Integrability and exact solutions of the nonautonomous mixed mKdV–sinh–Gordon equation

• The equation is proved to be Painleve-integrable in the case of positive and negative resonances, respectively.
• Soliton, negaton, positon and interaction solutions are introduced by means of the Hirota method and Wronskian representation.
• Wave velocities depend on the time-dependent variable coefficient.
• The singularities and asymptotic estimate of these solutions are discussed.
• At last, the superposition formulae for these solutions are also constructed.

In this paper, a nonautonomous mixed mKdV–sinh–Gordon equation with one arbitrary time-dependent variable coefficient is discussed in detail. It is proved that the equation passes the Painlevé test in the case of positive and negative resonances, respectively. Furthermore, a dependent variable transformation is introduced to get its bilinear form. Then, soliton, negaton, positon and interaction solutions are introduced by means of the Wronskian representation. Velocities are found to depend on the time-dependent variable coefficient appearing in the equation and this leads to a wide range of interesting behaviours. The singularities and asymptotic estimate of these solutions are discussed. At last, the superposition formulae for these solutions are also constructed.