Sinh-Gordon type equations and harmonic maps

Speaker(s)

Giannis Polychrou

Affiliation

Aristotle University of Thessaloniki

Language of the talk

English

Date

Oct. 10, 2024, 12:30 p.m.

Room

room 5070

Seminar

Seminar of Mathematical Physics Equations Group

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We study harmonic maps from a subset of the complex plane to a subset of the
hyperbolic plane. Harmonic maps are related to the elliptic sinh-Gordon equation
and a Bäcklund transformation is introduced, which connects solutions of the
elliptic sinh-Gordon and sine-Gordon equation. We develop this machinery in
order to construct new harmonic maps to the hyperbolic plane. Moreover, we study
a sinh-Gordon type equation that is connected with harmonic maps between
Pseudoriemannian surfaces. By suitable choice of the constants, this equation
turns into the hyperbolic and elliptic versions of the sine-Gordon and
sinh-Gordon equations on the real plane.  We construct solutions to this
equation via the method of functional separation. We prove that these are the
only families that have the property of functional separation and so we obtain a
classification. To this end, we construct new families of solutions for the
hyperbolic and elliptic versions of both sine and sinh Gordon equations in a
unified way. Finally, we apply the Hirota method to obtain the N-soliton
solutions for this equation.

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