New expansion algorithm of three Riccati equations and its applications in nonlinear mathematical physics equations

Based on the study of tanh function method and the coupled projective Riccati equation method, we propose a new algorithm to search for explicit exact solutions of nonlinear evolution equations. We use the higher-order Schr\ddot\rm odinger equation and mKdV equation to illustrate this algorithm. As a result, more new solutions are obtained, which include new solitary solutions, periodic solutions, and singular solutions.Some new solutions are illustrated in figures.