| **Proceedings of the Conference on Nonlinear Coherent Structures in Physics and Biology,** Heriot-Watt University, Edinburgh, July 95 | |

*The Dirac equation and
the Davey-Stewartson I equation:
deformation of the dromion solutions*

### M. Mañas

Departamento de Física Teórica,

Universidad Complutense, Spain

Abstract

Our contribution to this conference regards some of the recent results
obtained jointly with F. Guil, which can be found in the papers *
The Dirac Equation and Integrable Systems of KP Type* submitted to
J. Phys. A: Math. & Gen. and * Deformation of the Dromion and
Solitoff Solutions of the Davey-Stewartson I Equation* submitted to
Phys. Lett. A. We summarize these results: The propagator for the
2D heat equation in an arbitrary linear space give solutions of the
two-component Kadomtsev-Petviashvilii equations. This propagator is
subject to the Klein-Gordon equation and its right-derivatives are
required to be of rank-one, that imply that it can be expressed in
terms of solutions of the Dirac equation. Large families of solutions
of the two-component Kadomtsev-Petviashvilii equations are constructed
in terms of solutions of the heat and Dirac equations. Particular
attention is paid to the real reductions of the Davey-Stewartson type,
recovering in this way the dark line solitons and the multidromion
solutions.
A new solution of the Davey-Stewartson I equation is presented. This
is a one parameter deformation of the dromion solution, that is
contained in as particular example. The deformation corresponding to
the 1-dromion solution is studied in detail. We analyze its asymptotic
behaviour, appearing as a nonlinear superposition of the 1-dromion with
the a 2-line dark soliton type solution. The 1-solitoff degeneration
of the 1-dromion and its deformation is studied in a similar fashion.
The 1-line dark soliton solution appears as a deformation of the
1-solitoff solution.

Here we present only a part of these results, rather than a full papaer.

Conference paper.

*Last modified Mon Apr 8 16:24:59 GB-Eire 1996
(DBD)*