BÄCKLUND TRANSFORMATION AND DIVERSE EXACT EXPLICIT SOLUTIONS OF THE FRACTAL COMBINED KdV–mKdV EQUATION

Abstract

A fractal modification of the combined KdV–mKdV equation which plays a key role in various fields of physics is presented in this work for the first time. Aided by the fractal two-scale transform, the homogeneous balance method is employed to construct the fractal Bäcklund transformation. By means of the Bäcklund transformation, some new exact explicit solutions such as the algebraic solitary wave solution of rational function, single-soliton solution, double-soliton solutions, N-soliton solutions, singular traveling solutions and the periodic wave solutions of trigonometric function are obtained. Finally, some solutions are illustrated with different fractal orders in the form of the 3D plot, 3D density and 2D curves by assigning reasonable parameters with the help of Mathematica. The findings in this paper are expected to present some new insights into the fractal theory of the fractal PDEs.