Yongshuai Zhang, Deqin Qiu, Wei Liu
The Darboux transformation (DT) and Bäcklund transformation (BT) of the celebrated Tzitzéica equation are reconsidered. The 1st- and 2nd-order Darboux transformation and Bäcklund transformation of the Tzitzéica equation are linked by the fractional transformations of corresponding eigenfunctions. By using the formula of n -order new solution, the compact determinant representation of n -Bäcklund transformation of the Tzitzéica equation is obtained for the first time, to the best of our knowledge. As an application of the formula, a 3-order complex-valued solution for the equation is constructed, which exhibits abundant dynamical evolutions, including 3-soliton solution, breather-soliton solution, soliton-periodic wave solution, breather-periodic wave solution and so on.