Jacobi Elliptic Solutions of Fractional Phi-4, BBM, and KdV Equations: A Unified Framework
International Journal of Theoretical Physics, vol.65, no.9, 2026 (SCI-Expanded, Scopus)
- Publication Type: Article / Article
- Volume: 65 Issue: 9
- Publication Date: 2026
- Doi Number: 10.1007/s10773-026-06479-y
- Journal Name: International Journal of Theoretical Physics
- Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Aerospace Database, INSPEC, MathSciNet, zbMATH, Academic Search Ultimate (EBSCO), Engineering Source (EBSCO), Technology Collection (ProQuest)
- Keywords: BBM equation, Jacobi elliptic functions, KdV equation, Nonlinear wave equations, Phi-4 equation
- Yıldız Technical University Affiliated: Yes
Abstract
This study develops a unified analytical framework for constructing exact travelling-wave solutions of the conformable time-fractional Phi-4, Benjamin–Bona–Mahony (BBM), and Korteweg–de Vries (KdV) equations. The three equations are treated simultaneously under the same travelling-wave transformation and auxiliary differential equation, enabling a direct comparative analysis of cubic nonlinearity in the Phi-4 equation against the quadratic nonlinear dispersive structures of the BBM and KdV equations (a comparison that single-equation studies cannot provide). By incorporating eleven Jacobi elliptic structures, including reciprocal forms (ns, nc, nd) and ratio-type forms (sc, sd, cs) in addition to the standard sn, cn, and dn functions, the framework produces wave profiles (periodic cnoidal waves, soliton-approaching pulses, asymmetric structures, and singular limiting forms) that standard approaches restricted to three elliptic functions cannot capture. The conformable fractional order is shown to act as a temporal scaling parameter that modifies the phase velocity of each wave type without altering its spatial structure, a behavior verified comparatively across all three models. Residual-based substitution confirms that the derived solutions satisfy the governing equations exactly. The obtained closed-form profiles are intended to serve as analytical reference solutions for future numerical, semi-analytical, and stability studies of fractional nonlinear dispersive wave equations.