What counts as a solution to the KdV equation?

[hanson]

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 traveling 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?

 

[arildno]

Certainly.

Solitons are not the only solution to the KdV equation.

What IS surprising is that soliton solutions DO exist..

The soliton solution is gained by hypothesizing the existence of a non-dispersive solution of KdV; calculations then reveal that:

Insofar as such solutions exist, they need to have a Sech^2-profile.