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Local holomorphic Cauchy problem for soliton equations of parabolic type. (English. Russian original) Zbl 1163.35300
Dokl. Math. 77, No. 3, 332-335 (2008); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 420, No. 1, 14-17 (2008).
From the text: Soliton equations of the parabolic type are evolution equations on \(\mathbb C^2\) whose complexified versions are specified by the zero-curvature condition for a connection of the form \(U(x,t,z)dx_ V(x,t,z)dt\), where \(U\) is a polynomial of the first degree in the spectral parameter \(z\) and \(V\) is a polynomial of degree \(m\geq 2\) in \(z\). This class of equations includes the Korteweg-de Vries equation and the cubic nonlinear Schrödinger equation, together with their integrable modifications and hierarchies, but does not contain, for example, the sine-Gordon equation. We show that the local holomorphic Cauchy problem for the indicated equations with initial data at \(t=t_0\) is solvable if the scattering data of the initial conditions belong to a Gevrey class strictly lower than \(\frac1m\). Moreover, any local holomorphic solution to each of the indicated equations can be analytically extended to a globally meromorphic function of \(x\) for every fixed \(t\).

MSC:
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
35Q51 Soliton equations
35Q53 KdV equations (Korteweg-de Vries equations)
35Q55 NLS equations (nonlinear Schrödinger equations)
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