Abstract
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 Backlund 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.
Highlights
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]
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]
Lax pair of (2+1) Calogero-Bogoyavlenskii-Schiff equation is obtained by using the singular manifold method
Summary
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]. 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]. The singular manifold method is used to deduce the CBS Lax pair. Shaimaa Salem et al.: Similarity Solution of (2+1)-Dimensional Calogero-Bogoyavlenskii-Schiff Equation Lax Pair.
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