Inverse scattering transform for the Gerdjikov–Ivanov equation with nonzero boundary conditions

In this article, we focus on the inverse scattering transform for the Gerdjikov–Ivanov equation with nonzero boundary at infinity. An appropriate two-sheeted Riemann surface is introduced to map the original spectral parameter k into a single-valued parameter z . Based on the Lax pair of the Gerdjik...

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Bibliographic Details
Published in:Zeitschrift für angewandte Mathematik und Physik 2020-10, Vol.71 (5), Article 149
Main Authors: Zhang, Zechuan, Fan, Engui
Format: Article
Language:English
Subjects:
Boundary conditions
Eigenvalues
Engineering
Formulas (mathematics)
Inverse scattering
Mathematical analysis
Mathematical Methods in Physics
Matrix methods
Parameters
Riemann surfaces
Solitary waves
Theoretical and Applied Mechanics
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Summary:In this article, we focus on the inverse scattering transform for the Gerdjikov–Ivanov equation with nonzero boundary at infinity. An appropriate two-sheeted Riemann surface is introduced to map the original spectral parameter k into a single-valued parameter z . Based on the Lax pair of the Gerdjikov–Ivanov equation, we derive its Jost solutions with nonzero boundary. Further asymptotic behaviors, analyticity and the symmetries of the Jost solutions and the spectral matrix are in detail derived. The formula of N -soliton solutions is obtained via transforming the problem of nonzero boundary into the corresponding matrix Riemann–Hilbert problem. As examples of N -soliton formula, for N = 1 and N = 2 , respectively, different kinds of soliton solutions and breather solutions are explicitly presented according to different distributions of the spectrum. The dynamical features of those solutions are characterized in the particular case with a quartet of discrete eigenvalues. It is shown that distribution of the spectrum and non-vanishing boundary also affect feature of soliton solutions.
ISSN:0044-2275
1420-9039
DOI:10.1007/s00033-020-01371-z