Invariance Analysis, Exact Solution and Conservation Laws of (2 + 1) Dim Fractional Kadomtsev-Petviashvili (KP) System

: In this work, a Lie group reduction for a (2 + 1) dimensional fractional Kadomtsev-Petviashvili (KP) system is determined by using the Lie symmetry method with Riemann Liouville derivative. After reducing the system into a two-dimensional nonlinear fractional partial differential system (NLFPDEs), the power series (PS) method is applied to obtain the exact solution. Further the obtained power series solution is analyzed for convergence. Then, using the new conservation theorem with a generalized Noether’s operator, the conservation laws of the KP system are obtained.


Introduction
In the field of fractional order differential equations, prevalent advancement is currently speculated. The dominant use of multifarious projects which are masked by fractional differential equations (FDEs), lies in the field of nano-technology, bio-informatics, control system, chemical engineering, heat conduction, ion-acoustic wave, mechanical engineering, diffusion equations and, additionally, several other sciences. Because of its prodigious scope and applications in the various area of science and technology, congruent consideration has been given to the exact solutions of FDEs. There are many techniques that can be used to analyze NLFPDEs [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. The exact solution provides a proper understanding of the physical phenomena modeled by NLFPDEs. Finding exact solutions to NLFPDEs are quite difficult as compared to approximate solutions. The Lie symmetry method is one of the most powerful methods used to find the exact solution of NLFPDEs [15][16][17][18][19][20][21][22][23]. This technique is used to reduce the NLFPDEs into a lower dimension. The conservation laws can be investigated for nonlinear FPDEs, which are very important tool for the study of differential equations. Noether's theorem involves a methodology for constructing conservation laws, using symmetries associated with Noether's operator [19][20][21][22][24][25][26][27][28][29]. In general, there is no technique that provides specific solutions for the system. In recent years, many researchers have concentrated on the approximate analytical solutions to the FDE system and some methods have been developed. One of the most useful techniques for solving the linear system and non-linear system of fractional differential equations with a quick convergence rate and small calculation error is the fractional power series method.
Another major benefit is that this approach can be used directly, without requiring linearization, discretization, Adomian polynomials, etc., to the non-linear fractional PDE system. The power series method is applied to finding an exact solution in the form of a power series of a fractional differential equation. The (2 + 1) dimensional Kadomtsev-Petviashvili (KP) system [30,31] is given by which can also be written as the system In nonlinear wave theory, the KP system is one of the most universal models which arises as a reduction in the system with quadratic nonlinearity. This system has been broadly studied in terms of its mathematical association in recent years. The KP equation was originated by the two Soviet physicists, Boris Kadomtsev and Vladimir Petviashvili in [32]. The KP equation has been studied by many authors for integer-order or fractionorder derivatives by different methods in recent years. Exact traveling wave solutions have been analyzed in [31]. In [30], KP equation is studied for symmetry reduction using a loop algebra. In [33], KP solitary waves has been studied. Symmetries of the integer order KP equation have been studied in [34]. In [35], the Cauchy problem for the fractional KP equations has been discussed.
The main goal of this work is to analyze the fractional order KP system with arbitrary constant coefficients as This is a system of NLPDEs of fractional order, which depicts the evolution of nonlinear long waves with small amplitude. Here, u and w are dependent functions of x, y, t, and A 1 , A 2 , A 3 , A 4 are arbitrary constants. x and y are the longitudinal and transverse spatial coordinates, respectively.
In this work, the KP system (2) is considered for symmetry reduction. The exact solutions, in the form of power series, are obtained, and the conservation laws are investigated.
To find some new exact solutions to the system (2), we apply the Lie symmetry method to reduce the system into lower dimensions. The system is also studied for conservation laws by using the new conservation theorem [27]. The preliminary material is given in Section 2. In Section 3, the symmetry of system (2) is obtained via the classical Lie method. Through the corresponding generators, we reduce system (2) to lower-dimensional NLFPDEs. Some exact solutions are obtained, corresponding to the reduced equation, by using the power series method in Section 4. In Section 5 the obtained power series solutions are analyzed for convergence. Some conservation laws are investigated in Section 6. In the last section, the conclusion to the study is presented.

Preliminaries
In this section, we will discuss basic definitions and theories for Lie symmetry analysis. Definition 1. Riemann-Liouville fractional derivative [36,37] Let f : [a, b] ⊆ R −→ R, such that ∂ n f ∂t n is continuous and integrable for all n ∈ N ∪ {0} and n − 1 < α < n, then the Riemann-Liouville fractional derivative of order α > 0 is defined by where Γ(α) is the Euler's gamma function.

Symmetry Analysis
Consider the system of NLFPDEs as follows where ∂ α v ∂t α , ∂ β v ∂x β and ∂ γ v ∂x γ are the fractional derivatives of Riemann-Liouville (RL) type. Suppose that the Lie group of transformations are given by where being the group parameter and ξ, τ, µ, η (r) are the infinitesimals, are extended infinitesimals. In (10), D x and D t are total derivative operators. The α th , β th and γ th extended infinitesimals related to the RL fractional derivative are given in [38]. The associated vector field is The corresponding extended symmetry generator is as follows As the lower limit of RL fractional derivative [36,37,39] is fixed, we have

Symmetry Analysis of (2 + 1)-Dimensional Fractional Kadomtsev-Petviashvili System
Let us assume that the system (2) is invariant under group of transformations (9), then we have Therefore, using (9) in (14) the invariance criteria for (2) are obtained as Using the value of extended infinitesimals and collecting the coefficients of various powers of u and partial derivatives of u and w, we have where n ∈ N. Solving these equations simultaneously, we get the infinitesimals where C 1 is thw arbitrary constant. Thus, the corresponding vector field is Corresponding to vector field V, the characterisitc equation is written as After solving these equations, we get the symmetry variables and symmetry transformations where f and g are arbitrary functions.

Power Series Solution
In this section, we will obtain the power series solutions of NLFPDEs ( Therefore, from (27), we have (n + 1)a n+1,m z n 1 z m 2 , (n + 2)(n + 1)a n+2,m z n 1 z m 2 , Inserting (27) and (28) into (26), we have 6γ a n,m z n 1 , z m Comparing coefficients for n = m = 0, we have When n ≥ 0, m ≥ 0, but both are not simultaneously zero, we have a n+3,m = 1 A 3 (n + 3)(n + 2)(n + 1) In view of (31), we can obtain all coefficients a n,m (n ≥ 3, m ≥ 0) and b n,m (n, m ≥ 0) of the power series (27) for arbitrary chosen series ∞ ∑ m=0 a i,m (i = 0, 1, 2). Therefore, the system (2) has the exact power series solution, and the coefficient of the series depends on (31). Hence, we can write the power series (27) as 6γ a n,m Therefore, the required exact solution of the reduced form (26) is w(x, t) = ∞ ∑ n,m=0 ) .

Analysis of the Convergence
In this section, we will analyze the convergence of the power series solution (33) and (34).
Therefore, the series are the majorant series of the series f (z 1 , z 2 ) and g(z 1 , z 2 ), respectively.
Let us consider one particular case, Next, we investigate the convergence of the series P = P(z 1 , z 2 ) and R = R(z 1 , z 2 ).
Consider the implicit functional system as follows F, H are analytics in the neighbourhood of (0, 0, A 0 , N A 0 ). F(0, 0, A 0 , N A 0 ) = 0, G(0, 0, A 0 , N A 0 ) = 0, and the Jacobian determinant is Then, by the implicit function theorem [41], both power series are convergent. Hence, an exact solution of KP system (2) exists.

Conservation Laws
In this section, conservation laws of (2) will be constructed by using the new conservation theorem and the nonlinear self adjointness [27,29].
The conservation laws for (2) are introduced as where C t (x, y, t, u, w), C x (x, y, t, u, w) and C y (x, y, t, u, w) are conserved vectors of (2). The Euler-Lagrange operators given by where D i k represents the total derivative operator. (D α t ) * , (D β x ) * and (D γ y ) * are also the adjoint operators of the RL derivative operators [36,39] D γ t and D β x , respectively, given as follows where I n−α p , I m−β q and I k−γ r are the right-hand-side fractional integral operators of order n − α, m − β and k − γ, respectively, defined as follows where n = [α] + 1 where m = [β] + 1 where k = [γ] + 1 The formal Lagrangian of the system (2) is given by where T and Q are new dependent variables. The adjoint equations are defined by From (50) and (51), the adjoint equations are If, by substituting the values T = ϕ(x, y, t, u, w), Q = ψ(x, y, t, u, w), Equation (52) satisfies, with at least one of T, Q variable being non-zero, the system (2) is called the nonlinear self adjoint. Now, the derivative(s) of T = ϕ(x, y, t, u, w) with respect to x, are x w x + 3ϕ uu u x w xx + 3ϕ uww u x w 2 x + 3ϕ xu u xx +3ϕ uw (u x w xx + w x u xx ) + 3ϕ ww w x w xx + 3ϕ xxw w x + 3ϕ xxu u x + 3ϕ xw w xx +ϕ u u xxx + ϕ w w xxx + 3ϕ xuu u 2 x + 3ϕ xww w 2 Thus, the nonlinear self adjointness conditions are where λ i (i = 1, 2, 3, 4) are to be determined. Therefore, we have x w x + 3ϕ uu u x w xx +3ϕ ww w x w xx + 3ϕ uw (u x w xx + w x u xx ) + 3ϕ xxw w x + 3ϕ xxu u x + 3ϕ xw w xx + ϕ u u xxx + 3ϕ xuu u 2 x +3ϕ xww w 2 x + ϕ w w xxx + 3ϕ xu u xx + 3ϕ uww u x w 2 x + ϕ uuu u 3 x + ϕ www w 3 Collecting the coefficients of various powers of u, w and their derivatives on both sides of (56), and solving them simultaneously, we have λ i = 0, i = 1, 2, 3, 4, and ϕ = a(t, y)x 2 + b(t, y)x + c(t, y), ψ = ψ(x, y, t, u, w), where a(t, y), b(t, y) and c(t, y) are functions of t, y.
Corresponding with symmetry generators, the characteristic functions W 1 and W 2 , are defined by The fractional Noether's operator [29] is defined as Data Availability Statement: Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.