1996/05/22 by H.J.S. Dorren, Roel Snieder, Dorren, H. J. S. +1 · 1 citation
Earth and Planetary Sciences · Physics and Astronomy · #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #Seismic Waves and Analysis
paper · pdf · doi:10.48550/arxiv.solv-int/9605005
openalex publication_date 1996/05/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper the stability of the Korteweg-de Vries (KdV) equation is investigated. It is shown analytically and numerically that small perturbations of solutions of the KdV-equation introduce effects of dispersion, hence the perturbation propagates with a different velocity then the unperturbed solution. This effect is investigated analytically by formulating a differential equation for perturbations of solutions of the KdV-equation. This differential equation is solved generally using an Inverse Scattering Technique (IST) using the continuous part of the spectrum of the Schrödinger equation. It is shown explicitly that the perturbation consist of two parts. The first part represents the time-evolution of the perturbation only. The second part represents the interaction between the perturbation and the unperturbed solution. It is shown explicitly that singular non-dispersive solutions of the KdV-equation are unstable.