.2 +1 Dimensions Coupled Mkdv Equation, Darboux Transformation And Exact Solutions

The solitonian theory is an important part of the nonlinear science. There are many nonlinear partial differential equations that have solitonian properties in the pure and applied science. Therefore, solving solitonian equations(especial 2+1 dimensional equation) is important in the field.The properties and solutions of solitonian equations are studied in this thesis. We first consider a TD spectral problem. By deriving its hierarchy, a new 2+1-dimensional coupled MKdV equation(CMKdV) together with its Lax pairs is obtained. Based on the resulting Lax matrix, an iV-fold Darboux transformation associated with the TD spectral problem is proposed. The explicit solutions of the CMKdV equation are given by means of Darboux transformation for the Lax pairs.There are five sections in the paper. The first part is an introduction. In the second section, we study a TD spectral problemProm it, we derive a hierarchy of nonlinear evolutional equationsand its Lax pairs. Then, from the first two nontrivial memberswe get the (2+1) coupled MKdV equationIn Section Three, we directly construct the N-fold Darboux transformation of CMKdV equation. In fact, it is a gauge transformationwithof Lax pairs If satisfies the Lax pairs then, the new solutions of CMKdV equation can be obtained as follow:In Section Four, the exact solutions of CMKdV are obtained by means of the Darboux transformation; and several interesting figures of the solitonian solutions are plotted.In the last section, we make a conclusion of this paper.