They are therefore susceptible to solution by techniques resembling the inverse scattering transform which was originally developed to solve the Korteweg-de Vries ( KdV ) equation, a nonlinear partial differential equation which arises in the theory of solitons, and which is also completely integrable.
32.
Drinfeld introduced the notion of a quantum group ( independently discovered by Michio Jimbo at the same time ) and made important contributions to mathematical physics, including the ADHM construction of instantons, algebraic formalism of the Quantum inverse scattering method, and the Drinfeld Sokolov reduction in the theory of solitons.
33.
Two closely related methods : the Bethe ansatz approach, in its modern sense, based on the "'Yang Baxter equations "'and the "'quantum inverse scattering method "'provide quantum analogs of the "'inverse spectral methods " '.
34.
In 2006 he received with Martin Kruskal, Robert M . Miura, and John M . Greene the Leroy P . Steele Prize for their work on the inverse scattering transformation method for the solution of nonlinear differential equations ( special soliton modeling equations similar to the Korteweg de Vries equation ).
35.
Whereas linear inverse problems were completely solved from the theoretical point of view at the end of the nineteenth century, only one class of nonlinear inverse problems was so before 1970, that of inverse spectral and ( one space dimension ) inverse scattering problems, after the seminal work of the Russian mathematical school ( Marchenko ).
36.
Important applications of phase retrieval include X-ray crystallography, transmission electron microscopy and coherent diffractive imaging, for which M = 2 . ( Fienup 1982 : 2759 ) Uniqueness theorems for both 1-D and 2-D cases of the phase retrieval problem, including the phaseless 1-D inverse scattering problem, were proved by Klibanov and his collaborators ( see References ).
37.
Jost researched quantum-mechanical scattering theory ( also inverse scattering theory : Reconstruction of potentials from scattering data ) and the mathematical quantum field theory, where he in 1958 with the methods of Arthur Strong Wightman proved the PCT theorem and in 1957 introduced " the Jost-Dyson-representation " ( an integral representation of the expectancy value of the commutator of two field operators ).
