Exact solutions of the ( 2 + 1 )-dimensional Camassa – Holm Kadomtsev – Petviashvili equation

Abstract. This paper studies the (2 + 1)-dimensional Camassa–Holm Kadomtsev–Petviashvili equation. There are a few methods that will be utilized to carry out the integration of this equation. Those are the G′/G method as well as the exponential function method. Subsequently, the ansatz method will be applied to obtain the topological soliton solution of this equation. The constraint conditions, for the existence of solitons, will also fall out of these.

There are several tools of integration that was developed in the past few years [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15].They are the Adomian decomposition method, Fan's F -expansion method, variational iteration method, semi-inverse variational principle, homotopy perturbation method and several others.One of the major drawbacks of these methods is that it is not possible to extract an analytical structure of the soliton radiation.The Inverse Scattering Transform method that was studied earlier can however formulate the soliton solution as well as the radiation.In this paper, however, the G /G method [2][3][4] and the exponential function c Vilnius University, 2012 method [3] will be used to study the CH-KP equation.Finally, the ansatz method [1,6] will obtain topological soliton solution or the shock wave solutions to this equation.

Governing equation
The CH-KP equation that will be studied in this paper is given by [1,5,9,15]: and Equations ( 1) and ( 2) were studied by Wazwaz in 2005 [7].Later, Lai and Xu [8] studied the generalized forms of CH-KP (gCH-KP) equation, which are and Here, in Eqs. ( 1)-( 4), the dependent variable is u(x, y, t) and t is the temporal variable while x and y are the spatial variables.Also, n is the strength of the nonlinearity, and a > 0, k ∈ R.
In the next two sections the exponential function method and the G /G method are going to be applied to extract some solutions to Eqs. ( 1) and (2).Finally, in the next section, the topological 1-soliton solution will be obtained to the gCH-KP equation using the soliton ansatz method.

Exact solutions by exp exp exp-function method
In this section, initially a detailed overview of the exp-function method is presented.Subsequently, this method will be applied to (1) and ( 2) to obtain a few solutions.

Details of the method
We now present briefly the main steps of the exp-function method [3].A traveling wave hypothesis u = u(ξ) for ξ = x + ct + ly converts a partial differential equation Ψ(u, u t , u x , u y , u xx , u yy , u tx , . ..) = 0 into an ordinary differential equation Φ u, cu , u , lu , u , l 2 u , cu , . . .= 0. ( The exp-function method is based on the assumption that traveling wave solutions can be expressed in the following form where w, d, p and q are positive integers to be determined further, a n and b m are unknown constants.To determine the values of d and q, we balance the linear term of highest order in Eq. ( 5) with the highest order nonlinear term.Similarly, to determine the values of w and p, we balance the linear term of lowest order in Eq. ( 5) with the lowest order nonlinear term.

Application to the (2 + 1)-dimensional CH-KP equations
In this section, we use exp-function method to obtain new and more general exact solutions of the (2 + 1)-dimensional CH-KP equations.We consider Eqs. ( 1) and ( 2) and use the transformation where l and c are constants to be determined later.Substituting (7) into Eqs.( 1) and ( 2) yields two ordinary differential equations for u(ξ).Integrating twice and taking integration constant to zero, gives We make transformations respectively for Eqs. ( 8) and (9).Thus, Eqs. ( 8) and ( 9) is transformed respectively into the following ordinary differential equations: and According to the exp-function method, we assume that the solution of Eqs. ( 10) and ( 11) can be expressed in the form of (6).In order to determine values of w and p, in (6) we balance the linear term of highest order with the highest order nonlinear term in Eqs.(10) and (11).By simple calculation for both Eqs. ( 10) and ( 11), we have where c i are determined coefficients.Balancing highest order of exp-function in them gets 2w + 3p = 4w + p, which leads to p = w.
Similarly to determine values of d and q, we balance the linear term of lowest order with the lowest order nonlinear term in Eqs. ( 10) and (11).By simple calculation for both Eqs. ( 10) and ( 11), we have and where d i are determined coefficients.From ( 12) and ( 13), we obtain We can freely choose the values of w and d, but the final solution does not strongly depend upon the choice of values of w and d.So for simplicity, we set p = w = 1 and d = q = 1, then we have Substituting Eq. ( 14) into Eqs.( 10) and (11), and equating to zero the coefficients of all powers of exp(nξ) yields a set of algebraic equations for a First consider Eq. ( 10).Solving the system of algebraic equations by the help of Maple, we obtain two cases.

Exact solutions by G /G method
In this section, we first describe the G /G-expansion method, which will then be applied to construct the traveling wave solutions for the (2 + 1)-dimensional CH-KP equations.

Details of the method
Suppose that a nonlinear partial differential equation is given by where u(x, y, t) is an unknown function, F is a polynomial in u = u(x, y, t) and its partial derivatives, in which are involved the highest order derivatives and nonlinear terms.In the following, we give the main steps of the G /G-expansion method [2][3][4].
Step 1.The traveling wave variable where l and c are constants, permits us reducing Eq. ( 28) to an ODE for u = u(ξ) in the form Step 2. Suppose that the solution of (29) can be expressed by a polynomial in G /G as follows: where G = G(ξ) satisfies the second order linear differential equation in the form: where a i , c, l, λ and µ are constants to be determined later, a m = 0.The positive integer m can be determined by considering the homogeneous balance between the highest order derivatives and nonlinear terms appearing in (29).
Step 3. Substituting (30) into (29) and using (31), collecting all terms with the same power of G /G together, and then equating each coefficient of the (G /G) i to zero, yields a set of algebraic equations for a i , c, l, λ and µ.
Step 4. Since the general solutions of (31) is well known for us, then substituting a i , c, l, λ, µ and the general solutions of (31) into (30) we have more traveling wave solutions of the nonlinear partial differential Eq. (28).
Step 5. We solve the system with the aid of Maple.Depending on the sign of the discriminant ∆ = λ 2 − 4µ, the solutions of Eq. ( 31) are well known to us.So, as a final step, we can obtain exact solutions of the given Eq.(28).

Application to the (2 + 1)-dimensional CH-KP equations
In this section, we use G /G-expansion method to obtain new and more general exact solutions of the (2 + 1)-dimensional CH-KP equations.To apply the G /G-expansion method to the Eqs.( 3), (4), we consider Eqs. ( 8) and ( 9) as the converted from of them.
By transformations (34) and (35), Eqs. ( 8) and ( 9) are converted respectively into the following ordinary differential equations: and Nonlinear Anal.Model.Control, 2012, Vol.17, No. 3, 280-296 Suppose that the solutions of Eqs. ( 36) and (37) can be expressed by a polynomial in G /G as follows: where G = G(ξ) satisfies Eq. ( 32).Balancing W 4 with W W in both of Eqs. ( 36) and (37) gives Thus we can write Eq. ( 38) as where a 0 and a 1 are constants to be determined later.

Exact solutions by ansatz method
In this section, the gCH-KP equation will be studied.They are Eqs.( 3) and (4), respectively.These two equations will be respectively rewritten with arbitrary coefficients as and respectively.They will also be referred to as Forms-I and II, respectively, in the rest of this section.The ansatz method will be applied to retrieve the topological 1-soliton solutions to these two forms.It needs to be noted that the non-topological soliton solutions are already obtained in 2011 [1].Here, for (44) and (45) To start off, the hypothesis is given by [1,6] q(x, y, t) is picked, where Here A, B 1 and B 2 are free parameters, while v represents the velocity of the soliton.The value of the unknown exponent p will be determined as a function of m and n later during the derivation of the soliton solution.From the ansatz (46), we get and

Form-I
Substituting (47)-( 52) into (44), gives By using the relation (56), the expressions (58)-(60) reduces to Combining any two equations from (61)-(63) yields the the same value of v given by which shows the consistency of the method.
Equating the two values of the velocity v from (55) and (64) gives the free parameter A of the soliton pulse as , (65) which forces the constraint relation Thus, finally, the topological 1-soliton solution to the gCH-KP equation (Form-I) is given by q(x, y, t) = A tanh 2/(n−m−1) (B 1 x + B 2 y − vt) where the relation between the free parameters A, B 1 and B 2 is given by (65), and the velocity of the soliton is given by (55) or (64).Finally the conditions for the soliton solution (67) to exist are displayed in (57) and (66).