Dynamical analysis of (2+1)-dimensional Akbota-Myrzakulov-Tolkynay-Zhaidary equation: Bifurcations, chaos, and traveling wave solutions

Da Shi; Zhao Li; Da Shi; Zhao Li

doi:10.3934/math.20261036

AIMS Mathematics

2026, Volume 11, Issue 8: 25861-25873. doi: 10.3934/math.20261036

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Dynamical analysis of (2+1)-dimensional Akbota-Myrzakulov-Tolkynay-Zhaidary equation: Bifurcations, chaos, and traveling wave solutions

Da Shi ,
Zhao Li ^,

College of Computer Science, Chengdu University, Chengdu, 610106, China

Received: 04 July 2026 Revised: 06 August 2026 Accepted: 18 August 2026 Published: 20 August 2026
MSC : 34C23, 35B20, 35C05, 35C07

This article conducted a systematic dynamical study on the Akbota-Myrzakulov-Tolkynay-Zhaidary (AMTZ) equation, with a focus on its traveling wave solutions, bifurcations, and chaotic phenomena. First, the AMTZ equation was transformed into a nonlinear ordinary differential equation using traveling wave transformation. Based on the polynomial complete discriminant system method, various exact traveling wave solutions of the equation were obtained, including Jacobi elliptic function solutions, implicit solutions, and rational function solutions. Second, using the phase plane analysis method, the corresponding two-dimensional planar dynamical system and its dynamic behavior under small disturbances were studied, revealing the bifurcation characteristics of the equilibrium point. Finally, three-dimensional waveform diagrams, two-dimensional profile diagrams, and contour maps of typical solutions were provided through numerical simulations, visually demonstrating the rich wave propagation modes and spatial structures of the AMTZ equation.
- Akbota-Myrzakulov-Tolkynay-Zhaidary equation,
- exact solution,
- chaotic behavior,
- phase portrait
Citation: Da Shi, Zhao Li. Dynamical analysis of (2+1)-dimensional Akbota-Myrzakulov-Tolkynay-Zhaidary equation: Bifurcations, chaos, and traveling wave solutions[J]. AIMS Mathematics, 2026, 11(8): 25861-25873. doi: 10.3934/math.20261036

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Abstract

This article conducted a systematic dynamical study on the Akbota-Myrzakulov-Tolkynay-Zhaidary (AMTZ) equation, with a focus on its traveling wave solutions, bifurcations, and chaotic phenomena. First, the AMTZ equation was transformed into a nonlinear ordinary differential equation using traveling wave transformation. Based on the polynomial complete discriminant system method, various exact traveling wave solutions of the equation were obtained, including Jacobi elliptic function solutions, implicit solutions, and rational function solutions. Second, using the phase plane analysis method, the corresponding two-dimensional planar dynamical system and its dynamic behavior under small disturbances were studied, revealing the bifurcation characteristics of the equilibrium point. Finally, three-dimensional waveform diagrams, two-dimensional profile diagrams, and contour maps of typical solutions were provided through numerical simulations, visually demonstrating the rich wave propagation modes and spatial structures of the AMTZ equation.

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