  We have theoretically investigated Kelvin waves of quantized vortex lines in
trapped Bose-Einstein condensates. Counterrotating perturbation induces an
elliptical instability to the initially straight vortex line, driven by a
parametric resonance between a quadrupole mode and a pair of Kelvin modes of
opposite momenta. Subsequently, Kelvin waves rapidly decay to longer
wavelengths emitting sound waves in the process. We present a modified Kelvin
wave dispersion relation for trapped superfluids and propose a simple method to
excite Kelvin waves of specific wave number.
