Line Soliton Interactions for Shallow Ocean Waves and Novel Solutions with Peakon, Ring, Conical, Columnar, and Lump Structures Based on Fractional KP Equation

It is well known that the celebrated Kadomtsev-Petviashvili (KP) equation has many important applications. The aim of this article is to use fractional KP equation to not only simulate shallow ocean waves but also construct novel spatial structures. Firstly, the definitions of the conformable fractional partial derivatives and integrals together with a physical interpretation are introduced and then a fractional integrable KP equation consisting of fractional KPI and KPII equations is derived. Secondly, a formula for the fractional 
 
 n
 
 -soliton solutions of the derived fractional KP equation is obtained and fractional line one-solitons with bend, wavelet peaks, and peakon are constructed. Thirdly, fractional X-, Y- and 3-in-2-out-type interactions in the fractional line two- and three-soliton solutions of the fractional KPII equation are simulated for shallow ocean waves. Besides, a falling and spreading process of a columnar structure in the fractional line two-soliton solution is also simulated. Finally, a fractional rational solution of the fractional KP equation is obtained including the lump solution as a special case. With the development of time, the nonlinear dynamic evolution of the fractional lump solution of the fractional KPI equation can change from ring and conical structures to lump structure.


Introduction
As Ablowitz and Baldwin [1] pointed, the two-space and onetime dimensional equation governing unidirectional, maximally balanced, weakly nonlinear shallow water waves with weak transverse variation is the celebrated KP equation [2]: where g is gravity, h is constant mean height, η = ηðx, y, tÞ is wave high above h, γ = 1 − τ/3, τ = T/ðρgh 2 Þ is dimensionless surface tension coefficient, and ρ is density. Obviously, if η is not related to y, then Equation (1) reduces to the Korteweg-de Vries (KdV) equation [3]. So, KP Equation (1) can be regarded as an extension of the KdV equation to two-dimensional space.
In nondimensional form, KP Equation (1) can be rescaled as [1]: where σ = 1 and σ = −1, depending on the sign of γ, correspond to KPII and KPI equations, respectively. The KPI equation exists lump solutions rationally localized in all directions in the space, for example, those [4] obtained by Ma. At the same time, n-soliton solutions with a single phase, called line solitons, have been found for the KPII equation by the Hirota bilinear method [5].
Recently, Abolwitz and Baldwin [1] obtained line twoand three-soliton solutions with X-, Y-, and more complextype interactions of the KPII equation. Interestingly, such types of X, Y, and more complex (for example 3-in-2-out) interactions have been observed to appear frequently in shallow water on two relatively flat beaches. Three types (short stem, long stem with lower and higher incoming line solitons) of X-and typical Y-type ocean wave interactions observed by Abolwitz and Baldwin in Mexico and California occur daily, shortly before and after low tide. However, these complex interactions like the 3-in-2-out one with three incoming line solitons on one side and two outgoing line solitons on the other side are much less frequent than X-and Y-type interactions. If the distances in the open-ocean direction are large enough, the destructive tsunami waves can merge in similar ways to the initial formation of an X-or Y-type wave, such as the observations that happen in the tsunami induced by the 2011 Japanese Tohoku-Oki earthquake indicating there was a merging phenomenon from a cylindrical-wave-type interaction. For more details, refer to Ref. [1].
The rest of this paper is organized as follows. In Section 2, we present the conformable fractional partial derivatives and integrals and give a physical interpretation of the conformable fractional derivative. In Section 3, we derive fractional KP Equation (3) from the corresponding fractional Lax pair. In Section 4, we transform fractional KP Equation (3) into bilinear form and then not only construct its fractional line solitons with X-, Y-, and 3-in-2-out interactions but also simulate a falling and spreading process of a columnar structure in the fractional KPII equation. At the same time, we construct a fractional rational solution of fractional KP Equation (3) and employ it to form ring, conical, and lump structures of the fractional KPI equation. In Section 5, we conclude this paper.

Definitions and Physical Interpretation
We give, in this section, the definitions of the conformable fractional partial derivatives and integrals. Since these definitions are based on the similar definitions in Ref. [24], we add this reference when defining them.
Definition 1 [24]. For the given function uðx, tÞ: R × R ⟶ C, arbitrary constant a ∈ R, and fractional-order α ∈ ð0, 1, the conformable fractional partial derivative with respect to x has the definition: where p and q are coprime and positive integers guaranteeing ðx − aÞ α can return a unique real or complex value. Here, if q is an even number, then ðx − aÞ α is stipulated to locate in the first quadrant of the complex plane and has the smallest imaginary part for arbitrary x < a. Note that u ðαÞ x,a ðx, tÞj x=a = lim x→a + u ðαÞ x,a ðx, tÞ is a supplement of Definition 1. As a special case of Definition 1, we have the following Definition 2.
Definition 2 [24]. If function uðx, tÞ: R × R ⟶ R, arbitrary constant a ∈ R, and fractional-order α ∈ ð0, 1 are given, then we define the conformable fractional partial derivative with respect to x: where p and q are coprime and positive integers, and p is an even number. A supplement of Definition 2 is the stipulation u ðαÞ x,a ðx, tÞj x=a = lim x→a + u ðαÞ x,a ðx, tÞ: Definition 3 [24]. The conformable fractional integral of function uðx, tÞ: R × R ⟶ C is defined: where a ∈ R is an arbitrary constant, and ðx − aÞ α is stipulated to return a unique real or complex value.
Definition 4 [24]. Suppose that p is an even number and a ∈ R is an arbitrary constant, the conformable fractional integral of function uðx, tÞ: R × R ⟶ R is defined: We may say uðx, tÞ is α-differentiable with respect to x in the case that Equation (4) or (5) holds. By comparisons, we can see that Definitions 1-4 include those in [24,25] as special cases. The defined conformable partial derivatives and integrals in Definitions 1-4 have some basic properties, which are all similar to the ones in [24,25]. Only one of them is listed here, and the rest are omitted, that is, D α x,a uðx, tÞ = ðx − aÞ 1−α u x ðx, tÞ. In other words, D α x,a uðx, tÞ is the product of the rate of change of uðx, tÞ along the x-axis and ðx − aÞ 1−α , which is helpful to understand the physical meaning of conformable fractional partial derivatives.

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It is a challenging problem to explore the physical or geometric meaning of fractional derivatives and integrals. As Xue [26] pointed out, although fractional calculus was born more than 300 years ago, there are no widely accepted physical and geometric interpretations of fractional derivatives and fractional integrals. Inspired by Podlubny's work [27], we would like to give a physical explanation of the conformable fractional derivative T α ð f ÞðtÞ [25]. Suppose an object moves with variable acceleration, and its variable acceleration and velocity are aðtÞ and vðtÞ at time t, respectively. Furthermore, it is assumed that the relationship between the cosmic time [27] (real time) T = g t ðτÞ and the individual time [27] (inaccurate timing) τ satisfies With the above preparation, we can see that the real velocity v O ðtÞ of the moving object with the variable acceleration aðτÞ in the individual time measure at cosmic time T is from which the variable acceleration a O ðtÞ at time t can be derived as the following: Therefore, the α-order conformable fractional derivative T α ðvÞðtÞ of the velocity function vðtÞ is just the variable acceleration a O ðtÞ from the observer's perspective with accurate time. This is a physical interpretation of the conformable fractional derivative. For the special case α = 1, Equations (8) and (10) reduce g t ðτÞ = τ and a O ðtÞ = v ′ ðtÞ = aðtÞ, respectively. That is to say, when there is no difference between the cosmic time and the individual time, the corresponding variable accelerations are equal.

Lax Representation
For the Lax representation of fractional KP Equation (3), we have the following Theorem 5.

Theorem 5. Fractional KP Equation (3) is the Lax integrable, the Lax representation of which is
with the Lax pair Proof. It is easy to derive that On the other hand, we can verify the relation: Substituting Equations (13) and (14) into Equation Further taking α-order conformable partial derivative with respect to x, we finally transform Equation (16) into Equation (3). Thus, we complete the proof.

Fractional Line Soliton Solutions and Their Interactions
There are four parts in this section for the fractional line soliton solutions and their interactions of fractional KP Equation (3): one part is the bilinearization of Equation (3), and the other three parts are the simulations for X-, Y-, and 3-in-2-out-type interactions and peakon, ring, conical, columnar, and lump structures in the obtained fractional line one-, two-, and three-soliton solutions.

Bilinearization and Formula of Soliton Solutions.
We have the following Theorem 6 for the bilinear form of fractional KP Equation (3).

Theorem 6.
Let then fractional KP Equation (3) has the following bilinear form:

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where H α x,a , H α y,b , and H α t,c are the fractional versions [28] of the Hirota bilinear operator [5]: Proof. Substituting Equation (17) into fractional KP Equation (3), we convert Equation (3) into which can be rewritten as Equation (18) by using the fractional Hirota's bilinear operator (19). The proof ends.
With the help of the fractional bilinear form (18), we then obtain fractional n-soliton solutions of fractional KP Equation (3) as follows: where Σ μ=0,1 denotes all the possible combinations of each μ j = f0, 1g for j = 1, 2, ⋯, n, while j are constants. In particular, when n = 1, Equation (21) gives the fractional one-soliton solution of fractional KP Equation (3): with Compared with one-soliton of integer order α = 1, we can see from Figures 1-6 that the fractional ones of the corre-sponding fractional KPII equation have different profiles. Figure 1 shows there are different wave widths. But the essential difference is that every fractional soliton has a certain inclination in the opposite direction of its propagation, as if it is affected by the resistance of the carrier in the forward direction. Such a characteristic also appears in Figure 2. More interestingly, the fractional one-soliton in Figure 2 has a bend in the middle; such phenomena often occur on the edges of flat beaches. In addition to the same amplitude k 2 1 /2 and the similar bell profile known from Equation (23), to gain more insights in the dynamics of the one-solitons in Figure 1 mathematically, we insert a = 0, b = 2, c = 3, k 1 = 1, p 1 = 1/2, σ = 1, and ξ ð0Þ 1 = 0 into Equation (24); then, the velocity along the x -axis of the one-solitons can be derived by the expression If _ x < 0, such as t = −18, all these solitons, whether integer or fractional, propagate along the negative x-axis, and when t = −18, they all reach the negative x-axis. Figure 3 shows there are not only different wave widths but also amplitudes between the integer-order one-soliton and the fractional ones. Besides, a fractional one-soliton with α = 53/98 in Figure 4 appears two obvious wavelet peaks in  (24) return complex values according to Definition 1 for arbitrary x < a and y < b when α = p/q and q is an even number. More specifically, it is known that sec h 2 z ≤ 4 always holds for any z ∈ R, but it does not hold when z = x + iyðx, y ≠ 0 ∈ RÞ is a complex number. It is because that jsech zj 2 = |1/ðcosh 2 x − sin 2 yÞ | can reach any given positive number as long as x and y are selected appropriately. For example, jsech ð0:2 + 1:3iÞj 2 > 8:9. Figures 5 and 6 indicate that the uplift in the middle of the fractional line one-solitons appears some features of peakons, the peaks of which have discontinuous first derivatives of integer order. For the fractional one-soliton u with α = 6/ 11, the limit of u x ðx, y, tÞ at the point ð1,1:5,0Þ is infinity. The reason that the peakons appear in Figures 5 and 6 is due to Definition 2 which let Equation (24) perform the computation ðx − aÞ α = ½ðx − aÞ p 1/q and ðy − bÞ α = ½ðy − bÞ p 1/q when α = p/q and p is an even number.

Fractional X-, Y-, and 3-in-2-out-Type Interactions.
Assign n = 2 to Equation (21), we first write the fractional two-soliton solution of fractional KP Equation (3) as x,a ln 1 + e ξ 1 + e ξ 2 + e ξ 1 +ξ 2 +A 12 , ð26Þ where ξ 1 satisfies Equation (24), and Next, we use the fractional two-soliton solution (26) to simulate the X-and Y-type interactions [1] observed by Abolwitz and Baldwin. Since the phase shifts of the colliding 2-solitons in a two-dimensional KdV equation are determined by e A 12 in Equation (27), more details can be found in [29], we choose appropriate values of e A 12 to simulate the X-and Y-type interactions through the fractional KPII equation. Based on the selection of k 1 = 1/2, k 2 = 1/2, p 1 = 2/3, p 2 = −2/3 [1], and σ = 1 letting e A 12 ≈ 2:3, Figure 7 shows an X-type interaction formed by the fractional two-soliton solution (26) with α = 83/99. At the same time, it should be noted that a phase shift caused by the collision of two-solitons will form the so-called "stem," so the change of e A 12 will affect the length, height, and even disappearance of stem. By choosing k 1 = 1/2, k 2 = 1, p 1 = 3/4, p 2 = 1, and σ = 1 so that e A 12 = 0, Figure 8 simulates a Y-type interaction without a stem. In addition, two X-type interactions with long stem (e A 12 ≈ 5:0 × 10 9 ) and short stem (e A 12 ≈ 50:8) are shown in Figures 9 and 10, respectively. In Figure 11, we show another X-type interaction, the stem corresponding to e A 12 ≈ 5:0 × 1 0 −8 of which is lower than one incoming line soliton not as the ones having high stems in Figures 7, 9, and 10, while is almost equal to the other incoming line soliton in height.

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Finally, we employ the fractional three-soliton solution of fractional KP Equation (3) to simulate not only X-and Ytype interactions but also 3-in-2-out-type interaction [1].
2 = 0, respectively. In this case, for any fixed time t, the limit of the amplitude u of this fractional two-soliton always tends to zero as x ⟶ ±∞ and y ⟶ ±∞.

Fractional Rational Solution and Lump Structure.
Expanding Equation (18) yields Following Ma's work [4], we suppose that with where a i ði = 1, 2,⋯,9Þ are undetermined constants. Substituting Equations (32) Figure 17 shows the dynamic evolution of the fractional lump solution (38) by the selection of α = 51/98, a = 0:2, b = 0, c = 0, a 2 = −0:6, and a 6 = 0:8, where the spatial structure of |u| changes from ring to lump. Two profiles of the fractional lump solution (38) at time t = 0 are shown in Figure 18. With a different fractional order α = 61/98, Figure 19 shows another dynamic evolution of the fractional lump solution (38), where the spatial structure of |u| changes from cone to lump. Two different profiles of the fractional lump solution (38) at time t = 0 are shown in Figure 20.

Conclusions
Based on the definitions of the conformable fractional partial derivatives and integrals, fractional KP Equation (3) with Lax integrability is derived. Then, a formula (21) of the fractional n-soliton solutions and rational solution (37) of KP Equation (3) is obtained by employing the Hirota bilinear method with fractional bilinear operators. When the fractional order α = 1,

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Advances in Mathematical Physics most of the simulated fractional X-, Y-, and 3-in-2-out-type interactions and fractional lump structures deform to the ones which can be constructed in [1,4]. Even though, the selectivity of fractional order α results in the variation of velocity of soliton evolution propagation, the nonlinearity of motion trajectory, the inclination and steepness of appearance profile, the changeability of amplitude, the bend and wavelet peaks as well as peakon in the fractional line one-soliton, the falling and spreading process of a columnar structure in the fractional line two-soliton, and the ring and conical structures before lump structure generation are novel. This is due to the fractional partial derivatives contained in the fractional KP equation, which make the fractional soliton and lump solutions show different nonlinearity at different positions in the coordinate plane. In order to better understand the conformable fractional derivatives and integrals, this paper also attempts to give a physical explanation of the conformable fractional derivative from the perspective of variable acceleration. Extending fractional calculus to other important topics of nonlinear integrable systems is worth studying, such as soliton molecules [30], full reversal symmetric multiple soliton solutions [31], financial rogue wave [32], long-time asymptotic behavior [33], initial-boundary value problems [34], and high-dimensional hierarchies of evolution equations and their Hamiltonian structures [35].

Data Availability
The data in the manuscript are available from the corresponding authors upon reasonable request.