  A potential flux of high-energy neutrinos from the annihilation of dark
matter particles trapped within the Sun has been exploited to place indirect
limits on particle dark matter. In most models, the dark matter interacts
weakly, but the possibility of a dark matter particle with a large cross
section for elastic scattering with itself has been proposed in several
contexts. I study the consequences of such dark matter self-interactions for
the high-energy neutrino flux from annihilation within the Sun. The
self-interaction may allow dark matter in the halo to be captured within the
Sun by scattering off of previously-captured dark matter within the Sun. This
effect is not negligible in acceptable and accessible regions of parameter
space. Enhancements in the predicted high-energy neutrino flux from the Sun of
tens to hundreds of percent can be realized in broad regions of parameter
space. Enhancements as large as factors of several hundred may be realized in
extreme regions of the viable parameter space. Large enhancements require the
dark matter annihilation cross section to be relatively small, sigma*v <~
10^-27 cm^3/s. This phenomenology is interesting. First, self-capture is
negligible for the Earth, so dark matter self-interactions break the
correspondence between the solar and terrestrial neutrino signals. Likewise,
the correspondence between indirect and direct detection limits on scattering
cross sections on nuclei is broken by the self-interaction. These broken
correspondences may evince strong dark matter self-interactions. In some cases,
self-capture can lead to observable indirect signals in regions of parameter
space where limits on dark matter-nucleon cross sections from direct detection
would indicate that no such signal should be observable.
