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Multisolitons are the unique constrained minimizers of the KdV conserved quantities

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Abstract

We consider the following variational problem: minimize the \((n+1)\)st polynomial conserved quantity of KdV over \(H^n(\mathbb {R})\) with the first n conserved quantities constrained. Maddocks and Sachs (Comm Pure Appl Math 46:867–901, 1993) used that n-solitons are local minimizers for this problem in order to prove that n-solitons are orbitally stable in \(H^n(\mathbb {R})\).

Given n constraints that are attainable by an n-soliton, we show that there is a unique set of n amplitude parameters so that the corresponding multisolitons satisfy the constraints. Moreover, we prove that these multisolitons are the unique global constrained minimizers. We then use this variational characterization to provide a new proof of the orbital stability result from Maddocks and Sachs (Comm Pure Appl Math 46:867–901, 1993) via concentration compactness.

In the case when the constraints can be attained by functions in \(H^n(\mathbb {R})\) but not by an n-soliton, we discover new behavior for minimizing sequences.

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Acknowledgements

I was supported in part by NSF grants DMS-1856755 and DMS-1763074. I would also like to thank my advisors, Rowan Killip and Monica Vişan, for their guidance, and Jaume de Dios Pont for pointing out the proof method from [34] for Lemma 3.2. Finally, I would like to thank the referee for their thoughtful comments and corrections.

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  1. Department of Mathematics, University of California, Los Angeles, CA, 90095, USA

    Thierry Laurens

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Laurens, T. Multisolitons are the unique constrained minimizers of the KdV conserved quantities. Calc. Var. 62, 192 (2023). https://doi.org/10.1007/s00526-023-02534-2

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