Multisolitons are the unique constrained minimizers of the KdV conserved quantities

TL;DR

This paper proves that multisolitons uniquely minimize certain conserved quantities of the KdV equation under constraints, providing a new variational characterization and a novel proof of their orbital stability.

Contribution

It establishes the uniqueness of multisolitons as constrained minimizers of KdV conserved quantities and offers a new proof of their stability using variational methods.

Findings

Multisolitons are the unique global constrained minimizers of the KdV conserved quantities.

A new variational characterization of multisolitons is provided.

The paper offers a novel proof of orbital stability via concentration compactness.

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 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 of Maddocks and Sachs 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.