  We consider two types of the generalized Korteweg - de Vries equation, where
the nonlinearity is given with or without absolute values, and, in particular,
including the low powers of nonlinearity, an example of which is the Schamel
equation. We first prove the local well-posedness of both equations in a
weighted subspace of $H^1$ that includes functions with polynomial decay,
extending the result of Linares et al [39] to fractional weights. We then
investigate solutions numerically, confirming the well-posedness and extending
it to a wider class of functions that includes exponential decay. We include a
comparison of solutions to both types of equations, in particular, we
investigate soliton resolution for the positive and negative data with
different decay rates. Finally, we study the interaction of various solitary
waves in both models, showing the formation of solitons, dispersive radiation
and even breathers, all of which are easier to track in nonlinearities with
lower power.
