Method for solving certain nonlinear partial differential equations
In mathematics, the inverse scattering 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 formula_1 and a time variable formula_2. Subscripts or differential operators (formula_3) indicate differentiation. The function formula_4 is a solution of a nonlinear partial differential equation, formula_5, with initial condition (value) formula_6.
Requirements.
The differential equation's solution meets the integrability and Fadeev conditions:
Integrability condition:formula_7 
Fadeev condition: formula_8 
Differential operator pair.
The Lax differential operators, formula_9 and formula_10, are linear ordinary differential operators with coefficients that may contain the function formula_11 or its derivatives. The self-adjoint operator formula_9 has a time derivative formula_13 and generates a eigenvalue (spectral) equation with eigenfunctions formula_14 and time-constant eigenvalues (spectral parameters) formula_15.
 formula_16 and formula_17
The operator formula_10 describes how the eigenfunctions evolve over time, and generates a new eigenfunction formula_19 of operator formula_9 from eigenfunction formula_14 of formula_9. 
 formula_23
The Lax operators combine to form a multiplicative operator, not a differential operator, of the eigenfuctions formula_14. 
 formula_25 
The Lax operators are chosen to make the multiplicative operator equal to the nonlinear differential equation. 
 formula_26 
The AKNS differential operators, developed by Ablowitz, Kaup, Newell, and Segur, are an alternative to the Lax differential operators and achieve a similar result.
Direct scattering transform.
The direct scattering transform generates initial scattering data; this may include the reflection coefficients, transmission coefficient, eigenvalue data, and normalization constants of the eigenfunction solutions for this differential equation.
 formula_27
Scattering data time evolution.
The equations describing how scattering data evolves over time occur as solutions to a 1st order linear ordinary differential equation with respect to time. Using varying approaches, this first order linear differential equation may arise from the linear differential operators (Lax pair, AKNS pair), a combination of the linear differential operators and the nonlinear differential equation, or through additional substitution, integration or differentiation operations. Spatially asymptotic equations (formula_28) simplify solving these differential equations.
Inverse scattering transform.
The Marchenko equation combines the scattering data into a linear Fredholm integral equation. The solution to this integral equation leads to the solution, u(x,t), of the nonlinear differential equation.
Example: Korteweg–De Vries equation.
The nonlinear differential Korteweg–De Vries equation is
 formula_29 
Lax operators.
The Lax operators are:
 formula_30 and formula_31 
The multiplicative operator is:
 formula_32
Direct scattering transform.
The solutions to this differential equation 
 formula_33 
may include scattering solutions with a continuous range of eigenvalues (continuous spectrum) and bound-state solutions with discrete eigenvalues (discrete spectrum). The scattering data includes transmission coefficients formula_34, left reflection coefficient formula_35, right reflection coefficient formula_36, discrete eigenvalues formula_37, and left and right bound-state normalization (norming) constants. 
 formula_38 
 formula_39
Scattering data time evolution.
The spatially asymptotic left formula_40 and right formula_41 Jost functions simplify this step. 
formula_42
The dependency constants formula_43 relate the right and left Jost functions and right and left normalization constants. 
formula_44 
The Lax formula_10 differential operator generates an eigenfunction which can be expressed as a time-dependent linear combination of other eigenfunctions. 
formula_46 
formula_47 
The solutions to these differential equations, determined using scattering and bound-state spatially asymptotic Jost functions, indicate a time-constant transmission coefficient formula_48, but time-dependent reflection coefficients and normalization coefficients.
 formula_49
Inverse scattering transform.
The Marchenko kernel is formula_50.
formula_51 
The Marchenko integral equation is a linear integral equation solved for formula_52. 
 formula_53 
The solution to the Marchenko equation, formula_54, generates the solution formula_11 to the nonlinear partial differential equation. 
formula_56
Citations.
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