Aerospace companies and NASA have been exploring lots of different ways to deal with sonic booms, and most of their designs are similar: long, skinny aircraft with pointy noses designed to "mitigate" the noise problem. Researchers at Tohoku University are trying a completely different and awesomely retro-futuristic idea: a supersonic biplane that eliminates sonic booms entirely.
When an airplane is in flight, it's continuously pushing a series waves of air out of the way in front of it, the same way that a boat moving through water is pushing out a bow wave. These waves of air travel at the speed of sound, and as long as the airplane is going slower than that, the waves can get out of the way of each other and people on the ground will just hear a regular airplane noise when the plane passes overhead.
Sonic booms happen when an aircraft starts going fast enough that the waves of air (pressure, really) that it's producing can't outrun the aircraft anymore, and they all stack up on each other, forming a single shockwave of sound at the front of the plane which can be decidedly unpleasant for anyone on the ground who gets smacked with it. And then they get smacked again by another, trailing shockwave, formed by the negative pressure at the rear of the aircraft. This is where that distinctive "double boom" comes from.
Misora (the honorific name for "sky" in Japanese) is a conceptual design for an entirely new sort of supersonic aircraft, from the Institute of Fluid Science at Tohoku University. As you can see, it's a biplane, a type of aircraft that went out of style back in the 1930s since two wings create tons of drag, generally making high speeds difficult. If you're clever, though, you can arrange those two wings to reflect shock waves back at each other, taking the positive pressure shockwave and the negative pressure shockwave and zeroing them both out. Without shockwaves, you get supersonic airspeeds with no booms at all.
As far as getting to supersonic speeds with two wings, a group from MIT and Stanford has come up with a design that uses smooth inner-wing surfaces combined with bumpy wing edges to reduce drag so much that it should be possible to develop a supersonic biplane that can travel at Mach 5 while simultaneously using half as much fuel as a conventional supersonic aircraft. Mach 5, for the record, is nearly 4,000 mph, which is fast enough to make the hop from LA to New York, or New York to London, in under an hour.