Sensitivity, bifurcation behavior, stability and new traveling wave solutions for the stochastic Nizhnik-Novikov-Veselov system

Hafiz M. A. Siddiqui; Lotfi Jlali; A. Nazir; Syed T. R. Rizvi; Atef F. Hashem; Aly R. Seadawy; Hafiz M. A. Siddiqui; Lotfi Jlali; A. Nazir; Syed T. R. Rizvi; Atef F. Hashem; Aly R. Seadawy

doi:10.3934/math.20251179

AIMS Mathematics

2025, Volume 10, Issue 11: 26823-26843. doi: 10.3934/math.20251179

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Research article

Sensitivity, bifurcation behavior, stability and new traveling wave solutions for the stochastic Nizhnik-Novikov-Veselov system

1.
Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Lahore Pakistan
2.
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11566, Saudi Arabia
3.
Mathematics Department, Faculty of Science, Taibah University, Al-Madinah Al-Munawarah 41411, Saudi Arabia

Received: 16 August 2025 Revised: 27 October 2025 Accepted: 11 November 2025 Published: 19 November 2025
MSC : 35B10, 35C07, 35C08, 35Q35

Nonlinear stochastic models play a crucial role in describing complex wave phenomena in multidimensional physical systems. Motivated by this, we investigated the Stochastic Nizhnik-Novikov-Veselov (SNNV) equation to explore its solitary wave dynamics, chaotic behavior, bifurcation structures, sensitivity, and stability characteristics. We used the extended modified auxiliary equation mapping (EMAEM) method, an enhanced analytical framework with greater flexibility and broader solution structures versus conventional methods. With this approach, we derived new families of exact solitary wave solutions, including single, dark, and bright singular solitons. The proposed methods explain dynamical characteristics that were previously unexplored for the SNNV equation. The stochastic model will be converted into a dynamical system using the Galilean transformation. This approach enables exploration of its dynamical behavior via stochastic processes. We used Poincaré maps, phase portraits, and time-series trajectory simulations to establish its strong stochastic behavior, including chaotic behavior. To show that these methods are dynamically stable, we conducted a stability analysis using the Hamiltonian system framework. The proposed study will significantly advance the dynamic interpretation of chaos in the SNNV equation and establish the superiority of EMAEM approach for wave structure formation.
- stochastic Nizhnik-Novikov-Veselov (SNNV) system,
- exact solutions,
- integrability
Citation: Hafiz M. A. Siddiqui, Lotfi Jlali, A. Nazir, Syed T. R. Rizvi, Atef F. Hashem, Aly R. Seadawy. Sensitivity, bifurcation behavior, stability and new traveling wave solutions for the stochastic Nizhnik-Novikov-Veselov system[J]. AIMS Mathematics, 2025, 10(11): 26823-26843. doi: 10.3934/math.20251179

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Abstract

Nonlinear stochastic models play a crucial role in describing complex wave phenomena in multidimensional physical systems. Motivated by this, we investigated the Stochastic Nizhnik-Novikov-Veselov (SNNV) equation to explore its solitary wave dynamics, chaotic behavior, bifurcation structures, sensitivity, and stability characteristics. We used the extended modified auxiliary equation mapping (EMAEM) method, an enhanced analytical framework with greater flexibility and broader solution structures versus conventional methods. With this approach, we derived new families of exact solitary wave solutions, including single, dark, and bright singular solitons. The proposed methods explain dynamical characteristics that were previously unexplored for the SNNV equation. The stochastic model will be converted into a dynamical system using the Galilean transformation. This approach enables exploration of its dynamical behavior via stochastic processes. We used Poincaré maps, phase portraits, and time-series trajectory simulations to establish its strong stochastic behavior, including chaotic behavior. To show that these methods are dynamically stable, we conducted a stability analysis using the Hamiltonian system framework. The proposed study will significantly advance the dynamic interpretation of chaos in the SNNV equation and establish the superiority of EMAEM approach for wave structure formation.

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