Long-time asymptotics for the Korteweg-de Vries equation with integrable reflectionless initial data

TL;DR

This work analyzes the long-time behavior of the Korteweg-de Vries equation ${q_t = 6 q q_x - q_{xxx}}$ on ${\mathbb R}$ for integrable, reflectionless initial data. It develops a novel representation of reflectionless potentials via symmetric coupling problems for real entire functions, enabling long-time asymptotics without relying on decay beyond integrability. For finite spectral data, the solution exhibits the classical $N$-soliton decomposition with explicit phase shifts; for general reflectionless data with potentially infinite eigenvalues, the authors prove asymptotics and discuss convergence via a Lieb-Thirring inequality. Time evolution of the spectral data leads to generalized soliton solutions whose long-time behavior is captured by a precise sum of solitons, with phase shifts explicitly determined by the initial norming data. The approach provides an elementary, inverse-scattering-based route to multi-soliton asymptotics in the reflectionless, integrable setting and broadens the scope beyond finite-width spectral data.

Abstract

We show that solutions of the Korteweg-de Vries equation with reflectionless integrable initial data decompose into a (in general infinite) linear superposition of solitons after long enough time. The proof is based on a representation of reflectionless integrable potentials in terms of solutions to symmetric coupling problems for entire functions.

Theorems & Definitions (12)

• Lemma 2.1
• proof
• Example 2.2
• Example 2.3
• Corollary 2.4
• Theorem 2.5
• proof
• Theorem 3.1
• proof
• Remark 3.2