  We modify the Schr\"{o}dinger equation in a way that preserves its main
properties but makes use of higher order derivative terms. Although the
modification represents an analogy to the Doebner-Goldin modification, it can
differ from it quite distinctively. A particular model of this modification
including derivatives up to the fourth order is examined in greater detail. We
observe that a special variant of this model partially retains the linear
superposition principle for the wave packets of standard quantum mechanics
remain solutions to it. It is a peculiarity of this variant that a periodic
structure emerges naturally from its equations. As a result, a free particle,
in addition to a plane wave solution, can possess band solutions. It is argued
that this can give rise to well-focused particle trajectories. Owing to this
peculiarity, when interpreted outside quantum theory, the equations of this
modification could also be used to model pattern formation phenomena.
