Sine-Gordon solitons in AdS, dS and other hyperbolic spaces

We find infinitely many soliton-like solutions in a deformation of the sine-Gordon theory in $(d+1)$-dimensional $AdS_{d+1}$ (anti-de Sitter) spacetime for $d \geq 2$, as well as single solitonic solutions in $dS_{d+1}$ (de Sitter) and $\mathrm{H}{d+1}$ (Lobachevsky) spaces for $d \geq 1$ and in $AdS_2$. We also find a deformation of the kink solution in scalar field theory with a polynomial potential in $AdS_2$. The deformation of the sine-Gordon theory strikingly resembles the bosonic part of the flat-space supersymmetric sine-Gordon theory. In the infinite radius limit, single soliton solutions reduce to solitons in flat space. Meanwhile, the multisoliton solution of $AdS{d+1}$, $d\geq 2$ for certain values of the parameters reduces in the same limit to a single soliton solution boosted in the normal direction. However, there are also multisoliton solutions in $AdS_{d+1}$, $d \geq 2$ that do not have a flat space limit.