Group analysis and conservation laws of an integrable Kadomtsev – Petviashvili equation ∗

Abstract. In this paper, an integrable KP equation is studied using symmetry and conservation laws. First, on the basis of various cases of coefficients, we construct the infinitesimal generators. For the special case, we get the corresponding geometry vector fields, and then from known soliton solutions we derive new soliton solutions. In addition, the explicit power series solutions are derived. Lastly, nonlinear self-adjointness and conservation laws are constructed with symmetries.


Introduction
It is well known that Kadomtsev-Petviashvili (KP) equation is mainly used to describe the nonlinear wave phenomenon.It is first derived by physicists Boris B. Kadomtsev and Vladimir I. Petviashvili in 1970 [5].This equation play a very key role in the field of mathematical physics.There are many authors studied various versions of KP equation with different method.In [12], the authors studied (3 + 1)-dimensional generalized KP and BKP (Bogoyavlenskii-Kadomtsev-Petviashvili) equations using the multiple expfunction algorithm.The authors [7] investigated extended KP-like equation.The authors [22,23] considered mixed lump-kink solutions to the KP, BKP equation.In [10], the authors studied diversity of interaction solutions to the (2 + 1)-dimensional Itô equation.The authors [6], with the use of the normal form, derived an extended KP equation with higher-order correction.
Recently, the authors [20] derived a new integrable KP equation from pseudo-differential formalism perspective.Motivated by the above papers, we study the more general case of KP equation with arbitrary coefficients m t + am xxy + b∂ −2  x m yyy + cm x ∂ −1 x m y + emm y = 0.
For the special case, a = b = −1/2, c = −2, d = −4, this equation reduce the case in paper [20].In [21], the authors considered the multiple solitons of the special coefficients of the equation.Using the same transformation [21] m(x, y, t) = u xx (x, y, t), one can get here a, b, c and d are constants.
In this paper, we try to use the symmetry and conservation laws to study this equation.Symmetry [1][2][3][13][14][15][16][17][18][19] and conservation laws play a key roles in the fields of applied mathematics and physics.Ibragimov [4] give a new theorem to derive the conservation laws.Recently, Ma [8,9] studied the conservation laws by using symmetries and adjoint symmetries in details.The authors [11] investigated a few generalized KP and BKP equations via Hirota bilinear forms.The paper is divided as follows.In Section 2, we deal with differen cases for different coefficients, and then, the corresponding infinitesimal generators are derived.In Section 3, we consider the symmetry reductions and explicit solutions.In Section 4, first, the nonlinear self-adjointness are considered, and then conservation laws are derived with symmetries.
Therefore, based the obtained results, we can construct new exact solutions of KP equation for known exact solutions.We will show these results in the next section.
For various cases of coefficients, we get different vector fields.In the following, we try to derive symmetry reductions and exact solutions.
3 Symmetry reductions and exact solutions For this case, we let ξ = x + y − vt, where v is the speed of wave.We can get the reduced equation is Integral once, and let the integral constant equal to zero, one can arrive at In order to further simplify the equation, let h = f , which leads to the following results: So, now, if we get the h, we can get the exact solutions of the original equation.

Case B. Scalar reduction
For the case v 5 , we can get the invariant solutions and invariants are In this way, we get the reduced equation is In fact, we can further reduce this equation based on the symmetry analysis.We, however, for brevity, do not list all of them.

Soliton solutions via the known soliton solutions
For one-parameter groups, that is space-invariance, g(x, y, t, u) → g(x + ε, y, t, u), we can construct new exact solutions via the known soliton solutions.For example, for the single soliton solution [21], we have the following new exact solutions , so, the final soliton solutions of original equation is .
https://www.mii.vu.lt/NAFor two soliton solutions [21], For the invariant group g(x, y, t, u) → g(e −ε x, e −2ε y, e −4ε t, u), we can get new two soliton solutions are u(x, y, t) We can also construct other new explicit soliton solutions via other invariant group.Here, we do not list all of them.

The explicit power series solutions
Now, we deal with (2).Assume that (2) has the following solution: Putting ( 3) into ( 2), one has Nonlinear Anal.Model.Control, 24(1):34-46 Consider the case when n = 0, one leads to For this case, it requires that a = 0. Consider the general case n 1, one gets Therefore, we have the following results: At last, we get the explicit solutions of (1) Here c i (i = 0, 1, 2, 3) are arbitrary constants, one can get the other coefficients c n (n 4) from the similar way.
In order to provide the help for numerical results, we rewrite it in approximate form Remark: It is easily to verify the convergence, we do not give the proof for simplicity.
https://www.mii.vu.lt/NA 4 Nonlinear self-adjointness and conservation laws In this section, we consider the nonlinear self-adjointness and conservation laws of (1).We need to use the following results [4] Theorem 1.Every Lie point, Lie-Bäcklund and nonlocal symmetry provides a conservation law for (1) and the adjoint equation.Then the elements of conservation vector (C 1 , C 2 , C 3 ) are defined by the following expression: Based on the definition in [4], we get the adjoint equation of (1) as follows: It is easily found that this equation is not self-adjointness.In order to get the conditions, we let υ = F (u), It is clear that for this case F = c 1 y + c 2 , this equation is strictly self adjoint for all parameters.
Based on Theorem 1, we get Now, for the special case of w = −u t , we derive the following results:

Conclusions
In this paper, based on symmetries and conservation laws, we studied a new integrable KP equation.First, we considered the corresponding infinitesimal generators for different coefficients.In particular, for the special case, we get the geometric vector fields and get the corresponding invariant group.Then, based on the invariant group, some new soliton solutions are presented.In addition, the explicit power series solutions are derived.Meanwhile, the recursive relationship between the coefficients is found.Subsequently, nonlinear self-adjointness of this equation are presented.Particulary, strictly self-adjointness conditions is explained.Lastly, conservation laws are obtained.In future works, we will study the nonlocal symmetries, inverse scattering and other properties, also including other solutions using various method.