  Should it be a pebble hitting water surface or an explosion taking place
underwater, concentric surface waves inevitably propagate. Except for possibly
early times of the impact, finite amplitude concentric water waves emerge from
a balance between dispersion or nonlinearity resulting in solitary waves. While
stability of plane solitary waves on deep and shallow water has been
extensively studied, there are no analogous analyses for concentric solitary
waves. On shallow water, the equation governing soliton formation -- the nearly
concentric Korteweg-de Vries -- has been deduced before without surface
tension, so we extend the derivation onto the surface tension case. On deep
water, the envelope equation is traditionally thought to be the nonlinear
Schr\"{o}dinger type originally derived in the Cartesian coordinates. However,
with a systematic derivation in cylindrical coordinates suitable for studying
concentric waves we demonstrate that the appropriate envelope equation must be
amended with an inverse-square potential, thus leading to a Gross-Pitaevskii
equation instead.
  Properties of both models for deep and shallow water cases are studied in
detail, including conservation laws and the base states corresponding to
axisymmetric solitary waves. Stability analyses of the latter lead to singular
eigenvalue problems, which dictate the use of analytical tools. We identify the
conditions resulting in the transverse instability of the concentric solitons
revealing crucial differences from their plane counterparts. Of particular
interest here are the effects of surface tension and cylindrical geometry on
the occurrence of transverse instability.
