### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 028, 20 pages      arXiv:1409.5444      https://doi.org/10.3842/SIGMA.2015.028

### Darboux Transformations for $(2+1)$-Dimensional Extensions of the KP Hierarchy

Oleksandr Chvartatskyi a and Yuriy Sydorenko b
a) Mathematisches Institut, Georg-August-Universität Göttingen, 37073 Göttingen, Germany
b) Faculty of Mechanics and Mathematics, Ivan Franko National University of Lviv, 79000 Lviv, Ukraine

Received September 23, 2014, in final form March 27, 2015; Published online April 10, 2015

Abstract
New extensions of the KP and modified KP hierarchies with self-consistent sources are proposed. The latter provide new generalizations of $(2+1)$-dimensional integrable equations, including the DS-III equation and the $N$-wave problem. Furthermore, we recover a system that contains two types of the KP equation with self-consistent sources as special cases. Darboux and binary Darboux transformations are applied to generate solutions of the proposed hierarchies.

Key words: KP hierarchy; symmetry constraints; binary Darboux transformation; Davey-Stewartson equation; KP equation with self-consistent sources.

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