UNDERSTANDING THE SOLITON PHENOMENA: LEAPFROGGING THROUGH THE KORTEWEG-DE VRIES EQUATION FOR MULTI-SOLITONS SOLUTION
DOI:
https://doi.org/10.33003/fjs-2026-1006-5171Keywords:
Korteweg-de Vries Equation, Leapfrog, Multi-Solitons, Numerical StudyAbstract
This study applies the Leapfrog algorithm to numerically obtain multi-soliton solutions of the Korteweg–de Vries (KdV) equation. The KdV equation is a classical nonlinear partial differential equation that models wave motion in dispersive media and has been widely used to describe shallow water waves, plasmas, fiber optics, Josephson junctions, and electromagnetic wave propagation. Over the years, it has become a fundamental model for understanding how nonlinearity and dispersion interact in physical systems. One of its most remarkable features is the existence of solitons; stable, localized wave packets that travel without changing shape. These waves have attracted strong interest because of their unusual ability to interact with other solitons and still retain their identity. The Leapfrog algorithm is implemented in MATLAB and Python, two common scientific computing languages. This comparison evaluates the performance, accuracy, and efficiency of these implementations. The simulation results focus on solitons, especially their collision and interaction behaviours. The analytical and numerical solutions match, although the numerical results exhibit small fluctuations in soliton profiles after collisions. However, these fluctuations do not significantly affect their core properties. This supports the existence of soliton solutions and confirms that solitons collide without altering their characteristics or identities. The findings from this study deepen our understanding of the KdV equations and their soliton solutions and provide valuable insights into the computational aspects of solving differential equations in Python and MATLAB.
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