  This paper concentrates on a (1+1)-dimensional nonlinear Dirac (NLD) equation
with a general self-interaction, being a linear combination of the scalar,
pseudoscalar, vector and axial vector self-interactions to the power of the
integer $k+1$. The solitary wave solutions to the NLD equation are analytically
derived, and the upper bounds of the hump number in the charge, energy and
momentum densities for the solitary waves are proved in theory. The results
show that: (1) for a given integer $k$, the hump number in the charge density
is not bigger than $4$, while that in the energy density is not bigger than
$3$; (2) those upper bounds can only be achieved in the situation of higher
nonlinearity, namely, $k\in\{5,6,7,\cdots \}$ for the charge density and
$k\in\{3,5,7,\cdots\}$ for the energy density; (3) the momentum density has the
same multi-hump structure as the energy density; (4) more than two humps (resp.
one hump) in the charge (resp. energy) density can only happen under the linear
combination of the pseudoscalar self-interaction and at least one of the scalar
and vector (or axial vector) self-interactions. Our results on the multi-hump
structure will be interesting in the interaction dynamics for the NLD solitary
waves.
