Article
Physics, Mathematical
Yan Rybalko, Dmitry Shepelsky
Summary: The study focuses on the Cauchy problem of the integrable nonlocal focusing nonlinear Schrodinger equation and the large-t behavior of the solution. The solution splits the (x, t) plane into multiple sectors, with different asymptotic behaviors of decay or approaching constants along rays. The main technical tool used is the representation of the solution through an associated matrix Riemann-Hilbert problem and subsequent asymptotic analysis using the nonlinear steepest descent method.
COMMUNICATIONS IN MATHEMATICAL PHYSICS
(2021)
Article
Mathematics, Applied
Jian Li, Tiecheng Xia
Summary: The main focus of this paper is to investigate the long-time behavior of the defocusing generalized nonlinear Schrodinger equation with a decaying initial value. The Riemann-Hilbert method and the nonlinear steepest descent method by Deift-Zhou have provided significant contributions to obtain important results. Starting from the Lax pair of the defocusing generalized nonlinear Schrodinger equation, the associated oscillatory Riemann-Hilbert problem is obtained. Then, by utilizing the stationary point, the steepest descent contours, and the trigonometric decomposition of the jump matrix, the solvable Riemann-Hilbert problem is derived from the associated oscillatory Riemann-Hilbert problem. Based on the decaying initial value in Schwartz space, the Weber equation, and the standard parabolic cylinder function, the expression for the solution of the generalized nonlinear Schrodinger equation is given.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics, Applied
Mingming Chen, Xianguo Geng, Kedong Wang, Bo Xue
Summary: In this paper, the long-time asymptotic behavior of the general coupled nonlinear Schrodinger system with initial data in Schwartz space is analyzed using the nonlinear steepest descent method. A corresponding 3 x 3 matrix Riemann-Hilbert problem is constructed using the inverse scattering method. The solution of the system can be transformed into the solution of the matrix Riemann-Hilbert problem, which is solved explicitly in terms of the parabolic cylinder functions. The leading-order asymptotics of the solution of the Cauchy problem for the system are obtained.
COMMUNICATIONS ON PURE AND APPLIED ANALYSIS
(2022)
Article
Mathematics, Applied
Sitai Li, Peter D. Miller
Summary: We study the Cauchy problem for the Maxwell-Bloch equations, analyzing the behavior of the light-matter interaction through asymptotics. We present a Riemann-Hilbert problem that generates the unique causal solution to the problem, and identify self-similar solutions related to the Painleve-III equation. Furthermore, we reveal a boundary layer phenomenon in which the solution undergoes a sudden transition over a small propagation distance.
COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS
(2023)
Article
Mathematics, Applied
Xiu-Bin Wang, Shou-Fu Tian
Summary: In this paper, we propose an inverse scattering method for an integrable higher-order NLSE with zero boundary condition. By relating an appropriate Riemann-Hilbert problem to two cases of scattering data, we obtain exact formulae for N-th order position and soliton solutions in the form of determinants. Furthermore, by selecting specific free parameters, we determine remarkable characteristics of these solutions and discuss them graphically. These results can also be applied to other types of NLSEs and contribute to further exploration of nonlinear wave phenomena.
EAST ASIAN JOURNAL ON APPLIED MATHEMATICS
(2023)
Article
Mathematics, Applied
Mingming Chen, Xianguo Geng, Kedong Wang, Bo Xue
Summary: In this paper, we analyze the long-time asymptotic behavior for the general coupled nonlinear Schrödinger system with initial data in Schwartz space via the nonlinear steepest descent method.
COMMUNICATIONS ON PURE AND APPLIED ANALYSIS
(2022)
Article
Mathematics, Applied
Xuan Zhou, Engui Fan
Summary: In this paper, we consider the Cauchy problem for a nonlocal mKdV equation with nonzero boundary conditions. By performing spectral analysis on the Lax pair, we express the solution in terms of a Riemann-Hilbert problem. Applying the partial differential over bar-steepest descent method in a fixed solitonic region, we analyze the long-time asymptotic behavior of the solution and find that it can be characterized by solitons on the discrete spectrum and a leading order term on the continuous spectrum.
PHYSICA D-NONLINEAR PHENOMENA
(2022)
Article
Mathematics, Applied
Kedong Wang, Xianguo Geng, Mingming Chen
Summary: In this paper, the Cauchy problem of the positive flow short-pulse equation is studied using the Riemann-Hilbert approach. Multi-soliton formulas under the reflectionless case and the long-time asymptotic behavior in the solitonless sector region are obtained. The soliton classification of the positive flow short-pulse equation without reflection is given, and various Deift-Zhou contour deformations and the motivation behind them are discussed. The long-time asymptotics of the positive flow short-pulse equation in the solitonless sector region are obtained using the Deift-Zhou nonlinear steepest descent method.
PHYSICA D-NONLINEAR PHENOMENA
(2022)
Article
Mathematics
Taiyang Xu, Zechuan Zhang, Engui Fan
Summary: We investigate the long time asymptotics for the solutions to the Cauchy problem of defocusing modified Kortweg-de Vries (mKdV) equation with finite density initial data. Through the Riemann-Hilbert (RH) problem associated to the Cauchy problem, the long-time asymptotics in the solitonless regions for the defocusing mKdV equation are obtained. The compatibility between the leading term of the asymptotics and the background solution is shown, and the error terms are derived via rigorous analysis.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2023)
Article
Physics, Multidisciplinary
Alfredo Deano
Summary: In this paper, the large variable asymptotic expansions of tronquee solutions of the Painleve I equation are revisited. The Riemann-Hilbert approach and the method of steepest descent are used to obtain detailed information about the exponential-type contributions beyond the standard Poincare expansions for tronquee and tritronquee solutions by explicitly constructing an extra local parametrix around the recessive stationary point of the phase function, in terms of complementary error functions.
JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL
(2023)
Article
Computer Science, Information Systems
Ali Imran Sandhu, Abdulla Desmal, Hakan Bagci
Summary: An efficient nonlinear contrast source inversion scheme is proposed for electromagnetic imaging of sparse two-dimensional investigation domains, tackling non-linearity using the NLW iterations and incorporating a self-adaptive projected accelerated steepest descent (A-PASD) algorithm for enhanced efficiency. The results demonstrate the accuracy, efficiency, and applicability of the proposed scheme.
Article
Physics, Multidisciplinary
Yongshuai Zhang, Haibing Wu, Deqin Qiu
Summary: This paper investigates a revised Riemann-Hilbert problem for a derivative nonlinear Schrödinger equation with a vanishing boundary condition, where an integral factor is introduced to satisfy the normalization condition. In the case of no reflection, formulas for Nth-order solutions of the DNLS equation are constructed, including solitons and positons corresponding to simple poles and Nth-order poles of the RHP. The expressions for Nth-order solitons are derived using the Cauchy-Binet formula, and the second-order positon is explicitly expressed, along with graphical descriptions of the evolution of third-order and fourth-order positons.
THEORETICAL AND MATHEMATICAL PHYSICS
(2023)
Article
Engineering, Mechanical
Yong Chen, Xue-Wei Yan
Summary: This study investigates the Riemann-Hilbert problem and soliton solutions to the high-order nonlinear Schrodinger equation with a matrix version through an equivalent spectral problem. By utilizing inverse scattering, a pair of Jost solutions satisfying the asymptotic conditions and the matrix spectral problem are obtained, leading to the matrix Riemann-Hilbert problem. Different soliton solutions are theoretically and graphically presented based on the two types of zero structures of det(P+) in the case of reflection-less.
NONLINEAR DYNAMICS
(2022)
Article
Physics, Mathematical
Anne Boutet de Monvel, Jonatan Lenells, Dmitry Shepelsky
Summary: This paper investigates the Cauchy problem for the focusing nonlinear Schrodinger equation with initial data approaching different plane waves. The long-time asymptotics of the solution is determined based on the value of xi = x/t. Using the Riemann-Hilbert approach, the general situation is analyzed and different asymptotic scenarios are detected, particularly in the shock case B-1 < B-2, where genus 3 sectors are observed.
COMMUNICATIONS IN MATHEMATICAL PHYSICS
(2022)
Article
Mathematics
Yan Rybalko, Dmitry Shepelsky
Summary: This study focuses on the Cauchy problem of the nonlocal nonlinear Schrodinger equation and the long-time behavior of its solution, showing qualitatively different asymptotic forms in different quarter-planes. The representation of the solution in terms of an associated matrix Riemann-Hilbert problem is the main tool used for the analysis.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
