What I remember;
This book was set in a post apocalyptic near future.
Something referred to as "the wave" has decimated most of the country, possibly the world.
The US has broken up into different factions.
The (actual) President lives on Battery Island in New York.
New York is occupied by different gangs and entities.
I think this might be The Disappearance Trilogy by John Birmingham.
There are three books: Without Warning (2008), After America (2010), Angels of Vengeance (2011).
On March 14, 2003, a wave of energy washes over North America, wiping out most of the populations of Canada, the United States, Mexico, and Cuba. The source is unknown, but it appears that Seattle, Western Canada, Alaska, and Hawaii are unaffected by the disaster.
This wave of energy is referred to as the "Wave" throughout the series.
In Without Warning, James Kipper (formerly a Seattle City Council Member) is eventually appointed President of the USA after the former President and almost his entire possible lineage of successors are wiped out. The only surviving Cabinet member is not a natural born US citizen and therefore ineligible.
In After America, President Kipper goes to New York in 2007 wanting to rebuild. Of course, chaos reigns.
After a meeting with foreign dignitaries in Seattle, Kipper and Culver head to New York City, where salvage crews are working to clear the streets of debris, the Wave having made its victims disappear, so crew-less vehicles and aircraft crashed as a result. Due to the lack of people to maintain New York City as well as cities all over the contiguous United States, fires burned unchecked and New York City is now flooded in some areas. The U.S. military has also had to step in alongside the New York Militia to fight droves of pirates, looters, terrorists and organized gangs who have moved into the U.S. East Coast from South America, Africa, and Eastern Europe.
President Kipper holds a press meeting at Castle Clinton in Battery Park (not island) when the area is hit with Katyusha rockets from Muslim fighters on Ellis Island in an assassination attempt. So he was at Battery Park, though not living there.
