Given the initial value problems for the Korteweg-de Vries equation
$u_t + u_{xxx} = u u_x; \quad u(0,x) = u_0(x)$
I have read that they can be solved exactly by the inverse scattering method, but at the same time, I found on the web that there is a substantial literature on numerical solutions on KdV. Why are researcher so interested in developing numerical methods to solve these equations if they already know the analitic solution?
As with any PDE, boundary and initial data plays an important role. Just because we can solve a particular problem with an analytic solution, it is important to develop methods for more difficult situations where an analytic solution is not possible. I.e. numerics or analysis of solutions with various data.