have been conducted on the study of NLSEs with the polynomial law of nonlinearity. For instance, Seadawy et al. [47] established soliton solutions using the extended simplest equation method, and they also described this model [48], which includes the conservation principles for optical soliton (OS) with polynomial and triple power law nonlinearities, while Aziz et al. [49] discovered chirped soliton solutions by utilizing Jacobi elliptic functions. Furthermore, this model was discussed by Dieu-donne et al. [50], who examined the optical soliton (OS) solutions with two types of nonlinearities, which are triple power and polynomial laws. Sugati et al.[51] also examined this model, obtaining the traveling pulse solutions to the NLSE when it is treated with both spatio-temporal dispersion (STD) and group velocity dispersion (GVD).
In this paper, our aim is to investigate soliton solutions for two models of the NLSE that carry the polynomial law of nonlinearity (cubic-quintic-septic). We will use the variational principle based on finding Lagrangian, which we will then apply it with different trial functions that have one or two nontrivial variational parameters. Furthermore, we will employ another technique called the amplitude ansatz method to extract new solitary wave solutions.
The paper is categorized as follows. In Section 2, we discuss model-I of NLSE with polynomial law nonlinearity in terms of formulation of the variational principle and finding solitary wave solutions. In Section 3, we apply the same steps on model-II of NLSE with polynomial law nonlinearity. Finally, the work concludes in Section 4.
Nonlinear Schrödinger equation (NLSE) with the polynomial nonlinear law is [47]:
$ iΓt+aΓxx+b1Γ|Γ|2+b2Γ|Γ|4+b3Γ|Γ|6=0. $
|
(2.1) |
Such that $ \Gamma $ is a complex function on the form: $ \Gamma(x, t) = \Theta(x, t)+i\Psi(x, t) $. Since $ \Theta $ and $ \Psi $ are real functions of $ x $ and $ t $, furthermore $ |\Gamma|^2 = (\Theta+i\Psi)(\Theta-i\Psi) $. Using the variational approach, we will search for solutions to NLSE with the polynomial nonlinear law. As a result, we derive $ \Gamma $ with respect to $ x $ and $ t $ to investigate the existence of a Lagrangian and the invariant variational principle for this equation, which are expressed in the following way:
Let $ M $ and $ N $ are functionals in $ \Theta $ and $ \Psi $:
$ M(Θ,Ψ)=∂Θ∂t+a∂2Ψ∂x2+b1ΨΘ2+b1Ψ3+b2ΨΘ4+2b2Θ2Ψ3+b2Ψ5+b3ΨΘ6+3b3Θ4Ψ3+3b3Θ2Ψ5+b3Ψ7, $
|
(2.2) |
$ N(Θ,Ψ)=−∂Ψ∂t+a∂2Θ∂x2+b1Θ3+b1ΘΨ2+b2Θ5+2b2Ψ2Θ3+b2ΘΨ4+b3Θ7+3b3Ψ2Θ5+3b3Θ3Ψ4+b3ΘΨ6. $
|
(2.3) |
Put $ \Theta = \lambda\Theta $ and $ \Psi = \lambda\Psi $,
$ ∫10M(λΘ,λΨ)dλ=12∂Θ∂t+12a∂2Ψ∂x2+14b1ΨΘ2+14b1Ψ3+16b2ΨΘ4+13b2Θ2Ψ3+16b2Ψ5+18b3ΨΘ6+38b3Θ4Ψ3+38b3Θ2Ψ5+18b3Ψ7, $
|
$ ∫10N(λΘ,λΨ)dλ=−12∂Ψ∂t+12a∂2Θ∂x2+14b1Θ3+14b1ΘΨ2+16b2Θ5+13b2Ψ2Θ3+16b2ΘΨ4+18b3Θ7+38b3Ψ2Θ5+38b3Θ3Ψ4+18b3ΘΨ6. $
|
The consistency conditions are expressed as follows [52,53], where $ \Theta_x $, $ \Psi_x $, $ \Theta_t $ and $ \Psi_t $ stand for the partial derivatives of $ \Theta $ and $ \Psi $ with respect to variables $ x $ and $ t $. If the system of Eqs (2.2) and (2.3) satisfies the the previously mentioned conditions, then a functional integral $ J(\Theta, \Psi) $ can be written down using the formula given by Tonti [53]:
$ J(Θ,Ψ)=∫ω[14b1Θ4+16b2Θ6+18b3Θ8+12b1Θ2Ψ2+12b2Θ4Ψ2+12b3Θ6Ψ2+14b1Ψ4+12b2Θ2Ψ4+34b3Θ4Ψ4+16b2Ψ6+12b3Θ2Ψ6+18b3Ψ8+12ΨΘt−12ΘΨt+12aΘΘxx+12aΨΨxx]dω, $
|
where $ d\omega = dxdt $.
The terms $ \Theta\Theta_{xx} $ and $ \Psi\Psi_{xx} $ solve integration by parts and choosing the boundary on $ \Theta_x $ and $ \Psi_x $ to be such that the boundary terms vanish, we get
$ J(Θ,Ψ)=∫ω[14b1Θ4+16b2Θ6+18b3Θ8+12b1Θ2Ψ2+12b2Θ4Ψ2+12b3Θ6Ψ2+14b1Ψ4+12b2Θ2Ψ4+34b3Θ4Ψ4+16b2Ψ6+12b3Θ2Ψ6+18b3Ψ8+12ΨΘt−12ΘΨt−12aΘ2x−12aΨ2x]dω, $
|
(2.4) |
where $ d\omega = dxdt $.
We obtain the Lagrangian $ L $ as
$ L(Θ,Ψ)=14b1Θ4+16b2Θ6+18b3Θ8+12b1Θ2Ψ2+12b2Θ4Ψ2+12b3Θ6Ψ2+14b1Ψ4+12b2Θ2Ψ4+34b3Θ4Ψ4+16b2Ψ6+12b3Θ2Ψ6+18b3Ψ8+12ΨΘt−12ΘΨt−12aΘ2x−12aΨ2x. $
|
(2.5) |
We find the value of $ L $ in the Euler-Lagrange equations to validate our interpretations:
$ \frac{\partial L}{\partial \Theta}-\frac{\partial}{\partial t}\left(\frac{\partial L}{\partial \Theta_t}\right)-\frac{\partial}{\partial x}\left(\frac{\partial L}{\partial \Theta_{x}} \right) = 0, $ |
$ \frac{\partial L}{\partial \Psi}-\frac{\partial}{\partial t}\left(\frac{\partial L}{\partial \Psi_t}\right)-\frac{\partial}{\partial x}\left(\frac{\partial L}{\partial \Psi_{x}} \right) = 0. $ |
The resulting derivatives give us the system of Eqs (2.2) and (2.3).
We demonstrate the simplest example of the application of this technique, by taking the box-shaped initial pulse and an ansatz based on linear Jost functions in a single nontrivial variational parameter in three cases.
Case 1: We use the following ansatz for $ \Theta(x, t) $ and $ \Psi(x, t) $ functions:
$ Θ(x,t)={100exp(−μ(t−5)(x−5)),atx>5,t>5,(t+5)(x+5),at|x|<5,|t|<5,0,atx<−5,t<−5, $
|
(2.6) |
$ Ψ(x,t)={0,atx>5,t>5,(5−t)(5−x),at|x|<5,|t|<5,100exp(μ(t+5)(x+5)),atx<−5,t<−5. $
|
(2.7) |
We found the values of the Lagrangian $ L $ from substituting Eqs (2.6) and (2.7) into Eq (2.5) (see Figure 1):
$ J(\Theta, \Psi) = \int_{-\infty}^{-5} \int_{-\infty}^{-5} L \; dxdt + \int_{-5}^{5} \int_{-5}^{5} L \; dxdt + \int_{5}^{\infty} \int_{5}^{\infty} L \; dxdt, $ |
where
$ \int_{-\infty}^{-5} \int_{-\infty}^{-5} L \; dxdt = \int_{5}^{\infty} \int_{5}^{\infty} L \; dxdt = 0. $ |
By using Mathematica program we get the value of $ J(\Theta, \Psi) $ at interval $ -5 < x < 5, \, -5 < t < 5 $ as
$ J(Θ,Ψ)=−10000a3+1850000000b19+304000000000000b2441+1760000000000000000b3567+250003. $
|
(2.8) |
Case 2: Assume $ \Theta(x, t) $ and $ \Psi(x, t) $ for the Jost function as the following:
$ Θ(x,t)={20exp(−μ(t−5)(x−5)),atx>5,t>5,t+x+10,at|x|<5,|t|<5,0,atx<−5,t<−5, $
|
(2.9) |
$ Ψ(x,t)={0,atx>5,t>5,10−t−x,at|x|<5,|t|<5,20exp(μ(t+5)(x+5)),atx<−5,t<−5. $
|
(2.10) |
We found the values of the Lagrangian $ L $ from substituting by Eqs (2.9) and (2.10) into Eq (2.5), then we get
$ J(Θ,Ψ)=10063(−63a+882000b1+145800000b2+28120000000b3+630). $
|
(2.11) |
Case 3:
$ Θ(x,t)={12(exp(6π−t−x)−exp(−2π−t−x)),atx>π,t>π,sinh(t+x+2π),at|x|<π,|t|<π,0,atx<−π,t<−π, $
|
(2.12) |
$ ψ(x,t)={0,atx>π,t>π,sinh(2π−t−x),at|x|<π,|t|<π,12(exp(6π+t+x)−exp(−2π+t+x)),atx<−π,t<−π. $
|
(2.13) |
We found the values of the Lagrangian $ L $ from substituting by Eqs (2.12) and (2.13) into Eq (2.5), then we have
$ J(Θ,Ψ)=−1.02783×1010a+2.64108×1019b1+1.60864×1029b2+1.39506×1039b3+2.83012×106. $
|
(2.14) |
Case 1: We try the following Jost functions:
$ Θ(x,t)={12(exp(π2(3+2α+α2)−πt−πx)−exp(π2(1−2α−α2)−πt−πx)),atx>π,t>π,sinh((π+αt)(π+αx)),at0<x<π,0<t<π,sinh((t+π)(x+π)),at−π<x<0,−π<t<0,0,atx<−π,t<−π, $
|
(2.15) |
$ Ψ(x,t)={0,atx>π,t>π,sinh((π−t)(π−x)),at0<x<π,0<t<π,sinh((π−αt)(π−αx)),at−π<x<0,−π<t<0,12(exp(π2(3+2α+α2)+πt+πx)−exp(π2(1−2α−α2)+πt+πx)),atx<−π,t<−π. $
|
(2.16) |
This ansatz now contains nontrivial variational parameter $ \alpha $. Substituting Eqs (2.15) and (2.16) into Eq (2.5) (see Figure 2), then we obtain the functional integral at $ \alpha = -1 $ as follows:
$ J(Θ,Ψ)=−4.43569×107a+1.13554×1014b1+6.23167×1021b2+4.89178×1029b3. $
|
(2.17) |
Also, we can use $ \alpha = 2 $, then we get
$ ∫0−π∫0−πLdxdt=∫π0∫π0Ldxdt=−4.42653×1075a+5.59314×10148b1+5.89626×10224b2+8.85443×10300b3−2.66975×1036. $
|
The functional integral at $ \alpha = 2 $ becomes
$ J(Θ,Ψ)=−1.77562×1076a+4.12756×10150b1+4.35525×10226b2+6.5433×10302b3−5.33949×1036. $
|
(2.18) |
Case 2: We now try the following Jost function:
$ Θ(x,t)={12(exp(4π−t−x+2πα)−exp(−t−x−2πα)),atx>π,t>π,sinh(2π+αt+αx),at0<x<π,0<t<π,sinh(2π+t+x),at−π<t<0,−π<x<0,0,atx<−π,t<−π, $
|
(2.19) |
$ Ψ(x,t)={0,atx>π,t>π,sinh(2π−t−x),at0<x<π,0<t<π,sinh(2π−αt−αx),at−π<x<0,−π<t<0,12(exp(4π+t+x+2πα)−exp(t+x−2πα)),atx<−π,t<−π. $
|
(2.20) |
This ansatz contains nontrivial variational parameter $ \alpha $. Substituting Eqs (2.19) and (2.20) into Eq (2.5), then we get the functional integral at $ \alpha = 0.5 $ as follows:
$ J(Θ,Ψ)=−1.84004×107a+2.29576×1014b1+2.6187×1021b2+4.24151×1028b3+270561. $
|
(2.21) |
Also, we can take $ \alpha = -0.2 $, then we obtain
$ ∫0−π∫0−πLdxdt=∫π0∫π0Ldxdt=−13516.2a+2.22124×109b1+7.54018×1013b2+3.79568×1018b3+19035.7. $
|
The functional integral at $ \alpha = -0.2 $ becomes
$ J(Θ,Ψ)=−28484a+4.44353×109b1+1.50805×1014b2+7.59137×1018b3+38071.3. $
|
(2.22) |
We assume the Jost function by quadratic polynomials at interval $ |x| < 5, \; |t| < 5 $:
Case 1: We use the following Jost functions:
$ Θ(x,t)={(100+10000α)exp(−μ(t−5)(x−5)),atx>5,t>5,(t+5)(x+5)+α(5+t)2(5+x)2,at|x|<5,|t|<5,0,atx<−5,t<−5, $
|
(2.23) |
$ Ψ(x,t)={0,atx>5,t>5,(5−t)(5−x)+α(5−t)2(5−x)2,at|x|<5,|t|<5,(100+10000α)exp(μ(t+5)(x+5)),atx<−5,t<−5. $
|
(2.24) |
We found the values of Lagrangian calculation from Eqs (2.23) and (2.24) into Eq (2.5), and we get
$ J(Θ,Ψ)=5.55556×106α2−2.66667×107aα2−500000aα+555556α−3333.33a+8.65053×1030α8b3+7.81255×1029α7b3+1.97241×1023α6b2+3.11121×1028α6b3+1.38897×1022α5b2+7.14358×1026α5b3+6.1741×1015α4b1+4.13321×1020α4b2+1.03576×1025α4b3+3.12755×1014α3b1+6.67171×1018α3b2+9.72777×1022α3b3+6.14172×1012α2b1+6.18569×1016α2b2+5.79228×1020α2b3+5.61111×1010αb1+3.14172×1014αb2+2.00529×1018αb3+2.05556×108b1+6.89342×1011b2+3.10406×1015b3+8333.33. $
|
(2.25) |
We choose $ a = 3 $, $ b_1 = 2 $, $ b_2 = \frac{1}{2} $, and $ b_3 = \frac{1}{6} $, then the functional integral $ J(\Theta, \Psi) $ becomes
$ J(Θ,Ψ)=1.44175×1030α8+1.30209×1029α7+5.18545×1027α6+1.19067×1026α5+1.72648×1024α4+1.62163×1022α3+9.65689×1019α2+3.34372×1017α+5.17688×1014. $
|
(2.26) |
Derive Eq (2.26) with respect to $ \alpha $, then values of $ \alpha $ are:
$ α=−0.0136245,α=−0.012952−0.00218808i,α=−0.012952+0.00218808i,α=−0.0110959−0.00364364i,α=−0.0110959+0.00364364i,α=−0.0086518−0.00394467i,α=−0.0086518+0.00394467i. $
|
We substitute the exact roots of $ \alpha $ into Eq (2.26) give the following analytical equations:
$ J(Θ,Ψ)=4.83907×1010,J(Θ,Ψ)=3.84513×1010−4.55557×1010i,J(Θ,Ψ)=3.84513×1010+4.55557×1010i,J(Θ,Ψ)=−3.14336×109−1.06115×1011i,J(Θ,Ψ)=−3.14336×109+1.06115×1011i,J(Θ,Ψ)=−1.8701×1011−2.04551×1011i,J(Θ,Ψ)=−1.8701×1011+2.04551×1011i. $
|
(2.27) |
Case 2: We try the following Jost functions:
$ Θ(x,t)={(20+200α)exp(−μ(t−5)(x−5)),atx>5,t>5,(10+t+x)+α(t+5)2+α(x+5)2,at|x|<5,|t|<5,0,atx<−5,t<−5, $
|
(2.28) |
$ Ψ(x,t)={0,atx>5,t>5,(10−t−x)+α(5−t)2+α(5−x)2,at|x|<5,|t|<5,(20+200α)exp(μ(t+5)(x+5)),atx<−5,t<−5. $
|
(2.29) |
We found the values of Lagrangian calculation from Eqs (2.28) and (2.29) into Eq (2.5) (see Figure 3), and we have
$ J(Θ,Ψ)=50000α2−13333.3aα2−2000aα+15000α−100a+9.90698×1017α8b3+9.04639×1017α7b3+5.97273×1013α6b2+3.65565×1017α6b3+4.25597×1013α5b2+8.55415×1016α5b3+4.78095×109α4b1+1.28615×1013α4b2+1.27048×1016α4b3+2.39683×109α3b1+2.11556×1012α3b2+1.22953×1015α3b3+4.62698×108α2b1+2.00368×1011α2b2+7.59375×1013α2b3+4.08889×107αb1+1.03937×1010αb2+2.7454×1012αb3+1.4×106b1+2.31429×108b2+4.46349×1010b3+1000. $
|
(2.30) |
We put $ a = 6 $, $ b_1 = 3 $, $ b_2 = \frac{1}{4} $, and $ b_3 = \frac{1}{3} $, then the functional integral $ J(\Theta, \Psi) $ becomes
$ J(Θ,Ψ)=3.30233×1017α8+3.01546×1017α7+1.2187×1017α6+2.85245×1016α5+4.23816×1015α4+4.10379×1014α3+2.5364×1013α2+9.17853×1011α+1.49404×1010. $
|
(2.31) |
Derive Eq (2.31) with respect to $ \alpha $, then values of $ \alpha $ are:
$ α=−0.131984,α=−0.129184−0.0176052i,α=−0.129184+0.0176052i,α=−0.117682−0.0366915i,α=−0.117682+0.0366915i,α=−0.0866387−0.051817i,α=−0.0866387+0.051817i. $
|
We substitute the exact roots of $ \alpha $ into Eq (2.31) give the following analytical equations:
$ J(Θ,Ψ)=207360,J(Θ,Ψ)=−102453−293854i,J(Θ,Ψ)=−102453+293854i,J(Θ,Ψ)=−1.38256×106+542198i,J(Θ,Ψ)=−1.38256×106−542198i,J(Θ,Ψ)=−1.31177×107+1.75734×107i,J(Θ,Ψ)=−1.31177×107−1.75734×107i. $
|
(2.32) |
We inspect here the exact solutions of the nonlinear Schrödinger equation with the polynomial nonlinear law. This equation is defined as Eq (2.1).
Case 1: We suppose the ansatz function of the NLSE with the polynomial nonlinear law is in the form of a bright solitary wave solution
$ h1(x,t)=Asech(w(x−tv)),Γ(x,t)=Asech(w(x−tv))ei(kx−ωt), $
|
(2.33) |
where $ A, \text{w} $ and $ v $ are the amplitude, the pulse width, and velocity of soliton in normalized unites. Substituting from Eq (2.33) in Eq (2.1), and separating the real and imaginary parts, we obtain
$ −ak2v+avw2+vω+(A2b1v−2avw2)sech2(w(x−tv))+A4b2vsech4(w(x−tv))+A6b3vsech6(w(x−tv))=0, $
|
(2.34) |
$ (w−2akvw)tanh(w(x−tv))=0. $
|
(2.35) |
Equating the coefficients of the linearly independent terms to zero, we obtain the dynamical system in $ A, \text{w}, v, k, \omega, a, b_1, b_2, b_3 $ by solving this system we get:
Family I:
$ A=±w√b1kv,a=12kv,ω=k2−w22kv. $
|
(2.36) |
The sufficient conditions for solitary wave solution stability are
$ b1kv>0,kv≠0. $
|
(2.37) |
Family II:
$ A=±√k−2vωb1v,a=12kv,w=±√k(k−2vω), $
|
(2.38) |
provided that
$ k−2vωb1v>0,kv≠0,k(k−2vω)>0. $
|
(2.39) |
Family III:
$ A=±w√2ωb1k2−b1w2,a=ωk2−w2,v=k2−w22kω, $
|
(2.40) |
whenever
$ ωb1(k2−w2)>0,kω≠0,k2−w2≠0. $
|
(2.41) |
Family IV:
$ A=±√1b1(√v4ω2+v2w2v2−ω),a=√v4ω2+v2w2−v2ω2v2w2,k=√v4ω2+v2w2v+vω, $
|
(2.42) |
provided that
$ v4ω2+v2w2>0,1b1(√v4ω2+v2w2v2−ω)>0,v2w2≠0. $
|
(2.43) |
Then, the solutions of the NLSE with the polynomial nonlinear law as bright solitary wave solutions are (see Figures 4 and 5):
$ Γ11(x,t)=±w√b1kvsech(w(x−tv))ei(kx−ωt), $
|
(2.44) |
$ Γ12(x,t)=±√k−2vωb1vsech(w(x−tv))ei(kx−ωt), $
|
(2.45) |
$ Γ13(x,t)=±w√2ωb1k2−b1w2sech(w(x−tv))ei(kx−ωt), $
|
(2.46) |
$ Γ14(x,t)=±√1b1(√v4ω2+v2w2v2−ω)sech(w(x−tv))ei(kx−ωt). $
|
(2.47) |
Case 2: Another choice of the dark solitary wave solution of the NLSE with the polynomial nonlinear law is
$ h2(x,t)=A+Btanh(w(x−tv)),Γ(x,t)=(A+Btanh(w(x−tv)))ei(kx−ωt). $
|
(2.48) |
By replacement from Eq (2.48) in Eq (2.1) and separating the real and imaginary parts
$ −aAk2+A7b3+21A5b3B2+A5b2+35A3b3B4+10A3b2B2+A3b1+7Ab3B6+5Ab2B4+3Ab1B2+Aω+(−21A5b3B2−70A3b3B4−10A3b2B2−21Ab3B6−10Ab2B4−3Ab1B2)×sech2(w(x−tv))+(35A3b3B4+21Ab3B6+5Ab2B4)×sech4(w(x−tv))−7Ab3B6sech6(w(x−tv))+(−aBk2+7A6b3B+35A4b3B3+5A4b2B+21A2b3B5+10A2b2B3+3A2b1B+b3B7+b2B5+b1B3+Bω)×tanh(w(x−tv))+(−2aBw2−35A4b3B3−42A2b3B5−10A2b2B3−3b3B7−2b2B5−b1B3)×tanh(w(x−tv))sech2(w(x−tv))+(21A2b3B5+3b3B7+b2B5)tanh(w(x−tv))sech4(w(x−tv))−B7b3tanh(w(x−tv))sech6(w(x−tv))=0, $
|
(2.49) |
$ (2aBkw−Bwv)sech2(w(x−tv))=0 $
|
(2.50) |
Equating the coefficients of the linearly independent terms to zero, we deduce the coefficients $ A, B, \text{w}, v, k, \omega, a, b_1, b_2, b_3 $ in the form:
Family I:
$ A=0,B=±i√b23b3,b1=27b23ω+2b32−27ab23k29b2b3,w=±13b3√27b23ω−b32−27ab23k26a,v=12ak. $
|
(2.51) |
The sufficient conditions for dark solitary wave solution stability are
$ b2b3<0,b2b3≠0,ak≠0,27b23ω−b32−27ab23k2a>0. $
|
(2.52) |
Family II:
$ A=0,B=±i√b23b3,w=±13b3√3b1b2b3−b322a,v=12ak,ω=27ab23k2−2b32+9b1b3b227b23, $
|
(2.53) |
provided that
$ b2b3<0,3b1b2b3−b32a>0,ak≠0,b3≠0. $
|
(2.54) |
Family III:
$ A=0,B=±6√k2−2kvω+2w22b3kv,a=12kv,b1=3√b3(3k2−6kvω+4w2)3√4k2v2(k2−2kvω+2w2),b2=−33√b23(k2−2kvω+2w2)2kv, $
|
(2.55) |
such that
$ k2−2kvω+2w2b3kv>0,kv≠0,4k2v2(k2−2kvω+2w2)≠0. $
|
(2.56) |
Family IV:
$ A=0,B=±6√ak2+2aw2−ωb3,b1=(3ak2+4aw2−3ω)3√b3ak2+2aw2−ω,b2=−33√b23(ak2+2aw2−ω),v=k2+2w22k(ak2+2aw2), $
|
(2.57) |
whenever
$ ak2+2aw2−ωb3>0,ak2+2aw2−ω≠0,k(ak2+2aw2)≠0. $
|
(2.58) |
Then, the dark soliton solutions of the NLSE with the polynomial nonlinear law Eq (2.1) are (see Figure 6):
$ Γ21(x,t)=±i√b23b3tanh(w(x−tv))ei(kx−ωt), $
|
(2.59) |
$ Γ23(x,t)=±6√k2−2kvω+2w22b3kvtanh(w(x−tv))ei(kx−ωt), $
|
(2.60) |
$ Γ24(x,t)=±6√ak2+2aw2−ωb3tanh(w(x−tv))ei(kx−ωt). $
|
(2.61) |
The model II of the NLSE with polynomial nonlinearity is given by [49]:
$ iΨt+aΨxx+bΨxt+(k1|Ψ|2+k2|Ψ|4+k3|Ψ|6)Ψ=0, $
|
(3.1) |
where $ \Psi(x, t) $ denotes the complex valued function. The coefficients $ a $ and $ b $ indicate group velocity dispersion (GVD) and spatio-temporal dispersion (STD), respectively. The first term represents the linear evolution of pulses in nonlinear optical fibers. The final term represents the nonlinearity of the non-Kerr law. Since $ \Psi(x, t) $ is a complex function on the form $ \Psi(x, t) = u(x, t)+iv(x, t) $ such that $ u $ and $ v $ are real functions of $ x $ and $ t $, also $ |\Psi|^2 = (u+iv)(u-iv) $. We will use the variational technique to find solutions for the Eq (3.1), where we derive $ \Psi $ with respect to $ x $ and $ t $ to obtain the Lagrangian of this equation, which is expressed in the following way:
Let $ M $ and $ N $ are functionals in $ u $ and $ v $,
$ M(u,v)=∂u∂t+a∂2v∂x2+b∂2v∂x∂t+k1vu2+k1v3+k2vu4+2k2u2v3+k2v5+k3vu6+3k3v3u4+3k3u2v5+k3v7, $
|
(3.2) |
$ N(u,v)=−∂v∂t+a∂2u∂x2+b∂2u∂x∂t+k1u3+k1uv2+k2u5+2k2v2u3+k2uv4+k3u7+3k3v2u5+3k3u3v4+k3uv6. $
|
(3.3) |
The system of Eqs (3.2) and (3.3) satisfies the conditions mentioned in [52,53], then a functional integral $ J(u, v) $ can be written down using the formula given by Tonti [53]:
$ J(u,v)=∫ω[12auuxx+12avvxx+12buuxt+12bvvxt+12k3v2u6+34k3u4v4+12k2v2u4+12k3u2v6+12k2u2v4+12k1u2v2+18k3u8+16k2u6+14k1u4+18k3v8+16k2v6+14k1v4+12vut−12uvt]dω, $
|
(3.4) |
where $ d\omega = dxdt $.
Then the Lagrangian $ L $ is given by
$ L(u,v)=12auuxx+12avvxx+12buuxt+12bvvxt+12k3v2u6+34k3u4v4+12k2v2u4+12k3u2v6+12k2u2v4+12k1u2v2+18k3u8+16k2u6+14k1u4+18k3v8+16k2v6+14k1v4+12vut−12uvt. $
|
(3.5) |
As a necessary check to our calculations, we use the value of $ L $ in the Euler-Lagrange equations:
$ \frac{\partial L}{\partial u}-\frac{\partial}{\partial t}\left(\frac{\partial L}{\partial u_t}\right)+\frac{\partial^2}{\partial x^2}\left(\frac{\partial L}{\partial u_{xx}}\right)+\frac{\partial^2}{\partial x \partial t}\left(\frac{\partial L}{\partial u_{xt}}\right) = 0, $ |
$ \frac{\partial L}{\partial v}-\frac{\partial}{\partial t}\left(\frac{\partial L}{\partial v_t}\right)+\frac{\partial^2}{\partial x^2}\left(\frac{\partial L}{\partial v_{xx}}\right)+\frac{\partial^2}{\partial x \partial t}\left(\frac{\partial L}{\partial v_{xt}}\right) = 0, $ |
which yields the system of Eqs (3.2) and (3.3).
We provide an example of applying this technique by using a box-shaped initial pulse and a single nontrivial variational parameter based on linear Jost functions in three cases.
Case 1: We use the following ansatz for $ u(x, t) $ and $ v(x, t) $ functions:
$ u(x,t)={100exp(−μ(t−5)(x−5)),atx>5,t>5,(t+5)(x+5),at|x|<5,|t|<5,0,atx<−5,t<−5, $
|
(3.6) |
$ v(x,t)={0,atx>5,t>5,(5−t)(5−x),at|x|<5,|t|<5,100exp(μ(t+5)(x+5)),atx<−5,t<−5. $
|
(3.7) |
We found the values of the Lagrangian $ L $ from substituting Eqs (3.6) and (3.7) into Eq (3.5), and by using Mathematica program we get the value of $ J(u, v) $ at interval $ -5 < x < 5, \, -5 < t < 5 $ as
$ J(u,v)=2500b+2.05556×108k1+6.89342×1011k2+3.10406×1015k3+8333.33. $
|
(3.8) |
Case 2: Assume $ u(x, t) $ and $ v(x, t) $ for the Jost function as the following:
$ u(x,t)={20exp(−μ(t−5)(x−5)),atx>5,t>5,t+x+10,at|x|<5,|t|<5,0,atx<−5,t<−5, $
|
(3.9) |
$ v(x,t)={0,atx>5,t>5,10−t−x,at|x|<5,|t|<5,20exp(μ(t+5)(x+5)),atx<−5,t<−5. $
|
(3.10) |
We found the values of the Lagrangian $ L $ from substituting by Eqs (3.9) and (3.10) into Eq (3.5) (see Figure 7), then we have
$ J(u,v)=100063(88200k1+14580000k2+2812000000k3+63). $
|
(3.11) |
Case 3:
$ u(x,t)={12(exp(6π−t−x)−exp(−2π−t−x)),atx>π,t>π,sinh(t+x+2π),at|x|<π,|t|<π,0,atx<−π,t<−π, $
|
(3.12) |
$ v(x,t)={0,atx>π,t>π,sinh(2π−t−x),at|x|<π,|t|<π,12(exp(6π+t+x)−exp(−2π+t+x)),atx<−π,t<−π. $
|
(3.13) |
We found the values of the Lagrangian $ L $ from substituting by Eqs (3.12) and (3.13) into Eq (3.5), then we get
$ J(u,v)=1.02783×1010a+1.02783×1010b+2.64108×1019k1+1.60864×1029k2+1.39506×1039k3+2.83012×106. $
|
(3.14) |
Case 1: We try the following Jost functions:
$ u(x,t)={12(exp(π2(3+2α+α2)−πt−πx)−exp(π2(1−2α−α2)−πt−πx)),atx>π,t>π,sinh((π+αt)(π+αx)),at0<x<π,0<t<π,sinh((t+π)(x+π)),at−π<x<0,−π<t<0,0,atx<−π,t<−π, $
|
(3.15) |
$ v(x,t)={0,atx>π,t>π,sinh((π−t)(π−x)),at0<x<π,0<t<π,sinh((π−αt)(π−αx)),at−π<x<0,−π<t<0,12(exp(π2(3+2α+α2)+πt+πx)−exp(π2(1−2α−α2)+πt+πx)),atx<−π,t<−π. $
|
(3.16) |
This ansatz now contains nontrivial variational parameter $ \alpha $. Substituting Eqs (3.15) and (3.16) into Eq (3.5), then we obtain the functional integral at $ \alpha = -1 $ as follows:
$ J(u,v)=4.43568×107a+4.92254×107b+1.13554×1014k1+6.23167×1021k2+4.89178×1029k3. $
|
(3.17) |
Also, we can use $ \alpha = 2 $, then we get
$ ∫0−π∫0−πLdxdt=∫π0∫π0Ldxdt=4.42653×1075a+4.47679×1075b+5.59314×10148k1+5.89626×10224k2+8.85443×10300k3−2.66975×1036. $
|
The functional integral at $ \alpha = 2 $ becomes
$ J(u,v)=1.77562×1076a+1.78568×1076b+4.12756×10150k1+4.35525×10226k2+6.5433×10302k3−5.33949×1036. $
|
(3.18) |
Case 2: We now try the following Jost function:
$ u(x,t)={12(exp(4π−t−x+2πα)−exp(−t−x−2πα)),atx>π,t>π,sinh(2π+αt+αx),at0<x<π,0<t<π,sinh(2π+t+x),at−π<t<0,−π<x<0,0,atx<−π,t<−π, $
|
(3.19) |
$ v(x,t)={0,atx>π,t>π,sinh(2π−t−x),at0<x<π, 0<t<π,sinh(2π−αt−αx),at−π<x<0,−π<t<0,12(exp(4π+t+x+2πα)−exp(t+x−2πα)),atx<−π,t<−π. $
|
(3.20) |
This ansatz contains nontrivial variational parameter $ \alpha $. Substituting Eqs (3.19) and (3.20) into Eq (3.5) (see Figure 8), then we get the functional integral at $ \alpha = -0.2 $ as follows:
$ J(u,v)=28473.8a+28473.8b+4.44353×109k1+1.50805×1014k2+7.59137×1018k3+38071.3. $
|
(3.21) |
Also, we can take $ \alpha = 0.5 $ then we obtain
$ ∫0−π∫0−πLdxdt=∫π0∫π0Ldxdt=4.40169×106a+4.40169×106b+9.17623×1013k1+1.04745×1021k2+1.6966×1028k3+135281. $
|
The functional integral at $ \alpha = 0.5 $ becomes
$ J(u,v)=1.84004×107a+1.84004×107b+2.29576×1014k1+2.6187×1021k2+4.24151×1028k3+270561. $
|
(3.22) |
We assume the Jost function by quadratic polynomials at interval $ |x| < 5, |t| < 5 $:
Case 1: We use the following Jost functions:
$ u(x,t)={(100+10000α)exp(−μ(t−5)(x−5)),atx>5,t>5,(t+5)(x+5)+α(5+t)2(5+x)2,at|x|<5,|t|<5,0,atx<−5,t<−5, $
|
(3.23) |
$ v(x,t)={0,atx>5,t>5,(5−t)(5−x)+α(5−t)2(5−x)2,at|x|<5,|t|<5,(100+10000α)exp(μ(t+5)(x+5)),atx<−5,t<−5. $
|
(3.24) |
We found the values of Lagrangian calculation from Eqs (3.23) and (3.24) into Eq (3.5) (see Figure 9), and we get
$ J(u,v)=5.55556×106α2+1.33333×107aα2+250000aα+555556α+2.5×107α2b+555556αb+2500b+8.65053×1030α8k3+7.81255×1029α7k3+1.97241×1023α6k2+3.11121×1028α6k3+1.38897×1022α5k2+7.14358×1026α5k3+6.1741×1015α4k1+4.13321×1020α4k2+1.03576×1025α4k3+3.12755×1014α3k1+6.67171×1018α3k2+9.72777×1022α3k3+6.14172×1012α2k1+6.18569×1016α2k2+5.79228×1020α2k3+5.61111×1010αk1+3.14172×1014αk2+2.00529×1018αk3+2.05556×108k1+6.89342×1011k2+3.10406×1015k3+8333.33. $
|
(3.25) |
We put $ a = 2 $, $ b = 6 $, $ k_1 = 2 $, $ k_2 = 3 $, and $ k_3 = 4 $, then the functional integral $ J(u, v) $ becomes
$ J(u,v)=3.46021×1031α8+3.12502×1030α7+1.24449×1029α6+2.85747×1027α5+4.14318×1025α4+3.89131×1023α3+2.3171×1021α2+8.02211×1018α+1.24183×1016. $
|
(3.26) |
Derive Eq (3.26) with respect to $ \alpha $, then values of $ \alpha $ are:
$ α=−0.0136235,α=−0.0129514−0.00218597i,α=−0.0129514+0.00218597i,α=−0.0110962−0.00364064i,α=−0.0110962+0.00364064i,α=−0.00865263−0.00394263i,α=−0.00865263+0.00394263i. $
|
We substitute the exact roots of $ \alpha $ into Eq (3.26) give the following analytical equations:
$ J(u,v)=1.15381×1012,J(u,v)=9.16876×1011−1.08685×1012i,J(u,v)=9.16876×1011+1.08685×1012i,J(u,v)=−7.64569×1010−2.53327×1012i,J(u,v)=−7.64569×1010+2.53327×1012i,J(u,v)=−4.47645×1012−4.88381×1012i,J(u,v)=−4.47645×1012+4.88381×1012i. $
|
(3.27) |
Case 2: We try the following Jost functions:
$ u(x,t)={(20+200α)exp(−μ(t−5)(x−5)),atx>5,t>5,(10+t+x)+α(t+5)2+α(x+5)2,at|x|<5,|t|<5,0,atx<−5,t<−5, $
|
(3.28) |
$ v(x,t)={0,atx>5,t>5,(10−t−x)+α(5−t)2+α(5−x)2,at|x|<5,|t|<5,(20+200α)exp(μ(t+5)(x+5)),atx<−5,t<−5. $
|
(3.29) |
We found the values of Lagrangian calculation from Eqs (3.28) and (3.29) into Eq (3.5), and we have
$ J(u,v)=50000α2+13333.3aα2+2000aα+15000α+9.90698×1017α8k3+9.04639×1017α7k3+5.97273×1013α6k2+3.65565×1017α6k3+4.25597×1013α5k2+8.55415×1016α5k3+4.78095×109α4k1+1.28615×1013α4k2+1.27048×1016α4k3+2.39683×109α3k1+2.11556×1012α3k2+1.22953×1015α3k3+4.62698×108α2k1+2.00368×1011α2k2+7.59375×1013α2k3+4.08889×107αk1+1.03937×1010αk2+2.7454×1012αk3+1.4×106k1+2.31429×108k2+4.46349×1010k3+1000. $
|
(3.30) |
We put $ a = 3 $, $ b = -2 $, $ k_1 = 2 $, $ k_2 = 4 $, and $ k_3 = 6 $, then the functional integral $ J(u, v) $ becomes
$ J(u,v)=5.94419×1018α8+5.42784×1018α7+2.19363×1018α6+5.13419×1017α5+7.62802×1016α4+7.38564×1015α3+4.56427×1014α2+1.6514×1013α+2.68738×1011. $
|
(3.31) |
Derive Eq (3.31) with respect to $ \alpha $, then values of $ \alpha $ are:
$ α=−0.131961,α=−0.129142−0.0174743i,α=−0.129142+0.0174743i,α=−0.117682−0.0366515i,α=−0.117682+0.0366515i,α=−0.0866909−0.0518096i,α=−0.0866909+0.0518096i. $
|
We substitute the exact roots of $ \alpha $ into Eq (3.31) give the following analytical equations:
$ J(u,v)=3.53554×106,J(u,v)=−1.84827×106−5.13825×106i,J(u,v)=−1.84827×106+5.13825×106i,J(u,v)=−2.46359×107+1.0125×107i,J(u,v)=−2.46359×107−1.0125×107i,J(u,v)=−2.31965×108+3.16423×108i,J(u,v)=−2.31965×108−3.16423×108i. $
|
(3.32) |
We survey here the exact solutions of the nonlinear Schrödinger equation with polynomial law nonlinearity Eq (3.1).
Case 1: We suppose the ansatz function of the NLSE with polynomial law nonlinearity is in the form of a bright solitary wave solution.
$ h1(x,t)=Asech(w(x−tv)),Ψ(x,t)=Asech(w(x−tv))ei(μx−ωt), $
|
(3.33) |
where $ A, \text{w} $ and $ v $ are the amplitude, the pulse width, and velocity of soliton in normalized unites. Substituting from Eq (3.33) in Eq (3.1), and separating the real and imaginary parts, we obtain
$ avw2+bμvω−aμ2v−bw2+vω+(−2avw2+A2k1v+2bw2)×sech2(w(x−tv))+A4k2vsech4(w(x−tv))+A6k3vsech6(w(x−tv))=0, $
|
(3.34) |
$ (−2aμvw+bvwω+bμw+w)tanh(w(x−tv))=0. $
|
(3.35) |
Equating the coefficients of the linearly independent terms to zero, we obtain the dynamical system in $ A, \text{w}, v, \mu, \omega, a, b, k_1, k_2, k_3 $ by solving this system we get:
Family I:
$ A=±√2(μw2−vw2ω)k1μ2v+k1vw2,a=−v2ω2−w2v(μ2+w2)(vω−μ),b=2μvω+w2−μ2(μ2+w2)(μ−vω). $
|
(3.36) |
The sufficient conditions for solitary wave solution stability are:
$ μw2−vw2ωk1μ2v+k1vw2>0,v(μ2+w2)(vω−μ)≠0,(μ2+w2)(μ−vω)≠0. $
|
(3.37) |
Family II:
$ A=±√bμ2−bμvω+μ−2vωk1v,a=bμ+bvω+12μv,w=±√bμ3−bμ2vω+μ2−2μvωbvω+1−bμ, $
|
(3.38) |
provided that
$ bμ2−bμvω+μ−2vωk1v>0,bμ3−bμ2vω+μ2−2μvωbvω+1−bμ>0,μv≠0. $
|
(3.39) |
Family III:
$ A=±w√2ωbk1μ3+bk1μw2+k1μ2−k1w2,a=ω(b2μ2+b2w2+2bμ+1)bμ3+bμw2+μ2−w2,v=bμ3+bμw2+μ2−w2ω(bμ2+bw2+2μ), $
|
(3.40) |
whenever
$ ωbk1μ3+bk1μw2+k1μ2−k1w2>0,bμ3+bμw2+μ2−w2≠0,ω(bμ2+bw2+2μ)≠0. $
|
(3.41) |
Then, the solutions of the NLSE with polynomial law nonlinearity as bright solitary wave solutions are (see Figures 10 and 11):
$ Ψ11(x,t)=±√2(μw2−vw2ω)k1μ2v+k1vw2sech(w(x−tv))ei(μx−ωt), $
|
(3.42) |
$ Ψ12(x,t)=±√bμ2−bμvω+μ−2vωk1vsech(w(x−tv))ei(μx−ωt), $
|
(3.43) |
$ Ψ13(x,t)=±w√2ωbk1μ3+bk1μw2+k1μ2−k1w2sech(w(x−tv))ei(μx−ωt). $
|
(3.44) |
Case 2: Another choice of the dark solitary wave solution of the NLSE with polynomial law nonlinearity is
$ Ψ(x,t)=(A+Btanh(w(x−tv)))ei(μx−ωt). $
|
(3.45) |
By replacement from Eq (3.45) in Eq (3.1) and separating the real and imaginary parts
$ A7k3−aAμ2+21A5B2k3+A5k2+35A3B4k3+10A3B2k2+A3k1+Abμω+7AB6k3+5AB4k2+3AB2k1+Aω+(−21A5B2k3−70A3B4k3−10A3B2k2−21AB6k3−10AB4k2−3AB2k1)×sech2(w(x−tv))+(35A3B4k3+21AB6k3+5AB4k2)×sech4(w(x−tv))−7AB6k3sech6(w(x−tv))+(−aBμ2+7A6Bk3+35A4B3k3+5A4Bk2+21A2B5k3+10A2B3k2+3A2Bk1+bBμω+B7k3+B5k2+B3k1+Bω)tanh(w(x−tv))+(−2aBw2−35A4B3k3−42A2B5k3−10A2B3k2+2bBw2v−3B7k3−2B5k2−B3k1)tanh(w(x−tv))sech2(w(x−tv))+(21A2B5k3+3B7k3+B5k2)tanh(w(x−tv))sech4(w(x−tv))−B7k3tanh(w(x−tv))sech6(w(x−tv))=0, $
|
(3.46) |
$ (2aBμw−bBμwv−bBwω−Bwv)sech2(w(x−tv))=0. $
|
(3.47) |
Equating the coefficients of the linearly independent terms to zero, we deduce the coefficients $ A, B, \text{w}, v, \mu, \omega, a, b, k_1, k_2, k_3 $ in the form:
Family I:
$ A=0,B=±i√k23k3,a=k32v2ω+k32μv+54k23w2−27k23v2ω227k23v(μ2−2w2)(vω−μ),b=54k23μvω−27k23μ2−2k32μv−54k23w227k23(μ2−2w2)(μ−vω),k1=3k32μ2v−54k23vw2ω−4k32vw2+54k23μw29k2k3v(μ2−2w2). $
|
(3.48) |
The sufficient conditions for dark solitary wave solution stability are
$ k2k3<0,k23v(μ2−2w2)(vω−μ)≠0,k2k3v(μ2−2w2)≠0. $
|
(3.49) |
Family II:
$ A=0,B=±i√k23k3,a=27b2k23μ2−54b2k23w2+54bk23μ+bk32v+27k2327k23v(bμ2−2bw2+2μ),k1=3bk32μ2v−4bk32vw2+6k32μv+54k23w29k2k3v(bμ2−2bw2+2μ),ω=27bk23μ3−54bk23μw2+27k23μ2+2k32μv+54k23w227k23v(bμ2−2bw2+2μ), $
|
(3.50) |
provided that
$ k2k3<0,k2k3v(bμ2−2bw2+2μ)≠0,k23v(bμ2−2bw2+2μ)≠0. $
|
(3.51) |
Family III:
$ A=0,B=±i√k23k3,a=bμ+bvω+12μv,k1=27bk23μvω−27bk23μ2−27k23μ+54k23vω+4k32v18k2k3v,w=−√μ(27bk23μvω−27k23μ+54k23vω−2k32v−27bk23μ2)54bk23vω+54k23−54bk23μ, $
|
(3.52) |
whenever
$ k2k3<0,μv≠0,k2k3v≠0,μ(27bk23μvω−27k23μ+54k23vω−2k32v−27bk23μ2)54bk23vω+54k23−54bk23μ>0. $
|
(3.53) |
Family IV:
$ B=±6√aμ2v2ω−2av2w2ω−aμ3v+2aμvw2+v2ω2−2w2v(2w2−μ2)(vω−μ)(k3(μ+vω)(2w2−μ2)(μ−vω)),A=0,b=2aμv−1μ+vω,k1=1v2/3(μ+vω)3√2w2(av(vω−μ)+1)−v(aμ2(vω−μ)+vω2)×(3√(k3(μ+vω)(2w2−μ2)(μ−vω))(2w2−μ2)(vω−μ)×(3v(aμ2(vω−μ)+vω2)−4w2(av(vω−μ)+1))),k2=3k33√2w2(av(vω−μ)+1)−v(aμ2(vω−μ)+vω2)3√v(2w2−μ2)(vω−μ)(k3(μ+vω)(2w2−μ2)(μ−vω)), $
|
(3.54) |
such that
$ aμ2v2ω−2av2w2ω−aμ3v+2aμvw2+v2ω2−2w2v(2w2−μ2)(vω−μ)(k3(μ+vω)(2w2−μ2)(μ−vω))>0,μ+vω≠0,v2/3(μ+vω)3√2w2(av(vω−μ)+1)−v(aμ2(vω−μ)+vω2)≠0,3√v(2w2−μ2)(vω−μ)(k3(μ+vω)(2w2−μ2)(μ−vω))≠0. $
|
(3.55) |
Then, the dark soliton solutions of the NLSE with polynomial law nonlinearity Eq (3.1) are (see Figure 12)
$ Ψ21(x,t)=±i√k23k3tanh(w(x−tv))ei(μx−ωt), $
|
(3.56) |
$ Ψ24(x,t)=±6√aμ2v2ω−2av2w2ω−aμ3v+2aμvw2+v2ω2−2w2v(2w2−μ2)(vω−μ)(k3(μ+vω)(2w2−μ2)(μ−vω))×tanh(w(x−tv))ei(μx−ωt). $
|
(3.57) |
In this paper, we discussed the two models of the nonlinear Schrödinger equation (NLSE) with polynomial law nonlinearity by effective and understandable techniques, such as the variational principle method based on the Lagrangian and the amplitude ansatz method. We found the functional integral and the Lagrangian of these models. Meanwhile, the solutions of the proposed equations were extracted by choice of different ansatz functions based on the Jost function, and they are continuous at all intervals. Firstly, the Jost function was approximated by piecewise linear ansatz function with a single nontrivial variational parameter in three cases from a region of a rectangular box. Then, the Jost function was approximated by the piecewise ansatz function containing the hyperbolic function in two cases of the two-box potential and was also approximated by quadratic polynomials with two free parameters rather than a piecewise linear ansatz function. Finally, this trial function had been approximated by the tanh function. Besides, we applied the amplitude ansatz method to obtain the new soliton solutions of the offered equations that contain bright soliton, dark soliton, bright-dark solitary wave solutions, rational dark-bright solutions, and periodic solitary wave solutions. The conditions for the stability of solutions were conducted. Graphical models, such as 2D, 3D, and contour plots, were induced using appropriate parameter values. These solutions have vital applications in applied sciences and might provide valuable support for investigators and physicists to solve more complex physical phenomena.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
The authors extend their appreciation to the Deputyship for Research and Innovation, Ministry of Education in Saudi Arabia for funding this research work through project number 445-9-693. Furthermore, the authors would like to extend their appreciation to Taibah University for its supervision support.
The authors declare that they have no competing interests.
[1] |
T. Alperovich and A. Sopasakis, Modeling highway traffic with stochastic dynamics, J. Stat. Phys, 133 (2008), 1083-1105. doi: 10.1007/s10955-008-9652-6
![]() |
[2] | S. Amin, et al., Mobile century - Using GPS mobile phones as traffic sensors: A field experiment, in 15th World Congress on Intelligent Transportation Systems, New York, Nov., 2008. |
[3] |
A. Aw and M. Rascle, Resurrection of second order models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916-938. doi: 10.1137/S0036139997332099
![]() |
[4] | M. Bando, Hesebem K., A. Nakayama, A. Shibata and Y. Sugiyama, Dynamical model of traffic congestion and numerical simulation, Phys. Rev. E, 51 (1995), 1035-1042. |
[5] |
A. M. Bayen and C. G. Claudel, Lax-Hopf based incorporation of internal boundary conditions into Hamilton-Jacobi equation. Part I: Theory, IEEE Trans. Automat. Contr., 55 (2010), 1142-1157. doi: 10.1109/TAC.2010.2041976
![]() |
[6] |
A. M. Bayen and C. G. Claudel, Convex formulations of data assimilation problems for a class of Hamilton-Jacobi equations, SIAM J. Control Optim., 49 (2011), 383-402. doi: 10.1137/090778754
![]() |
[7] |
N. Bellomo and C. Dogbe, On the modeling of traffic and crowds: A survey of models, speculations, and perspectives, SIAM Rev., 53 (2011), 409-463. doi: 10.1137/090746677
![]() |
[8] |
F. Berthelin, P. Degond, M. Delitala and M. Rascle, A model for the formation and evolution of traffic jams, Arch. Ration. Mech. Anal., 187 (2008), 185-220. doi: 10.1007/s00205-007-0061-9
![]() |
[9] |
S. Blandin, G. Bretti, A. Cutolo and B. Piccoli, Numerical simulations of traffic data via fluid dynamic approach, Appl. Math. Comput., 210 (2009), 441-454. doi: 10.1016/j.amc.2009.01.057
![]() |
[10] | S. Blandin, A. Coque and A. Bayen, On sequential data assimilation for scalar macroscopic traffic flow models, Physica D, (2012), 1421-1440. |
[11] |
S. Blandin, D. Work, P. Goatin, B. Piccoli and A. Bayen, A general phase transition model for vehicular traffic, SIAM J. Appl. Math., 71 (2011), 107-127. doi: 10.1137/090754467
![]() |
[12] |
R. Borsche, M. Kimathi and A. Klar, A class of multiphase traffic theories for microscopic, kinetic and continuum traffic models, Comp. Math. Appl., 64 (2012), 2939-2953. doi: 10.1016/j.camwa.2012.08.013
![]() |
[13] |
C. Chalons and P. Goatin, Transport-equilibrium schemes for computing contact discontinuities in traffic flow modeling, Commun. Math. Sci., 5 (2007), 533-551. doi: 10.4310/CMS.2007.v5.n3.a2
![]() |
[14] |
G. Q. Chen, C. D. Levermore and T. P. Liu, Hyperbolic conservation laws with stiff relaxation terms and entropy, Comm. Pure Appl. Math., 47 (1994), 787-830. doi: 10.1002/cpa.3160470602
![]() |
[15] |
R. M. Colombo, Hyperbolic phase transitions in traffic flow, SIAM J. Appl. Math., 63 (2003), 708-721. doi: 10.1137/S0036139901393184
![]() |
[16] | R. M. Colombo and P. Goatin, Traffic flow models with phase transitions, Flow Turbulence Combust., 76 (2006), 383-390. |
[17] |
R. M. Colombo, F. Marcellini and M. Rascle, A 2-phase traffic model based on a speed bound, SIAM J. Appl. Math., 70 (2010), 2652-2666. doi: 10.1137/090752468
![]() |
[18] |
R. Courant, K. Friedrichs and H. Lewy, Über die partiellen Differenzengleichungen der mathematischen Physik, Mathematische Annalen, 100 (1928), 32-74. doi: 10.1007/BF01448839
![]() |
[19] |
C. F. Daganzo, The cell transmission model: A dynamic representation of highway traffic consistent with the hydrodynamic theory, Transp. Res. B, 28 (1994), 269-287. doi: 10.1016/0191-2615(94)90002-7
![]() |
[20] |
C. F. Daganzo, Requiem for second-order fluid approximations of traffic flow, Transp. Res. B, 29 (1995), 277-286. doi: 10.1016/0191-2615(95)00007-Z
![]() |
[21] | C. F. Daganzo, Fundamentals of Transportation and Traffic Operations, Emerald Group Pub Ltd, 1997. |
[22] |
C. F. Daganzo, In traffic flow, cellular automata = kinematic waves, Transp. Res. B, 40 (2006), 396-403. doi: 10.1016/j.trb.2005.05.004
![]() |
[23] | L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, 1998. |
[24] | S. Fan, Data-fitted Generic Second Order Macroscopic Traffic Flow Models, Dissertation, Temple University, 2013. |
[25] | S. Fan, B. Piccoli and B. Seibold, The Collapsed Generalized Aw-Rascle-Zhang Model of Traffic Flow, in preparation, 2014. |
[26] | S. Fan and B. Seibold, A comparison of data-fitted first order traffic models and their second order generalizations via trajectory and sensor data, in 93rd Annual Meeting of Transportation Research Board, paper number 13-4853, Washington DC, 2013. |
[27] | S. Fan and B. Seibold, Effect of the choice of stagnation density in data-fitted first- and second-order traffic models, arXiv:1308.0393, 2013. |
[28] | Website, http://www.fhwa.dot.gov/publications/research/operations/06137. |
[29] | Website, http://ops.fhwa.dot.gov/trafficanalysistools/ngsim.htm. |
[30] |
M. R. Flynn, A. R. Kasimov, J.-C. Nave, R. R. Rosales and B. Seibold, Self-sustained nonlinear waves in traffic flow, Phys. Rev. E, 79 (2009), 056113, 13 pp. doi: 10.1103/PhysRevE.79.056113
![]() |
[31] |
M. Fukui and Y. Ishibashi, Traffic flow in 1D cellular automaton model including cars moving with high speed, J. Phys. Soc. Japan, 65 (1996), 1868-1870. doi: 10.1143/JPSJ.65.1868
![]() |
[32] | M. Garavello and B. Piccoli, Traffic Flow on Networks, American Institute of Mathematical Sciences, 2006. |
[33] |
P. Goatin, The Aw-Rascle vehicular traffic flow model with phase transitions, Math. Comput. Modeling, 44 (2006), 287-303. doi: 10.1016/j.mcm.2006.01.016
![]() |
[34] | S. K. Godunov, A difference scheme for the numerical computation of a discontinuous solution of the hydrodynamic equations, Math. Sbornik, 47 (1959), 271-306. |
[35] |
J. M. Greenberg, Extension and amplification of the Aw-Rascle model, SIAM J. Appl. Math., 62 (2001), 729-745. doi: 10.1137/S0036139900378657
![]() |
[36] |
J. M. Greenberg, Congestion redux, SIAM J. Appl. Math., 64 (2004), 1175-1185. doi: 10.1137/S0036139903431737
![]() |
[37] | B. D. Greenshields, A study of traffic capacity, Proceedings of the Highway Research Record, 14 (1935), 448-477. |
[38] |
A. Harten, P. D. Lax and B. van Leer, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev., 25 (1983), 35-61. doi: 10.1137/1025002
![]() |
[39] |
D. Helbing, Improved fluid-dynamic model for vehicular traffic, Phys. Rev. E, 51 (1995), 3164-3169. doi: 10.1103/PhysRevE.51.3164
![]() |
[40] |
D. Helbing, Traffic and related self-driven many-particle systems, Reviews of Modern Physics, 73 (2001), 1067-1141. doi: 10.1103/RevModPhys.73.1067
![]() |
[41] | R. Herman and I. Prigogine, Kinetic Theory of Vehicular Traffic, Elsevier, New York, 1971. |
[42] |
M. Herty and R. Illner, Analytical and numerical investigations of refined macroscopic traffic flow models, Kinet. Relat. Models, 3 (2010), 311-333. doi: 10.3934/krm.2010.3.311
![]() |
[43] |
M. Herty and L. Pareschi, Fokker-Planck asymptotics for traffic flow models, Kinet. Relat. Models, 3 (2010), 165-179. doi: 10.3934/krm.2010.3.165
![]() |
[44] |
R. Illner, A. Klar and T. Materne, Vlasov-Fokker-Planck models for multilane traffic flow, Commun. Math. Sci., 1 (2003), 1-12. doi: 10.4310/CMS.2003.v1.n1.a1
![]() |
[45] |
R. J. Karunamuni and T. Alberts, A generalized reflection method of boundary correction in kernel density estimation, Canad. J. Statist., 33 (2005), 497-509. doi: 10.1002/cjs.5550330403
![]() |
[46] | A. R. Kasimov, R. R. Rosales, B. Seibold and M. R. Flynn, Existence of jamitons in hyperbolic relaxation systems with application to traffic flow, in preparation, 2014. |
[47] |
B. S. Kerner and P. Konhäuser, Cluster effect in initially homogeneous traffic flow, Phys. Rev. E, 48 (1993), R2335-R2338. doi: 10.1103/PhysRevE.48.R2335
![]() |
[48] |
B. S. Kerner and P. Konhäuser, Structure and parameters of clusters in traffic flow, Phys. Rev. E, 50 (1994), 54-83. doi: 10.1103/PhysRevE.50.54
![]() |
[49] |
A. Klar and R. Wegener, Kinetic derivation of macroscopic anticipation models for vehicular traffic, SIAM J. Appl. Math., 60 (2000), 1749-1766. doi: 10.1137/S0036139999356181
![]() |
[50] | J.-P. Lebacque, Les modeles macroscopiques du traffic, Annales des Ponts., 67 (1993), 24-45. |
[51] | J.-P. Lebacque, S. Mammar and H. Haj-Salem, Generic second order traffic flow modelling, in Transportation and Traffic Theory (eds. R. E. Allsop, M. G. H. Bell and B. G. Heydecker), Proc. of the 17th ISTTT, Elsevier, 2007, 755-776. |
[52] |
M. J. Lighthill and G. B. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads, Proc. Roy. Soc. A, 229 (1955), 317-345. doi: 10.1098/rspa.1955.0089
![]() |
[53] |
T. P. Liu, Hyperbolic conservation laws with relaxation, Comm. Math. Phys., 108 (1987), 153-175. doi: 10.1007/BF01210707
![]() |
[54] | Website, http://data.dot.state.mn.us/datatools. |
[55] | http://traffic.berkeley.edu. |
[56] | K. Nagel and M. Schreckenberg, A cellular automaton model for freeway traffic, J. Phys. I France, 2 (1992), 2221-2229. |
[57] | P. Nelson and A. Sopasakis, The Chapman-Enskog expansion: A novel approach to hierarchical extension of Lighthill-Whitham models, in Proceedings of the 14th International Symposium on Transportation and Trafic Theory (ed. A. Ceder), Jerusalem, 1999, 51-79. |
[58] |
G. F. Newell, Nonlinear effects in the dynamics of car following, Operations Research, 9 (1961), 209-229. doi: 10.1287/opre.9.2.209
![]() |
[59] |
G. F. Newell, A simplified theory of kinematic waves in highway traffic II: Queueing at freeway bottlenecks, Transp. Res. B, 27 (1993), 289-303. doi: 10.1016/0191-2615(93)90039-D
![]() |
[60] |
E. Parzen, On estimation of a probability density function and mode, Ann. Math. Statist., 33 (1962), 1065-1076. doi: 10.1214/aoms/1177704472
![]() |
[61] | H. J. Payne, Models of freeway traffic and control, Proc. Simulation Council, 1 (1971), 51-61. |
[62] | H. J. Payne, FREEFLO: A macroscopic simulation model of freeway traffic, Transp. Res. Rec., 722 (1979), 68-77. |
[63] |
W. F. Phillips, A kinetic model for traffic flow with continuum implications, Transportation Planning and Technology, 5 (1979), 131-138. doi: 10.1080/03081067908717157
![]() |
[64] |
L. A. Pipes, An operational analysis of traffic dynamics, Journal of Applied Physics, 24 (1953), 274-281. doi: 10.1063/1.1721265
![]() |
[65] |
M. Rascle, An improved macroscopic model of traffic flow: Derivation and links with the Lightill-Whitham model, Math. Comput. Modelling, 35 (2002), 581-590. doi: 10.1016/S0895-7177(02)80022-X
![]() |
[66] |
P. I. Richards, Shock waves on the highway, Operations Research, 4 (1956), 42-51. doi: 10.1287/opre.4.1.42
![]() |
[67] |
M. Rosenblatt, Remarks on some nonparametric estimates of a density function, Ann. Math. Statist., 27 (1956), 832-837. doi: 10.1214/aoms/1177728190
![]() |
[68] |
S. Sakai, K. Nishinari and S. IIda, A new stochastic cellular automaton model on traffic flow and its jamming phase transition, J. Phys. A: Math. Gen., 39 (2006), 15327-15339. doi: 10.1088/0305-4470/39/50/002
![]() |
[69] |
B. Seibold, M. R. Flynn, A. R. Kasimov and R. R. Rosales, Constructing set-valued fundamental diagrams from jamiton solutions in second order traffic models, Netw. Heterog. Media, 8 (2013), 745-772. doi: 10.3934/nhm.2013.8.745
![]() |
[70] |
F. Siebel and W. Mauser, On the fundamental diagram of traffic flow, SIAM J. Appl. Math., 66 (2006), 1150-1162. doi: 10.1137/050627113
![]() |
[71] | B. Temple, Systems of conservation laws with coinciding shock and rarefaction curves, Contemp. Math., 17 (1983), 143-151. |
[72] | R. Underwood, Speed, Volume, and Density Relationships: Quality and Theory of Traffic Flow, Technical Report, Yale Bureau of Highway Traffic, 1961. |
[73] | G. B. Whitham, Linear and Nonlinear Waves, John Wiley and Sons, New York, 1974. |
[74] | D. Work, S. Blandin, O.-P. Tossavainen, B. Piccoli and A. Bayen, A traffic model for velocity data assimilation, Appl. Math. Res. Express., 1 (2010), 1-35. |
[75] |
H. M. Zhang, A non-equilibrium traffic model devoid of gas-like behavior, Transp. Res. B, 36 (2002), 275-290. doi: 10.1016/S0191-2615(00)00050-3
![]() |
1. | Feifei Yang, Xinlin Song, Zhenhua Yu, Dynamics of a functional neuron model with double membranes, 2024, 188, 09600779, 115496, 10.1016/j.chaos.2024.115496 | |
2. | Xin-Yi Gao, Hetero-Bäcklund transformation, bilinear forms and multi-solitons for a (2+1)-dimensional generalized modified dispersive water-wave system for the shallow water, 2024, 92, 05779073, 1233, 10.1016/j.cjph.2024.10.004 | |
3. | Shuang Li, Xing‐Hua Du, The analysis of traveling wave structures and chaos of the cubic–quartic perturbed Biswas–Milovic equation with Kudryashov's nonlinear form and two generalized nonlocal laws, 2024, 0170-4214, 10.1002/mma.10462 | |
4. | Atallah El-shenawy, Mohamed El-Gamel, Amir Teba, Simulation of the SIR dengue fever nonlinear model: A numerical approach, 2024, 11, 26668181, 100891, 10.1016/j.padiff.2024.100891 | |
5. | Syed T. R. Rizvi, Aly R. Seadawy, Nighat Farah, Sarfaraz Ahmad, Ali Althobaiti, The interactions of dark, bright, parabolic optical solitons with solitary wave solutions for nonlinear Schrödinger–Poisson equation by Hirota method, 2024, 56, 1572-817X, 10.1007/s11082-024-07008-z | |
6. | Jicheng Yu, Yuqiang Feng, Lie symmetries, exact solutions and conservation laws of (2+1)-dimensional time fractional cubic Schrödinger equation, 2024, 1425-6908, 10.1515/jaa-2024-0072 | |
7. | Muhammad Arshad, Aly R. Seadawy, Aliza Mehmood, Khurrem Shehzad, Lump Kink interactional and breather-type waves solutions of (3+1)-dimensional shallow water wave dynamical model and its stability with applications, 2024, 0217-9849, 10.1142/S0217984924504025 | |
8. | Noor Alam, Ali Akbar, Mohammad Safi Ullah, Md. Mostafa, Dynamic waveforms of the new Hamiltonian amplitude model using three different analytic techniques, 2024, 0973-1458, 10.1007/s12648-024-03426-7 | |
9. | Thabet Abdeljawad, Asma Al-Jaser, Bahaaeldin Abdalla, Kamal Shah, Manel Hleili, Manar Alqudah, Coherent manipulation of bright and dark solitons of reflection and transmission pulses through sodium atomic medium, 2024, 22, 2391-5471, 10.1515/phys-2024-0058 | |
10. | Mohammed Aldandani, Syed T. R. Rizvi, Abdulmohsen Alruwaili, Aly R. Seadawy, Discussion on optical solitons, sensitivity and qualitative analysis to a fractional model of ion sound and Langmuir waves with Atangana Baleanu derivatives, 2024, 22, 2391-5471, 10.1515/phys-2024-0080 | |
11. | Subhashish Tiwari, Ajay Vyas, Vijay Singh, G. Maity, Achyutesh Dixit, Investigation of wave propagation characteristics in photonic crystal micro-structure containing circular shape with varied nonlinearity, 2024, 56, 1572-817X, 10.1007/s11082-024-07421-4 | |
12. | Aydin Secer, Ismail Onder, Handenur Esen, Neslihan Ozdemir, Melih Cinar, Hasan Cakicioglu, Selvi Durmus, Muslum Ozisik, Mustafa Bayram, On Stochastic Pure-Cubic Optical Soliton Solutions of Nonlinear Schrödinger Equation Having Power Law of Self-Phase Modulation, 2024, 63, 1572-9575, 10.1007/s10773-024-05756-y | |
13. | M. Aamir Ashraf, Aly R. Seadawy, Syed T. R. Rizvi, Ali Althobaiti, Dynamical optical soliton solutions and behavior for the nonlinear Schrödinger equation with Kudryashov’s quintuple power law of refractive index together with the dual-form of nonlocal nonlinearity, 2024, 56, 1572-817X, 10.1007/s11082-024-07096-x | |
14. | Setu Rani, Sachin Kumar, Raj Kumar, Dynamical Study of Newly Created Analytical Solutions, Bifurcation Analysis, and Chaotic Nature of the Complex Kraenkel–Manna–Merle System, 2024, 23, 1575-5460, 10.1007/s12346-024-01148-z | |
15. | Syed Oan Abbas, Aly R. Seadawy, Sana Ghafoor, Syed T. R. Rizvi, Applications of variational integrators to couple of linear dynamical models discussing temperature distribution and wave phenomena, 2024, 0217-9849, 10.1142/S0217984924504359 | |
16. | Kashif Ali, Aly. R. Seadawy, Syed T. R. Rizvi, Noor Aziz, Ali Althobaiti, Dynamical properties and travelling wave analysis of Rangwala–Rao equation by complete discrimination system for polynomials, 2024, 56, 1572-817X, 10.1007/s11082-024-06894-7 | |
17. | Engy A. Ahmed, Rasha B. AL-Denari, Aly R. Seadawy, Numerical solution, conservation laws, and analytical solution for the 2D time-fractional chiral nonlinear Schrödinger equation in physical media, 2024, 56, 1572-817X, 10.1007/s11082-024-06828-3 | |
18. | Prasanta Chatterjee, Nanda Kanan Pal, Gaston M. N'Guérékata, Asit Saha, Multiple tsunami waves of the fractional geophysical KdV equation, 2024, 0003-6811, 1, 10.1080/00036811.2024.2426238 | |
19. | Sheikh Zain Majid, Muhammad Imran Asjad, Sachin Kumar, Taseer Muhammad, Dynamical Study with Exact Travelling Waves with High Amplitude Solitons to Clannish Random Walker’s Parabolic Equation, 2025, 24, 1575-5460, 10.1007/s12346-024-01175-w | |
20. | Ghauss ur Rahman, J. F. Gómez-Aguilar, Energy balance approach to an optical solitons of (2+1)-dimensional fourth-order Korteweg–de Vries equation with two contemporary integration norms using a new mapping method, 2024, 0924-090X, 10.1007/s11071-024-10659-y | |
21. | Huilin Cui, Yexuan Feng, Zhonglong Zhao, The Integrability and Several Localized Wave Solutions of a Generalized (2+1)-Dimensional Nonlinear Wave Equation, 2025, 24, 1575-5460, 10.1007/s12346-024-01176-9 | |
22. | Hassan Khaider, Achraf Azanzal, Abderrahmane Raji, Global well-posedness and analyticity for the fractional stochastic Hall-magnetohydrodynamics system in the Besov–Morrey spaces, 2024, 2193-5343, 10.1007/s40065-024-00488-7 | |
23. | Brij Mohan, Sachin Kumar, Raj Kumar, On investigation of kink-solitons and rogue waves to a new integrable (3+1)-dimensional KdV-type generalized equation in nonlinear sciences, 2024, 0924-090X, 10.1007/s11071-024-10792-8 | |
24. | Khurrem Shehzad, Jun Wang, Muhammad Arshad, Ali Althobaiti, Aly R. Seadawy, Analytical solutions of (3+1)-dimensional modified KdV–Zakharov–Kuznetsov dynamical model in a homogeneous magnetized electron–positron–ion plasma and its applications, 2025, 22, 0219-8878, 10.1142/S0219887824503146 | |
25. | Yousef Jawernah, Qasem M Tawhari, Musaad S. Aldhabani, Ali H. Hakami, Hussain Gissy, An analytical examination of bright and dark kink solitons in Conformable Pochhammer-Chree equation arising in elastic medium, 2025, 16, 20904479, 103539, 10.1016/j.asej.2025.103539 |