In most of the resources that I have read, integrable systems described by a PDE posses a zero-curvature equation $$ \partial_t U - \partial_x V + [U,V] = 0 $$ which gives rise to the monodromy matrix $$ T = \mathcal{P} \exp \int\limits_0^{L} \mathrm{d} x \; U $$ Then the following quantities $$ I_n(\lambda) = \mathrm{Tr} (T^n(t,\lambda)) $$ are independent in time and so the system has an infinite amount of conserved charges, as required. However, to proof this, it is always (at least in all the notes and books I've read) assumed that all the fields are periodic in $x$ with period $L$ such that $$ V(0,t,\lambda) = V(L,t,\lambda) $$ However, I really don't understand what is the justification for this assumption. For the KdV equation, for instance, it seems clear to me that $$ u(x+L,t) \neq u(x,t) \forall x $$ which can be seen by simply looking at a plot of the (for instance 3-soliton solution) of the KdV equation

I might be misunderstanding something really simple because I've read many resources and they all make this assumption of periodicity without justifying it, but I hope someone can explain it to me.