  The first full amplitude analysis of $B^+\to J/\psi \phi K^+$ with
$J/\psi\to\mu^+\mu^-$, $\phi\to K^+K^-$ decays is performed with a data sample
of 3 fb$^{-1}$ of $pp$ collision data collected at $\sqrt{s}=7$ and $8$ TeV
with the LHCb detector. The data cannot be described by a model that contains
only excited kaon states decaying into $\phi K^+$, and four $J/\psi\phi$
structures are observed, each with significance over $5$ standard deviations.
The quantum numbers of these structures are determined with significance of at
least $4$ standard deviations. The lightest has mass consistent with, but width
much larger than, previous measurements of the claimed $X(4140)$ state.
