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<p>This article presents a comprehensive analytical and dynamical investigation of the recently formulated integrable Nurshuak-Tolkynay-Myrzakulov-I (NTM-I) system, which arises in the context of nonlinear wave propagation across several branches of physics, including fluid dynamics, nonlinear optics, quantum field theory, and plasma physics. First, we reduce the coupled partial differential equations to an ordinary differential equation using a traveling wave transformation. Bifurcation structures are then examined via a Galilean transformation, allowing for the identification of equilibrium states and their transitions under parameter variation. To probe the chaotic regime of the dynamical system, we introduce external perturbations of trigonometric, hyperbolic, and elliptic type. The resultant dynamics are analyzed using qualitative and quantitative tools such as phase portraits, Poincaré maps, time series, return maps, Lyapunov exponents and multistability, all of which demonstrate the model’s complex attractor behavior. In the integrable framework, we further employ the Jacobi elliptic function expansion method to derive analytical exact solutions. These soliton solutions are expressed in terms of Jacobi elliptic functions <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\Upsilon (\rho , m)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Υ</mi> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo>,</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and reduce to trigonometric or hyperbolic forms under specific limits of the modulus parameter <i>m</i>. The resulting wave structures include bright, dark, periodic, smooth periodic, shock wave, and W-shaped solitons. A detailed graphical analysis in terms of 3D density surfaces, 2D comparative profiles, and polar plots is performed in order to illustrate the evolution and stability as well as the parametric dependence of these waveforms. This work not only enriches the theoretical understanding of the NTM-I system but also gives a useful system for investigating the coexistence of integrability and chaos in nonlinear dynamical models.</p>

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Bifurcation, Chaos, and Solitary Wave Propagation in the Integrable NTM-I System via the Jacobi Elliptic Function Expansion

  • 期刊论文

  • 发表日期：

    2026年2月23日

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  • Ghulam Hussain Tipu,
  • Kainat Anwar,
  • Fengping Yao,
  • Faisal Javed

摘要

This article presents a comprehensive analytical and dynamical investigation of the recently formulated integrable Nurshuak-Tolkynay-Myrzakulov-I (NTM-I) system, which arises in the context of nonlinear wave propagation across several branches of physics, including fluid dynamics, nonlinear optics, quantum field theory, and plasma physics. First, we reduce the coupled partial differential equations to an ordinary differential equation using a traveling wave transformation. Bifurcation structures are then examined via a Galilean transformation, allowing for the identification of equilibrium states and their transitions under parameter variation. To probe the chaotic regime of the dynamical system, we introduce external perturbations of trigonometric, hyperbolic, and elliptic type. The resultant dynamics are analyzed using qualitative and quantitative tools such as phase portraits, Poincaré maps, time series, return maps, Lyapunov exponents and multistability, all of which demonstrate the model’s complex attractor behavior. In the integrable framework, we further employ the Jacobi elliptic function expansion method to derive analytical exact solutions. These soliton solutions are expressed in terms of Jacobi elliptic functions \(\Upsilon (\rho , m)\) Υ ( ρ , m ) and reduce to trigonometric or hyperbolic forms under specific limits of the modulus parameter m. The resulting wave structures include bright, dark, periodic, smooth periodic, shock wave, and W-shaped solitons. A detailed graphical analysis in terms of 3D density surfaces, 2D comparative profiles, and polar plots is performed in order to illustrate the evolution and stability as well as the parametric dependence of these waveforms. This work not only enriches the theoretical understanding of the NTM-I system but also gives a useful system for investigating the coexistence of integrability and chaos in nonlinear dynamical models.