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.