Article
Mathematics
Mateusz Piorkowski
Summary: In this paper, the asymptotic behavior of solutions to the Korteweg-de Vries equation with steplike initial data, leading to shock waves, is studied. An alternative approach is presented, which involves the direct comparison of resolvents related to the corresponding Riemann-Hilbert problems, instead of the usual argument involving a small norm Riemann-Hilbert problem. The motivation for this approach arises from the absence of an invertible holomorphic outer parametrix solution for our problem at certain discrete times.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Shengyang Yuan, Jian Xu
Summary: This paper studies the initial value problem for the negative order integrable KdV equation using the Riemann-Hilbert problem method. The solutions of the NKdV equation are constructed based on the asymptotic behavior of the spectral variable at the singularity point λ = & INFIN;. The one-soliton and two-soliton solutions are discussed in detail.
APPLIED MATHEMATICS LETTERS
(2022)
Article
Mathematics, Applied
Mateusz Piorkowski, Gerald Teschl
Summary: This paper takes a closer look at the relationship between one-gap solutions of the Korteweg-de Vries equation and the Riemann-Hilbert problem. By reformulating it as a scalar Riemann-Hilbert problem on the torus, the authors deduce the vector-valued and singular matrix-valued solutions using Jacobi theta functions. The results are compared with those in recent literature.
ANALYSIS AND MATHEMATICAL PHYSICS
(2022)
Article
Mathematics
Christophe Charlier, Tom Claeys, Giulio Ruzza
Summary: This paper investigates the uniform asymptotics for deformed Airy kernel determinants, which have important mathematical and physical significance in finite temperature free fermion models and the narrow wedge solution of the Kardar-Parisi-Zhang equation.
JOURNAL OF FUNCTIONAL ANALYSIS
(2022)
Article
Mathematics, Applied
Alexei Rybkin
Summary: In the context of the Korteweg-de Vries equation, we propose a continuous version of the binary Darboux transformation based on the Riemann-Hilbert problem. Our method provides a new explicit formula for perturbing the negative spectrum of step-type potentials while preserving the rest of the scattering data. This extends the applicability of previous formulas for inserting/removing bound states to arbitrary sets of negative spectrum.
STUDIES IN APPLIED MATHEMATICS
(2023)
Article
Mathematics, Applied
Deqin Qiu, Yongshuai Zhang
Summary: This paper provides a revised Riemann-Hilbert problem for the Kundu equation with zero boundary condition. By assuming the reflection coefficient having one pair of Nth order poles, the RHP is solved and the expression of the integral factor involved in the solution is obtained. An explicit formula for the Nth order bound-state soliton of the Kundu equation is also presented.
APPLIED MATHEMATICS LETTERS
(2023)
Article
Mathematics, Applied
Iryna Egorova, Johanna Michor, Gerald Teschl
Summary: We show how the inverse scattering transform can be used as a convenient tool to derive the long-time asymptotics of the KdV shock problem in the soliton region. In particular, we improve the results previously obtained via the nonlinear steepest decent approach both with respect to the decay of the initial conditions as well as the region where they are valid.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2022)
Article
Physics, Multidisciplinary
Yongshuai Zhang, Nan Wang, Deqin Qiu, Jingsong He
Summary: This study applies the Riemann-Hilbert problem (RHP) to the Kundu equation with zero boundary condition. By solving the revised RHP, a pair of differential equations are obtained, and the explicit formula for the N-th order soliton is derived. The formula does not involve unsatisfactory integrals, unlike existing representations in literature.
Article
Engineering, Mechanical
Minmin Wang, Yong Chen
Summary: This paper studies the inverse scattering transformation for a novel nonlocal Lakshmanan-Porsezian-Daniel (LPD) equation, and provides multiple solutions and diverse soliton patterns.
NONLINEAR DYNAMICS
(2023)
Article
Physics, Mathematical
Nalini Joshi, Pieter Roffelsen
Summary: By solving a Riemann-Hilbert problem for a q-difference Painleve equation (qP(IV)), a bijective correspondence between transcendental solutions of qP(IV) and corresponding data on an associated q-monodromy surface was established. Additionally, the moduli space of qP(IV) was explicitly constructed.
COMMUNICATIONS IN MATHEMATICAL PHYSICS
(2021)
Article
Engineering, Mechanical
Mengtao Xu, Nan Liu, Chunxiao Guo
Summary: The article discusses the application of the inverse scattering transform method to solving the six-order nonlinear Schrodinger equation with zero boundary condition. By solving the associated Riemann-Hilbert problem, exact formulas for N-soliton and multiple-poles soliton solutions are obtained, along with displaying the dynamic behaviors of various solitons.
NONLINEAR DYNAMICS
(2021)
Article
Mathematics, Applied
Yuan Li, Shou-Fu Tian, Jin-Jie Yang
Summary: This study systematically investigates general n-component nonlinear Schrodinger equations using the Riemann-Hilbert method. It focuses on multi-soliton solutions, analyzing phenomena such as elastic collision, soliton reflection, and parallel propagation. The study also proposes conjectures about the dynamic behaviors of N-soliton solutions, aiming to enhance understanding of soliton interactions in various fields such as nonlinear optics and plasma physics.
STUDIES IN APPLIED MATHEMATICS
(2022)
Article
Physics, Applied
Han-Dong Guo, Tie-Cheng Xia, Li-Ning Tong
Summary: In this study, the integrable Lakshmanan-Porsezian-Daniel (LPD) equation originating in nonlinear fiber is investigated using the Riemann-Hilbert (RH) approach. The formula for general N-soliton solutions is obtained by solving a special RH problem with reflectionless conditions. The localized structures and dynamic behaviors of the resulting solution are illustrated and discussed. Additionally, the collapse of higher-order soliton solutions is observed, indicating that they are not simple nonlinear superpositions of basic soliton solutions.
MODERN PHYSICS LETTERS B
(2022)
Article
Engineering, Mechanical
Minmin Wang, Yong Chen
Summary: The general soliton solutions and higher-order soliton solutions for the nonlocal generalized Sasa-Satsuma (SS) equation are explored. The study derived a novel nonlocal generalized SS equation, considered conservation laws, and obtained symmetry properties and nonlocal constraints. The N-soliton formula and higher-order soliton formulas were constructed using the Riemann-Hilbert problem and nonlocal properties. The study also exhibited and explored new patterns and unusual dynamical behaviors of the soliton solutions.
NONLINEAR DYNAMICS
(2022)
Article
Physics, Applied
Weifang Weng, Zhenya Yan
Summary: This paper presents general triple-pole multi-soliton solutions for the focusing mKdV equation with nonzero boundary conditions and triple zeros of analytical scattering coefficients using the inverse scattering transform. The obtained solutions can also be degenerated to triple-pole soliton solutions with zero boundary conditions. Moreover, the analysis includes representative reflectionless potentials containing triple-pole multi-dark-anti-dark solitons and breathers.
MODERN PHYSICS LETTERS B
(2021)
Article
Mathematics, Applied
Iryna Egorova, Markus Holzleitner, Gerald Teschl
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2016)
Article
Physics, Mathematical
Aleksey Kostenko, Gerald Teschl
LETTERS IN MATHEMATICAL PHYSICS
(2016)
Article
Mathematics, Applied
Jonathan Eckhardt, Gerald Teschl
Article
Mathematics, Applied
Isaac Alvarez-Romero, Gerald Teschl
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2017)
Article
Mathematics
Tom Koornwinder, Aleksey Kostenko, Gerald Teschl
ADVANCES IN MATHEMATICS
(2018)
Article
Mathematics
Jonathan Eckhardt, Aleksey Kostenko, Gerald Teschl
JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK
(2018)
Article
Mathematics
Elena Kopylova, Gerald Teschl
Summary: We derive dispersion estimates for solutions of a one-dimensional discrete Dirac equations with a potential. In particular, we improve our previous result, weakening the conditions on the potential. Moreover, we provide new results concerning scattering for the corresponding perturbed Dirac operators, showing that the reflection and transmission coefficients belong to the Wiener algebra.
MATHEMATISCHE NACHRICHTEN
(2022)
Article
Mathematics, Applied
Iryna Egorova, Johanna Michor, Gerald Teschl
Summary: We show how the inverse scattering transform can be used as a convenient tool to derive the long-time asymptotics of the KdV shock problem in the soliton region. In particular, we improve the results previously obtained via the nonlinear steepest decent approach both with respect to the decay of the initial conditions as well as the region where they are valid.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2022)
Article
Mathematics
Iryna Karpenko, Dmitry Shepelsky, Gerald Teschl
Summary: This paper aims to develop the Riemann-Hilbert approach for the modified Camassa-Holm equation with non-zero boundary conditions. It presents detailed properties of spectral functions associated with the initial data and obtains a representation for the solution of the Cauchy problem in terms of an associated RH problem.
MONATSHEFTE FUR MATHEMATIK
(2023)
Proceedings Paper
Mathematics, Applied
Markus Holzleitner, Aleksey Kostenko, Gerald Teschl
NON-LINEAR PARTIAL DIFFERENTIAL EQUATIONS, MATHEMATICAL PHYSICS, AND STOCHASTIC ANALYSIS: THE HELGE HOLDEN ANNIVERSARY VOLME
(2018)
Article
Mathematics, Applied
Iryna Egorova, Johanna Michor, Gerald Teschl
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS
(2018)
Article
Biochemical Research Methods
Clemens Ager, Karl Unterkofler, Pawel Mochalski, Susanne Teschl, Gerald Teschl, Chris A. Mayhew, Julian King
JOURNAL OF BREATH RESEARCH
(2018)
Article
Mathematics
Jonathan Eckhardt, Fritz Gesztesy, Helge Holden, Aleksey Kostenko, Gerald Teschl
ANNALES DE L INSTITUT FOURIER
(2017)
Article
Mathematics, Applied
Isaac Alvarez-Romero, Gerald Teschl
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS
(2017)
Article
Mathematics, Applied
Iryna Egorova, Johanna Michor, Gerald Teschl
JOURNAL OF MATHEMATICAL PHYSICS ANALYSIS GEOMETRY
(2018)
