The Estevez–Prada Equation: A Comprehensive Review with a Focus on High-Dimensional Wave Dynamics
Abstract
Nonlinear evolution equations in three spatial dimensions underpin the modelling of wave propagation in plasmas, optical fibres, oceans, and complex fluids. The Estevez–Prada equation (EPE), introduced by Estévez and Prada in the early 2000s as a higher-dimensional extension of integrable systems, has recently re-emerged as a versatile model of anisotropic, high-dimensional wave dynamics. This review provides a comprehensive and critical synthesis of the EPE literature up to 2026, with an explicit focus on its (3+1)-dimensional form. We describe the mathematical origins of the equation, examine its Painlevé integrability, and organise the extensive family of analytical methods that have been applied to it—ranging from the Hirota bilinear method, Darboux and Bäcklund transformations, and Lie symmetry analysis to the tanh–coth, (G′/G)-expansion, Jacobi elliptic function, and modified extended direct algebraic methods. We then classify the wave structures that these methods generate, including kinks, anti-kinks, dark and bright solitons, singular and periodic-singular waves, kink–antikink interactions, lumps, rogue waves, and breathers. A critical analysis identifies recurring methodological weaknesses, including solution redundancy, limited physical validation, and inconsistent verification, and documents clear trends toward variable-coefficient, fractional, stochastic, and data-driven formulations. A comparative synthesis of 133 verified primary sources is presented, and open problems and future research directions are highlighted. The review concludes that the EPE is a mathematically rich and physically relevant benchmark for high-dimensional nonlinear wave theory, but that its quantitative validation and standardisation remain incomplete. Keywords: Estevez–Prada equation; high-dimensional nonlinear systems; Painlevé integrability; solitons; exact solutions; fractional and stochastic models; physics-informed neural networks