  Spectra of pions, which are known as the pseudo-Goldstone bosons of
spontaneous chiral symmetry breaking, as well as their relationship with chiral
phase transition and pion superfluidity phase transition, have been
investigated in the framework of soft-wall AdS/QCD. In chiral limit, it is
proved both numerically and analytically that pions are massless Goldstone
bosons even at finite temperature, which was usually considered as an
assumption in soft-wall models. Above $T_c$, at which chiral condensate
$\langle \bar{q}q\rangle$ vanishes, the spectra of pions and scalar mesons
merge together, showing the evidence of the restored chiral symmetry in
hadronic spectrum level. Extending to finite quark mass, pion masses increase
with quark mass. Further, it is more interesting to observe that the pole
masses of pions decrease with temperature below $T_c$, which agrees with the
analysis in Phys.Rev.Lett.88(2002)202302. Meanwhile, symmetry restoration above
$T_c$ could be seen in the spectra of scalar and pseudo-scalar mesons. With
finite temperature and isospin chemical potential $\mu_I$, it is shown that the
masses of charged pions would split. The mass of positive charged pion $\pi^+$
decreases almost linearly to zero when $\mu_I$ grows to $\mu_{I}^c$, where pion
condensation starts to form. This reveals the Goldstone nature of $\pi^+$ after
pion superfluidity transition, which are closely related to the experimental
observation.
