Similarity Solution of (2+1)-Dimensional Calogero-Bogoyavlenskii-Schiff Equation Lax Pair

In this paper, we discussed and studied the solutions of the (2+1)-dimensional Calogero-Bogoyavlenskii-Schiff (CBS) equation. The Calogero-Bogoyavlenskii-Schiff equation describes the propagation of Riemann waves along the y-axis, with long wave propagating along the x-axis. Lax pair and Bäcklund transformation of the Calogero-Bogoyavlenskii-Schiff equation are derived by using the singular manifold method (SMM). The optimal Lie infinitesimals of the Lax pair are obtained. The detected Lie infinitesimals contain eight unknown functions. These functions are optimized through the commutator table. The eight unknown functions are evaluated through the solution of a set of linear differential equations, in which solutions lead to optimal Lie vectors. The CBS Lax pair is reduced by using the optimal Lie vectors to a system of ordinary differential equations (ODEs). The solitary wave solutions of Calogero-Bogoyavlenskii-Schiff equation Lax pair’s show soliton and kink waves. The obtained similarity solutions are plotted for different arbitrary functions and compared with previous analytical solutions. The comparison shows that we derive new solutions of Calogero-Bogoyavlenskii-Schiff equation by using the combination of two methods, which is different from the previous findings.


Introduction
Derivation of the Lax pairs of a nonlinear partial differential equation (NLPDE) needs first the study of its integrability, such as, the existence of a sufficiently large number of conservation laws or symmetries [1][2][3][4]. Many methods are used for studying the integrability of nonlinear partial differential equations. Among them the singular manifold method based on Painlevè analysis [5][6][7], homogeneous balance method [8][9][10][11], Weiss, Tabor and Carnevale (WTC) method [12], symbolic computation method [13] and Bäcklund transformation (BT) [14]. We here derive the Lax pair for Calogero-Bogoyavlenskii-Schiff (CBS) equation [15][16][17][18][19]; This equation describes the (2+1) dimensional interaction of Riemann wave propagating along the y-axis with long wave propagating along the x-axis [15][16][17][18][19]. CBS equation was investigated from various perspectives, such as the classical and non-classical methods. Through several symmetry reductions, exact solutions of the CBS equation were derived [20], while a variety of exact solutions using the improved (G`/G)-expansion method were presented [21][22][23], the symbolic computation method [24,25], the exponential expansion method [26], the improved tanh-coth method [27], the symmetry method [28], the Hirota's bilinear method to derive its multiple front solutions [29]. Here the singular manifold method is used to deduce the CBS Lax pair. Then we proceed to a similarity reduction of this Lax pair to a system of ordinary differential equations obtain optimal similarity solutions and compare our results with previous work on CBS equation. The organization of this paper is as follows: In Section 2 the Lax pair is deduced for CBS equation. In Section 3 the similarity solutions for this Lax pair are deduced. Finally we present the conclusions in section 4.

Singular Manifolds Method
In this section, the Singular Manifold Method is applied to find the BT and Lax pair for the (2+1) dimensional CBS equation (1). Singular Manifold Method is an inverse solution [30][31][32] of nonlinear partial differential equations having a series form; Where ( , , ) is an Eigen function and α is a real number obtained from the dominant behavior analysis.

Bäcklund Transformation of CBS Equation
Replacing for (2) into (1), the dominant behavior analysis yields α=1, in this case the series expansion (2) reduces to: This is the Bäcklund transformation of the Calogero-Bogoyavlenskii-Schiff equation. Substitute from (3) into (1), then equating the coefficients of the similar powers of to zero yields; Coefficient of ; Replacing for u 0 in (3) reduces it to;

Lax Pair of CBS Equation
Equation (1) Lax pair's is deduced by substituting (5) into (1) and equating the coefficients of the similar powers of to zero giving; Coefficient of =0; Then defined new variables V, R and Z as follows; V = , R = ) and Z = .

Lie Infinitesimals of CBS Equation
The Lie infinitesimals of the CBS Lax pair (19) and (20) The arbitrary functions f i (t), i=1.8, are optimized through the commutative products listed in Table 1. This leads to a system of ordinary differential equations in the unknown functions f i (t) reported here; Solving this system of ODE's (25), leads to the values of functions f i (t), i=1…8, listed below; According to these values the Lie vectors (21) to (24) is rewritten as: Vectors V 1 to V 4 are used to reduce and solve the Lax system (19) and (20).
Vector V 1 is used to reduce the system of equations (19) and (20) Where F 1 is an arbitrary function of y, F 2 is an arbitrary function of (y, t) and ; is a constant of integration. These , in ( Figure 1(a, b)) for t=0.1 and in (Figure 2 (a, b)), for t=1.    Where F 3 is an arbitrary function of (x, t) and F 4 is an arbitrary function of (t  (33) and (34) are plotted for ; = 1 as depicted in (Figure 3(a, b)), for t=0.1 and in (Figure 4(a, b)), for t=1.

Comparison with Previous Works
We do then compare results obtained using vectors V 1 and V 2 , (31)- (34) with previous solutions of (2+1)-dimensional CBS equation as in the following.
It's clear from this comparison that we derive a new solution of Calogero-Bogoyavlenskii-Schiff equation by using a new method different from the previous findings.

Conclusion
Lax pair of (2+1) Calogero-Bogoyavlenskii-Schiff equation is obtained by using the singular manifold method. The detected Lie infinitesimals for the CBS Lax pair's contains eight unknown functions that are specialized by the aide of the commutator table. These functions are evaluated through the solution of a set of linear differential equations. Their solutions lead to optimal Lie vectors. The CBS Lax pair is reduced by using the optimal Lie vectors to a system of ODEs. New solutions for CBS equation are obtained and plotted for different arbitrary functions, reveal some solitary waves in the form of soliton and kink waves. The obtained solutions are compared with previous works. The comparison reveals that, the derived solutions are new and the detection of the Lax pair solution's is effective in exposure traveling wave solutions of nonlinear evolution equations.