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Lie symmetry analysis and one-dimensional optimal system for the generalized 2+1 Kadomtsev-Petviashvili equation

We classify the Lie point symmetries for the 2+1 nonlinear generalized Kadomtsev-Petviashvili equation by determine all the possible f(u) functional forms where the latter depends. For each case the one-dimensional optimal system is derived; a necessary analysis to find all the possible similarity transformations which simplify the equation. We demonstrate our results by constructing static and travel-wave similarity solutions. In particular the latter solutions satisfy a second-order nonlinear ordinary differential equation which can be solved by quadratures.

Lie at the end of the 19th century, and it is described in his work on the theory of transformations groups [10][11][12].
The main novelty of Lie's theory is that the transformations groups which leave invariant a differential equation, can be used to simplify the given equation. In particular, Lie symmetries are applied to the simplification process of a differential equation by means of reduction. There are differences in the application of Lie symmetries between ordinary differential equations (ODEs) and partial differential equations (PDEs). For PDEs the application of a Lie point symmetry through the so-called similarity transformation leads to a differential equation with less independent variables and of the same order. Oppositely, in the case of ODEs the application of a Lie symmetry reduces the order of the given differential equation by one [1,13].
The application of the theory of transformations groups in differential equations is not restricted to the application of the similarity transformation. Lie symmetries can be used to determine algebraic equivalent systems as also to provide linearization criteria for nonlinear differential equations [14][15][16]. In addition, Lie symmetries are applied in order to construct conservation laws [17][18][19]; to determine new solutions from old solutions [20] and many other applications [21].
The plethora of results which can be obtained by the Lie symmetries for nonlinear differential equations have led to the algebraic classification problem for differential equations. The first algebraic classification scheme was performed by L.V. Ovsiannikov in 1982, who classified all the forms of the 1+1 nonlinear PDE u t − (f (u) u x ) x = 0 where the latter equation admits Lie symmetries [22]. In terms of nonlinear wave equations there are various studies on the group properties, Ames et al. classified the Lie point symmetries for the nonlinear differential equation u tt − (f (u) u x ) x = 0. Applications of Lie symmetries in Shallow-water equations are presented in [24][25][26][27][28][29][30][31]; while applications of other subjects of applied mathematics and mathematical physics are presented in [32][33][34][35][36][37][38][39][40][41][42][43] and references therein.
In this work we focus on the algebraic classification problem for the 2+1 nonlinear generalized Kadomtsev-Petviashvili (KP) equation [44] u or equivalent where f (u) is an arbitrary nonlinear function, u = u (t, x, y) , v = v (t, x, y), while parameter ε can be normalized to ε = ±1 and it measures the traverse dispersion effects on weakly nonlinear waves.
KP equation is recovered for the linear function f (u) and it can be seen as the extension of the Korteveg-de Vreis equation in higher dimensions. Nowadays KP equation is the standard model for the description of weakly nonlinear waves of small amplitude in various physical situations [45][46][47]. The KP equation is a well-known integrable equation which has been used as a source of integrable equations, for more details see [48].
In [49] it was found that the KP equation can be reduced to into the Painlevé transcendental equation of the first kind by using the Lie invariants. The Lie symmetries and the possible reductions of the KP equations were studied also by S.-Y. Lou in [50]; while recently the Lie point symmetries of the KP equation with timedependent coefficients have been determined in [51], a similar analysis with and time-and space-dependent coefficients was performed in [52]. For other integrable hierarchies of PDEs we refer the reader in [53] In the following Sections we shall determine the forms of the unknown nonlinear function f (u) where the 2+1 nonlinear generalized KP equation (1), (2) admits Lie point symmetries. For the different functions f (u) we determine the one-dimensional optimal system of the admitted Lie point symmetries by the generalized KP equation. The determination of the optimal system is necessary in order to understand the possible reductions of the differential equation.
For the one-dimensional system we calculate the corresponding invariants which define the similarity transformations to reduce the differential equation. The results are presented in a tabular list. Moreover, we shall present two examples where we show how to apply the Lie invariants and determine similarity transformations.
We shall see that for the arbitrary functional form of f (u) for the static solution and the travel-wave solution the generalized KP equation (1), (2) can be solved by quadratures. While for some specific functional forms of f (u) the solution of the original system is described by well-known one-dimensional Newtonian systems such is the Ermakov-Pinney equation. The outline of the paper follows.
In Section 2, we present the main results of our analysis, where we determine the Lie point symmetries for the 2+1 nonlinear generalized KP equation (1), (2) for specific forms of f (u). In particular we determine the Lie point symmetries for arbitrary function f (u), where additional symmetries exist when f (u) = u k + f 0 and For each of the cases, the one-dimensional optimal system is calculated. In Section 3, we determine the Lie invariants for all the one-dimensional systems. This invariants can be used to find similarity transformations in order to the generalized KP equation and construct similarity solutions. The similarity transformations are applied to find static similarity solutions or travel-wave solutions. In Appendices A and B we present the basic properties and definitions for the Lie theory and the one-dimensional optimal system, while in Appendix C we extend our analysis and we present the Lie point symmetries for the 3+1 nonlinear generalized KP equation [44]. Finally in Section 4, we discuss our results and we draw our conclusions.

Classification of Lie symmetries
In this section we solve the algebraic classification problem for the 2+1 nonlinear general KP equation of our consideration by finding all the nonlinear functions f (u) in which equations (1), (2) admit Lie point symmetries. In each case the one-dimensional optimal system is derived. The Lie theory and the definition of the one-dimensional optimal system are presented in Appendices A and B respectively.

Arbitrary function f (u)
For the arbitrary function f (u) the 2+1 generalized KP equations (1), (2) admit the following Lie point symmetries where function β (t) is arbitrary.
The symmetry vector X β indicates that there are infinity number of solutions of the form v (t, which solves the KP equation. However it does not play any role in the determination of the exact solutions, hence we shall omit it.
As far as the rest of the symmetry vectors are concerned, i.e. the vector fields X 1 , X 2 , X 3 and X 4 , we calculate the commutators which are presented in Table 1. The admitted Lie algebra is the A 4,3 in the Morozov-Mubarakzyanov classification scheme [54][55][56][57], for more details we refer the reader in the review article [58].

One-dimensional optimal system
In order to determine the one-dimensional optimal system, the adjoint representation and the invariants of the adjoint action should be determined. The adjoint representation of the symmetry vectors {X 1 , X 2 , X 3 , X 4 } is presented in Table 2.
The invariants φ (a i ) of the adjoint action are determined by the set of differential equations where C k ij are the structure constants of the Lie algebra. Therefore, from (5) and Table 1 we end up with the system of first-order partial differential equations from where we infer φ = φ (a 1 , a 4 ), that is, the invariants of the adjoint action are the a 1 and a 4 .
We define the generic symmetry vector and with the use of Table 2 and of the invariants of the adjoint representation as given in Table 2 we have the following possible cases Case 1: a 1 = 0, a 2 = 0. The generic symmetry vector is which gives the one-dimensional optimal system Case 2: a 1 = 0, a 2 = 0. The generic symmetry vector is from where we infer the additional one-dimensional algebras Case 3: a 1 = 0, a 2 = 0. The generic symmetry vector is where now the additional one-dimensional algebras are Case 4: a 1 a 2 = 0. In the generic case the additional one-dimensional Lie algebra is found to be Hence, the one-dimensional optimal system for the 2+1 generalized KP equation (1), (2) for arbitrary function f (u) consists by the Lie algebras When f (u) is a power law function, that is, f (u) = u k + f 0 the admitted Lie symmetries for equation (1), (2) are where again β (t) is an arbitrary function and X 5 is an extra Lie point symmetry. We observe that X 5 is a scaling symmetry. The commutators of the admitted Lie point symmetries are given in Table 3. The admitted Lie point symmetries form the A 5, 37 Lie algebra in the Patera et al. classification scheme [59].

One-dimensional optimal system
The invariants of the adjoint action are determined by the system of first-order differential equations The latter system provides that φ = φ (a 5 ), which means that a 5 is the unique invariant.
Indeed when a 5 = 0 we find the one-dimensional optimal system of the case where f (u) is arbitrary. However, for a 5 = 0 the additional one-dimensional algebra is found to be the {X 5 }.
In order to demonstrate it, let us consider the generic symmetry vector then by using the he adjoint representation of the symmetry vectors {X 1 , X 2 , X 3 , X 4 , X 5 } as presented in Table   4 we find where for a 5 kε = a 4 it becomes We continue by considered the adjoint transformation and for 2a 5 In addition we find and finally 2.3 Exponential f (u) = e σu + f 0 The last case where f (u) is an exponential function, that is, f (u) = e σu +f 0 , the admitted Lie point symmetries by equation (1), (2) are where β (t) is an arbitrary function. Remark that the additional Lie point symmetry is theX 5 while the symmetry vector X 6 is included into the infinity number of symmetries X β . However, in this case it is important to consider it separately in order to define the closed algebra of the symmetry vectors X 1 , X 2 , X 3 , X 4 ,X 5 , X 6 .
From the commutator of Table 5 we infer that the six Lie symmetries form the Lie algebra {A 5,37 ⊗ s A 1 }, where ⊗ s denotes semi-direct product of the two Lie algebras, namely A 5,37 and A 1 , see for details [59].

One-dimensional optimal system
In order to find the one-dimensional optimal system for the case where f (u) is an exponential function. To do that we need the Adjoint representation which is presented in Tables 6 and 7. We apply the same procedure as before, for the power-law potential from where we find that the additional one-dimensional algebras is again the vector field X 5 .
The question which is raised, is about the one-dimensional optimal system when the infinity number of symmetries, i.e. X β , is included. Recall that we should reduce the equation first from a partial differential equation into an ordinary differential equation and the application of X β does not perform such process. For that reason we have not included it in the presentation.
We continue our analysis by applying the Lie point symmetries in order to determine the similarity transformations and when it is feasible and to specify similarity solutions.

Similarity transformations
The main application of the Lie symmetries is that similarity transformations can be defined which can be used to simplify the differential equation. As far as partial differential equations are concerned the similarity transformations are applied to reduce the number of indepedent variables. On the contrary, in the case of ordinary differential equations the application of similarity transformations lead to a differential equation of lower-order. In the ideal scenario, where the admitted Lie point symmetries are sufficient to reduce a partial differential equation into an ordinary differential equation and the latter equation into an algebraic equation, or into another well-known integrable equation, with well-known solutions; we shall say that we have found a similarity solution for the original problem.
However, the application of a similarity transformation to a given differential equation leads to a new differential equation where it has different algebraic properties, that is, it admits different Lie symmetries.
There is a criterion in which the Lie point symmetries of the original equation are also point symmetries of the reduced equation. Consider the Lie point symmetries X 1 , X 2 with commutator [X 1 , X 2 ] = cX 2 where c may be zero. Then reduction by X 1 in the original equation results that X 2 being a nonlocal symmetry for the reduced equation; while reduction by X 2 results in X 1 being an inherited Lie symmetry of the reduced differential equation [60]. It is possible the reduced equation to admit extra Lie point symmetries, these are called hidden symmetries and can be used to perform further reduction [61].
Before we proceed with the application of the Lie symmetries to determine similarity solutions for the 2+1 nonlinear generalized KP equation, we calculate the Lie invariants which correspond to all the above one-dimensional Lie algebras. The Lie invariants are presented in Table 8.

Similarity solutions
We continue by applying some of the Lie invariants presented in Table 8 in order to determine similarity solutions for the 2+1 nonlinear generalized KP equation.

Static solution
The application of the Lie symmetry vector X 1 , leads to the time-independent equation where u = u (x, y) and v = v (x, y); that is, the solution which will be determined will be a static solution.
For arbitrary function f (u) the latter equation admits the Lie symmetry vectors X 2 , X 3 and X v = ∂ v . The latter vector fields are reduced symmetries while X v is the static symmetry vector X β . Additional symmetry vectors exist when f (u) = u k and f (u) = e σu . The additional Lie symmetries are the X 5 andX 5 vector fields for f 0 = 0, respectively. We remark that for f 0 = 0 there are not additional Lie point symmetries, that is because the vector fields X 5 andX 5 become nonlocal symmetries.
Further, reduction of the system (24), (25) with the application of the lie symmetry X 2 leads to the system εv y = 0, u y = 0 with the trivial solution v = v 0 and u = u 0 . On the other hand, reduction with the use of the symmetry vector X 3 leads to the third-order nonlinear ODE where v = v 0 . Equation (26) can be integrated as follows The latter equation is autonomous and can easily be integrated by quadratures. Indeed, equation (27) becomes As far as the classification problem for equation (27) is concerned, that it is well-known and was performed by Sophus Lie more than a century ago [10].
In particular there are four different families of potentials. (A) For arbitrary function F (u) equation (27) admits the symmetry vector ∂ x . (B) When F (u) = (a + βu) n or F (u) = e γu , n = 0, 1, −3 equation (27) (27) is maximally symmetric and admits eight Lie point symmetries. However, that case is not the subject of study of this analysis. We note that in the case (B) the additional symmetry is a reduced symmetry and it is described by the vector fields X 5 andX 5 .
Reduction with the Lie symmetry {X 2 − γX 3 } leads to the system where z = y + cx. The latter system is reduced in the form of equation (26).
Consider reduction of the original system with the symmetry vector {X 1 + γX 2 }, then it follows where z = x − γt. The latter system is in the form of the static system (24), (25), where someone replaces f (u) → f (u) − γ and x → z. Hence the above analysis is also applied and in that case The same results follow and for the rest of the reductions which provide travel-wave solutions; therefore we omit the presentation of the rest reductions which lead to travel-wave solutions.

Conclusions
In this work, we considered a generalization of the 2+1 KP equation which has been used for the study of weakly An important observation is that for the two different functions f (u) the two generalized KP equations has a common subalgebra, namely A 5,37 which means that they share a common reduction process, more general than that for arbitrary function f (u).
For all the different cases of f (u) we derived the one-dimensional optimal system and we calculated all the possible similarity transformations which can be applied to reduce the differential equation. We demonstrated our results by applying the similarity transformations to determine analytic solutions which are static or travelwaves. Surprisingly, we determined that for both types of solutions and after a further reduction we end up with a similar second-order ordinary differential equation, of the form which can be solved by quadratures.
Therefore, we conclude that the generalized 2+1 KP equation can be reduced to a classical Newtonian system, with a central force. That is an important result since we can see the dynamics of nonlinear waves reduce to that of classical system under the proper frame, that is, a proper similarity transformation. In a future work we plan to investigate in details the physical applications of these solutions.

Acknowledgements
The author wishes to thank Dr. Alex Giacomini and the Universidad Austral de Chile (UACh) for the hospitality provided while this work was performed.

A Lie symmetries
Consider the system of differential equations H(x i , u A , u A ,i , u A ,ij ) ≡ 0 where x i denotes the independent variables and u A are the dependent variables.
Under the action of the one-parameter point infinitesimal transformation with infinitesimal generator the system of differential equations H( or equivalently where L describes the Lie derivative with respect to the vector field X [n] . Vector field X [n] nth-extension of X in the jet space x i , u A , u A ,i , u A ,ij is given by the following expression where η [n] is defined as If condition (38) is true, then the generator X of the infinitesimal transformation (34)-(35) is called a Lie point symmetry of the system of differential equations H(x i , u A , u A ,i , u A ,ij ) . The Lie invariants which correspond to a given Lie point symmetries X are found by solving the following The characteristic functions W [0] x k , u , W [1] x k , u, u i and W [2] x k , u, u ,i , u ij which solve the latter Lagrange system are called the n − th invariants of the Lie symmetry vector X.

B One-dimensional optimal system
Let assume the n-dimensional Lie algebra G n , with elements X 1 , X 2 , ... X n . We shall say that the two generic vector fields where the Adjoint operator is defined as Hence, in order to perform a complete classification for the similarity solutions of a given differential equation we should determine all the one-dimensional indepedent symmetry vectors of the Lie algebra G n . The onedimensional independent symmetry vectors form the so-called one-dimensional optimal system [1].
When f (u) = u k + f 0 the additional Lie point symmetry is while when f (u) = e σu + f 0 the extra Lie point symmetry of the 3+1 generalized KP equation is