Long-wave KdV hierarchy approximation of the NLS hierarchy with nonzero boundary conditions

Robert Wegner

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arXiv · 2606.10687

Long-wave KdV hierarchy approximation of the NLS hierarchy with nonzero boundary conditions

Robert Wegner

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Abstract

We study the approximation of certain renormalized conserved quantities for the NLS hierarchy with nonzero boundary conditions, in the long-wave regime, by the energies of the KdV hierarchy. We extend this to all $n \in \mathbb{N}$ by proving an approximation result for the transmission coefficient of the Lax operator of the NLS hierarchy, which is a Dirac operator in the nonrelativistic regime, by the transmission coefficient of a Schr\"odinger operator, which is the Lax operator of the KdV hierarchy. This yields a formal approximation result between the hierarchies, which we quantify using energy methods and previously established well-posedness results.

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Robert Wegner. 2026-06-09. Long-wave KdV hierarchy approximation of the NLS hierarchy with nonzero boundary conditions. https://arxiv.org/abs/2606.10687

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