A new (n+1)-dimensional generalized Kadomtsev–Petviashvili equation: integrability characteristics and localized solutions

Searching for higher-dimensional integrable models is one of the most significant and challenging issues in nonlinear mathematical physics. This paper aims to extend the classic lower-dimensional integrable models to arbitrary spatial dimension. We investigate the celebrated Kadomtsev–Petviashvili (KP) equation and propose its (n+1)-dimensional integrable extension. Based on the singularity manifold analysis and binary Bell polynomial method, it is found that the (n+1)-dimensional generalized KP equation has N-soliton solutions, and it also possesses the Painlevé property, Lax pair, Bäcklund transformation as well as infinite conservation laws, and thus the (n+1)-dimensional generalized KP equation is proven to be completely integrable. Moreover, various types of localized solutions can be constructed starting from the N-soliton solutions. The abundant interactions including overtaking solitons, head-on solitons, one-order lump, two-order lump, breather, breather-soliton mixed solutions are analyzed by some graphs.

The KP equation formulated by Kadomtsev and Petviashvili in 1970 [25], as a two spatial dimensional analog of the classic Korteweg-de Vries equation, can well simulate nonlinear phenomena in fluid physics, plasma physics, Bose-Einstein condensates, optics, etc. Due to the physical and mathematical significance, KP equation attracts much attention of scholars. By using the singularity manifold analysis, Dorizzi et al. [26] and Wazwaz et al. [27] proved the integrability of the KP equation with constant and variable coefficients. Tian and Gao derived some families of travelling wave solutions of the KP equation [28]. Based on the Hirota's bilinear method, the lump solution, the lump and one stripe soliton and resonance stripe soliton solutions were constructed for the KP equation [29]. Using the binary Darboux transformation, Guo et al. presented the higher-order rogue wave solutions of the KPI equation [30]. Very recently, multiple lump solution, periodic soliton solution and other types of explicit solutions were constructed by using different methods [31,32].
A number of research achievements have been made for the (3+1)-dimensional KP equation and its variant version. Alagesan et al. [33] and Xu [34] investigated the integrability and derived its Bäcklund transformation, soliton solutions and dromions. Su et al. presented the nonautomous solitons and Wronskian solutions for the (3+1)-dimensional KP equation with variable coefficients [35]. Kumar et al. presented the group invariant solutions using optimal system of Lie subalgebra [36]. The N-solitons, breathers and multiwave interaction solutions have been constructed by using different methods [37][38][39][40]. However, these higher-dimensional extensions do not keep their original integrable characteristics since the Painlevé property, N-soliton, Lax pair, symmetry structure as well as infinite conservation laws are no longer conserved.
The question that naturally arises is whether the well-known KP equation can be extended to three spatial dimensions or more higher-dimensional space. More importantly, the extended KP equations in higher dimensions are expected to possess the same integrable properties as the classic KP equation in (2+1) dimensions. This paper aims to give an affirmative answer to this question. For this purpose, we propose a new generalized KP equation in n+1 dimensions where n ≥ 2, u is a differentiable function with respect to spatial variables x 1 , x 2 , · · · , x n and time variable t, the subscripts represent the partial derivatives, and α, β, γ and σ i (i = 1, · · · , n) are constant parameters. In the same direction, Ma et al. extended the KP equation in n+1 dimensions and presented a class of lump solutions generated from quadratic functions [41]. Subsequently, Chen et al. derived the multi-lump or lump-type solutions [42]. However, this extended KP equation is not integrable when n ≥ 3.
(2) when = i and = 1, (2) is exactly the KPI equation and KPII equation, respectively. The change in sign of 2 is related to the magnitudes of gravity and surface tension. When Note that another (3+1)-dimensional KP equation investigated in Ref. [33] fails the integrability test due to the existence of second-order dispersion term u zz . The remaining parts of the paper are arranged as follows. In Sect. 2, the singularity manifold analysis is conducted to prove that equation (1) is Painlevé integrable based on the WTC method. In Sect. 3, we employ the binary Bell polynomial approach to investigate several integrable features of Eq. (1), including the bilinear representation, N-soliton solution, the bilinear Bäcklund transformation, Lax pair and infinite conservation. In Sect. 4, some localized solutions and interactions of multiple waves are analyzed by graphs. Finally, some brief conclusions are given in Sect. 5.

Integrability test for Eq. (1)
Generally speaking, if a partial differential equation passes the Painlevé test, this equation is a prime candidate for being completely integrable. One may use different methods to check whether nonlinear partial differential equations pass the Painlevé test, including the ARS algorithm, the WTC method, Conte invariant method, and so on [5]. In this section, we first utilize the singularity manifold analysis to perform the integrability test for Eq. (1).
Equation (1) is said to have the Painlevé property if its solution can be written as Laurent series and it is "single-valued" in the neighborhood of singularity manifold φ. Note that in (4), both φ and u j ( j = 0, 1, · · · ) are analytic functions of {x 1 , x 2 , · · · , x n , t}.
Subsequently, we need to determine the resonance points at which the coefficients in (4) are arbitrary. To this end, inserting u = u 0 φ −2 + u j φ j−2 into (1), and vanishing the coefficients of φ j−6 , one obtains the general recursion relation about u j . It easily follows from the recursion relation that the four resonant points occur at j = −1, 4, 5 and 6.
Finally, one should verify the compatibility conditions for each non-negative resonant point. For this purpose, inserting the truncated series into (1) and equating the coefficients of φ with different powers, it is obtained as The compatibility conditions corresponding to the remained resonant points 4, 5, 6 are listed as follows: Combining the u 0 , u 1 , u 2 and u 3 values given by (6), the above three compatibility conditions are proven to be satisfied identically. In other words, u 4 , u 5 and u 6 in (5) are arbitrary functions. Therefore, it is concluded that the new (n+1)-dimensional KP equation (1) with general form possesses the Painlevé property without any constraints between the parameters α, β, γ and σ i (i = 1, · · · , n).

Bell polynomials method for Eq. (1)
As pointed out by Lambert and his coworkers, there exist close connections between Bell polynomials and Hirota's operators [4]. Inspired by this, they developed a unified scheme to investigate integrable properties for some classic soliton equations. The binary Bell polynomial method provides a direct and effective framework to construct Bäcklund transformation, Lax pair and other integrable characteristics in a systematic way, which is widely applied in a great number of NLEEs with physical interests [43][44][45][46][47].

Bäcklund transformation and Lax pair
To construct the Bäcklund transformation, it is assumed that q = 2 ln F andq = 2 ln G are two different solutions of equation (8). In addition, introducing w and v where R is given by Then, to write (14) as the Y -polynomial, we suppose where λ and δ are constant parameters. Then, after some calculations, Eq. (14) can be rewritten as By setting δ 2 = γ /(3β), equation (16) is equivalent to Finally, combining (13), (15) and (17), one obtains With proper transformation, the Y -polynomials can be reduced to Hirota's differential operators. Based on this, the above Y -polynomial system (18) yields the bilinear BT of Eq. (1), which reads where To derive the Lax pair, we take Starting from the bilinear Bäcklund transformation (18), through the linearizing technique, the Lax pair of (1) is as follows, where δ 2 = γ /(3β), and λ is an arbitrary constant parameter. The two equations in (21) are compatible provided that q satisfies Eq. (8), i.e., ϕ x 1 x 1 t = ϕ t x 1 x 1 . In other words, the system (21) is just the Lax pair of equation (1).
As a particular case of equation (1) in (3+1) dimensions, its bilinear Bäcklund transformation and the associated Lax pair may be easily derived. Here, we denote x 1 = x, x 2 = y, x 3 = z, (19) yields the bilinear Bäcklund transformation of equation (3) with the form, The corresponding Lax representation of (3) is given by Under the constraint δ 2 = γ /(3β), the compatibility condition ϕ x xt = ϕ t x x implies that q xt +βq x x x x +3βq 2 x x +γ q xy +σ 1 q x x +σ 2 q xy +σ 3 q xz = 0, which indicates that the system (23) is exactly the Lax pair of equation (3).

Infinite conservation laws
As discussed in Sect. 3.1-3.2, equation (1) is both Painlevé integrable and Lax integrable. The existence of infinite conservation laws is also an essential and significant integrable property. In this section, starting from the coupled Y -polynomials system (18), we will derive the infinite conservation laws of equation (1) in arbitrary spatial dimension. First, by introducing η = (q x 1 − q x 1 )/2, the functions v and w are related by Inserting it into the coupled Y -polynomials system (18) leads to Suppose that η =η + and λ = 2 , equation (25) is reduced tō Next, we expand the functionη as the series with the form, Inserting the series (28) into (26) and collecting the coefficients with the same power of , we have the explicit recursion formulae for the conversed densities I n Finally, combining (27) and (28), one obtains the infinite conservation laws of the (n+1)-dimensional generalized KP equation (1), where I j is presented by (29). Moreover, the recursion formulae of the fluxes F n is obtained as Other fluxes G n are given by , , In conclusion, equation (1) has N-soliton solutions, and it also possesses the Painlevé property, Lax pair, bilinear BT as well as infinite conservation laws, thus it is concluded that the proposed (n+1)-dimensional generalized KP equation (1) is completely integrable.

Multiple solitons and localized solutions of equation (3)
In this part, the (3+1)-dimensional KP equation (3) is chosen to illustrate the interactions between multiple solitons and localized solutions more intuitively. For convenience, we set x 1 = x, x 2 = y, x 3 = z, k j1 = k j , k j2 = l j , k j3 = m j , combining (11)- (12), the Nsoliton solution of equation (3) is written as where φ is given by where ρ=0,1 means a summation of possible combinations for ρ j = 0 and 1. The remained parameters k j , l j , m j and η j0 ( j = 1, · · · , N ) are arbitrary constants.

Two solitons and localized solutions
Taking N = 2 in (32), equation (31) with (32) leads to the two-soliton solution. Due to the existence of arbitrary parameters k j , l j and m j ( j = 1, 2) in (32), there are various types of interactions of multiple waves. Case 1. Two-soliton solution Figure 1 displays an overtaking interactions between two solitons. Here we suppose that l j > 0 and m j > 0 ( j = 1, 2), other parameter choices in (32) only need satisfy ω j < 0( j = 1, 2) and a 12 > 0 to ensure that both two solitary wave are right-going along the positive x-, y-and z-axis. Figure 1a depicts the evolution of two solitons along x direction. It is easily observed that these two solitons move along the positive x direction with different speeds, and the soliton with smaller amplitude moves faster and it overtakes the soliton with larger amplitude. In Fig. 1b and c, two solitons move along the positive y and z direction with different speeds, and the soliton with larger amplitude moves faster and it overtakes the soliton with smaller amplitude. Figure 2 shows the head-on collisions between two solitons. Here we suppose that l j > 0 and m j < 0 ( j = 1, 2), other parameter choices in (32) only need satisfy ω 1 < 0, ω 2 > 0 and a 12 > 0 to ensure that one solitary wave is left-going and the other solitary wave is right-going. In Fig. 2a, the smaller amplitude soliton propagates along the positive x direction, and the larger amplitude soliton moves along the negative x direction, they still keep their original wave speeds and shapes after the head-on interactions. Similarly, Fig. 2b and c displays the evolutions of two solitons along with y and z directions, and there also exist head-on interactions between two solitons. Case 2. One-order lump solution By employing the long wave limit method, the lump solutions can be constructed from solitons with even orders. Starting from the above two soliton solution, one can obtain the one-order lump solution. Taking the limit k j → 0, e η j0 = −1 and k 1 /k 2 = O(1) ( j = 1, 2), along with (32), we have where In order to rewrite (33) as quadratic functions, we set where Together with the transformation (31), we get the oneorder lump solution of Eq. (3) as displayed in Fig. 3. Note that γ = 0, ν 1 = 0, and the parameters choices in (32) should ensure the value of φ in (35) to be positive. Case 3. One-order breather solution If taking k 1 = k 2 = μ 1 , l 1 = l * 2 = ρ 1 + iν 1 , where Substituting (37) into (31), one may obtain the periodic soliton solution of equation (3). Figure 4a-c depicts the evolution of one-order breather solution in x − y plane, x − z plane and y − z plane, respectively.

Four solitons and localized solutions
Taking N = 4 in (32), equation (31) leads to the four-soliton solution of the (3+1)-dimensional generalized KP equation (3). For the sake of simplicity, we only consider two types of interactions among multiple waves. Case 1. Four-soliton solution Figure 8 shows the overtaking collisions among four solitons with different velocities. Here we suppose that l j > 0 and m j > 0 ( j = 1, · · · , 4), and other parameter choices in (32) only need satisfy the conditions to ensure that three solitary waves with different velocities move along the same direction. Figure 8a depicts the evolution of four solitons propagating along x direction. It is easily observed that the shortest soliton moves fastest and overtakes three other soliton. After interactions, both of these four solitons keep their original speed and profile. Figure 8b and c depicts the evolution of four solitons propagating along y and z directions.  (31), (39) and (40). The parameters are selected as μ 1 = 0.4, ρ = 0.75, ν 1 = 1.2, κ 1 = 0.25, λ 1 = 1.5, μ 2 = −1.5,  Figure 9 shows the head-on collisions among four solitons. Here we suppose that k j > 0 and l j > 0 ( j = 1, · · · , 4), and other parameter choices in (32) only need satisfy the conditions   where ϑ j = x + l j y + m j z + ω j t, ω j = − (σ 1 + γ l 2 j + σ 2 l j + σ 3 m j ), j = 1, · · · , 4, B js = 12β γ (l j − l s ) 2 , 1 ≤ j < s ≤ 4.
By setting l 2 = l * 1 , m 2 = m * 1 , l 4 = l * 3 , m 4 = m * 3 , we obtain the two-order lump solution of equation (3). Figure 10a-c depicts the evolutions of two-order lump waves in the x − y plane, x − z plane and y − z plane. Note that γ = 0, l j = l s , and the parameters choices in (32) should ensure the value of B js in (42) to be positive.

Conclusions
Searching for higher-dimensional integrable models is a significant and challenging issue in nonlinear mathematical physics. Our previous work presented some novel integrable models in (3+1) and (4+1) dimensions, and their integrable features as well as exact solutions are explored from different viewpoints. This paper aims to extend the classic lower-dimensional integrable modes to arbitrary spatial dimension.
Due to the physical and mathematical significance, we investigate the celebrated KP equation and propose its (n+1)-dimensional integrable extension. By employing the singularity manifold analysis, the (n+1)dimensional KP equation is shown to be Painlevé integrable without any constrains of parameters. The binary Bell polynomial method is successfully used to the proposed (n+1)-dimensional KP equation, and as a result, the N-soliton, Bäcklund transformation, Lax pair and infinite conservation laws are constructed systematically. Therefore, the extended KP equation in arbitrary spatial dimensions also possesses the same integrable properties as the classic KP equation in (2+1) dimensions. In addition, abundant interaction structures like overtaking and head-on solitons, one-order lump, twoorder lump, breather, and breather-soliton mixed solutions are analyzed by some graphs.
For the (n+1)-dimensional KP equation (1), more abundant localized solutions like higher-order lump and rogue wave solutions can be constructed by the bilinear neural network method [48,49]. The research framework developed in this work may be applicable to some other lower-dimensional integrable models with great physical interests. Future research is expected to construct more and more integrable models in arbitrary spatial dimension.