New Auxiliary Equation Method Reveals Solitons and Chaos in Time-Fractional Long-Wave Equation

A team of applied mathematicians has unveiled a comprehensive analytical study of the time-fractional Symmetric Regularised Long-Wave (SRLW) equation, an important nonlinear model used to describe weakly nonlinear ion-acoustic and space-charge waves in plasmas as well as shallow-water wave propagation. The research, published in Frontiers in Physics by Moazzma Inayat and Muhammad Abbas of the University of Sargodha in Pakistan, Alina Alb Lupas of the University of Oradea in Romania, independent researcher Ahmet Bekir, and Hissah Albaqami of Taif University in Saudi Arabia, combines a new auxiliary equation method with conformable fractional derivatives to extract families of exact soliton solutions, and then subjects the underlying dynamical system to bifurcation analysis, chaos testing, and sensitivity analysis.

The SRLW equation, first derived by Seyler and Fenstermacher in 1984, is a symmetric regularised variant of the long-wave equation that captures the interplay between nonlinearity and dispersion in weakly nonlinear wave systems. It has been applied to model solitary waves, ion-acoustic waves in plasma, and shallow sea waves. In the new study, the authors reformulate the classical integer-order model with a conformable fractional derivative in time, an operator introduced by R. Khalil and colleagues that extends differentiation to non-integer orders while retaining familiar properties such as the product and chain rules. The fractional order acts as a memory parameter: when it equals one, the classical model is recovered, while values below one introduce hereditary effects that allow the present state of the wave to depend on its full history.

The mathematical engine of the work is the New Auxiliary Equation Method (NAEM), a systematic ansatz-based technique for solving nonlinear partial differential equations. The researchers first apply a travelling-wave transformation that converts the fractional SRLW equation into an ordinary differential equation, balancing the highest-order derivative term against the highest-degree nonlinear term to determine the polynomial degree of the solution expansion. The ansatz is then expressed as a finite series in powers of a function that satisfies an auxiliary differential equation with real constant parameters. Substituting the series into the reduced equation and setting all like coefficients to zero yields an algebraic system, which the authors solve with Mathematica to determine the unknown constants.

Depending on the discriminant conditions of the auxiliary equation, the method generates twenty distinct solution families, from which the authors extract trigonometric, hyperbolic, and rational solution clusters for the SRLW model. The physically significant outcomes include bell-shaped and anti-bell-shaped solitons, which correspond to localised solitary waves of positive and negative elevation respectively, and kink and anti-kink solutions, which describe transitions between two stable states. The authors draw explicit physical analogies: kink solutions can be interpreted as magnetic domain walls in ferromagnetic materials, while anti-bell solutions resemble depression solitary waves observed in stratified fluids, internal solitary waves in lakes and coastal seas, and depression waves in stratified ocean waters.

To visualise how the fractional parameter shapes the waves, the team produced two-dimensional profiles, contour plots, and three-dimensional surface diagrams for various fractional orders and parameter values. The graphics demonstrate that the fractional order directly modulates the amplitude, width, propagation speed, and decay rate of the solitons, offering a tunable family of wave profiles that the fixed integer-order model cannot reproduce. This flexibility, the authors argue, makes the conformable fractional framework a more faithful description of complex media with memory and nonlocal interactions, from anomalous transport to long-range temporal correlations.

Beyond constructing solutions, the study converts the SRLW equation into a planar dynamical system via a Galilean transformation, reducing the wave problem to a first-order system with a Hamiltonian constant. Analysing the equilibrium points of this system, the researchers classify them as saddle points, centre points, or cuspidal points depending on the sign of the Jacobian determinant. Across four parameter regimes defined by the signs of the system coefficients, the phase portraits alternate between configurations of one centre and one saddle point, revealing how the qualitative behaviour of the wave system flips as parameters cross critical thresholds.

Bifurcation analysis then maps how these qualitative changes unfold. The authors show that when two unstable equilibria and one stable equilibrium collide and annihilate, the stability of the equilibrium points can interchange in a way that destroys two stable points and produces a single unstable one. Such transitions, they note, are essential for forecasting state changes in nonlinear systems and have direct relevance to fluid dynamics, plasma physics, optical fibres, and biological wave propagation, where small parameter adjustments can trigger abrupt shifts from order to instability.

To probe chaos, the team introduced a periodic perturbation term into the planar system, with an amplitude and a frequency that can be varied independently. The results trace a clear route to chaos: at low forcing amplitude and frequency the system exhibits regular periodic behaviour, increasing the frequency and intensity produces quasi-periodic patterns with multiple incommensurate frequencies, and at higher amplitude and frequency the dynamics become fully chaotic, displaying the irregular, unpredictable trajectories characteristic of deterministic chaos. Time series and phase portraits confirm this progression, illustrating how external disturbances can destabilise otherwise coherent soliton dynamics.

Finally, sensitivity analysis examines how tiny differences in initial conditions propagate through the system. By running the dynamical system with closely spaced initial values and plotting the diverging trajectories in different colours, the authors identify which parameters dominate the system’s response and where behaviour is most fragile. This analysis helps pinpoint stability regions, quantify the impact of uncertainty, and improve model reliability by focusing attention on the critical parameters that control soliton evolution.

The authors emphasise that the study’s novelty lies in pairing the conformable fractional derivative with the new auxiliary equation method, distinguishing it from conventional approaches such as the exp-function method, the Jacobi elliptic function method, and various expansion techniques, which they compare in a detailed table highlighting each method’s mathematical mechanism and limitations. Looking ahead, they propose extending the framework to higher-dimensional and coupled fractional wave equations, adding stability analysis and numerical simulation, and applying it to more intricate models in fluid dynamics, plasma physics, and nonlinear optics. The work offers a unified recipe, they conclude, for both finding exact soliton solutions and decoding the dynamical behaviour, chaos, and sensitivity of fractional nonlinear wave models.

Subject of Research: Analytical soliton solutions and dynamical analysis of the time-fractional symmetric regularised long-wave equation

Article Title: Bifurcation, chaotic behavior, sensitivity analysis, and optical soliton solutions to the time-fractional symmetric regularised-long-wave equation

Article References: Bifurcation, chaotic behavior, sensitivity analysis, and optical soliton solutions to the time-fractional symmetric regularised-long-wave equation. (n.d.). https://doi.org/10.3389/fphy.2026.1941838

Image Credits: AI Generated

DOI: 10.3389/fphy.2026.1941838

Keywords: solitons, fractional calculus, nonlinear waves, bifurcation analysis, chaos, plasma physics, SRLW equation, Bifurcation, chaotic, behavior, sensitivity, analysis