Exploration of nonclassical symmetries and exact solutions to the (4+1)-dimensional Boiti–Leon–Manna–Pempinelli equation

Abstract

Abstract This paper presents a complete nonclassical symmetry analysis of the nonlinear integrable model known as the (4 + 1)-dimensional Boiti–Leon–Manna–Pempinelli (4D-BLMP) equation. The analysis is divided into two parts. The first part involves constructing systems of nonlinear partial differential equations for the determining equations based on the dimensions of the model. Five distinct cases of these systems are examined and solutions to these systems are found, leading to the creation of various new nonclassical symmetries. The second part focuses on classifying the developed unknown functions using the constructed nonclassical symmetries and their invariant formulations. These classified functions are then applied to obtain a range of new explicit exact solutions to the model. The paper also includes a graphical analysis of the dynamical behavior of these solutions, taking into account special parameter values. The results highlight the existence of various wave structures in the 4D-BLMP equation, setting it apart from other models that lack non-singular complexiton solutions. The analysis of higher-dimensional nonlinear integrable equations is essential because such models capture complex wave phenomena arising in mathematical physics, fluid dynamics, and optical systems. In particular, understanding their exact and nonclassical solutions provides deeper insight into the underlying dynamics and supports the development of effective analytical and numerical techniques.