This study investigates the dynamical behavior of a three-dimensional extended Kadomtsev-Petviashvili-Boussinesq (eKP-BO) equation, a higher-dimensional wave system that unifies Kadomtsev-Petviashvili-type weak dispersion with Boussinesq-type bidirectional propagation. Using the -expansion method, we systematically construct diverse classes of exact solutions of hyperbolic, trigonometric, and rational types. These analytical results enrich the family of admissible solitary and lump-type wave structures and provide closed-form benchmarks for validating numerical simulations. Uncover the underlying dynamics by reducing the governing equation to a planar system. This can then be analyzed through phase portraits, bifurcation diagrams, Poincaré sections, and time series. The results will reveal equilibrium structures such as centers, saddles, and cusps, as well as transitions from regular oscillations to period-doubling and fully developed chaos under period forcing. The study will also highlight both the advantages and limitations of the exponential expansion method, focusing on its ability to generate rich explicit solutions while noting its dependence on balance conditions. In sum, the results shed light on the nonlinear dispersive wave dynamics of the eKP-BO model in the unified framework and identify several potential future research paths, such as fractional-order models, stochastic perturbations, and hybrid analytical-numerical methods.
The most effective mathematical modeling tools for expressing the spatial and temporal variations of physical quantities like heat, fluid velocity, electromagnetic fields, or wave amplitudes are partial differential equations (PDEs). However, linear PDEs have long been studied and solved. At the same time, nonlinear partial differential equations (NLPDEs) are a considerably larger and more challenging family of problems that appear practically everywhere in natural applications. The PDE is deemed nonlinear if the unknown function or its derivatives manifest in a nonlinear form, such as nonlinear powers, derivative products, or nonlinear functions of the solution. This nonlinearity greatly complicates both the theoretical analysis and the numerical solution of the equations.
NLPDEs usually lack closed-form solutions and rely on sophisticated mathematical and computational tools, unlike linear PDEs, which usually permit the superposition of solutions and may be solved analytically using techniques like variable separation or Fourier transforms. Numerous scientific and engineering domains encounter NLPDEs. Viscous fluid motion in fluid dynamics is governed by the Navier-Stokes equations, which are inherently nonlinear. In general relativity, the structure of spacetime is described by Einstein's field equations, which are NLPDEs. Nonlinear wave equations describe a wide range of phenomena, from shallow water waves (by the KdV equation) to optical pulses in fiber optics (via the nonlinear Schrödinger equation).
More recently, much has been going on in the field of applications of enhanced analytical and numerical tools to nonlinear evolution equations, a testament to their flexibility and richness in physical connotations. In particular, Asjad et al. treated the Nizhnik-Novikov-Veselov equation and obtained exact invariant travelling-wave soliton solutions with dynamical analyses. Majid et al. investigated the Riemann wave equation and indicated definite soliton structures and their sensitivity, while Asjad et al. performed an in-depth sensitivity analysis of solutions to the nonlinear Landau-Ginzburg-Higgs equation using the generalized projective Riccati method. Related developments have also occurred in plasma and atmospheric dynamics, with Alqurashi et al. demonstrating solitary wave patterns with chaotic behavior in tropical and mid-latitude regions. Also, Ullah et al. applied the Sardar-subequation method to nonlinear optical models and obtained new wave structures, while they illuminated the dynamics of nonlinear optics via various analytical methods. All these studies collectively emphasize the increasing importance of strong solution methods and serve as a strong motivation for the present research, where the exp()-expansion method is employed to address the extended KP-Boussinesq equation. In conjunction with these developments, there have been other developments in studying nonlinear dynamical models. For example, analytical solutions of the modified Benjamin-Bona-Mahony equation were reported with applications in nonlinear optics. The fractional Gross-Pitaevskii model was also investigated, in which its complex dynamical behavior and solutions were demonstrated. In addition, bifurcation, multistability, and soliton dynamics in the noisy potential Korteweg-de Vries equation were explored, which exhibit the inner complexity of nonlinear evolution systems. These studies further stimulate the present work and highlight the general significance of exploring exact solutions and dynamical behaviors of extended nonlinear PDEs.
The Kadomtsev-Petviashvili (KP) hierarchy is a prominent work in solitary waves theory that describes interesting higher-dimensional wave occurrences in NLPDEs. There are some excellent scientific properties and uses for the KP hierarchy. Numerous extended KP hierarchies that preserve the original KP hierarchy's integrable structures have been created and extensively examined in the literature.
Numerous KP Eq. (1.1) and Boussinesq (BO) Eq. (1.2) that occur in (1+1) dimensions and higher dimensions have been the subject of extensive research. To investigate the stability of the well-known Korteweg-de Vries equation in two-dimensional media, the classic KP Eq. (1.1) was developed. A vast amount of research has been done on the KP equation, which describes a wide range of physical phenomena, including fluid dynamics, dust acoustic waves, weakly nonlinear quasi-unidirectional waves, and many more. In addition, the KP equation admits a rich variety of exact solutions, each characterized by distinct structures and dynamical features.
The underlying dynamics and solution structures of the (3+1)-dimensional KP-like equations with constant coefficients are examined in this work. The -expansion technique is used to derive a large number of exact wave solutions systematically. These consist of lump wave solutions and N-soliton solutions.
The following is the standard integrable KP equation:
It has a weak dispersion term and a quadratic nonlinearity term , admitting weakly dispersive waves. Eq. (1.1) is used in many physical contexts and describes the phenomenon of small surface tension relative to the gravitational force in fluid dynamics. However, the BO equation says
where the height of a fluid's free surface is represented by the real-valued, sufficiently often differentiable function , and partial derivatives are indicated by subscripts. Small-amplitude dispersive waves traveling in both left and right directions in shallow water are modeled by the Boussinesq Eq. (1.2). A great deal of research has been done to investigate a large number of exact solutions to the two well-known Eqs. (1.1) and (1.2).
Additionally, the following extended KP (eKP) equation was put out:
where is a differentiable function, and are non-zero constants, and and are added to the standard KP Eq. (1.1). The KP equation can also be extended as follows:
Finally, three extensions of (2+1)-dimensional dispersive equations are provided in Ref. as:
and
were examined, where and c are arbitrary parameters and is a differentiable function with respect to the spatial variables x and y and the temporal variable t.
The following three new (3+1)-dimensional equations have been proposed very recently in.
which will be called the first extended KP equation (1eKP), second extended KP equation (2eKP), and extended KP-Boussinesq equation (eKP-BO), respectively, where , and ) are non-zero parameters. Notice that the last term in each equation of (1.8)-(1.10) is the only term that changes between these equations, while the other terms in each equation remain unchanged.
This paper focuses on (3+1)-dimensional Kadomtsev-Petviashvili-Boussinesq (eKP-BO) Eq. (1.10) where , and are non-zero arbitrary parameters, and is a differentiable function concerning the spatial variables x, y, and z. It is clear that the terms in Eq. (1.8) and in Eq. (1.9), respectively, are replaced by the term in Eq. (1.10) as well. The conventional Boussinesq equation typically has the terms , and in Eq. (1.10) as well.
The extended Kadomtsev-Petviashvili-Boussinesq equation is chosen because it naturally couples the weakly dispersive character of KP-type models with the bidirectional propagation of Boussinesq-type systems. This unified structure makes it particularly relevant for describing higher-dimensional nonlinear waves in shallow-water hydrodynamics, plasma oscillations, and nonlinear optical media. While the KP and Boussinesq equations have been extensively studied in isolation, their combined extension has not previously been investigated in a comprehensive way that includes both explicit soliton solutions and qualitative dynamical analysis. For this reason, the present work provides new insight into how solitary and lump-type waveforms interact with bifurcation structures and chaotic regimes within a single modeling framework.
We construct numerous soliton solutions to demonstrate the rich analytical structure of the eKP-BO equation. In addition, a class of lump-type solutions is explored for various parameter regimes, highlighting the diversity and flexibility of exact wave structures admitted by the model. These results not only deepen the understanding of the underlying nonlinear dynamics but also serve as valuable benchmarks for validating numerical algorithms. Given their complexity and relevance, such equations are widely used in computational mathematics to test, calibrate, and verify methods for solving NLPDEs, with implications for both theoretical development and applied modeling.
To analyze such nonlinear wave models, a variety of solution techniques have been developed, including the -function method, Hirota bilinear formalism, and -expansion approaches. In this study, we adopt the -expansion method because of its systematic structure and flexibility. The method is capable of producing different families of exact solutions, hyperbolic, trigonometric, and rational, within a unified framework, and the resulting expressions can be verified symbolically. These properties make it a reliable tool for constructing closed-form benchmarks that can later be used to validate numerical simulations and to interpret qualitative dynamics. A limitation of the method is that it relies on balancing the highest-order nonlinear and derivative terms, so it may not capture all possible nonlinear structures or multi-soliton interactions. Nevertheless, within its scope, it provides a direct and efficient algorithm for generating explicit solitary and lump-type solutions.
Compared with alternative approaches, the -expansion method has several advantages for the present model. Unlike the -method, which typically yields only hyperbolic solutions, or the Hirota bilinear method, which requires a suitable bilinearization, the -expansion offers a more versatile route to multiple waveform types without changing the underlying ansatz. In addition, in contrast to the -expansion, which can be limited in generating rational forms, the present method naturally accommodates hyperbolic, trigonometric, and rational families in one framework. This versatility aligns well with the objectives of our work, where exact solution construction is combined with qualitative bifurcation and chaos analysis of the same higher-dimensional equation.
In addition, recent studies demonstrate how chaos and stability analyses enrich diverse nonlinear systems, including advection-diffusion-reaction models, the stochastic Davey-Stewartson equation, embedded carbon nanotubes, and the Klein-Fock-Gordon equation.
The structure of the paper is as follows: the mathematical model and the -expansion method are presented, also several exact analytical solutions are given in Section 2. In Section 3, graphical illustrations of the soliton solutions are offered. The bifurcation phenomena of the reduced system are discussed in Section 4 through phase portraits. Section 5 analyzes the chaotic dynamics and sensitivity of the system to external perturbations. Finally, the main findings and potential directions for future research are summarized in Section 6.