Travelling wave solutions for the MKdV-sine-Gordon and the MKdV-sinh-Gordon equations by using a variable separated ODE method

In this work we formally derive exact travelling wave solutions for the modified KdV-sine-Gordon (MKdV-sine-Gordon) equation and the modified KdV-sinh-Gordon equation (MKdV-sinh-Gordon). The proposed analysis depends mainly on a variable separated ODE method. Two distinct sets of exact solitary wave solutions, that possess distinct physical structures, are formally derived for each equation. The derived solutions include others’ results and introduce entirely new solutions.