The Hilbert problem for first-order linear elliptic systems on a Riemann surface with boundary

2000 ◽  
Vol 36 (4) ◽  
pp. 559-566
Author(s):  
I. A. Bikchantaev
Keyword(s):  
Riemann Surface ◽  
Hilbert Problem ◽  
Elliptic Systems ◽  
Order Linear ◽  
First Order ◽  
Riemann Surface With Boundary

First-order elliptic systems degenerated at the boundary

1993 ◽  
Vol 123 (6) ◽  
pp. 1203-1212
Author(s):  
Abduhamid Dzhuraev
Keyword(s):  
Boundary Conditions ◽  
Explicit Formula ◽  
Elliptic Systems ◽  
Bounded Solutions ◽  
Order Linear ◽  
Multiply Connected ◽  
First Order

SynopsisIn this paper we state that the bounded solutions of general first order linear elliptic systems of two equations in bounded multiply-connected plane domains, degenerated at the boundary, are determined in domains without any boundary conditions, provided the boundary is not characteristic for this system. The explicit formula for calculating the index of the system is derived.

scholarly journals The Riemann–Hilbert approach to focusing Kundu–Eckhaus equation with non-zero boundary conditions

2020 ◽  
Vol 34 (30) ◽  
pp. 2050332
Author(s):  
Li-Li Wen ◽  
En-Gui Fan
Keyword(s):  
Riemann Surface ◽  
Boundary Conditions ◽  
Hilbert Problem ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Spectral Matrix ◽  
First Order ◽  
Dynamical Features ◽  
Jost Solutions ◽  
Riemann Hilbert Problem

In this paper, we investigate the focusing Kundu–Eckhaus equation with non-zero boundary conditions. An appropriate two-sheeted Riemann surface is introduced to map the spectral parameter [Formula: see text] into a single-valued parameter [Formula: see text]. Starting from the Lax pair of Kundu–Eckhaus equation, two kinds of Jost solutions are constructed. Further, their asymptotic, analyticity, symmetries as well as spectral matrix are analyzed in detail. It is shown that the solution of the Kundu–Eckhaus equation with non-zero boundary conditions can be characterized with a matrix Riemann–Hilbert problem. Then a formula of [Formula: see text]-soliton solutions is derived by solving the Riemann–Hilbert problem. As applications of the [Formula: see text]-soliton formula, the first-order explicit soliton solutions with different dynamical features are obtained and analyzed.

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