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.

The intrinsic nonlinearity of environmental physical problems has sparked a great deal of interest in nonlinear dynamical models among researchers. These models use differential equations to demonstrate a wide domain of physical phenomena and have implications in different technological and scientific fields, involving optics, ocean engineering, quantum mechanics, fluid dynamics, mechanical engineering, cosmology, and others. Differential equations, a branch of mathematics, are utilized to represent the evolution of physical systems over space or time.

Exact solutions are extremely significant across physics, mathematics, and engineering because they offer comprehensive and precise representations of systems without relying on approximations. These solutions provide profound understanding of the underlying behavior of models and phenomena and are frequently presented in closed, analytical forms. For instance, the precise solution to the equations of motion under gravity in physics explains planetary orbits and identifies the fundamental principles that underlie them. This clarity makes it possible to comprehend theory more thoroughly, which can result in more extensive generalizations and discoveries. When exact forms are too difficult or impossible to get, numerical and approximation approaches are frequently used, and exact answers are also essential standards for evaluating their accuracy. Approximations are compared to known exact results to help researchers assess how reliable their models are. A level of generality that numerical solutions could obfuscate is also provided by precise solutions, which offer symbolic expressions that illustrate the interactions between various variables and factors. They are also crucial instruments for theoretical development, aiding in the creation of new frameworks and the testing of hypotheses. Because they don't require recalculation, exact solutions are frequently computationally efficient, saving time and money.

Integrable models play an essential role in mathematical physics, specifically in the study of exactly solvable problems and nonlinear systems. These models are unique because they offer exact analytical solutions, frequently in the form of closed expressions. In general, a system is said to be integrable if it has as many conserved quantities (integrals of motion) as degrees of freedom, which usually permits the full solution. This property makes integrable models effective tools for comprehending intricate dynamics in a mathematically controlled and rigorous way. One of the most impressive characteristics of integrable systems is their capacity to elaborate nonlinear phenomena without resorting to approximations. For example, solitons, localized, stable waves that retain their shape during propagation and after interactions, emerge naturally from integrable equations such as the nonlinear Schrödinger equation and the Korteweg-de Vries (KdV) equation. These solutions have significant applications ranging from quantum field theory to fluid dynamics and optics. Furthermore, integrable models are frequently idealized frameworks that allow for the study of more complex, non-integrable systems.

The (4+1)-dimensional Boiti-Leon-Manna-Pempinelli (BLMP) equation is an essential extension of traditional integrable systems into higher dimensions. Originally presented in lower dimensions, the BLMP equation is part of a class of nonlinear partial differential equations (NLPDEs) that explain intricate wave processes and possess rich mathematical characteristics, including soliton solutions and integrability. Its extension to five dimensions four spatial and one temporal improves its ability to system more realistic physical models, especially in the fields like nonlinear optics, plasma physics, and fluid dynamics, where multidimensional effects cannot be ignored. The investigation of integrable systems has long been central to comprehending nonlinear wave process and their characteristics in distinct scientific fields, including plasma physics, nonlinear optics, and fluid dynamics. In this paper, the 4D-BLMP equation, describing complex interactions in a system with four spatial and one temporal dimension, is considered in the following form:

The importance of the 4D-BLMP equation lies mainly in its integrable feature, which means it satisfies a large variety of exact solutions. These exact solutions are more than simply mathematical oddities; they reflect stable, localized structures that mimic the actual behavior of nonlinear media in the real world and can spread over time without altering shape. Understanding the movement and interaction of energy, information, or disturbances in complex settings is aided by the study of such solutions. Xu and Wazwaz introduced the aforementioned nonlinear model, which has been examined using a variety of methodologies. Xu and Wazwaz used the Bell polynomial approach to obtain the aforementioned novel model's bilinear representation, bilinear Bäcklund transformation, Lax pair, and infinite conservation laws. They also showed that the model had the Painlevé property.

It is worth highlighting that when , , then Eq. (1) reduces to the following -dimensional BLMP equation:

Additionally, for , , and , Eq. (1) reduces to the following -dimensional BLMP equation:

The ability to obtain exact solutions for such systems is significant for both practical applications and theoretical understanding. Specifically, the higher-dimensional nature of the equation presents novel effects that do not appear in lower-dimensional models, including higher-order singularities of intricate soliton interactions and a greater variety of wave phenomena's. In lower-dimensional environments, these structures can result in more detailed representations of physical systems that were previously unclear. The integrability of these equations enables researchers to find explicit solutions, offering insights into the nonlinear behavior of waves that would otherwise be challenging to analyze. The 4D-BLMP equation is related to a larger class of nonlinear PDEs that describe the dynamics of wave-like phenomena. There are different techniques to addressing nonlinear differential models, which involves inverse scattering method, Bäcklund transformation, the Darboux transform and Hirota bilinear form, applying the linear superposition principle and utilizing symbolic computations to obtain rational wave solutions. The study of NLPDEs often involves finding solutions that are invariant under certain transformations. One of the most significant approaches for solving PDEs using symmetry methods is Lie's classical symmetry analysis, which involves finding continuous transformations that leave the equation invariant. However, while the classical Lie symmetry method has proven effective for many types of PDEs, it may not always be sufficient for solving more complex or nonlinear equations, particularly in higher dimensions. To overcome these limitations, several generalizations of Lie's classical method have been introduced, including the generalized conditional symmetries method, the method of heir equations, the direct method, the nonclassical symmetry reduction method, and the B-determining equations method. Among these methods, the nonclassical symmetries method has gained significant popularity and is widely regarded as one of the most effective approaches for solving nonlinear PDEs.

The concept of nonclassical symmetry, often referred to as conditional symmetry, was originally formulated by Bluman and Cole. Unlike the standard Lie symmetry approach, this technique allows for a wider range of admissible transformations by enforcing invariance not only of the governing equation itself but also of the corresponding invariant surface condition together with its differential consequences. In contrast to the classical method, which focuses exclusively on the invariance of the PDE itself, the nonclassical framework requires simultaneous invariance of both the equation and the constraint manifold defined by the invariant surface condition. This additional requirement often leads to reductions and solutions that cannot be obtained by standard Lie techniques. Over the past decades, the nonclassical method has proven invaluable in the analysis of integrable and nonlinear models, particularly in higher-dimensional settings where the complexity of the system limits the effectiveness of classical symmetry methods. In the context of the 4D-BLMP equation, the nonclassical approach provides a systematic procedure for generating determining equations whose solutions yield new symmetry transformations. These, in turn, allow the derivation of a wide range of exact solutions, thereby deepening our understanding of the nonlinear dynamics and rich wave structures associated with the model. Moreover, the practical implementation of the nonclassical method often relies on symbolic computational tools such as Maple and specialized packages like SADE, which significantly streamline the derivation and solution of the determining equations. By combining rigorous theoretical analysis with advanced computational techniques, the nonclassical symmetry method offers a powerful framework for uncovering novel solution families such as solitary waves, complexitons, and multi-dimensional structures that are otherwise inaccessible through classical approaches.

The remaining content of our manuscript is structured as follows: "Generalized symmetries" describes the comprehensive analysis of nonclassic symmetry to the proposed 4D-BLMP equation. "Invariant structures via conditional symmetries, determination of auxiliary functions, analytical solutions, and illustrative plots" provides the symmetry reductions, invariant solutions, as well as graphical descritions of the gained solutions. Finally, we illustrate our concluding remark in "Conclusion".