The elliptic sinh-Gordon equation in a semi-strip

2017 ◽  
Vol 8 (1) ◽  
pp. 533-544 ◽  
Author(s):  
Guenbo Hwang
Keyword(s):  
Gordon Equation ◽  
Hilbert Problem ◽  
Lax Pair ◽  
Inverse Scattering Transform ◽  
Boundary Values ◽  
Spectral Functions ◽  
Scattering Transform ◽  
The Matrix ◽  
Fokas Method ◽  
Riemann Hilbert Problem

Abstract We study the elliptic sinh-Gordon equation posed in a semi-strip by applying the so-called Fokas method, a generalization of the inverse scattering transform for boundary value problems. Based on the spectral analysis for the Lax pair formulation, we show that the spectral functions can be characterized from the boundary values. We express the solution of the equation in terms of the unique solution of the matrix Riemann–Hilbert problem whose jump matrices are defined by the spectral functions. Moreover, we derive the global algebraic relation that involves the boundary values. In addition, it can be verified that the solution of the elliptic sinh-Gordon equation posed in the semi-strip exists if the spectral functions defined by the boundary values satisfy this global relation.

scholarly journals A new approach to integrable evolution equations on the circle

2021 ◽  
Vol 477 (2245) ◽  
pp. 20200605
Author(s):  
A. S. Fokas ◽  
J. Lenells
Keyword(s):  
Evolution Equations ◽  
Hilbert Problem ◽  
Inverse Scattering Transform ◽  
Effective Solution ◽  
Initial Value ◽  
New Approach ◽  
Scattering Transform ◽  
Integrable Evolution ◽  
Periodic Setting ◽  
Riemann Hilbert Problem

We propose a new approach for the solution of initial value problems for integrable evolution equations in the periodic setting based on the unified transform. Using the nonlinear Schrödinger equation as a model example, we show that the solution of the initial value problem on the circle can be expressed in terms of the solution of a Riemann–Hilbert problem whose formulation involves quantities which are defined in terms of the initial data alone. Our approach provides an effective solution of the problem on the circle which is conceptually analogous to the solution of the problem on the line via the inverse scattering transform.

The N-soliton solutions to the Hirota and Maxwell–Bloch equation via the Riemann–Hilbert approach

2021 ◽  
pp. 2150153
Author(s):  
Jian Li ◽  
Tiecheng Xia ◽  
Hanyu Wei
Keyword(s):  
Physical Meaning ◽  
Hilbert Problem ◽  
Three Dimensional ◽  
Soliton Solutions ◽  
Bloch Equation ◽  
Lax Pair ◽  
Integrable Equation ◽  
Motion Trajectory ◽  
The Matrix ◽  
Riemann Hilbert Problem

In this paper, we study the [Formula: see text]-soliton solutions for the Hirota and Maxwell–Bloch equation with physical meaning. From the Lax pair and Volterra integral equations, the Riemann–Hilbert problem of this integrable equation is constructed. By solving the matrix Riemann–Hilbert problem with the condition of no reflecting, the [Formula: see text]-soliton solutions for the Hirota and Maxwell–Bloch equation are obtained explicitly. Finally, we simulate the three-dimensional diagram of [Formula: see text] with 2-soliton solutions and the motion trajectory of [Formula: see text]-axis in the case of different [Formula: see text].

scholarly journals On the Modular Operator of Mutli-component Regions in Chiral CFT

2021 ◽  
Author(s):  
Stefan Hollands
Keyword(s):  
Field Theory ◽  
Integral Equation ◽  
Hilbert Problem ◽  
Matrix Elements ◽  
New Approach ◽  
Conformal Field ◽  
The Matrix ◽  
Kms Condition ◽  
Modular Operator ◽  
Riemann Hilbert Problem

AbstractWe introduce a new approach to find the Tomita–Takesaki modular flow for multi-component regions in general chiral conformal field theory. Our method is based on locality and analyticity of primary fields as well as the so-called Kubo–Martin–Schwinger (KMS) condition. These features can be used to transform the problem to a Riemann–Hilbert problem on a covering of the complex plane cut along the regions, which is equivalent to an integral equation for the matrix elements of the modular Hamiltonian. Examples are considered.

ASYMPTOTIC EXPANSIONS FOR RIEMANN–HILBERT PROBLEMS

2008 ◽  
Vol 06 (03) ◽  
pp. 269-298 ◽  
Author(s):  
W.-Y. QIU ◽  
R. WONG
Keyword(s):  
Asymptotic Expansion ◽  
Complex Analysis ◽  
Elementary Proof ◽  
Hilbert Problem ◽  
Jump Condition ◽  
Similar Type ◽  
Matrix Solution ◽  
Jump Matrix ◽  
The Matrix ◽  
Riemann Hilbert Problem

Let Γ be a piecewise smooth contour in ℂ, which could be unbounded and may have points of self-intersection. Let V(z, N) be a 2 × 2 matrix-valued function defined on Γ, which depends on a parameter N. Consider a Riemann–Hilbert problem for a matrix-valued analytic function R(z, N) that satisfies a jump condition on the contour Γ with the jump matrix V(z, N). Assume that V(z, N) has an asymptotic expansion, as N → ∞, on Γ. An elementary proof is given for the existence of a similar type of asymptotic expansion for the matrix solution R(z, N), as n → ∞, for z ∈ ℂ\Γ. Our method makes use of only complex analysis.

Riemann–Hilbert approach and N-soliton solutions for the Chen–Lee–Liu equation

2019 ◽  
Vol 33 (02) ◽  
pp. 1950002 ◽  
Author(s):  
Ming-Jun Xu ◽  
Tie-Cheng Xia ◽  
Bei-Bei Hu
Keyword(s):  
Soliton Solution ◽  
Hilbert Problem ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Potential Matrix ◽  
Riemann Hilbert Problem

In this paper, we construct the Riemann–Hilbert problem to the Lax pair of Chen–Lee–Liu (CLL) equation. As far as we know, many researchers have studied various equations with Riemann–Hilbert method before, but no one compared the N-soliton solutions calculated by different symmetries of potential matrix. Using different symmetries of potential matrix, we get two N-soliton solution formulae of the CLL equation. The interesting thing is that we find the equivalence of these two N-soliton solutions.

scholarly journals Riemann-Hilbert problem associated with the vector Lakshmanan-Porsezian-Daniel model in the birefringent optical fibers

2021 ◽  
Author(s):  
Beibei Hu ◽  
Ji Lin ◽  
Ling Zhang
Keyword(s):  
Optical Fibers ◽  
Hilbert Problem ◽  
Ultrashort Pulses ◽  
Initial Boundary ◽  
Transformation Method ◽  
Initial Boundary Value Problems ◽  
Spectral Functions ◽  
Birefringent Optical Fiber ◽  
Initial Boundary Value ◽  
Riemann Hilbert Problem

In this paper, we investigate vector Lakshmanan-Porsezian-Daniel (VLPD) model which can be used to describe the ultrashort pulses in the birefringent optical fiber. Based on the unified transformation method, the Riemann-Hilbert problem is introduced and initial-boundary value problems of the VLPD model are studied. By solving the formulated matrix Riemann-Hilbert problem, the potential function solutions of the VLPD model can be reconstructed. Moreover, that the spectral functions are not independent but meet the so-called global relation is shown.

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