Hi all. I am getting confused about the notion of "a solution" to a differential equation. Let's consider the KdV equation, ut+uux+uxxx=0. So, if the initial condition is a Sech^2 pulse, then the solution would be a travelling wave solution and this is the well known solitary wave. So, what if I arbitrarily use another initial condition? say, a Sech^3 pulse or anything? This initial profile shall be also governed by the KdV equation and the evolution of this strange initial profile shall be still a solution of the KdV equation, right? just that we cannot find the analytical or exact solution? Say if I use an excellent numerical scheme to see the evolution of this Sech^3, theoretically, the evolution generated using this arbitrary initial condition shall be called a solution of the KdV, right?