dorsal/arxiv
View SchemaSolutions of the Kpi Equation with Smooth Initial Data
| Authors | M. Boiti, F. Pempinelli, A. Pogrebkov |
|---|---|
| Categories | |
| ArXiv ID | solv-int/9306004 |
| URL | https://arxiv.org/abs/solv-int/9306004 |
| DOI | 10.1088/0266-5611/10/3/001 |
Abstract
The solution $u(t,x,y)$ of the Kadomtsev--Petviashvili I (KPI) equation with given initial data $u(0,x,y)$ belonging to the Schwartz space is considered. No additional special constraints, usually considered in literature, as $\int\!dx\,u(0,x,y)=0$ are required to be satisfied by the initial data. The problem is completely solved in the framework of the spectral transform theory and it is shown that $u(t,x,y)$ satisfies a special evolution version of the KPI equation and that, in general, $\partial_t u(t,x,y)$ has different left and right limits at the initial time $t=0$. The conditions of the type $\int\!dx\,u(t,x,y)=0$, $\int\!dx\,xu_y(t,x,y)=0$ and so on (first, second, etc. `constraints') are dynamically generated by the evolution equation for $t\not=0$. On the other side $\int\!dx\!\!\int\!dy\,u(t,x,y)$ with prescribed order of integrations is not necessarily equal to zero and gives a nontrivial integral of motion.
{
"annotation_id": "364f7cdf-c77c-4e7e-b8d2-6cd6b9d4cc6d",
"date_created": "2026-03-02T18:02:48.236000Z",
"date_modified": "2026-03-02T18:02:48.236000Z",
"file_hash": "3195c4a504580858cd724374ce7a4c549de76995529e47c7847c19344c84da84",
"private": false,
"record": {
"abstract": "The solution $u(t,x,y)$ of the Kadomtsev--Petviashvili I (KPI) equation with\ngiven initial data $u(0,x,y)$ belonging to the Schwartz space is considered. No\nadditional special constraints, usually considered in literature, as\n$\\int\\!dx\\,u(0,x,y)=0$ are required to be satisfied by the initial data. The\nproblem is completely solved in the framework of the spectral transform theory\nand it is shown that $u(t,x,y)$ satisfies a special evolution version of the\nKPI equation and that, in general, $\\partial_t u(t,x,y)$ has different left and\nright limits at the initial time $t=0$. The conditions of the type\n$\\int\\!dx\\,u(t,x,y)=0$, $\\int\\!dx\\,xu_y(t,x,y)=0$ and so on (first, second,\netc. `constraints\u0027) are dynamically generated by the evolution equation for\n$t\\not=0$. On the other side $\\int\\!dx\\!\\!\\int\\!dy\\,u(t,x,y)$ with prescribed\norder of integrations is not necessarily equal to zero and gives a nontrivial\nintegral of motion.",
"arxiv_id": "solv-int/9306004",
"authors": [
"M. Boiti",
"F. Pempinelli",
"A. Pogrebkov"
],
"categories": [
"solv-int",
"nlin.SI"
],
"doi": "10.1088/0266-5611/10/3/001",
"title": "Solutions of the Kpi Equation with Smooth Initial Data",
"url": "https://arxiv.org/abs/solv-int/9306004"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "b156a412-895b-42c5-9a8d-d12734067d45",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}