The following are examples of integrable systems arising from the AKNS system (check out AKNS paper here and a short Wikipedia description)

  1. Non-Linear Schrodinger equation
  2. Korteweg-de Vries equation
  3. modified Korteweg-de Vries equation

I am paying attention the last one (mKdV) that has the form $$u_t\pm3u^2u_x+u_{xxx} = 0$$ I know that the $\pm$ represent two cases that change fundamentally change the dynamics of the system. I would like intuition: 1) on how to recognize whether this is the focusing or defocusing case, and 2) the effect of the non-linearity.

I know there are focusing and defocusing cases for the first two equations too, but how does the sign of the non-linearity show this? and what is the intuitive effect of focusing/defocusing? Is there a stability result associated to each case?


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