The Inverse Scattering Transform

Abstract

• 1967: Gardner, Greene, Kruskal, and Miura discover a strange way to solve the initial-value problem for the Korteweg-de Vries (KdV) equation: u t + uu x + u xxx = 0, u(x, 0) = u 0 (x), x ∈ R. Pretend that for each t, the function V (x) := −u(x, t) is taken as a potential energy coefficient in the linear (time-independent) Schrödinger equation: −6φ (x) + V (x)φ(x) = Eφ(x) where E is the energy (eigenvalue). By direct calculations, GGKM showed that various quantities associated with this linear problem are either indpendent of t (L 2 energy eigenvalues E) or evolve in an elementary fashion (the so-called reflection coefficient) as u evolves according to KdV. Therefore, if the appropriate data can be calculated from given initial conditions and if the inverse problem of finding the potential V given the data can be solved, one has an algorithm for solving the KdV initial-value problem. • 1968: Lax finds an operator-theoretic way to generalize the method of GGKM, introducing Lax pairs. • 1971: Zakharov and Shabat apply Lax's method to solve the cubic nonlinear Schrödinger equation. • 1974: Ablowitz, Kaup, Newell, and Segur write an influential paper emphasizing an analogy between the solution methods of GGKM, Lax, and ZS, and the solution of linear PDE by Fourier transform. They introduce the term "inverse-scattering transform". The inverse-scattering transform is different in the details for different equations. Hence there is no such thing as "the" inverse-scattering transform. We will explore the easiest case today. The inverse-scattering transform for the defocusing nonlinear Schrödinger equation Lax pair and zero-curvature representation. The defocusing nonlinear Schrödinger (NLS) equation (1) iψ t + 1 2 ψ xx − |ψ| 2 ψ = 0 for a complex-valued field ψ(x, t) can be viewed as a compatibility condition for the simultaneous linear equations of a Lax pair: (2) ∂w ∂x = Uw, U = U(x, t, λ) := −iλ ψ ψ * iλ and (3) ∂w ∂t = Vw, V = V(x, t, λ) := −iλ 2 − i 1 2 |ψ| 2 λψ + i 1 2 ψ x λψ * − i 1 2 ψ * x iλ 2 + i 1 2 |ψ| 2. In other words, the defocusing NLS equation is equivalent to the zero-curvature condition ∂U ∂t − ∂V ∂x + [U, V] = 0 (the left-hand side is independent of λ and vanishes when the defocusing NLS equation holds for ψ). We want to solve general initial-value problems, say of the form in which we seek ψ(x, t) for x ∈ R and t ≥ 0, such that ψ(x, t) is smooth and decays to zero for large |x| and satisfies a specified initial condition ψ(x, 0) = ψ 0 (x) , for some given function ψ 0 (x).