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Remarks on the local version of the inverse scattering method. (English. Russian original) Zbl 1351.35096
Proc. Steklov Inst. Math. 253, 37-50 (2006); translation from Tr. Mat. Inst. Steklova 253, 46-60 (2006).
Summary: 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.

MSC:
35P25 Scattering theory for PDEs
37K15 Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems
35Q53 KdV equations (Korteweg-de Vries equations)
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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