the inverse scattering transform
The Fourier transform is a miracle for LINEAR PDEs: it turns a hard differential equation into simple algebra by switching to the frequency picture, where each mode just evolves on its own. For decades it seemed nonlinear equations could have no such trick — until, around 1967, a small group found one for the KdV equation. The inverse scattering transform is essentially a 'nonlinear Fourier transform': it linearizes certain special nonlinear PDEs by passing to the right curved coordinates, evolving them trivially there, and transforming back.
Here is the scheme for KdV, u_t = 6 u u_x - u_xxx. The key insight is to treat the unknown u(x, t) as the POTENTIAL in a Schrodinger eigenvalue problem -psi_xx + u psi = lambda psi. From u you compute its 'scattering data': the discrete eigenvalues (which turn out to encode the solitons) and the reflection coefficient (which encodes the dispersive radiation) — this is the DIRECT scattering step, the analogue of taking a Fourier transform. The magic is that as u evolves by KdV in time, this scattering data evolves in a trivial, explicit, LINEAR way: the eigenvalues stay constant and the reflection coefficient just picks up a simple time factor. Finally you run INVERSE scattering — reconstruct the potential u(x, t) from its evolved scattering data, via the Gelfand-Levitan-Marchenko integral equation. Three steps: transform to scattering data, evolve trivially, transform back. The constant eigenvalues are exactly why solitons keep their identities through collisions.
Why is this a triumph? It gives EXACT solutions of a genuinely nonlinear PDE — explicit multi-soliton formulas — and explains the conservation laws and particle-like collisions structurally rather than by luck. It works for a special family of 'integrable' equations (KdV, nonlinear Schrodinger, sine-Gordon, and a handful of others), each linked to its own scattering problem. The honest limitation is precisely that specialness: integrability is rare and fragile. Most nonlinear PDEs are NOT solvable this way, which is why inverse scattering, for all its beauty, sits beside (not in place of) the estimates-and-inequalities machinery used for everything else.
For KdV, feed the initial profile u(x, 0) into the Schrodinger operator -d^2/dx^2 + u. Each bound state (negative eigenvalue) you find corresponds to exactly one soliton in the eventual solution, with its eigenvalue fixing the soliton's height and speed. Because KdV makes the eigenvalues time-INDEPENDENT, you can read off how many solitons will emerge and how fast, just from the initial data — before solving anything.
A nonlinear Fourier transform: solve KdV via a Schrodinger scattering problem.
The inverse scattering transform is powerful but NOT general — it works only for the rare 'integrable' equations that possess a Lax pair. For the overwhelming majority of nonlinear PDEs there is no such exact method, and one falls back on estimates, weak solutions and numerics.