A variational characterization of 2-soliton profiles for the KdV equation

TL;DR

This work establishes a variational characterization of 2-soliton profiles for the KdV equation by showing that, under fixed values of the conserved quantities $E_2$ and $E_3$, the global minimizers of the fourth conserved functional $E_4$ are precisely 1- or 2-soliton profiles, with a complete description of when minimizers exist and how minimizing sequences behave. The authors employ a profile decomposition to reduce the problem to a finite-dimensional optimization over soliton parameters, proving that minimizers are exactly the corresponding soliton configurations and that minimizing sequences converge in $H^2(\mathbb{R})$ to the soliton set. A stability result for the 2-soliton family in $H^2$ follows as a corollary, aligning with and complementing later multisoliton stability frameworks. The paper also discusses connections to broader N-soliton variational characterizations and situates its approach relative to inverse-scattering-based methods. Overall, it provides a simple, variational route to identifying global minimizers for $N=2$ and clarifies the role of profile decomposition in multisoliton variational problems.

Abstract

We use profile decomposition to characterize 2-soliton solutions of the KdV equation as global minimizers to a constrained variational problem involving three of the polynomial conservation laws for the KdV equation.

Theorems & Definitions (59)

• Lemma 2.1
• proof
• Proposition 2.2
• proof
• Definition 2.3
• Theorem 2.4: Al
• Corollary 2.5
• proof
• Theorem 2.6
• Corollary 2.7