dorsal/arxiv
View SchemaA stability analysis for the Korteweg-de Vries equation
| Authors | H. J. S. Dorren, R. K. Snieder |
|---|---|
| Categories | |
| ArXiv ID | solv-int/9605005 |
| URL | https://arxiv.org/abs/solv-int/9605005 |
Abstract
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 Schr\"{o}dinger 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.
{
"annotation_id": "a503805a-310f-4cf9-952f-ea73bedb3018",
"date_created": "2026-03-02T18:02:50.570000Z",
"date_modified": "2026-03-02T18:02:50.570000Z",
"file_hash": "a7a949bf4b5a9b233c2b8ab7afa50769903567a53a76293bad764abbd25c743f",
"private": false,
"record": {
"abstract": "In this paper the stability of the Korteweg-de Vries (KdV) equation is\ninvestigated. It is shown analytically and numerically that small perturbations\nof solutions of the KdV-equation introduce effects of dispersion, hence the\nperturbation propagates with a different velocity then the unperturbed\nsolution. This effect is investigated analytically by formulating a\ndifferential equation for perturbations of solutions of the KdV-equation. This\ndifferential equation is solved generally using an Inverse Scattering Technique\n(IST) using the continuous part of the spectrum of the Schr\\\"{o}dinger\nequation. It is shown explicitly that the perturbation consist of two parts.\nThe first part represents the time-evolution of the perturbation only. The\nsecond part represents the interaction between the perturbation and the\nunperturbed solution. It is shown explicitly that singular non-dispersive\nsolutions of the KdV-equation are unstable.",
"arxiv_id": "solv-int/9605005",
"authors": [
"H. J. S. Dorren",
"R. K. Snieder"
],
"categories": [
"solv-int",
"nlin.SI"
],
"title": "A stability analysis for the Korteweg-de Vries equation",
"url": "https://arxiv.org/abs/solv-int/9605005"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "7e457079-bb9d-4a4e-92df-55c71369b56e",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}