Well-Posedness of an Integrable Generalization of the Nonlinear Schr?dinger Equation on the Circle |
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Authors: | Athanasios S Fokas A Alexandrou Himonas |
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Institution: | 1. Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, CB3 0WA, UK 2. Department of Mathematics, University of Notre Dame, Notre Dame, IN, 46556, USA
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Abstract: | It is shown that the periodic initial value problem (i.v.p.) for a novel integrable generalization of the nonlinear Schrödinger equation (igNLS) is well-posed in Sobolev spaces with exponent greater than 3/2. The proof is based on a Galerkin-type approximation method. When 1/?? is not an integer, then a mollified version of the i.v.p. is solved first by applying the fundamental ODE theorem in Banach spaces. Then, deriving appropriate energy estimates, it is shown that the family of the approximate solutions thus obtained has a convergent subsequence, which at the limit gives a solution to the igNLS equation. Finally, again using energy estimates, it is shown that this solution is unique and that the data-to-solution map is continuous. When 1/?? is an integer then well-posedness is proved for the nonlocal version of this equation in the corresponding homogeneous Sobolev spaces. |
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