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We carry out a series of high resolution ($1024\times 1024$) hydrodynamicalsimulations to investigate the orbital evolution of Jupiter and Saturn embeddedin a gaseous protostellar disk. Our work extends the results in the classicalpapers of Masset & Snellgrove (2001) and Morbidelli & Crida (2007) by exploringvarious surface density profiles ($\sigma$), where $\sigma \proptor^{-\alpha}$. The stability of the mean motion resonances(MMRs) caused by theconvergent migration of the two planets is studied as well. Our results showthat:(1) The gap formation process of Saturn is greatly delayed by the tidalperturbation of Jupiter. These perturbations cause inward or outward runawaymigration of Saturn, depending on the density profiles on the disk. (2) Theconvergent migration rate increases as $\alpha$ increases and the type of MMRsdepends on $\alpha$ as well. When $0<\alpha<1$, the convergent migration speedof Jupiter and Saturn is relatively slow, thus they are trapped into 2:1 MMR.When $\alpha>4/3$, Saturn passes through the $2:1$ MMR with Jupiter and iscaptured into the $3:2$ MMR. (3) The $3:2$ MMR turns out to be unstable whenthe eccentricity of Saturn ($e_s$) increases too high. The critical value abovewhich instability will set in is $e_s \sim 0.15$. We also observe that the twoplanets are trapped into $2:1$ MMR after the break of $3:2$ MMR. This processmay provide useful information for the formation of orbital configurationbetween Jupiter and Saturn in the Solar System.

On the orbital evolution of a giant planet pair embedded in a gaseous disk. I: Jupiter-Saturn configuration