Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Vectorial Darboux Transformations for the Kadomtsev-Petviashvili
  Hierarchy   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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The categories of this paper are: physics
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The categories of this paper are: q-alg
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The categories of this paper are: nucl-ex
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The categories of this paper are: ao-sci
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The categories of this paper are: chem-ph
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The categories of this paper are: math
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The categories of this paper are: bayes-an
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The categories of this paper are: mtrl-th
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The categories of this paper are: q-fin
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The categories of this paper are: hep-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: cond-mat
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The categories of this paper are: nucl-th
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The categories of this paper are: adap-org
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The categories of this paper are: stat
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The categories of this paper are: math-ph