dorsal/arxiv
View SchemaVectorial Darboux Transformations for the Kadomtsev-Petviashvili Hierarchy
| Authors | Q. P. Liu, M. Manas |
|---|---|
| Categories | |
| ArXiv ID | solv-int/9705012 |
| URL | https://arxiv.org/abs/solv-int/9705012 |
Abstract
We consider the vectorial approach to the binary Darboux transformations for the Kadomtsev-Petviashvili hierarchy in its Zakharov-Shabat formulation. We obtain explicit formulae for the Darboux transformed potentials in terms of Grammian type determinants. We also study the $n$-th Gel'fand-Dickey hierarchy introducing spectral operators and obtaining similar results. We reduce the above mentioned results to the Kadomtsev-Petviashvili I and II real forms, obtaining corresponding vectorial Darboux transformations. In particular for the Kadomtsev-Petviashvili I hierarchy we get the line soliton, the lump solution and the Johnson-Thompson lump, and the corresponding determinant formulae for the non-linear superposition of several of them. For Kadomtsev-Petviashvili II apart from the line solitons we get singular rational solutions with its singularity set describing the motion of strings in the plane. We also consider the I and II real forms for the Gel'fand-Dickey hierarchies obtaining the vectorial Darboux transformation in both cases.
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"abstract": "We consider the vectorial approach to the binary Darboux transformations for\nthe Kadomtsev-Petviashvili hierarchy in its Zakharov-Shabat formulation. We\nobtain explicit formulae for the Darboux transformed potentials in terms of\nGrammian type determinants. We also study the $n$-th Gel\u0027fand-Dickey hierarchy\nintroducing spectral operators and obtaining similar results. We reduce the\nabove mentioned results to the Kadomtsev-Petviashvili I and II real forms,\nobtaining corresponding vectorial Darboux transformations. In particular for\nthe Kadomtsev-Petviashvili I hierarchy we get the line soliton, the lump\nsolution and the Johnson-Thompson lump, and the corresponding determinant\nformulae for the non-linear superposition of several of them. For\nKadomtsev-Petviashvili II apart from the line solitons we get singular rational\nsolutions with its singularity set describing the motion of strings in the\nplane. We also consider the I and II real forms for the Gel\u0027fand-Dickey\nhierarchies obtaining the vectorial Darboux transformation in both cases.",
"arxiv_id": "solv-int/9705012",
"authors": [
"Q. P. Liu",
"M. Manas"
],
"categories": [
"solv-int",
"hep-th",
"nlin.SI"
],
"title": "Vectorial Darboux Transformations for the Kadomtsev-Petviashvili Hierarchy",
"url": "https://arxiv.org/abs/solv-int/9705012"
},
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