Abstract
The article is focused on the analysis of a mathematical model known as the Nizhnik–Novikov–Veselov (SNNV) system, which incorporates a specialized mathematical tool called the truncated M-fractional derivative (TMD). This model has broad applications in various scientific fields and considered an isotropic Lax extension of the one-dimensional Kdv equation which has diverse applications in the areas of warm ions, shallow water waves, electromagnetic signals, and river irrigation flows. To extract the solutions of the SNNV system, a modified version of the extended tanh-function method which is known as amended extended tanh-function method (AETF) is brought into play to derive solitary wave solutions for the SNNV system. These formulated solutions are categorized as trigonometric, exponential, and hyperbolic functions. For physical exemplification and knowledge of the dynamic physical features of the governing model, some reported solutions have been deliberated graphically by selecting appropriate parametric values. The computed outcomes have implications in diverse areas of applied sciences such as quantum mechanics, magneto-electrodynamics, heavy ion collisions, optical fiber technology and also opens doors to practical applications in diverse domains. The application of the AETF approach in this study demonstrates its efficiency and competence in scrutinizing and extracting soliton solutions in nonlinear partial differential equation.
Similar content being viewed by others
Data availibility
As this research did not involve the creation or examination of any datasets, data sharing is not relevant in this context.
References
Abd Elaziz, M., Yousri, D., Al-qaness, M.A., AbdelAty, A.M., Radwan, A.G., Ewees, A.A.: A Grunwald–Letnikov based Manta ray foraging optimizer for global optimization and image segmentation. Eng. Appl. Artif. Intell. 98, 104105 (2021)
Abdelwahab, A.M., Mekheimer, K.S., Ali, K.K., El-Kholy, A., Sweed, N.S.: Numerical simulation of electroosmotic force on micropolar pulsatile bloodstream through aneurysm and stenosis of carotid. Waves Random Complex Med. 21, 1–32 (2021)
Ahmad, J., Mohyud-Din, S.T.: An efficient algorithm for some highly nonlinear fractional PDEs in mathematical physics. PLoS ONE 9(12), 109127 (2014)
Ahmad, J., Rani, S., Turki, N.B., Shah, N.A.: Novel resonant multi-soliton solutions of time fractional coupled nonlinear Schrodinger equation in optical fiber via an analytical method. Res. Phys. 52, 106761 (2023a)
Ahmad, J., Mustafa, Z., Turki, N.B., Shah, N.A.: Solitary wave structures for the stochastic Nizhnik–Novikov–Veselov system via modified generalized rational exponential function method. Res. Phys. 52, 106776 (2023b)
Al-Amr, M.O., Rezazadeh, H., Ali, K.K., Korkmazki, A.: N1-soliton solution for Schrödinger equation with competing weakly nonlocal and parabolic law nonlinearities. Commun. Theor. Phys. 72(6), 065503 (2020)
Al-Askar, F.M., Cesarano, C., Mohammed, W.W., El-Morshedy, M.: Solitary wave solutions of the fractional-stochastic quantum Zakharov–Kuznetsov equation arises in quantum magneto plasma. Mathematics 11(2), 488 (2023)
Ali, M., Alquran, M., Jaradat, I.: Asymptotic-sequentially solution style for the generalized Caputo time-fractional Newell-Whitehead-Segel system. Adv. Differ. Equ. 2019(1), 1–9 (2019)
Ali, M., Alquran, M., Jaradat, I.: Explicit and approximate solutions for the conformable-Caputo time-fractional diffusive Predator-Prey model. Int. J. Appl. Comput. Math. 7(3), 90 (2021)
Ali, A., Ahmad, J., Javed, S.: Exact soliton solutions and stability analysis to (3+1)-dimensional nonlinear Schrödinger model. Alex. Eng. J. 76, 747–756 (2023)
Almusawa, H., Jhangeer, A.: Nonlinear self-adjointness, conserved quantities and Lie symmetry of dust size distribution on a shock wave in quantum dusty plasma. Commun. Nonlinear Sci. Numer. Simul. 114, 106660 (2022)
Almusawa, H., Ali, K.K., Wazwaz, A.M., Mehanna, M.S., Baleanu, D., Osman, M.S.: Protracted study on a real physical phenomenon generated by media inhomogeneities. Res. Phys. 31, 104933 (2021)
Alquran, M.: The amazing fractional Maclaurin series for solving different types of fractional mathematical problems that arise in physics and engineering. Partial Differ. Equ. Appl. Math. 7, 100506 (2023)
Alquran, M., Jaradat, I.: Identifying combination of Dark-Bright Binary-Soliton and Binary-Periodic Waves for a new two-mode model derived from the (2+1)-dimensional Nizhnik–Novikov–Veselov equation. Mathematics 11(4), 861 (2023)
Arora, S., Mathur, T., Agarwal, S., Tiwari, K., Gupta, P.: Applications of fractional calculus in computer vision: a survey. Neurocomputing 489, 407–428 (2022)
Cattani, C., Sulaiman, T.A., Baskonus, H.M., Bulut, H.: On the soliton solutions to the Nizhnik–Novikov–Veselov and the Drinfel’d-Sokolov systems. Opt. Quant. Electron. 50, 1–11 (2018)
Cesarano, C., Aly, E.S., Mohammed, W.W., Al-Askar, F.M.: The soliton solutions of the stochastic shallow water wave equations in the sense of beta-derivative. Mathematics 11(6), 1338 (2023)
Chandra, S., Abbas, S.: Analysis of fractal dimension of mixed Riemann-Liouville integral. Numer. Algorithms 91(3), 1021–1046 (2022)
Chen, W., Sun, H., Li, X.: Fractional derivative modeling in mechanics and engineering. Beijing, China, Springer (2022)
Choi, J.H., Kim, H.: Coupled fractional traveling wave solutions of the extended Boussinesq-Whitham-Broer-Kaup-type equations with variable coefficients and fractional order. Symmetry 13(8), 1396 (2021)
Dan, J., Sain, S., Ghose-Choudhury, A., Garai, S.: Application of the Kudryashov function for finding solitary wave solutions of NLS type differential equations. Optik 224, 165519 (2020)
Duhe, J.F., Victor, S., Melchior, P., Abdelmounen, Y., Roubertie, F.: Fractional derivative truncation approximation for real-time applications. Commun. Nonlinear Sci. Numer. Simul. 119, 107096 (2023)
El-shamy, O., El-barkoki, R., Ahmed, H.M., Abbas, W., Samir, I.: Exploration of new solitons in optical medium with higher-order dispersive and nonlinear effects via improved modified extended tanh function method. Alex. Eng. J. 68, 611–618 (2023)
Ghanbari, B.: Abundant soliton solutions for the Hirota–Maccari equation via the generalized exponential rational function method. Mod. Phys. Lett. B 33(09), 1950106 (2019)
Ghanbari, B., Baleanu, D.: New optical solutions of the fractional Gerdjikov–Ivanov equation with conformable derivative. Front. Phys. 8, 167 (2020)
Ghanbari, B., Baleanu, D.: Applications of two novel techniques in finding optical soliton solutions of modified nonlinear Schrödinger equations. Res. Phys. 44, 106171 (2023)
Ghanbari, B., Gómez-Aguilar, J.F.: New exact optical soliton solutions for nonlinear Schrödinger equation with second-order spatio-temporal dispersion involving M-derivative. Mod. Phys. Lett. B 33(20), 1950235 (2019)
Ilhan, E., Kiymaz, I.O.: A generalization of truncated M-fractional derivative and applications to fractional differential equations. Appl. Math. Nonlinear Sci. 5(1), 171–188 (2020)
Jhangeer, A., Hussain, A., Junaid-U-Rehman, M., Baleanu, D., Riaz, M.B.: Quasi-periodic, chaotic and travelling wave structures of modified Gardner equation. Chaos Solitons Fractals 143, 110578 (2021)
Jhangeer, A., Almusawa, H., Hussain, Z.: Bifurcation study and pattern formation analysis of a nonlinear dynamical system for chaotic behavior in traveling wave solution. Res. Phys. 37, 105492 (2022)
Khater, M., Ghanbari, B.: On the solitary wave solutions and physical characterization of gas diffusion in a homogeneous medium via some efficient techniques. Eur. Phys. J. Plus 136(4), 1–28 (2021)
Khater, M.M., Muhammad, S., Al-Ghamdi, A., Higazy, M.: Abundant wave structures of the fractional Benjamin-Ono equation through two computational techniques. J. Ocean Eng. Sci. 1–5 (2022)
Kudryashov, N.A.: Solitary wave solutions of hierarchy with non-local nonlinearity. Appl. Math. Lett. 103, 106155 (2020)
Leake, C., Johnston, H., Mortari, D.: The multivariate theory of functional connections: Theory, proofs, and application in partial differential equations. Mathematics 8(8), 1303 (2020)
Lou, S.Y.: On the coherent structures of the Nizhnik–Novikov–Veselov equation. Phys. Lett. A 277(2), 94–100 (2000)
Manafian, J., Ilhan, O.A., Avazpour, L., Alizadeh, A.A.: N-lump and interaction solutions of localized waves to the (2+1)-dimensional asymmetrical Nizhnik–Novikov–Veselov equation arise from a model for an incompressible fluid. Math. Methods Appl. Sci. 43(17), 9904–9927 (2020)
Manukure, S., Chowdhury, A., Zhou, Y.: Complexiton solutions to the asymmetric Nizhnik–Novikov–Veselov equation. Int. J. Mod. Phys. B 33(11), 1950098 (2019)
Mathanaranjan, T.: Optical solitons and stability analysis for the new (3+1)-dimensional nonlinear Schrodinger equation. J. Nonlinear Opt. Phys. Mater. 32(2), 2350016 (2023)
Mekheimer, K.S., Abo-Elkhair, R.E., Ali, K.K., Keshta, M.G.: Entropy generation and curvature effect on peristaltic thrusting of (Cu-Al2O3) hybrid nanofluid in resilient channel: Nonlinear analysis. Heat Trans. 50(8), 7918–7948 (2021)
Mohamed, M.S., Akinyemi, L., Najati, S.A., Elagan, S.K.: Abundant solitary wave solutions of the Chen-Lee-Liu equation via a novel analytical technique. Opt. Quant. Electron. 54(3), 141 (2022)
Mohammed, W.W., El-Morshedy, M.: The influence of multiplicative noise on the stochastic exact solutions of the @@@@ system. Math. Comput. Simul. 190, 192–202 (2021)
Mohammed, W.W., El-Morshedy, M.: The influence of multiplicative noise on the stochastic exact solutions of the Nizhnik–Novikov–Veselov system. Math. Comput. Simul. 190, 192–202 (2021)
Mohammed, W.W., El-Morshedy, M., Moumen, A., Ali, E.E., Benaissa, M., Abouelregal, A.E.: Effects of M-truncated derivative and multiplicative noise on the exact solutions of the breaking soliton equation. Symmetry 15(2), 288 (2023)
Omame, A., Abbas, M., Onyenegecha, C.P.: A fractional-order model for COVID-19 and tuberculosis co-infection using Atangana–Baleanu derivative. Chaos Solitons Fractals 153, 111486 (2021)
Osman, M.S., Ali, K.K.: Optical soliton solutions of perturbing time-fractional nonlinear Schrödinger equations. Optik 209, 164589 (2020)
Rafiq, M.H., Jhangeer, A., Raza, N.: The analysis of solitonic, supernonlinear, periodic, quasiperiodic, bifurcation and chaotic patterns of perturbed Gerdjikov-Ivanov model with full nonlinearity. Commun. Nonlinear Sci. Numer. Simul. 116, 106818 (2023)
Rani, A., Ashraf, M., Shakeel, M., Mahmood-Ul-Hassan, Q., Ahmad, J.: Analysis of some new wave solutions of DNA-Peyrard-Bishop equation via mathematical method. Mod. Phys. Lett. B 36(21), 2250047 (2022)
Raza, N., Jhangeer, A., Arshed, S., Inc, M.: The chaotic, supernonlinear, periodic, quasiperiodic wave solutions and solitons with cascaded system. Waves in random and complex media, 1-15 (2021)
Shaikh, T.S., Baber, M.Z., Ahmed, N., Iqbal, M.S., Akgül, A., El Din, S.M.: Investigation of solitary wave structures for the stochastic Nizhnik–Novikov–Veselov (SNNV) system. Res. Phys. 48, 106389 (2023)
Sousa, J.V.C., de Oliveira, E.C.: A new truncated \(M\)-fractional derivative type unifying some fractional derivative types with classical properties. arXiv preprint arXiv:1704.08187 (2017)
Tian, H., Niu, Y., Ghanbari, B., Zhang, Z., Cao, Y.: Integrability and high-order localized waves of the (4+ 1)-dimensional nonlinear evolution equation. Chaos Solitons Fractals 162, 112406 (2022)
Tremblay, R.: Fractional derivatives of logarithmic singular functions and applications to special functions. Montes Taurus J. Pure Appl. Math. 3(1), 7–37 (2021)
Tuan, N.H., Mohammadi, H., Rezapour, S.: A mathematical model for COVID-19 transmission by using the Caputo fractional derivative. Chaos Solitons Fractals 140, 110107 (2020)
Wu, G., Guo, Y.: New complex wave solutions and diverse wave structures of the (2+ 1)-dimensional asymmetric Nizhnik–Novikov–Veselov equation. Fractal Fract. 7(2), 170 (2023)
Zhao, Z., He, L.: Resonance y-type soliton and hybrid solutions of a (2+1)-dimensional asymmetrical Nizhnik–Novikov–Veselov equation. Appl. Math. Lett. 122, 107497 (2021)
Zulfiqar, A., Ahmad, J.: Exact solitary wave solutions of fractional modified Camassa–Holm equation using an efficient method. Alex. Eng. J. 59(5), 3565–3574 (2020)
Zulfiqar, A., Ahmad, J.: New optical solutions of conformable fractional perturbed Gerdjikov–Ivanov equation in mathematical nonlinear optics. Res. Phys. 21, 103825 (2021)
Acknowledgements
Not Applicable.
Funding
The authors confirm that they do not receive any financial compensation.
Author information
Authors and Affiliations
Contributions
J.A. Resources, Supervision, Validation, acquisition. S.R. Conceptualization, Methodology, Writing—original draft, Formal analysis, Validation. Software.
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no known financial or interpersonal conflicts that would have appeared to have an impact on the research presented in this study.
Ethical approval
The authors of this research have unanimously agreed and granted their consent for the publication of their work.
Consent for publication
All authors have provided their approval and consent for the publication of this research work.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Ahmad, J., Rani, S. Exploring stochastic dynamics with different wave structures for the Nizhnik–Novikov–Veselov system and their applications. Opt Quant Electron 56, 453 (2024). https://doi.org/10.1007/s11082-023-05899-y
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11082-023-05899-y