Phase portraits, detection of chaos, and solitary wave solutions of the first extended (3+1)-dimensional KP equation for nonlinear optical systems

Ejaz Hussain; Nida Raees; Ali. H. Tedjani; Yakup Yildirim; Taha Radwan; Ejaz Hussain; Nida Raees; Ali. H. Tedjani; Yakup Yildirim; Taha Radwan

doi:10.3934/math.2026770

AIMS Mathematics

2026, Volume 11, Issue 6: 18917-18942. doi: 10.3934/math.2026770

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Phase portraits, detection of chaos, and solitary wave solutions of the first extended (3+1)-dimensional KP equation for nonlinear optical systems

1.
Institute of Mathematics, University of the Punjab, Quaid-e-Azam Campus, Lahore 54590, Pakistan
2.
Centre for High Energy Physics, University of the Punjab, Quaid-e-Azam Campus, Lahore 54590, Pakistan
3.
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11564, Saudi Arabia
4.
Department of Computer Engineering, Biruni University, Istanbul 34010, Turkey
5.
Mathematics Research Center, Near East University, 99138 Nicosia, Cyprus
6.
Department of Management Information Systems, College of Business and Economics, Qassim University, Buraydah 51452, Saudi Arabia

Received: 11 February 2026 Revised: 14 June 2026 Accepted: 18 June 2026 Published: 26 June 2026
MSC : 26A48, 26A51, 33B10, 37K40, 39B62

This investigation gives a comprehensive dynamical and analytical analysis of the first extended (3+1)-dimensional Kadomtsev-Petviashvili (eKP) equation, which is featured in the fields of non-linear optics and wave propagation. By the method of traveling wave transformation, the nonlinear partial differential equation is transformed into a planar dynamical system, and detailed phase plane and bifurcation analysis can be done. We analyze the qualitative characteristics of equilibrium points, i.e., centers, saddles, and cusps, for different physical conditions. An external periodic disturbance is introduced to study some complicated dynamics, and we observe chaos using a number of diagnostic tools, such as phase portraits, time series, return maps, Lyapunov exponents, and multistability analysis. A sensitivity study indicates that the dynamics of the waves depend greatly upon the initial condition, and this reveals the non-linear, unpredictable nature of the system. In parallel with the dynamical study, we obtained exact analytical solutions to the eKP equation with the help of a bilinear form. We applied various newly developed analytical techniques to obtain exact solutions, such as homoclinic solutions, multiwave solutions, and M-type rational solutions. We obtained homoclinic breather waves, M-type and rational waves, and multi-wave interactions, which exhibit localized oscillating states, stable rogue-wave solutions, and wave-number coupling. Plots show the robustness and toughness of these solutions. Merging dynamic insights and precise solutions of the extended KP model allows a better understanding of the complex nonlinear behavior, opening new horizons in soliton theory as well as applications in the fields of nonlinear optics, fluid mechanics, and complex wave systems.
- mathematical modeling,
- extended (3+1)-dimensional Kadomtsev-Petviashvili (eKP) equation,
- bifurcation analysis,
- chaotic structure,
- multi-stability analysis,
- return map,
- sensitivity analysis
Citation: Ejaz Hussain, Nida Raees, Ali. H. Tedjani, Yakup Yildirim, Taha Radwan. Phase portraits, detection of chaos, and solitary wave solutions of the first extended (3+1)-dimensional KP equation for nonlinear optical systems[J]. AIMS Mathematics, 2026, 11(6): 18917-18942. doi: 10.3934/math.2026770

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Abstract

This investigation gives a comprehensive dynamical and analytical analysis of the first extended (3+1)-dimensional Kadomtsev-Petviashvili (eKP) equation, which is featured in the fields of non-linear optics and wave propagation. By the method of traveling wave transformation, the nonlinear partial differential equation is transformed into a planar dynamical system, and detailed phase plane and bifurcation analysis can be done. We analyze the qualitative characteristics of equilibrium points, i.e., centers, saddles, and cusps, for different physical conditions. An external periodic disturbance is introduced to study some complicated dynamics, and we observe chaos using a number of diagnostic tools, such as phase portraits, time series, return maps, Lyapunov exponents, and multistability analysis. A sensitivity study indicates that the dynamics of the waves depend greatly upon the initial condition, and this reveals the non-linear, unpredictable nature of the system. In parallel with the dynamical study, we obtained exact analytical solutions to the eKP equation with the help of a bilinear form. We applied various newly developed analytical techniques to obtain exact solutions, such as homoclinic solutions, multiwave solutions, and M-type rational solutions. We obtained homoclinic breather waves, M-type and rational waves, and multi-wave interactions, which exhibit localized oscillating states, stable rogue-wave solutions, and wave-number coupling. Plots show the robustness and toughness of these solutions. Merging dynamic insights and precise solutions of the extended KP model allows a better understanding of the complex nonlinear behavior, opening new horizons in soliton theory as well as applications in the fields of nonlinear optics, fluid mechanics, and complex wave systems.

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