  We show that a number of nonlocal nonlinear equations including the
Ablowitz-Musslimani and the Yang variant of the nonlocal nonlinear
Schr\"od-inger (NLS) equation, nonlocal modified Korteweg de Vries (mKdV)
equation as well as the nonlocal Hirota equation admit novel kinklike and
pulselike superposed periodic solutions. Further, we show that the nonlocal
mKdV equation, in addition also admits the superposed (hyperbolic)
kink-antikink solution. Besides, we show that while the nonlocal
Ablowitz-Musslimani variant of the NLS admits complex (parity-time reversal or)
PT-invariant kink and pulse solutions, neither the local NLS nor the Yang
variant of the nonlocal NLS admits such solutions. Finally, except for the Yang
variant of the nonlocal NLS, we show that the other three nonlocal equations
admit both the kink and pulse solutions in the same model.
