ORIGINAL RESEARCH article
Front. Phys.
Sec. Interdisciplinary Physics
Exact Soliton Solutions of the Modified Simplified Camassa Holm and Modified Benjamin Bona Mahony Equations via the Subsidiary ODE Method
Provisionally accepted
- College of Mathematics and Computer Science, Yan’an University, Yan’an, Shaanxi 716000, China, Yanan, China
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In this article, the main objective is the analytical investigation of the simplified modified Camassa Holm (SMCH) equation and the modified Benjamin Bona Mahony (BBM) equation. The SMCH equation plays an important role in modeling shallow water wave dynamics, nonlinear dispersive phenomena and the propagation of solitons in fluid mechanics. The BBM equation is frequently used to describe long surface gravity waves in nonlinear dispersive media and serves as a useful alternative to the standard Korteweg–de Vries (KdV) equation in mathematical physics. To construct exact analytical soliton solutions for these nonlinear models, the subsidiary ordinary differential equation (sub-ODE) method is employed. Through an appropriate wave transformation, the governing partial differential equations are reduced to nonlinear ordinary differential equations. Our mathematical technique yields several types of soliton wave shapes, including \textcolor{red}{bright}, dark, solitary and periodic solitons. Bright solitons depict localized wave peaks, whereas dark solitons reflect intensity drops against a continuous background. The resulting analytical solutions, represented in hyperbolic and trigonometric functions exhibit complex nonlinear behaviours such as periodic and singular patterns. These soliton structures exhibit the complex dynamics and stability of nonlinear waves propagating in dispersive mediums. The graphical demonstration of their propagation in three-dimensional, two-dimensional and contour forms is presented for suitable parameter values.
Keywords: Bright soliton solutions, Dark soliton solutions, Exact soliton solutions, Integrable Systems, Nonlinear evolution equations, Nonlinear partial differential equations (NLPDEs), periodic solutions, Sub-ODE method
Received: 21 Oct 2025; Accepted: 03 Dec 2025.
Copyright: © 2025 Ting. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
* Correspondence: Zhang Ting
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