Often your passport will have your maiden name, too. There might even be an empty field in your current passport. If in doubt for a Visa that you already have: call the embassy. Most likely they will tell you to use the name that is in the passport (surprise) until you get the passport changed. If you get a new passport, your will have to get a new visa obviously. Your new passport will likely contain your old name, too. Ask a friend that is married about their passpoe
rt. If you just get a sticker/stamp/print in your passport with the new name then it's not a problem. Essentially, consider a Visa to be tied go your passport, and assume that until you get your passport changed you have to use your old name when traveling. It is fairly standard to become married and change the name, you know... some people even get married abroad. Authorities know how to process this, and the usual procedure seems to be: use the old name until you get a new passport.
As for publications:
You can't change "printed" copies, but you can make sure all your webpages and your CV clarify the name change: "Jane Doe (née Obama)" such that when someone searches for your old name + affiliation they do get a pointer to your new name. That should be your key objective: if someone googles for the publication and your old name, they should find your new identity.
As for impact: of course some people will not associate these publications easily with your name, unless you do some good follow-up publications or get to know otherwise. But neither for your PhD nor for your scientific career this will matter much on the long run. Obviously your PhD committee will learn about the name change... and you can still add your old publications to your scholar profile as well as list them in your CV (again, you can emphasize the name change by giving your maiden name).