Pulse profile variability associated with the glitch of PSR J1048$-$5832

PSR J1048$-$5832 (B1046$-$58) is a Vela-like pulsar that has exhibited multiple glitch events. In this study, we analyze the timing data spanning nearly 16 years, acquired from both the Fermi Gamma-ray Space Telescope and the Parkes 64 m radio telescope. As a result, a total of five glitches are detected within this dataset. Among them, a previously unknown small glitch is newly found at MJD 56985(9) (November 24, 2014), making it the smallest glitch recorded from this source so far. The increments of the spin frequency and its first derivative are $\Delta \nu \approx 2.2(3) \times 10^ {-8} $ Hz, and $\Delta \dot{\nu} \approx 3(2) \times 10^ {-15}$ s$^{-2}$, respectively. Significant changes in the integrated normalized mean pulse profile are detected following three of the five glitch events, notably in the radio band. Although no evidence of a correlation is found between the spin-down rate and profile evolution, the jump phenomenon of $W_{55}$ (pulse width at the 55% peak amplitude) after the glitch in the narrow mode suggests that the glitch may influence the profile change. We discuss the influence of glitches on the pulsar's emission properties in terms of platelet motion by a crustquake and also put constraints on the equation of state from the moment of inertia and response timescales of involved superfluid layers inside the neutron star.