Dual nationality with different names

[Kyunam 0]

I'm undergraduate student currently living in South Korea and planning to apply graduate school in the U.S. (sorry for my bad english, although I'm Korean-American, english isn't my first language)
Since I have dual nationality, I never applied for any kinds of Visa and always used my U.S. passport whenever I visit the U.S.
Problem is, my name on Korean passport and U.S. passport are different. Which means, every documents such as a transcripts, college degree, GRE test result, etc will be posted as my Korean name, which is different from my American name.
What should I do in this case? Should I change my Korean name in family court? or is there any documentation that can prove both names indicate identical person?