Periodicity of KdV equation in relation to zero-curvature equation 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.
 A: The fact that integrable evolutionary partial differential systems have infinitely many integrals of motion (conserved quantities) is indeed of (differenial-)algebraic nature and is not related to the choice of boundary conditions. In the KdV case this can be established in a very simple fashion using the (generalized) Miura transformation, see e.g. here. For more general results on constructing conservation laws from zero-curvature-type representations, see e.g. this survey in the case of two independent variables or this paper and references therein for the case of dispersionless (i.e. first-order quasilinear homogeneous) integrable systems. 
One general idea here is that if we have the (generalized) Lax equations $$L(\psi)=0, \quad \psi_t=A(\psi),$$ where $L$ and $A$ are linear differential operators involving the spectral parameter $\lambda$ then the conservation laws, and hence the integrals of motion, for the associated integrable system can be obtained from the formal expansion of $\psi$ with respect to $\lambda$.     
