An amplitude analysis of the charmless three-body decay $B^+ \to K^+ \pi^+ \pi^-$ is presented, based on $pp$ collision data recorded with the LHCb detector at $\sqrt{s}=7$ and $8\,\text{TeV}$, corresponding to an integrated luminosity of $3\,\text{fb}^{-1}$.

The dominant $K^+\pi^-$ and $\pi^+\pi^-$ S-wave contributions pose the greatest modelling challenge.  Results are presented with three complementary approaches to these contributions: a physically motivated isobar model incorporating advanced dispersive $\pi\pi$ scattering amplitudes, the K-matrix formalism, and a quasi-model-independent method in which magnitudes and phases representing individual S-waves are free to vary in bins of the $K^+\pi^-$ and $\pi^+\pi^-$ mass. The results from all three approaches are consistent.

Improved treatment of the scalar sector resolves several long-standing puzzles. Relaxing assumptions related to the $K^*_0(1430)^0$ resonance reduces its branching fraction by a factor of 4-8, bringing it into agreement with QCD factorisation expectations.  As a consequence of the improved S-wave modelling, the ad-hoc $f_X(1300) \to \pi\pi$ component, introduced in previous amplitude analyses of $B^+ \to K^+ \pi^+ \pi^-$ decays, is no longer required. 

Six $CP$-violating phenomena are established or observed for the first time. These include short-distance $CP$ violation in $B^+ \to \rho(770)^0 K^+$, $CP$ violation mediated by long-distance final-state interactions in $B^+ \to f_2(1270) K^+$, the first evidence for $CP$ violation involving a spin-3 resonance, and significant effects in the interference between different $\pi\pi$ partial waves (S-P and S-D).  

The results offer new insight into strong dynamics as well as the origin of $CP$ violation in $B \to K\pi\pi$ decays.