In mathematics, the inverse scattering transform (or nonlinear Fourier transform) is a method that solves the initial value problem for a nonlinear partial differential equation using mathematical methods related to wave scattering. The direct scattering transform describes how a function scatters waves or generates bound-states. The inverse scattering transform uses wave scattering data to construct the function responsible for wave scattering. The direct and inverse scattering transforms are analogous to the direct and inverse Fourier transforms which are used to solve linear partial differential equations. Using a pair of differential operators, a 3-step algorithm may solve nonlinear differential equations; the initial solution is transformed to scattering data (direct scattering transform), the scattering data evolves forward in time (time evolution), and the scattering data reconstructs the solution forward in time (inverse scattering transform). This algorithm simplifies solving a nonlinear partial differential equation to solving 2 linear ordinary differential equations and an ordinary integral equation, a method ultimately leading to analytic solutions for many otherwise difficult to solve nonlinear partial differential equations. The inverse scattering problem is equivalent to a Riemann–Hilbert factorization problem, at least in the case of equations of one space dimension. This formulation can be generalized to differential operators of order greater than two and also to periodic problems. In higher space dimensions one has instead a "nonlocal" Riemann–Hilbert factorization problem (with convolution instead of multiplication) or a d-bar problem.
History The inverse scattering transform arose from studying solitary waves. J.S. Russell described a "wave of translation" or "solitary wave" occurring in shallow water. First J.V. Boussinesq and later D. Korteweg and G. deVries discovered the Korteweg-deVries (KdV) equation, a nonlinear partial differential equation describing these waves. Later, N. Zabusky and M. Kruskal, using numerical methods for investigating the Fermi–Pasta–Ulam–Tsingou problem, found that solitary waves had the elastic properties of colliding particles; the waves' initial and ultimate amplitudes and velocities remained unchanged after wave collisions. These particle-like waves are called solitons and arise in nonlinear equations because of a weak balance between dispersive and nonlinear effects. Gardner, Greene, Kruskal and Miura introduced the inverse scattering transform for solving the Korteweg–de Vries equation. Lax, Ablowitz, Kaup, Newell, and Segur generalized this approach which led to solving other nonlinear equations including the nonlinear Schrödinger equation, sine-Gordon equation, modified Korteweg–De Vries equation, Kadomtsev–Petviashvili equation, the Ishimori equation, Toda lattice equation, and the Dym equation. This approach has also been applied to different types of nonlinear equations including differential-difference, partial difference, multidimensional equations and fractional integrable nonlinear systems.
Description
Nonlinear partial differential equation The independent variables are a spatial variable x {\displaystyle x} and a time variable t {\displaystyle t} . Subscripts or differential operators ( ∂ x , ∂ t {\textstyle \partial _{x},\partial _{t}} ) indicate differentiation. The function u ( x , t ) {\displaystyle u(x,t)} is a solution of a nonlinear partial differential equation, u t + N ( u ) = 0 {\textstyle u_{t}+N(u)=0} , with initial condition (value) u ( x , 0 ) {\textstyle u(x,0)} .
Requirements The differential equation's solution meets the integrability and Fadeev conditions:
Integrability condition: ∫ − ∞ ∞ | u ( x ) | d x < ∞ {\displaystyle \int _{-\infty }^{\infty }\ |u(x)|\ dx\ <\infty }
Fadeev condition: ∫ − ∞ ∞ ( 1 + | x | ) | u ( x ) | d x < ∞ {\displaystyle \int _{-\infty }^{\infty }\ (1+|x|)|u(x)|\ dx\ <\infty }
Differential operator pair The Lax differential operators, L {\textstyle L} and M {\textstyle M} , are linear ordinary differential operators with coefficients that may contain the function u ( x , t ) {\textstyle u(x,t)} or its derivatives. The self-adjoint operator L {\textstyle L} has a time derivative L t {\textstyle L_{t}} and generates a eigenvalue (spectral) equation with eigenfunctions ψ {\textstyle \psi } and time-constant eigenvalues (spectral parameters) λ {\textstyle \lambda } .
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