Geodesic
Structure of Equations Solvable by the Inverse Scattering Transform for the Schr\"odinger Operator
Teoretičeskaâ i matematičeskaâ fizika, Tome 134 (2003) no. 1, pp. 110-123

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We propose a new approach for deriving nonlinear evolution equations solvable by the inverse scattering transform. The starting point of this approach is consideration of the evolution equations for the scattering data generated by solutions of an arbitrary nonlinear evolution equation that rapidly decrease as $x\to\pm\infty$. Using this approach, we find all nonlinear evolution equations whose integration reduces to investigation of the scattering-data evolution equations that are differential equations (in either ordinary or partial derivatives). In this case, the evolution equations for the scattering data themselves are linear and moreover solvable in the finite form.
Keywords: inverse scattering transform, integrable systems, Lax representation, operator representation. 
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     author = {V. K. Mel'nikov},
     title = {Structure of {Equations} {Solvable} by the {Inverse} {Scattering} {Transform} for the {Schr\"odinger} {Operator}},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {110--123},
     publisher = {mathdoc},
     volume = {134},
     number = {1},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2003_134_1_a9/}
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V. K. Mel'nikov. Structure of Equations Solvable by the Inverse Scattering Transform for the Schr\"odinger Operator. Teoretičeskaâ i matematičeskaâ fizika, Tome 134 (2003) no. 1, pp. 110-123. http://geodesic.mathdoc.fr/item/TMF_2003_134_1_a9/