The study of nonlinear waves has been a captivating subject for many researchers for centuries. With recent advancements in mathematical methods, computational techniques, and experimental demonstrations, the investigation of nonlinear waves has gained increasing attention [1]. Described by nonlinear partial differential equations (PDEs), the study of nonlinear waves encompasses a wide range of disciplines, including mathematics, physics and engineering. As computational power and data analysis methods continue to improve, researchers are exploring new techniques to leverage the potential of nonlinear waves in various fields such as telecommunications, optics, and material sciences [1–3]. The exploration of these waves requires a combination of precise mathematical modeling, solving equations, analyzing their dynamics, and verifying the results through experiments.

Solitons are unique self-sustaining waves that maintain their shape and velocity as they propagate. Unlike other waves that tend to spread out or change form, solitons maintain their characteristics even as they move through various systems, including water, plasma, and optical fibers. Mathematically, solitons are solutions to nonlinear PDEs in which the nonlinearities of the underlying physical systems balance the dispersive effects. These solutions exhibit notable properties such as energy conservation and unchanging waveforms [4]. Although first discovered in water waves, solitons have since been observed in many other areas of study, including plasma physics, nonlinear optics, and even some biological systems. However, solitons are just one type of nonlinear waves with distinct properties, and exploring their dynamics holds tremendous promise for a multitude of applications, including telecommunications, optics, Bose-Einstein condensates, water waves, and biological systems [1–6].

The study of nonlinear waves is a fascinating field that encompasses various models, including the Korteweg-de Vries (KdV), Kadomtsev–Petviashvili (KP), sine-Gordon (sG), Boussinesq, and nonlinear Schrödinger (NLS) equations [1–4]. The KP equation, introduced by Boris Borisovich Kadomtsev (1928–1998) and Vladimir Iosifovich Petviashvili (1936–1993), models the propagation of shallow-water waves in the presence of weakly nonlinear restoring forces and frequency dispersions [7]. This is a two-dimensional integrable generalization of the classical KdV equation, which describes the propagation of unidirectional shallow-water waves. Both KP and KdV equations are examples of completely integrable systems and admit a range of solutions, including N-soliton solutions, Lax pairs, solvable inverse scattering transform (IST), Painlevé integrability, and exact analytical solutions for localized, interacting, periodic, elliptic, and quasi-periodic nonlinear waves. Additionally, they have an infinite number of conservation laws [8–10].

The KdV equation, introduced by Dutch mathematicians Diederik Johannes Korteweg (1848–1941) and Gustav de Vries (1866–1934) in 1895, is a mathematical model that effectively describes the long-term evolution of dispersive wave phenomena. In the KdV equation, the nonlinear influence of wave steepening is balanced by the dispersion effect [11]. The KP equation is an extension of the KdV equation and includes wave dynamics in the transverse direction. The KdV and KP equations can be written as follows, respectively [1–11]: u t − 6 u u x x + u x x x = 0 , (1) u t − 6 u u x x + u x x x x + 3 σ 2 u y y = 0 . (2) The wave functions u = u(x, t) and u = u(x, y, t) are both scalar fields, where x, y, and t ∈ R represent the longitudinal spatial coordinate, transverse spatial coordinate, and temporal variable, respectively. The subscripts indicate partial derivatives with respect to the associated variables. The KP equation is commonly used to model water waves with weak surface tension (KP II equation, with σ = 1) or waves in thin films with strong surface tension (KP I equation, with σ = i) [3, 12, 13]. In addition to water waves [14–17], the KP equation has been applied as a model for other physical systems, such as nonlinear optics [18], ferromagnetic media [19], plasma physics [20], Bose-Einstein condensates [21], and many more.

Ma et al. (2021) developed an extended yet integrable KP equation and obtained soliton, breather, and lump interaction solutions using the Hirota bilinear form [22]. In a separate study, they extended the integrable KP equation to a 3D case and observed abundant dynamic behaviors, such as fusing and splitting phenomena in lump wave interactions [23]. Recently, Rao et al. (2021) have studied resonant collisions among localized lumps and line solitons of the KPI equation, resulting in the discovery of “rogue lumps,” which are completely localized in both space and time and possess the features of two-dimensional rogue waves. Analytically, these solutions are constructed using the KP hierarchy reduction method and Hirota’s bilinearization procedure, and the phenomena of resonant interactions and rogue lumps are expected to persist for other nonlinear evolution equations with two spatial dimensions [24].

Whereas solitons in one-dimensional systems have already garnered significant attention from researchers due to their intriguing complexity, solitons in higher-dimensional settings often display more complex dynamics arising from the interplay between different types of nonlinearity, dispersive effects, geometric factors, and varying nonlocalities. Carr and Brand (2008) explored the extension of 1D solitons into 2D and 3D in a book chapter, highlighting that trapped BECs can stabilize these solitons and lead to the emergence of new nonlinear objects and topological solitons, depending on whether the nonlinearity is repulsive or attractive [25]. Mihalache (2021) provided a comprehensive overview of recent theoretical and experimental studies on localized structures in optical and matter-wave media, covering various physical contexts such as light bullets, solitons, and rogue waves [26]. Malomed’s (2016) review on multidimensional solitons discussed the stabilization of soliton states in both 2D and 3D physical settings. In 2D, the solitons form a stable ground state, whereas in 3D, they form metastable solitons [27]. In his latest monograph, published in 2022, Malomed addressed the potential benefits of extending solitons from 1D to higher-dimensional systems while highlighting the challenges, particularly regarding stability. These challenges open up many interesting and unresolved problems in the area of nonlinear waves [28].

The NLS equation is a well-known integrable system that was initially proposed as a nonlocal nonlinear evolution equation. This equation has an infinite number of conservation laws in the context of Hamiltonian dynamical systems, making it integrable and also exhibiting parity-time ( PT ) reversal symmetry in its self-induced potential [29, 30]. The feature of nonlocality in the NLS equation can appear in the spatial, temporal, or both spatial and temporal variables. Several other integrable nonlocal nonlinear evolution equations have been identified, which can be obtained through simple symmetry reductions of the general Ablowitz–Kaup–Newell–Segur (AKNS) hierarchy that is solvable by the inverse scattering transformation [31]. In (1 + 1)-dimensions, other nonlinear evolution models such as the derivative NLS, modified Korteweg-de Vries (mKdV), and sine-Gordon equations also exhibit nonlocality. In (2 + 1)-dimensions, other PDE models such as the three-wave interaction, generalized NLS, and Davey–Stewartson (DS) equations display this nonlocallity property too [32].

Research on nonlocal models in the NLS equation and its related models, such as the Gross-Pitaevskii (GP) equation, has gained momentum in the third decade of this century. For example, Li et al. (2020) presented one- and two-bright-soliton solutions of the generalized nonlocal GP equation with an arbitrary time-dependent linear potential obtained through the improved Hirota method, which can describe the dynamics of soliton solutions in quasi-one-dimensional BEC and may be useful for understanding physical phenomena in nonlinear nonlocal soliton equations [33]. Interestingly, Yu et al. (2021) combined the feature of both locality and nonlocality in the NLS equation and obtained its one- and two-soliton solutions using Hirota’s bilinearization transformation. This mixed local–nonlocal nonlinear system can be useful for understanding physical phenomena in nonlinear wave propagation and applied in fields such as nonlinear optics and meteorology [34]. Recently, Li et al. (2023) presented a novel application of Darboux transformation for solving the (2 + 1)-dimensional nonlocal NLS equation with reverse time field q(x, y, − t), deriving various soliton solutions on a background of kink waves, which can be extended to other nonlocal nonlinear or higher-dimensional soliton equations [35].

Apart from the above NLS type nonlocal systems, the KP equation has been the subject of investigation for nonlocal models and their corresponding dynamics, with rich variations of wave motion observed in lump solutions, line breathers, and periodic normal breathers [36]. These nonlocal KP equations were inspired by Lou’s Alice–Bob (AB) system, which explored two-place physics and multi-place physics [37–39]. Lou and Huang’s pioneering work in 2017 established that an extension to shifted parity and delayed time reversal was advantageous for seeking group-invariant solutions for such models [38].

Recent studies in the published literature have employed the Hirota bilinear technique to explore various extended KP equations and nonlocal AB systems, highlighting the integrability of these systems. Despite the variety of solutions obtained—including solitons, breathers, lumps, and rogue waves—they share a common approach. Among the novel solutions discovered, some exhibit intriguing dynamical behaviors, such as semi-rational solutions with elastic interactions and symmetry-breaking solitons. For instance, Manukure et al. (2018) provided conditions for the rational localization of solutions and presented lump solutions for an extended KP equation [40]. Fei et al. (2019) derived the nonlocal AB-Boussinesq system for surface gravity waves, investigated its residual and finite symmetries, and obtained several exact solutions [41]. Wu and Lou (2019) also found solitons, breathers, and lump solutions using shifted parity and delayed time reversal to explore the exact solutions of the AB-KP equation [42].

Guo et al. conducted two studies using the Hirota bilinear technique to investigate an extended KP equation. In their 2020 study, they obtained exact analytical expressions for the higher-order soliton and breather solutions of the equation and discovered novel semi-rational solutions that exhibit elastic interactions [43]. In their 2021 study, they constructed multiple-order line rogue wave solutions using the Hirota bilinear method and a symbolic computation approach [44]. Shen et al. (2020) investigated the nonlocal AB Benjamin-Ono system using parity and time-reversal symmetry reduction and obtained breather, lump, and symmetry-breaking soliton solutions via an extended Bäcklund transformation and Hirota bilinear form [45]. Cao et al. (2021) studied another nonlocal ABKP system and proposed a constant dependence in the Bäcklund transformation to solve the system using the Hirota bilinear form. They found that this constant affects the symmetry-breaking characteristics of the solutions and obtained various analytical solutions, including single solitons, breathers, lumps, entangled lumps, and a pair of stripe solutions [46]. Recently, Dong et al. (2023) presented an integrable AB system for the (1 + 1)-dimensional Boussinesq equation with shifted parity and delayed time-reversal symmetry. To obtain explicit solutions, including line solitons, breathers, and lumps, the authors introduced an extended Bäcklund transformation [47].

In this study, we aim to investigate the nonlocal version of the (2 + 1)-dimensional extended Kadomtsev-Petviashivili (eKP) equation introduced by Manukure et al. (2018), which they solved to obtain the lump wave solution using the Hirota bilinear method with quadratic polynomials [40]. The eKP equation is given by u t + 6 u u x + u x x x x − u y y + α u t t + β u y t = 0 , (3) where x, y, and t represent the two spatial dimensions and time variables, respectively. It features second-order temporal and spatio-temporal dispersion coefficients α and β, respectively. While the equation is non-integrable for arbitrary α and β values, it becomes Painlevé integrable when α = −β ²/4 [22]. The literature reports various exotic nonlinear wave solutions for this model, such as solitons and breathers [43], multi-line rogue waves [44], and interacting wave structures [22]. Although the local eKP Eq. 3 has been extensively studied, there has been little investigation of its nonlocal counterpart, which is the focus of our research. We seek to utilize the AB approach to construct exotic nonlinear wave solutions, such as solitons and lumps, for the nonlocal version of the eKP Eq. 3 and examine their evolutionary dynamics under nonlocal effects to uncover physically interesting wave phenomena. Our objective is to contribute to the understanding of the nonlocal eKP equation and provide insights into its potential applications in various fields.

The remainder of this article is organized as follows. In Section 2, we derive the nonlocal version of the eKP Eq. 3 and provide its Bäcklund transformation and bilinear formalism. Section 3 presents one- and two-soliton solutions, including their propagation characteristics and symmetric and asymmetric interactions, as well as bound states and breathers. In Section 4, we introduce a rational and well-localized lump wave solution. Finally, we conclude with a summary of our findings in the final section.

By implementing the AB physics approach, the nonlocal counterpart of the considered Eq. 3 can be obtained through the substitution u ( x , y , t ) = 1 2 P ( x , y , t ) + Q ( x , y , t ) : α P t t + Q t t + β P y t + Q y t + P x t + Q x t − P y y + Q y y + 3 P x + Q x 2 + 3 P + Q P x x + Q x x + P x x x x + Q x x x x = 0 . (4)

Further, it can be explicitly written in the following two-coupled equations: α P t t + β P y t + P x t − P y y + P x x x x + 3 2 P x + Q x 2 + 3 P P x x + Q x x + H P , Q = 0 , (5a) α Q t t + β Q y t + Q x t − Q y y + Q x x x x + 3 2 P x + Q x 2 + 3 Q P x x + Q x x − H P , Q = 0 , (5b) where Q ( x , y , t ) is related to P ( x , y , t ) through Q ( x , y , t ) = P s x P s y P d t = P ( − x + p 0 , − y + q 0 , − t + r 0 ) , P _s T _d are parity transformation and time reversal, respectively, and H(P, Q) is an arbitrary function of both P and Q, which can have infinitely many choices. Moreover, we choose the following form for H(P, Q): H P , Q = 3 4 P x 2 − Q x 2 − P − 3 Q P x x + Q − 3 P Q x x . (6)

Finally, we obtain the required Alice-Bob nonlocal version of the eKP Eq. 3, which we refer to as the extended nonlocal KP (enKP) model, given as follows: 4 ( α P t t + β P y t + P x t − P y y + P x x x x ) + 3 ( P x + Q x ) ( 3 P x + Q x ) + 3 ( P + Q ) ( 3 P x x + Q x x ) = 0 , (7a) 4 ( α Q t t + β Q y t + Q x t − Q y y + Q x x x x ) + 3 ( P x + Q x ) ( P x + 3 Q x ) + 3 ( P + Q ) ( P x x + 3 Q x x ) = 0 . (7b)

It is interesting to check the integrability of the model, for which we have performed the Painlevé analysis [48] for the above Eqs 7a, 7b in the local setting for simplicity. From the result, we found that the above coupled Eqs 7a, 7b becomes non-integrable not only for arbitrary α and β parameters (as expected) but also for β = −α ²/4, for which the single-component eKP Eq. 3 is integrable. We do not present the details of the Painlevé analysis for the sake of space and its not-so-interesting mathematical equations. One can also view the above enKP model Eqs 7a, 7b as a particular form of coupled KP systems. For example, the enKP system Eqs 7a, 7b can be obtained as a special case from the following generalized version of coupled KP type equation (similar to the equations studied in Refs. [39, 49]) for an appropriate choice of γ _j , j = 0, 1 …10, coefficients: γ 0 P t t + ( γ 1 P + γ 2 Q ) t + ( γ 3 P + γ 4 Q ) x x x + ( γ 5 P + γ 6 Q ) P x + ( γ 7 P + γ 8 Q ) Q x x + γ 9 P y y + γ 10 P y t = 0 , (8a) γ 0 Q t t + ( γ 1 Q + γ 2 P ) t + ( γ 3 Q + γ 4 P ) x x x + ( γ 5 Q + γ 6 P ) Q x + ( γ 7 Q + γ 8 P ) P x x + γ 9 Q y y + γ 10 Q y t = 0 . (8b)

To be precise, our enKP model Eqs 7a, 7b can be reduced from the above Eqs 8a, 8b for the choice γ ₀ = 4α, γ ₁ = γ ₃ = 4, γ ₂ = γ ₄ = 0, γ ₅ = γ ₆ = 9, γ ₇ = γ ₈ = 3, γ ₉ = −4, and γ ₁₀ = 4β. In order to solve our intended enKP model Eqs 7a, 7b, we adopt Hirota’s bilinearization method [50–54], which is one of the efficient analytical techniques available for solving different classes of nonlinear equations. For this purpose, we consider the following superposed variable transformation: P = 2 ( ln ⁡ F ) x x + A ( ln ⁡ F ) x , (9a) Q = 2 ( ln ⁡ F ) x x − A ( ln ⁡ F ) x , (9b) where F(x, y, t) is the required function to be determined while A is an arbitrary real constant. It is clear to find that one can obtain the standard bilinearizing transformation when A = 0. However, we utilize the superposed bilinear transformation with A ≠ 0 to deduce the required bilinear equation(s). Note that the superposed bilinearization Eqs 9a, 9b is quite different from the standard bilinearization procedure [50–54], and there are a few recent reports on similar superposed techniques in the literature [37–39, 46]. On applying the above bilinearizing transformation to Eqs 7a, 7b and after its simplification, we get the following bilinear equation in a standard D-operator form: D x D t + D x 4 − D y 2 + α D t 2 + β D y D t F ⋅ F = 0 , (10) where D _j , j = x, y, t, are standard Hirota derivatives [50]. Further, the above bilinear equation can be written in its equivalent expanded form as below for a clear inference of Hirota derivatives. F F x t − F x F t + F F x x x x − 4 F x F x x x x + 3 F x x 2 − F F y y − F y 2 + α F F t t − F t 2 + β F F y t − F y F t = 0 . (11) Thus from the above bilinear Eq. 10 or its equivalent form Eq. 11, one can find a number of exact solutions to the considered enKP Eqs 7a, 7b through different forms of explicit F(x, y, t) computed for appropriate initial seed solutions.

In this section, we construct exact soliton solutions of the considered enKP model Eqs. 7a, 7b and explore their propagation as well as interaction dynamics in detail for different choices of parameters.

Lump solutions are localized rational nonlinear waves that generally evolve in higher dimensional systems. This prototype structure gained much attention because of its occurrence in several classes of physical systems. Though the lump can be obtained through the long-wave limit by the reduction of N-soliton solution, there are still several nonlinear models for which the construction of lump solutions is tedious because of their non-bilinearizable nature. To overcome this critical issue, Ma has recently proposed a positive quadratic function approach [36, 55], which can be successfully applied to both integrable and nonintegrable equations admitting either bilinear or multi-linear form. Even though the present enKP model Eqs 7a, 7b admits a compact bilinear equation, we construct a lump solution adopting Ma’s approach to test the applicability of the alternate tool [36]. In order to find such a lump solution, we choose the following positive quadratic function for F ( x , y , t ) [58]: F lump = χ 1 ( x , y , t ) 2 + χ 2 ( x , y , t ) 2 + θ 7 , (18a) where χ 1 = θ 1 x − p 0 2 + θ 2 y − q 0 2 + θ 3 t − r 0 2 , (18b) χ 2 = θ 4 x − p 0 2 + θ 5 y − q 0 2 + θ 6 t − r 0 2 . (18c) Here θ _j , j = 1, 2, …, 7, are the real parameters to be determined. On substituting F _lump given by Eq. 18a into the bilinear equation and solving the resulting equation, we obtain θ 1 = θ 3 ( θ 2 2 − θ 5 2 ) + 2 θ 2 θ 5 θ 6 θ 3 2 + θ 6 2 − β θ 2 − α θ 3 , (18d) θ 2 = 2 θ 2 θ 3 θ 5 − θ 2 2 θ 6 + θ 5 2 θ 6 θ 3 2 + θ 6 2 − β θ 5 − α θ 6 , (18e) θ 7 = 3 ( θ 3 2 + θ 6 2 ) ( θ 3 θ 5 − θ 2 θ 6 ) 2 θ 2 4 − 2 β θ 2 2 θ 3 + α 2 θ 3 4 + ( α θ 6 2 + β θ 5 θ 6 − θ 5 2 ) 2 + θ 3 2 ( ( 2 α + β 2 ) θ 5 2 + 2 α β θ 5 θ 6 + 2 α 2 θ 6 2 ) + 2 θ 2 θ 3 ( α β ( θ 3 2 + θ 6 2 ) − β θ 5 2 − 4 α θ 5 θ 6 ) + θ 2 2 ( ( β 2 − 2 α ) θ 3 2 + 2 θ 5 2 − 2 β θ 5 θ 6 + ( 2 α + β ) 2 θ 6 2 ) 2 , (18f) where the constraint condition θ ₃ θ ₅ − θ ₂ θ ₆ ≠ 0 has to be satisfied. Further, we can obtain the explicit lump form using a variable transformation. The evolutionary dynamics of the obtained lump waves are provided in Figure 10. The received lump waves have four arbitrary parameters with the restriction θ ₃ θ ₅ − θ ₂ θ ₆ ≠ 0. It is evident from the demonstration that the lump solution approaches zero when (x, y) approaches infinity. The left and middle panels show asymmetric singly localized bright-dark type lump waves, appearing with the peak bump on the right and left sides, respectively, for non-vanishing background A ≠ 0. Meanwhile, it is interesting to observe a bright symmetric lump structure shown in the right panel for A = 0.

Apart from the above first-order lump solution, the occurrence and extraction of higher-order lump solutions to address local and nonlocal characteristics of the present systems involve tedious mathematical derivations, which we have skipped in this work, considering the length of the manuscript. One can study such higher-order lumps and their interaction dynamics with other localized nonlinear waves as open questions and shall form a separate project.

To summarize, the study investigates the dynamics of nonlinear waves in a (2 + 1)-dimensional extended nonlocal Kadomtsev– Petviashvili (enKP) model. The enKP model was obtained from the KP equation using the Alice–Bob (AB) approach but was found to be non-integrable according to the Painlevé test in its local setting. Using a superposed blinearization technique, we obtained explicit soliton and lump wave solutions that highlight the nonlocal effects on wave structures. The study also explored the wave dynamics of two-soliton solutions, including elastic collisions and breather solutions, and discussed their evolutionary dynamics with graphical demonstrations.

The results of our study demonstrated the impact of nonlocality on the formation of nonlinear waves, resulting in both symmetrical and asymmetrical wave patterns. A single soliton solution highlights the emergence of symmetrical and asymmetrical waves from a combination of bell-type and kink-anti-kink solitons due to the nonlocal background and symmetrical bell-type solitons. Our analysis of the two-soliton solution revealed a range of fascinating wave dynamics, including elastic collisions that can occur as head-on or oblique interactions. By adjusting the soliton parameters (k _j and l _j , where j = 1, 2), we discovered soliton bound states undergoing periodic attraction and repulsion without intersecting.

We have made a significant discovery in our study by obtaining an explicit form of breather solution from the reduction of the two-soliton solution for specific values of k 2 = k 1 * and l 2 = l 1 * . This breather solution encompasses various forms, such as inclined or oblique breathers, localized breathers in both the x and y directions, rational line solitons, and periodic solitons/structures. To provide a complete understanding of the solution, we have briefly analyzed its evolutionary dynamics and presented clear graphical illustrations. Furthermore, we obtained a lump wave solution through the use of a quadratic function as a seed solution and discussed its behavior. The results of the study contribute to the understanding of localized waves in nonlocal nonlinear models and have the potential to be generalized to other integrable and non-integrable nonlocal models.