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Sensitive visualization, traveling wave structures and nonlinear self-adjointness of Cahn–Allen equation

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Abstract

This study covers the traveling wave analysis of the two-mode Cahn–Allen (TMCA) equation. The TMCA model plays the role of transmitting information into two different locations and preserving its physical properties. A wave transformation converts the considered model into an ordinary differential equation (ODE). By applying the two analytical techniques, the extended \(\tanh -\coth\) technique and the extended \(\left( \frac{G^{\prime }}{G^{2}}\right)\)-expansion scheme, we have obtained a new type of the wave structures like singular, kink and periodic. To present the significance of the acquired outcomes, we have shown the graphical behaviors of the results by taking some suitable values of the involved parameters. The acquired ODE can be transformed into a dynamical system by applying the Galilean transformation. To examine the sensitive behavior, the dynamical system is expressed in a non-autonomous form, and the effects are analyzed by considering various initial conditions. The considered model admits the two-dimensional Lie algebra. The nonlinear self-adjointness of the considered model is checked using a new conservation theorem. The results show that the considered model is not self-adjoint and is made self-adjoint by computing a new dependent variable. Finally, the conserved quantities corresponding to each symmetry generator are computed.

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Acknowledgements

This research is funded by “Researchers Supporting Project number (RSPD2024R733), King Saud University, Riyadh, Saudi Arabia.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Huanghuai University, Zhumadian, 463000, China

    Yingzi Guan

  2. Department of Mathematics, Quaid-i-Azam University, Islamabad, 45320, Pakistan

    Naseem Abbas

  3. Abdus Salam School of Mathematical Sciences, Government College University, 68-B New Muslim Town, Lahore, 54600, Pakistan

    Akhtar Hussain

  4. Department of Mathematics, Government College Women University Faisalabad, Faisalabad, 38000, Pakistan

    Samara Fatima

  5. Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh, 11451, Saudi Arabia

    Shah Muhammad

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  1. Yingzi Guan
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Guan, Y., Abbas, N., Hussain, A. et al. Sensitive visualization, traveling wave structures and nonlinear self-adjointness of Cahn–Allen equation. Opt Quant Electron 56, 994 (2024). https://doi.org/10.1007/s11082-024-06729-5

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