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<p>In this research, we derive the analytical solutions of nonlinear diffusion equation using the Lie symmetry approach and ansatz transformations. By using ansatz function approach, we derived Rogue waves (RWs), periodic waves (PWs), multi-waves (MWs), lump solutions and some other interaction solutions of the proposed model. These solutions are crucial for applications in modeling traffic flow dynamics and biological invasion fronts, where the evolution of the system is governed from nonlinear interactions of waves. Traffic flow models utilize nonlinear partial differential equations to describe both vehicle movement and density. These equations can display rich dynamics, including traffic jams and shock formation. Likewise, ecological models describing biological invasions can also have wave-like solutions, which may reflect the spreading of a species into a new habitat. Using various parameter values, illustrations are provided to show how the generated solutions behave. A quantitative analysis indicates that for rogue wave solutions, the maximum amplitude increases linearly with the control parameter <i>a</i>, demonstrating that greater nonlinearity increases the occurrence of these extreme wave events. Lastly, to the best of our knowledge, this is the first study in the literature to use ansatz transformations and Lie symmetry analysis technique to obtain these solutions for nonlinear diffusion equation.</p>

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Lie symmetry analysis, rogue waves, periodic waves and various analytical solutions of non-linear diffusion equation with applications

  • 期刊论文

  • 发表日期：

    2025年9月13日

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  • Sajawal Abbas Baloch,
  • Ali Raza,
  • Sibusiso Moyo

摘要

In this research, we derive the analytical solutions of nonlinear diffusion equation using the Lie symmetry approach and ansatz transformations. By using ansatz function approach, we derived Rogue waves (RWs), periodic waves (PWs), multi-waves (MWs), lump solutions and some other interaction solutions of the proposed model. These solutions are crucial for applications in modeling traffic flow dynamics and biological invasion fronts, where the evolution of the system is governed from nonlinear interactions of waves. Traffic flow models utilize nonlinear partial differential equations to describe both vehicle movement and density. These equations can display rich dynamics, including traffic jams and shock formation. Likewise, ecological models describing biological invasions can also have wave-like solutions, which may reflect the spreading of a species into a new habitat. Using various parameter values, illustrations are provided to show how the generated solutions behave. A quantitative analysis indicates that for rogue wave solutions, the maximum amplitude increases linearly with the control parameter a, demonstrating that greater nonlinearity increases the occurrence of these extreme wave events. Lastly, to the best of our knowledge, this is the first study in the literature to use ansatz transformations and Lie symmetry analysis technique to obtain these solutions for nonlinear diffusion equation.