I am considering the sine-Gordon equation $u_{xt}=\sin u$. It is well known that it has 1-soliton solutions, known as kinks and antikinks.
How exactly are they defined?
As far as I know, a kink is a solution of the form $$ u(x,t)=-4\arctan(\exp(\lambda x+\frac{t}{\lambda}+\mu))=4\arctan(C\exp(\lambda x+\frac{t}{\lambda})),\quad C:=-\exp(\mu),\quad \lambda>0\tag{1} $$ while an antikink is a solution of the form $$ u(x,t)=-4\arctan(\exp(\lambda x+\frac{t}{\lambda}+\mu))=4\arctan(C\exp(\lambda x+\frac{t}{\lambda})),\quad C:=-\exp(\mu),\quad \lambda<0.\tag{2} $$
If I am not mistaken, solution (1) is decreasing, while (2) is increasing.
That's a little bit confusing because in the literature, a kink usually seems to be identified with the 1-soliton solution with $\lambda>0$ but increasing, while an antikink is usually the 1-soliton with $\lambda<0$ but decreasing.