Negaton and complexiton solutions of sine-Gordon equation

The negaton and complexiton solutions are introduced for the sine-Gordon equation by means of the Darboux transformation (DT). The mutual interactions of multinegaton and interactions of negaton–soliton and negaton–positon are discussed in detail. The superposition formulae for negations and complexitons are constructed. Noticing the relation of the sine-Gordon equation and the sinh-Gordon equation, we also obtain the real-valued singular negaton and complexiton solution for the sinh-Gordon equation.