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Progress in Initial-Boundary Value Problems for Nonlinear Evolution Equations and the Fokas Method

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Chaos, Fractals and Complexity (COSA-Net 2022)

Part of the book series: Springer Proceedings in Complexity ((SPCOM))

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Abstract

The unified transform method (UTM), also known as the Fokas method [13], provides a novel approach for solving initial-boundary value problems (ibvp) for linear and integrable nonlinear partial differential equations. In particular, it gives solution formulas for forced linear ibvp. This motivated the initiation of a new program (by Fokas and collaborators) for studying the well-posedness in Sobolev spaces of ibvp for nonlinear evolution equations by employing this method, which is analogous to the way well-posedness of initial value problems (ivp) are studied. Using the Fokas formula we are able to derive linear estimates in Sobolev, Hadamard and Bourgain spaces. Then, using as an iteration map the one defined by the UTM formula when the forcing is replaced by the nonlinearity, and utilizing the linear estimates in combination with the multilinear estimates suggested by the nonlinearity we show that this map is a contraction in appropriate solution spaces. For the Korteweg-de Vries (KdV) and Nonlinear Schrödinger (NLS) equations this program has been implemented for ibvp in one-space dimension [16, 17, 29] and significant progress has been made in higher dimensions [24]. In this review we will try to present key points of this remarkable story. It is based on collaborative work with A. Fokas, D. Mantzavinos and F. Yan.

Dedicated to Professor Athanassios S. Fokas

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Acknowledgements

The author was partially supported by a grant from the Simons Foundation (#524469 to Alex Himonas).

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Authors and Affiliations

  1. University of Notre Dame, Notre Dame, IN, 46556, USA

    A. Alexandrou Himonas

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  1. A. Alexandrou Himonas
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Correspondence to A. Alexandrou Himonas .

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Editors and Affiliations

  1. Mathematics, University of Patras, Patras, Greece

    Tassos Bountis

  2. Geology and Geoenvironment, National and Kapodistrian University of Athens, Athens, Greece

    Filippos Vallianatos

  3. Institute of Nanoscience and Nanotechnology, National Center for Scientific Research Demokritos, Agia Paraskevi, Athens, Greece

    Astero Provata

  4. Electrical and Computer Engineering, Aristotle University of Thessaloniki, Thessaloniki, Greece

    Dimitris Kugiumtzis

  5. School of Applied Mathematical and Physical Sciences, National Technical University of Athens, Athens, Greece

    Yannis Kominis

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Himonas, A.A. (2023). Progress in Initial-Boundary Value Problems for Nonlinear Evolution Equations and the Fokas Method. In: Bountis, T., Vallianatos, F., Provata, A., Kugiumtzis, D., Kominis, Y. (eds) Chaos, Fractals and Complexity. COSA-Net 2022. Springer Proceedings in Complexity. Springer, Cham. https://doi.org/10.1007/978-3-031-37404-3_19

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