Investigating Novel Optical Soliton Solutions and qualitative analysis of the traveling waves for a Kolmogorov-Petrovsky-Piskunov Equation

Articles in Press

Document Type : Research Paper

Authors

1 Department of Mathematics, Faculty of Science and Art, Nevşehir Hacı Bektaş Veli University, Nevşehir, 50300, Türkiye.

2 $Department of Mathematics, Sikkim Manipal Institute of Technology, Sikkim Manipal University, Majitar, Sikkim 737136, India.

3 1. Mathematics Education Program, Faculty of Education and Arts, Sohar University, Sohar 311, Oman.\\ 2. Department of Mathematics, Faculty of Science, AL-Azhar University. Nasr City, P.N.Box: 11884- Cairo, Egypt.

Abstract

The main target of this article is to obtain new and several analytical and numerical traveling wave solutions of the Kolmogorov- Petrovsky-Piskunov (KPP) equation. Furthermore qualitative analysis of the traveling waves for the equation is presented through phase plane analysis. The genetics model introduced the equation for the diffusion of a beneficial gene throughout a population. Subsequently, it was used with several chemical, biological, and physical models. To address the inherent complexities associated with the nonlinear equation, the authors employed highly effective techniques: Firstly ( G′/ωG′+G+r )-expansion technique is implemented to create some alternative exact solutions of the equation. A strong and popular method for getting exact solutions of nonlinear partial differential equations (PDEs) is the ( G′/ωG′+G+r ) method. Next, a collocation approach
based on the septic B-spline approximation has been introduced and put into practice for the numerical solution of the equation taking various test problem parameter values into consideration. The appropriate solutions for two test problems are found by computing the L2 and L∞ error norms, which highlights the significance of the procedure and demonstrates its applicability and credibility. The numerical findings are inferred to match the analytical answers well, suggesting that the existing B-spline collocation algorithm is a strong and appealing algorithm. The results are tabulated and reported both modally and in terms of productivity of the procedure. Analytical and numerical results make the methods more convenient and systematically handle the nonlinear solution process. Qualitative analysis of the traveling waves for the Kolmogorov-Petrovsky-Piskunov Equation
is presented through phase plane analysis. The results produced from both analytical and
numerical methods demonstrate the great utility of this study for scientists tasked with
identifying characteristics and features of nonlinear processes across a variety of scientific
domains.

Keywords

Main Subjects

Computational Methods for Differential Equations

Articles in Press, Accepted Manuscript
Available Online from 19 December 2025

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History

  • Receive Date: 15 March 2025
  • Revise Date: 19 August 2025
  • Accept Date: 15 December 2025

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