I owned a CJ7 jeep with a rear diff that was welded together by the previous owner. It was very hard to make sharp turns (one wheel has to slip) and I could tell that the engine was working harder during that process.
Some ATV's have solid rear axles.
The diff helps you turn sharper. I think sharp turns are necessary in my lawn, so I wouldn't do it to my riding mower.
You would need to pull the rear axle apart and weld the spider gears together.
http://www.rockmodified.com/Tech/traction/traction.htm
How about installing some knobby ATV tires on the rear for better traction?
Or you could set the mower to cut higher. Although it is counter intuitive, the higher you mow, the less frequently you have to mow. I used to be a scalper, but now I am up to a 2 inch setting and love it.