  The dynamics of a wobbling kink in a two-component coupled $\phi^4$ scalar
field theory (with an excited orthogonal shape mode) is addressed. For this
purpose, the vibration spectrum of the second order small kink fluctuation is
studied in order to find the corresponding vibration modes associated to the
first (longitudinal) and second (orthogonal) field components. By means of this
analysis, it was found that the number of possible shape modes depends on the
value of the coupling constant. It is notable that when one of the orthogonal
field shape modes is initially triggered, the unique shape mode of the
longitudinal field is also activated. This coupling causes the kink to emit
radiation with twice the frequency of excited mode in the first field
component. Meanwhile, in the orthogonal channel we find radiation with two
different frequencies: one is three times the frequency of the orthogonal
wobbling mode and another is the sum of the frequencies of the longitudinal
shape mode and the triggered mode. All the analytical results obtained in this
study have been successfully contrasted with those obtained through numerical
simulations.
