Inverse scattering transform
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.
[2]: 66–67 The direct and inverse scattering transforms are analogous to the direct and inverse Fourier transforms which are used to solve linear partial differential equations.
[2]: 66–67 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).
[2]: 66–67 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.
[2]: 72 The inverse scattering problem is equivalent to a Riemann–Hilbert factorization problem, at least in the case of equations of one space dimension.
[3] This formulation can be generalized to differential operators of order greater than two and also to periodic problems.
[4] In higher space dimensions one has instead a "nonlocal" Riemann–Hilbert factorization problem (with convolution instead of multiplication) or a d-bar problem.
The inverse scattering transform arose from studying solitary waves.
Boussinesq and later D. Korteweg and G. deVries discovered the Korteweg-deVries (KdV) equation, a nonlinear partial differential equation describing these waves.
[5] 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.
[5] These particle-like waves are called solitons and arise in nonlinear equations because of a weak balance between dispersive and nonlinear effects.
[5] Gardner, Greene, Kruskal and Miura introduced the inverse scattering transform for solving the Korteweg–de Vries equation.
[5][7][8] This approach has also been applied to different types of nonlinear equations including differential-difference, partial difference, multidimensional equations and fractional integrable nonlinear systems.
is a solution of a nonlinear partial differential equation,
[2]: 72 The differential equation's solution meets the integrability and Fadeev conditions:[2]: 40 The Lax differential operators,
, are linear ordinary differential operators with coefficients that may contain the function
and generates a eigenvalue (spectral) equation with eigenfunctions
and time-constant eigenvalues (spectral parameters)
[1]: 4963 The Lax operators are chosen to make the multiplicative operator equal to the nonlinear differential equation.
[1]: 4963 The AKNS differential operators, developed by Ablowitz, Kaup, Newell, and Segur, are an alternative to the Lax differential operators and achieve a similar result.
[1]: 4964 [9][10] 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.
[2]: 39–48 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.
) simplify solving these differential equations.
[2]: 48–57 The nonlinear differential Korteweg–De Vries equation is [11]: 4 The Lax operators are:[2]: 97–102 The multiplicative operator is: The solutions to this differential equation 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
, and left and right bound-state normalization (norming) constants.
Jost functions simplify this step.
[1]: 4967 The solutions to these differential equations, determined using scattering and bound-state spatially asymptotic Jost functions, indicate a time-constant transmission coefficient
to the nonlinear partial differential equation.
