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Preprint, Working Paper, ... |
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| Title: |
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The Szegö Cubic Equation |
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| Author(s): |
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Patrick Gérard ( ) 1, Sandrine Grellier () 2 |
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| Abstract: |
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We consider the following Hamiltonian equation on the $L^2$ Hardy space on the circle, $$i\partial _tu=\Pi(|u|^2u)\ ,$$ where $\Pi $ is the Szegö projector. This equation can be seen as a toy model for totally non dispersive evolution equations. We display a Lax pair structure for this equation. We prove that it admits an infinite sequence of conservation laws in involution, and that it can be approximated by a sequence of finite dimensional completely integrable Hamiltonian systems. We establish several instability phenomena illustrating the degeneracy of this completely integrable structure. We also classify the traveling waves for this system. |
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| Fulltext language: |
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English |
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| Keyword(s): |
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Nonlinear Schrödinger equations – Integrable Hamiltonian systems – Lax pairs – Hankel operators |
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| Classification: |
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35B15, 37K10, 47B35 |
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| ANR Project: |
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| Project Id |
ANR EDP dispersives, ANR AHPI |
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