KdV suggests a connection between waves in shallow water and the potential in the Schrödinger equation. What is the intuitive explanation? The KdV equation
$$v_t+\frac{1}{4}v_{xxx}-\frac{3}{2}vv_x=0$$
was originally invented to model waves in shallow water.
However, it is well known that it also has applications in quantum mechanics. In particular, if we consider the Schrödinger operator
$$ H:=-\dfrac{d^2}{dx^2}+v,$$
its spectrum remains invariant if we evolve the potential $v$ according to the KdV equation. 
I understand mathematically why this works, but I don't really grasp the intuition behind this. Why is it true that if we treat the potential $v$ like a wave in shallow water, the allowed energies of the quantum mechanical system remain the same?  
 A: I'm not sure what intuition you are seeking in similarities of mathematical modeling... It's like intuition about the similar beat of two very different pieces of music? I fear it is all in the math.
That is, the KdV being a solvable equation with the prototypical "magical" soliton solution $v(x,t)=-2c \operatorname{sech} ^2 (\sqrt{c}(x-ct))$, this shape being protected by an infinity of conservation laws, it applies to shallow water, and thus evinces solitary waves. 
Purely formally, for a notional parameter "time", not real time, if a Schroedinger potential happens (!?) to also obey this equation, then you know how to deform it, i.e. to find a one parameter (t) family of potentials which have the same spectrum as it.  How? Presumably you know that given the KdV, you may define an antihermitean operator 
$$
B=-4\partial_x^3 +3(v\partial_x + \partial_x ~v),
$$
which combines with the Sturm-Liouville operator (Hermitean hamiltonian!)
$$
H=-\partial_x^2 +v,
$$
to yield the celebrated Lax equation of compatibility,
$$
H_t=[B,H],
$$
which is supposed to remind you of the Heisenberg equation of motion (but for real time, not this fake parameter; here, B plays the role of the hamiltonian, and H the role of the operator).
One may then solve the equation for a unitary U,
$$
U_t= BU,
$$
so that $B=U_t U^{-1}$, and then the Lax equation leads you to the equivalent t -dependent hermitean Hamiltonian,
$$
H(t,x)=U(t,x) H(0,x)U(t,x)^{-1}.
$$
This means that, defining a deformed wavefunction through $\psi_t=B\psi$,
$$
\psi(t,x)= U(t,x) \psi(0,x),
$$
the spectrum is preserved through this equivalence transformation of the Hilbert space,
$$
(H(0,x)-\lambda)\psi(0,x)=0 ~~\Longrightarrow (H(t,x)-\lambda)\psi(t,x)=0.
$$
Wow! Effectively, the potential has flowed w.r.t. this fake time parameter to a different one, which however has preserved its "shape": the set of eigenvalues that more or less characterize it (avoiding fussing... a long story). (Gardner, Greene, Kruskal and Miura, 1967). The isospectral flow and the shape-preservation of the solitons are two aspects of the same infinity of conserved integrals, higher symmetry, of the KdV.
You might think this is "Gee-wizz academic", but, no! It actually finds applications in isospectral supersymmetric potential constructions,  specifically reflectionless, in "real life"! 

Note added. In the half century since the introduction of the above-hinted inverse scattering method, the KdV has become the mainstay of the integrability branch of applied mathematics, and so with diverse applications to fluid mechanics,  beyond surface gravity waves, to internal solitons in the ocean subsurface currents; plasma physics; nonlinear acoustics of bubbly liquids; voidage slugs in fluidized beds, and magma flow and conduit waves in geophysics; the Great Red Spot of Jupiter, etc... 
My impression is that it is the first recourse to any study of nonlinearity, just as the harmonic oscillator basically underlies a huge chunk of quantum mechanics. So, to be somewhat deconstructive, the connection is not between shallow water waves and the Schroedinger equation spectrum, but, rather, the isospectral flow of special  Sturm-Liouville operators' intimate connection to the simplest integrable PDE, the KdV, which underlies that culture. Each has dozens of applications and ramifications, but the depth of the connection is mathematical, not a rough and ready bridge between models of physical systems... (Lightning may strike me, in this physics forum for admitting that!). 
