Geodesic
Remarks on the Local Version of the Inverse Scattering Method
Informatics and Automation, Complex analysis and applications, Tome 253 (2006), pp. 46-60

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It is very likely that all local holomorphic solutions of integrable $(1+1)$-dimensional parabolic-type evolution equations can be obtained from the zero solution by formal gauge transformations that belong (as formal power series) to appropriate Gevrey classes. We describe in detail the construction of solutions by means of convergent gauge transformations and prove an assertion converse to the above conjecture; namely, we suggest a simple necessary condition for the existence of a local holomorphic solution to the Cauchy problem for the evolution equations under consideration in terms of scattering data of initial conditions. 
@article{TRSPY_2006_253_a3,
     author = {A. V. Domrin},
     title = {Remarks on the {Local} {Version} of the {Inverse} {Scattering} {Method}},
     journal = {Informatics and Automation},
     pages = {46--60},
     publisher = {mathdoc},
     volume = {253},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2006_253_a3/}
}
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A. V. Domrin. Remarks on the Local Version of the Inverse Scattering Method. Informatics and Automation, Complex analysis and applications, Tome 253 (2006), pp. 46-60. http://geodesic.mathdoc.fr/item/TRSPY_2006_253_a3/