  Dispersion relations govern wave behaviors, and tailoring them is a grand
challenge in wave manipulation. We demonstrate inverse design of phononic
dispersion using non-local interactions on one-dimensional spring-mass chains.
For both single-band and double-band cases, we can achieve any valid dispersion
curves with analytical precision. We further employ our method to design
phononic crystals with multiple ordinary (roton/maxon) and higher-order
(undulation) critical points and investigate their wave packet dynamics.
