JDS Myoko is a third ship of the Kongo-class Aegis destroyer.
She appeared in the 2012 movie "Battleship".
Yes, the JMSDF has four Kongo Class AEGIS destroyers which are like the US Arliegh Burke flight II vessels:
DDG-173 Kongō, launched September 26, 1991, and commissioned March 25, 1993
DDG-174 Kirishima, launched 19, August 1993, and commissioned March 16, 1995
DDG-175 Myōkō, launched October 5, 1994, and commissioned March 14 1996
DDG-176 Chōkai, launched August 27, 1996, and commissioned March 20, 1998
Add to that the two current Atagos, which are larger and more akin to the Flight IIA Burkes with a helicopter hanger:
DDG-177 Atago, launched August 24, 2005, and commissioned March 15, 2007
DDG-178 Ashigara, launched 30, August 2006, and commissioned March 13, 2008
And the additional two Atagos that the JMSDF has announced:
DDG-179 Unnamed to start building in 2016 and be commissioned in 2018
DDG-180 Unnamed to start building in 2017 and be commissioned in 2019
By 2020 the JMSDF is going to have eight AEGIS destroyers altogether. The most of any nation except for the US Navy.
The first Kongo is probably going to have to be decommissioned in the early 2030s and it will be interesting to see what the JMSDF does at that time. A Burke II derivative? or more Atago vessels?
Here's a nice pic of the Atago, DDG-177.