New breather solutions of the model describing few-optical-cycle solitons beyond the slowly varying envelope approximation

The modified Korteweg-de Vries-sine-Gordon equation provides a general (1+1)-dimensional model describing few-cycle-pulse soliton propagation in two-component nonlinear media. For this model, we give its solutions in Wronskian form, which is convenient to describe interactions between solitons, breathers and their limit forms. Among these solutions, we get a stationary breather solution. Besides, the limit breather solution can provide two identical breathers traveling along the trajectories that are similar to logarithm curves rather than straight lines. As a reduction, we can also obtain solutions to the modified Korteweg-de Vries equation and the sine-Gordon equation.