Abstract
1 Introduction
One of the several models that is being lately addressed is the concatenation of three well known equations that are frequently visible in the field of nonlinear fiber optics. They are the Lakshmanan–Porsezian–Daniel (LPD) model, nonlinear Schrödinger’s equation (NLSE) and Sasa–Satsuma equation (SSE). This concatenated model first appeared during 2014 and was studied by others to date [
The current paper will take these studies a bit further along. This concatenation model is being addressed with the inclusion of spatio-temporal dispersion (STD) as well as the pre-existing chromatic dispersion (CD) and the self-phase modulation (SPM) that comes with Kerr law. The advantage of the inclusion of STD is that the velocity of the soliton can be controlled. This can be advantageously used to our benefit namely to control the Internet bottleneck effect that is a growing problem with an ever-increasing demand for faster Internet. This technological marvel is being utilized for the concatenation model for the initial once. The soliton solutions are first revealed with the usage of two algorithmic approaches. Subsequently, the conservation laws are derived, and the corresponding conserved quantities are identified After providing a brief overview of the model, the results and the respective mathematical analysis are exhibited.
The concatenation model with STD is formulated as [
The wave profile, including its spatial and temporal derivatives, can be described by the complex function q(x, t). The linear temporal evolution of solitons is given by the first term, while a and b are the CD and STD coefficients and finally c represents SPM. The concatenation model is the conjoined version of three familiar, frequently visible models. For λ1 = λ2 = 0, the model collapses to NLSE, while λ1 = 0 or λ2 = 0 give the familiar SSE or LPD equation respectively.
2 Integration algorithms: A recapitulation
Let us examine a governing model with the structure of,
The wave transformation:
2.1 Enhanced Kudryashov’s method
The fundamental principles of the methodology can be summarized as follows:
Step-1: The simplified equation
Step-2: The solutions derived from equations
Step-3: Inserting
2.2 General projective Riccati’s equation method
The algorithmic approach to the general projective Riccati’s equation method is listed here as follows:
Step-1: The explicit solution for the reduced equation
The functions ϕ(ξ) and ψ(ξ) satisfy,
Step-2: The solutions to
Step-3: Upon substituting the expressions given by equations
3 Application to the concatenation model
We can express the solution to equation
The evolution of the soliton speed can be obtained from the imaginary part as follows:
Equation
The soliton velocity given by
3.1 Enhanced Kudryashov’s scheme
Balancing U″″ with U5 in equation
Plugging
Result-1:
As a consequence, the optoelectronic wave field comes out as,
These solitons are also addressed together with the parametric restriction (L1 − L6)2 + 8(3L2 + 2L4)L5 > 0.
Result-2:
Thus, the wave profile stands as,
Choosing η = ±f, (L1 − 2L6)(L6 + ϱ) + 2(4L4 − L2)L5 > 0 and L5(L6 +
3.2 General projective Riccati’s equation approach
Balancing U″″ with U5 in equation
Plugging
Result-1:
In this case, the nonlinear waveform turns out to be,
Result-2:
Therefore, the optoelectronic wave field appears as,
Result-3:
As a result, the nonlinear wave profile shapes up as,
Result-4:
Hence, the nonlinear waveform can be expressed as,
4 Conservation laws
Suppose (Tt, Tx) is a conserved vector associated with the conservation law,
This implies that each multiplier Q results in a conserved vector through a Homotopy operator. E = 0 is the differential equation and Tt, Tx are the conserved densities and fluxes, respectively.
The concatenation model with STD, gives the following conservation laws: For, If, in addition, Conserved Hamiltonian (H) density is presented as below,
The expression for the bright 1-soliton solution, provided in equation
5 Conclusion
In this paper, the concatenation model is revisited with the incorporation of STD alongside the existing CD. The SPM is with Kerr law of nonlinearity. The rational expression for the soliton velocity placed us at an advantage of controlling the Internet bottleneck effect that is responsible of slowing down the traffic flow across the globe. Such an engineering marvel is being applied to the concatenation model for the first time and this gives a true flavor of novelty to the current paper. The results of the paper are indeed encouraging and are applicable to various additional avenues. One would next need to study this technological aspect in birefringent fibers followed by dispersion-flattened fibers. This would lead to the departure from the lab to a situation where rubber meets the road. Additional effects such as stochasticity, time-dependent coefficients to the model are yet to be explored. These would lead to several novelties that would be sequentially disseminated all across the board after aligning the results with pre-existing reports [
References
[1] A. Ankiewicz, N. Akhmediev. Higher-order integrable evolution equation and its soliton solutions.
[2] A. Ankiewicz, Y. Wang, S. Wabnitz, N. Akhmediev. Extended nonlinear Schrödinger equation with higher-order odd and even terms and its rogue wave solutions.
[3] A. Biswas, J. Vega-Guzman, A.H. Kara, S. Khan, H. Triki, O. Gonzalez-Gaxiola, L. Moraru, P.L. Georgescu. Optical solitons and conservation laws for the concatenation model: undetermined coefficients and multipliers approach.
[4] A. Biswas, J. Vega-Guzman, Y. Yildirim, L. Moraru, C. Iticescu, A.A. Alghamdi. Optical solitons for the concatenation model with differential group delay: undetermined coefficients.
[5] O. González-Gaxiola, A. Biswas, J. Ruiz de Chavez, A.A. Alghamdi. Bright and dark optical solitons for the concatenation model by the Laplace–Adomian decomposition scheme(2023).
[6] A. Kukkar, S. Kumar, S. Malik, A. Biswas, Y. Yildirim, S.P. Moshokoa, S. Khan, A.A. Alghamdi. Optical soliton for the concatenation model with Kudryashov’s approaches.
[7] H. Triki, Y. Sun, Q. Zhou, A. Biswas, Y. Yıldırım, H.M. Alshehri. Dark solitary pulses and moving fronts in an optical medium with the higher-order dispersive and nonlinear effects.
[8] M.-Y. Wang, A. Biswas, Y. Yıldırım, L. Moraru, S. Moldovanu, H.M. Alshehri. Optical solitons for a concatenation model by trial equation approach.
[9] N.A. Kudryashov, A. Biswas, A.G. Borodina, Y. Yildirim, H.M. Alshehri. Painleve analysis and optical solitons for a concatenated model.
[10] Y. Yıldırım, A. Biswas, L. Moraru, A.A. Alghamdi. Quiescent optical solitons for the concatenation model with nonlinear chromatic dispersion.
[11] Q. Zhou, Z. Huang, Y. Sun, H. Triki, W. Liu, A. Biswas. Collision dynamics of three-solitons in an optical communication system with third-order dispersion and nonlinearity.
[12] Q. Zhou, H. Triki, J. Xu, Z. Zeng, W. Liu, A. Biswas. Perturbation of chirped localized waves in a dual-power law nonlinear medium.
[13] Q. Zhou, M. Xu, Y. Sun, Y. Zhong, M. Mirzazadeh. Generation and transformation of dark solitons, anti-dark solitons and dark double-hump solitons.
[14] Q. Zhou, Y. Zhong, H. Triki, Y. Sun, S. Xu, W. Liu, A. Biswas. Chirped bright and kink solitons in nonlinear optical fibers with weak nonlocality and cubic-quantic-septic nonlinearity.
[15] Y. Zhong, H. Triki, Q. Zhou. Analytical and numerical study of chirped optical solitons in a spatially inhomogeneous polynomial law fiber with parity-time symmetry potential.
[16] C.C. Ding, Q. Zhou, H. Triki, Y. Sun, A. Biswas. Dynamics of dark and anti-dark solitons for the x-nonlocal Davey–Stewartson II equation.
[17] N.A. Kudryashov. Model of propagation pulses in an optical fiber with a new law of refractive indices.
[18] N.A. Kudryashov. Highly dispersive optical solitons of the generalized nonlinear eighth-order Schrödinger equation.
[19] M. Bayram. Optical bullets with Biswas-Milovic equation having Kerr and parabolic laws of nonlinearity.
[20] T.L. Belyaeva, M.A. Agüero, V.N. Serkin. Nonautonomous solitons of the novel nonlinear Schrödinger equation: Self-compression, amplification, and the bound state decay in external potentials.
[21] V.N. Serkin, A. Ramirez, T.L. Belyaeva. Nonlinear-optical analogies to the Moses sea parting effect: Dark soliton in forbidden dispersion or nonlinearity.
[22] L. Tang. Phase portraits and multiple optical solitons perturbation in optical fibers with the nonlinear Fokas–Lenells equation(2021).
[23] M.-Y. Wang. Optical solitons of the perturbed nonlinear Schrödinger equation in Kerr media.
[24] M.-Y. Wang. Highly dispersive optical solitons of perturbed nonlinear Schrödinger equation with Kudryashov’s sextic-power law nonlinear.
[25] M.-Y. Wang. Optical solitons with perturbed complex Ginzburg–Landau equation in Kerr and cubic–quintic–septic nonlinearity.
[26] T.Y. Wang, Q. Zhou, W.J. Liu. Soliton fusion and fission for the high-order coupled nonlinear Schrödinger system in fiber lasers.
[27] A. Secer. Stochastic optical solitons with multiplicative white noise via Itô calculus.
[28] H. Wang, Q. Zhou, W. Liu. Exact analysis and elastic interaction of multi-soliton for a two-dimensional Gross-Pitaevskii equation in the Bose–Einstein condensation.
[29] A.-M. Wazwaz. Bright and dark optical solitons for (3 + 1)-dimensional Schrödinger equation with cubic–quintic–septic nonlinearities.
[30] A.-M. Wazwaz. Bright and dark optical solitons of the (2 + 1)-dimensional perturbed nonlinear Schrödinger equation in nonlinear optical fibers.
[31] E.M.E. Zayed, M. El-Horbaty, M.E.M. Alngar, M. El-Shater. Dispersive optical solitons for stochastic Fokas–Lenells equation with multiplicative white noise.
[32] Q. Zhou. Influence of parameters of optical fibers on optical soliton interactions.
[33] Q. Zhou, Y. Sun, H. Triki, Y. Zhong, Z. Zeng, M. Mirzazadeh. Study on propagation properties of one-soliton in a multimode fiber with higher-order effects.
[34] Q. Zhou, Z. Luan, Z. Zeng, Y. Zhong. Effective amplification of optical solitons in high power transmission systems.
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