Painlevé-Type Asymptotics for n-Component Coupled Higher-Order Nonlinear Schrödinger Equation in Transition Region

Bao, Yixin; Zhang, Chaosheng; Chen, Wenxia

doi:10.3390/math14111976

This is an early access version, the complete PDF, HTML, and XML versions will be available soon.

Open AccessArticle

Painlevé-Type Asymptotics for n-Component Coupled Higher-Order Nonlinear Schrödinger Equation in Transition Region

by

Yixin Bao

¹,

Chaosheng Zhang

^1,2,*

and

Wenxia Chen

^1,2

¹

School of Mathematical Sciences, Jiangsu University, Zhenjiang 212013, China

²

Nonlinear Scientific Research Center, Faculty of Science, Jiangsu University, Zhenjiang 212013, China

^*

Author to whom correspondence should be addressed.

Mathematics 2026, 14(11), 1976; https://doi.org/10.3390/math14111976

Submission received: 25 April 2026 / Revised: 25 May 2026 / Accepted: 2 June 2026 / Published: 3 June 2026

(This article belongs to the Special Issue Theory and Applications of Integrable Systems in Mathematical Physics)

Download Versions Notes

Abstract

In this paper, we investigate the long-term asymptotic behavior of solutions to the Cauchy problem for the n-component coupled higher-order nonlinear Schrödinger (nC-HNLS) equation on a line with decaying initial data. Through the application of nonlinear steepest descent methods to an associated

(n + 1) \times (n + 1)

matrix Riemann–Hilbert (RH) problem, we find that, within the sector defined by

|\frac{x}{t} - \frac{1}{3 ε}| t^{\frac{2}{3}} ⩽ C

, where

C > 0

is a constant, the asymptotics can be characterized in relation to the solution to a coupled modified Painlevé II equation. This relationship is connected to a corresponding

(n + 1) \times (n + 1)

matrix RH problem.

Keywords: n-component coupled higher-order nonlinear Schrödinger equation; long-run asymptotic behavior; Painlevé II equation; nonlinear steepest descent method

Share and Cite

MDPI and ACS Style

Bao, Y.; Zhang, C.; Chen, W. Painlevé-Type Asymptotics for n-Component Coupled Higher-Order Nonlinear Schrödinger Equation in Transition Region. Mathematics 2026, 14, 1976. https://doi.org/10.3390/math14111976

AMA Style

Bao Y, Zhang C, Chen W. Painlevé-Type Asymptotics for n-Component Coupled Higher-Order Nonlinear Schrödinger Equation in Transition Region. Mathematics. 2026; 14(11):1976. https://doi.org/10.3390/math14111976

Chicago/Turabian Style

Bao, Yixin, Chaosheng Zhang, and Wenxia Chen. 2026. "Painlevé-Type Asymptotics for n-Component Coupled Higher-Order Nonlinear Schrödinger Equation in Transition Region" Mathematics 14, no. 11: 1976. https://doi.org/10.3390/math14111976

APA Style

Bao, Y., Zhang, C., & Chen, W. (2026). Painlevé-Type Asymptotics for n-Component Coupled Higher-Order Nonlinear Schrödinger Equation in Transition Region. Mathematics, 14(11), 1976. https://doi.org/10.3390/math14111976

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Menu

Painlevé-Type Asymptotics for n-Component Coupled Higher-Order Nonlinear Schrödinger Equation in Transition Region

Abstract

Share and Cite

Article Metrics

Article Access Statistics

Further Information

Guidelines

MDPI Initiatives

Follow MDPI