  Nonlinearity of magneto-dynamics is typically described by a single constant,
$\mathcal{N}$, with positive and negative values indicating repulsion and
attraction of magnons, respectively. In thin magnetic films with easy-plane
magnetic anisotropy, magnon attraction is typically achieved for an in-plane
magnetization. At sufficient stimulus, e.g. via application of spin transfer
torque, the attraction can give rise to self-localized magnetic solitons, such
as spin wave bullets, which shrink as their amplitude increases. In contrast,
for an oblique magnetization above a certain critical angle, the repulsion of
magnons only allows for propagating modes, which expand when pumped more
strongly. Here we demonstrate, both analytically and using micromagnetic
simulations, that such a dichotomic description is inadequate for magnonic edge
modes, which naturally appear in confined magnetic systems. In particular, we
demonstrate that the confinement potential of such modes is nonlinear in nature
and its contribution makes $\mathcal{N}$ non-monotonically dependent on their
amplitude. As a prominent example, edge modes show compression and expansion
for negative and positive $\mathcal{N}$, yet remain localized. In striking
contrast to the extended geometries, edge magnons might also repeal even for an
in-plane magnetization.
