Multiple complex and multiple real soliton solutions for the integrable sine-Gordon equation

In this work, we study the integrable sine-Gordon equation which is related to the short pulse equation (SPE). We show that this equation passes the Painlevé test to confirm its integrability. We introduce new complex forms of the simplified Hirota's method to formally derive multiple complex soliton solutions for this equation. Our results show that the complex simplified Hirota's method explicitly constructs new multiple complex soliton solutions for more integrable equations.