  The nonlinear, or warped, resolvent recently explored by Giselsson and
B\`ui-Combettes has been used to model a large set of existing and new monotone
inclusion algorithms. To establish convergent algorithms based on these
resolvents, corrective projection steps are utilized in both works. We present
a different way of ensuring convergence by means of a nonlinear momentum term,
which in many cases leads to cheaper per-iteration cost. The expressiveness of
our method is demonstrated by deriving a wide range of special cases. These
cases cover and expand on the forward-reflected-backward method of
Malitsky-Tam, the primal-dual methods of V\~u-Condat and Chambolle-Pock, and
the forward-reflected-Douglas-Rachford method of Ryu-V\~u. A new primal-dual
method that uses an extra resolvent step is also presented as well as a general
approach for adding momentum to any special case of our nonlinear
forward-backward method, in particular all the algorithms listed above.
