Nonlinear Integrable Equations and Nonlinear Fourier Transform
نویسندگان
چکیده
In this paper we study nonlocal functionals whose kernels are homogeneous generalized functions. We also use such functionals to solve the Kortewegde Vries (KdV), the nonlinear Schrödinger (NLS) and the Davey-Stewartson (DS) equations. The solution of certain integrable equations in terms of formal power series was obtained in [4], [5]. In these papers the solution was expressed in a formal power series involving scattering data. In this paper in addition to developing techniques for multiplying and inverting nonlocal functionals we also: (a) Give the correct version of these series by giving meaning to the relevant kernels , see (2.10) and (3.18)). (b) We invert these series to obtain scattering data in terms of initial data. (c) Prove the convergence of these series. (d) We extend these results to equations in two space dimensions.
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