New exact solutions of the Tzitzéica type equations arising in nonlinear optics using a modified version of the improved $$\tan \left( {\varPhi \left( \xi \right)/2} \right)$$ tan Φ ξ / 2 -expansion method

Abstract

The paper deals with the Tzitzeica type nonlinear evolution equations arising in nonlinear optics and their new exact solutions. First, through the use of the Painleve transformation and Lie symmetry method, the Tzitzeica, Dodd–Bullough–Mikhailov, and Tzitzeica–Dodd–Bullough equations are converted to nonlinear ordinary differential equations (NODEs), and then, a modified version of the improved $$\tan \left( {\varPhi \left( \xi \right)/2} \right)$$ -expansion method, proposed by the authors, is adopted to generate new exact solutions of the reduced equations. The method truly recommends a reliable and capable technique to produce new exact solutions of a variety of nonlinear partial differential equations (NPDEs).