This paper focuses on the dynamical analysis of the advection-diffusion-reaction equation under various conditions that highlight the system's sensitivity and potential for chaotic behavior. Traveling wave solutions for the underlying equation are derived using a novel modified [Formula: see text] expansion method based on the traveling wave transformation. A broad spectrum of exact traveling wave solutions, including solitons, kinks, periodic solutions, and rational solutions, is obtained. These solutions are recognized as having significant potential applications in fields such as engineering and plasma physics. The proposed method is demonstrated to successfully generate various exponential solutions, such as bright, dark, single, rational, and periodic solitary wave solutions. MATLAB simulations were carried out to visualize the results, producing 3D, 2D, and contour graphs that emphasize the impact of the advection-diffusion-reaction equation. Furthermore, the Galilean transformation is applied to derive the corresponding planar dynamical system, enabling deeper insights into its dynamical behavior. Sensitivity analysis is performed to evaluate the system's response to different initial conditions, with symmetrical properties and equilibrium points being represented through phase portraits. The chaotic behavior of the planar dynamical system under the influence of an external force is also examined. It is revealed that the system exhibits periodic, quasi-periodic, and chaotic processes, with significant increases in intensity and frequency being observed. Additionally, we apply Poincaré maps and Lyapunov exponent to analyze the behavior of the dynamical system by different initial conditions.

Department of Mathematics University of Management and Technology Lahore Pakistan

IT4Innovations VSB Technical University of Ostrava Ostrava Czech Republic

Jadara University Research Center Jadara University Irbid Jordan

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Katzourakis, N. An introduction to viscosity solutions for fully nonlinear PDE with applications to calculus of variations in L (Springer, Berlin, 2014).

Showalter, R. E. Monotone operators in Banach space and nonlinear partial differential equations Vol. 49 (American Mathematical Soc, USA, 2013).

Logan, J. D. An introduction to nonlinear partial differential equations Vol. 89 (Wiley, New York, 2008).

Helal, M. A. Soliton solution of some nonlinear partial differential equations and its applications in fluid mechanics. Chaos, Solitons & Fractals13(9), 1917–1929 (2002).

Wan, P., Manafian, J., Ismael, H. F. & Mohammed, S. A. Investigating One-, Two-, and Triple-Wave Solutions via Multiple Exp-Function Method Arising in Engineering Sciences. Adv. Math. Phys.2020(1), 8018064 (2020).

Esen, H., Ozdemir, N., Secer, A. & Bayram, M. Traveling wave structures of some fourth-order nonlinear partial differential equations. J. Ocean Eng. Sci.8(2), 124–132 (2023).

Bulut, H., Sulaiman, T. A. & Demirdag, B. Dynamics of soliton solutions in the chiral nonlinear Schrödinger equations. Nonlinear Dyn.91, 1985–1991 (2018).

Zhao, X. H. et al. Solitons, Bäcklund transformation and Lax pair for a (2+ 1)-dimensional Davey-Stewartson system on surface waves of finite depth. Waves Random Complex Media28(2), 356–366 (2018).

Si-Liu, X., Jian-Chu, L. & Lin, Y. Exact soliton solutions to a generalized nonlinear Schrödinger equation. Commun. Theor. Phys.53(1), 159 (2010).

Khan, K., Akbar, M. A. & Islam, S. R. Exact solutions for (1+ 1)-dimensional nonlinear dispersive modified Benjamin-Bona-Mahony equation and coupled Klein-Gordon equations. Springerplus3, 1–8 (2014). PubMed PMC

Ege, S. M. & Misirli, E. The modified Kudryashov method for solving some fractional-order nonlinear equations. Adv. Diff. Equ.2014, 1–13 (2014).

Yang, X. F., Deng, Z. C. & Wei, Y. A Riccati-Bernoulli sub-ODE method for nonlinear partial differential equations and its application. Adv. Diff. Equ.2015, 1–17 (2015).

Biswas, A., Ekici, M., Sonmezoglu, A. & Alshomrani, A. S. Optical solitons with Radhakrishnan-Kundu-Lakshmanan equation by extended trial function scheme. Optik160, 415–427 (2018).

Helal, M. A. Soliton solution of some nonlinear partial differential equations and its applications in fluid mechanics. Chaos, Solitons & Fractals13(9), 1917–1929 (2002).

Samir, I., Abd-Elmonem, A. & Ahmed, H. M. General solitons for eighth-order dispersive nonlinear Schrödinger equation with ninth-power law nonlinearity using improved modifed extended tanh method. Opt. Quantum Electron.55(5), 470 (2023).

Rezazadeh, H., Korkmaz, A., Achab, A. E., Adel, W. & Bekir, A. New travelling wave solution-based new Riccati Equation for solving KdV and modified KdV Equations. Appl. Math. Nonlinear Sci.6(1), 447–458 (2021).

Yokus, A. et al. Study on the applications of two analytical methods for the construction of traveling wave solutions of the modifed equal width equation. Open Phys.18(1), 1003–1010 (2020).

Li, C. & Chen, A. Numerical methods for fractional partial differential equations. Int. J. Comput. Math.95(6–7), 1048–1099 (2018).

Sawada, K. & Kotera, T. A method for finding N-soliton solutions of the KdV equation and KdV-like equation. Prog. Theor. Phys.51(5), 1355–1367. 10.1143/PTP.51.1355 (1974).

Da Prato, G. Kolmogorov equations for stochastic PDEs (Springer Science & Business Media, Berlin, 2004).

Younis, M. Soliton Solutions of Fractional Order KdV-Burger’s Equation. J. Adv. Phys.3(4), 325–328 (2014).

Kudryashov, N. A. Exact solitary waves of the Fisher equation. Phys. Lett. A342(1–2), 99–106 (2005).

Hauke, G. A simple subgrid scale stabilized method for the advection-diffusion-reaction equation. Comput. Methods Appl. Mech. Eng.191(27–28), 2925–2947 (2002).

Hauke, G. A simple subgrid scale stabilized method for the advection-diffusion-reaction equation. Comput. Methods Appl. Mech. Eng.191(27–28), 2925–2947 (2002).

Kaya, B. & Gharehbaghi, A. Implicit solutions of advection diffusion equation by various numerical methods. Aust. J. Basic Appl. Sci.8(1), 381–391 (2014).

Jannelli, A., Ruggieri, M. & Speciale, M. P. Analytical and numerical solutions of time and space fractional advection-diffusion-reaction equation. Commun. Nonlinear Sci. Numer. Simul.70, 89–101 (2019).

Savovic, S., Drljaca, B., & Djordjevich, A. A comparative study of two diferent fnite diference methods for solving advection-diffusion reaction equation for modeling exponential traveling wave in heat and mass transfer processes. Ricerche Mat. 1–8 (2022).

Chakrabarty, A. K., Roshid, M. M., Rahaman, M. M., Abdeljawad, T. & Osman, M. S. Dynamical analysis of optical soliton solutions for CGL equation with Kerr law nonlinearity in classical, truncated M-fractional derivative, beta fractional derivative, and conformable fractional derivative types. Res. Phys.60, 107636 (2024).

Umer, M., & Olejnik, P. Symmetry-Optimized Dynamical Analysis of Optical Soliton Patterns in the Flexibly Supported Euler-Bernoulli Beam Equation: A Semi-Analytical Solution Approach. Symmetry16(7), 849 (2024).

Raza, N., Jaradat, A., Basendwah, G. A., Batool, A. & Jaradat, M. M. M. Dynamic analysis and derivation of new optical soliton solutions for the modified complex Ginzburg-Landau model in communication systems. Alex. Eng. J.90, 197–207 (2024).

ur Rahman, M., Sun, M., Boulaaras, S., & Baleanu, D. Bifurcations, chaotic behavior, sensitivity analysis, and various soliton solutions for the extended nonlinear Schrödinger equation. Bound. Value Probl.2024(1), 15 (2024).

Yin, Y. H. & Lü, X. Dynamic analysis on optical pulses via modified PINNs: Soliton solutions, rogue waves and parameter discovery of the CQ-NLSE. Commun. Nonlinear Sci. Numer. Simul.126, 107441 (2023).

Cui, X. Q., Wen, X. Y. & Liu, X. K. Dynamical analysis of multi-soliton and breather solutions on constant and periodic backgrounds for the (2+ 1)-dimensional Heisenberg ferromagnet equation. Nonlinear Dyn.111(24), 22477–22497 (2023).

Kaur, L. & Wazwaz, A. M. Dynamical analysis of soliton solutions for space-time fractional Calogero-Degasperis and Sharma-Tasso-Olver equations. Rom. Rep. Phys74, 108 (2022).

Akgül, A. et al. Regularity and wave study of an advection-diffusion-reaction equation. Sci. Rep.14(1), 20776 (2024). PubMed PMC

Khan, M. A. U. et al. Soliton solutions of nonlinear coupled Davey-Stewartson Fokas system using modified auxiliary equation method and extended -expansion method. Sci. Rep.14, 21949 (2024). PubMed PMC

Behera, S. Analytical Solutions Interms of Solitonic Wave Profiles of Phi-four Equation.

Tsega, E. G. Numerical Solution of Two-Dimensional Nonlinear Unsteady Advection-Diffusion-Reaction Equations with Variable Coefficients. Int. J. Math. Math. Sci.2024(1), 5541066 (2024).

Karjanto, N. Modeling Wave Packet Dynamics and Exploring Applications: A Comprehensive Guide to the Nonlinear Schrödinger Equation. Mathematics12(5), 744 (2024).

Ibrahim, S., Sulaiman, T. A., Yusuf, A., Ozsahin, D. U. & Baleanu, D. Wave propagation to the doubly dispersive equation and the improved Boussinesq equation. Opt. Quant. Electron.56(1), 20 (2024).

Savović, S., Ivanović, M. & Min, R. A Comparative Study of the Explicit Finite Difference Method and Physics-Informed Neural Networks for Solving the Burgers’ Equation. Axioms12(10), 982. 10.3390/axioms12100982 (2023).

Savović, S., Ivanović, M., Drljača, B. & Simović, A. Numerical Solution of the Sine-Gordon Equation by Novel Physics-Informed Neural Networks and Two Different Finite Difference Methods. Axioms13(12), 872. 10.3390/axioms13120872 (2024).

Gu, Y. & Liao, L. Closed form solutions of Gerdjikov-Ivanov equation in nonlinear fiber optics involving the beta derivatives. Int. J. Mod. Phys. B36(19), 2250116 (2022).

Gu, Y., & Yuan, W. Closed form solutions of nonlinear space-time fractional Drinfel’d-Sokolov-Wilson equation via reliable methods. Math. Methods Appl. Sci. (2021).

Gu, Y., Peng, L., Huang, Z. & Lai, Y. Soliton, breather, lump, interaction solutions and chaotic behavior for the (2+ 1)-dimensional KPSKR equation. Chaos, Solitons & Fractals187, 115351 (2024).

Gu, Y., Jiang, C. & Lai, Y. Analytical Solutions of the Fractional Hirota-Satsuma Coupled KdV Equation along with Analysis of Bifurcation, Sensitivity and Chaotic Behaviors. Fractal Fract.8(10), 585 (2024).