On vortex solutions and links between the Weierstrass system and the complex sine-Gordon equations. (English) Zbl 1040.35090
Summary: The connection between the complex sine and sinh-Gordon equations associated with a Weierstrass type system and the possibility of construction of several classes of multivortex solutions is discussed in detail. We perform the Painlevé test and analyse the possibility of deriving the Bäcklund transformation from the singularity analysis of the complex sine-Gordon equation. We make use of the analysis using the known relations for the Painlevé equations to construct explicit formulae in terms of the Umemura polynomials which are \(\tau\)-functions for rational solutions of the third Painlevé equation. New classes of multivortex solutions of a Weierstrass system are obtained through the use of this proposed procedure. Some physical applications are mentioned in the area of the vortex Higgs model when the complex sine-Gordon equation is reduced to coupled Riccati equations.
MSC:
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 37K20 | Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions |
Keywords:
massless fermion model; motion of a vortex filament; complex sine and sinh-Gordon equations; Painlevé test; Bäcklund transformationReferences:
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