arXiv subjects

Liming Ling

Publications and source records attributed to Liming Ling.

At least 19 recordsLinked to original sources

arXiv2026

Coexistence of two distinct rogue wave patterns in the coupled nonlinear Schr\"odinger equation

This paper investigates the asymptotic behavior of high-order vector rogue wave (RW) solutions of the coupled nonlinear Schr\"odinger (CNLS) equation in the presence of multiple large internal parameters. We report several new high-order RW patterns in the CNLS system, including double-sector, double-heart, and mixed sector-heart configurations. The main novelty is that each RW pattern contains two distinct regions in which two different fundamental first-order RWs coexist simultaneously, potentially appearing as bright (eye-shaped) versus four-petaled or dark (anti-eye-shaped) forms. These two regions are respectively associated with the simple root structures of two different Adler--Moser polynomials: each region consists of well-separated first-order RWs in one-to-one correspondence with the simple roots of the associated polynomial. In addition, by tuning certain free parameters, the two regions of the RW pattern can be shifted to arbitrary locations in the $ (x,t) $-plane. This flexibility, together with the rich simple-root structures of Adler--Moser polynomials, enables the systematic generation of a much broader family of structured RW patterns in the CNLS equation.

nlin.SI↗

arXiv2025

Focusing mKdV equation: Two-phase solutions and their stability analysis

In this work, we primarily focus on the two-phase solutions and their stability to the focusing mKdV equation. By employing the algebro-geometric approach in combination with an effective integration method, we construct explicit two-phase solutions and their corresponding wave-functions expressed in terms of the Riemann theta function. The spectral stability of two-phase solutions is examined via a modified squared-eigenfunction approach, and their stability with respect to subharmonic perturbations is further analyzed under spectrally unstable conditions. In addition, the orbital stability of the two-phase solutions is investigated. To the best of our knowledge, this study provides the first rigorous stability theory for the two-phase solutions of the focusing mKdV equation.

nlin.SI↗

arXiv2025

Nonlinear stability of vector multi-solitons in coupled NLS and modified KdV equations

We prove that the $N$-solitons, including breathers and multi-hump solitons, of the coupled nonlinear Schrödinger (CNLS) equations are nonlinearly stable in the Sobolev space $H^{N}$. Moreover, $(N_{1},N_{2})$-solitons of the coupled modified Korteweg--de Vries (CmKdV) equations are shown to be nonlinearly stable in the Sobolev space $H^{2N_{1}+N_{2}}$. The number of negative eigenvalues of the second variation of the Lyapunov functional is $N$ for $N$-solitons of the CNLS equations, and $N_{1}+\lfloor (N_{2}+1)/2 \rfloor$ for $(N_{1},N_{2})$-solitons of the CmKdV equations, which is obtained by exploiting integrable properties. The stability of solitons for the classical NLS and mKdV equations also follows from the same method. In addition, we show that solutions to the linearized spectral problem of the mixed flow equation can be constructed from solutions of the stationary zero curvature equations in a large class of Lie algebras.

nlin.SI↗

arXiv2025

On the N-elliptic localized solutions to the derivative nonlinear Schrödinger equation and their asymptotic analysis

We parameterize the elliptic function solutions to the derivative nonlinear Schrödinger (DNLS) equation with four independent parameters and generate two equivalent forms of N-elliptic localized solutions to the DNLS equation through the Darboux-Bäcklund transformation. The N-elliptic localized solutions are expressed as (the derivative of) the ratios of determinants with entries in terms of Weierstrass sigma functions. Moreover, the asymptotic behaviors of both forms of N-elliptic localized solutions are analyzed along and between the propagation directions as $t \rightarrow \pm\infty$, which verify that the collisions between elliptic-solutions are elastic. We prove that the solution tends to a simple elliptic localized solution along each propagation direction. Between the propagation directions, the solution asymptotically approaches a shifted background. Furthermore, we establish sufficient conditions for strictly elastic collisions. The dynamic behaviors of the solutions are systematically investigated, with analytical results visualized through graphical illustrations. The asymptotic analysis of these solutions confirms that they exhibit the behavior predicted by the generalized soliton resolution conjecture on the elliptic function background.

math-ph↗

arXiv2025

Long-time asymptotics of the coupled nonlinear Sch\"odinger equation in a weighted Sobolev space

We study the Cauchy problem for the focusing coupled nonlinear Schr\"odinger (CNLS) equation with initial data $\mathbf{q}_0$ lying in the weighted Sobolev space and the scattering data having $n$ simple zeros. Based on the corresponding $3\times3$ matrix spectral problem, we deduce the Riemann-Hilbert problem (RHP) for CNLS equation through inverse scattering transform. We remove discrete spectrum of initial RHP using Darboux transformations. By applying the nonlinear steepest-descent method for RHP introduced by Deift and Zhou, we compute the long-time asymptotic expansion of the solution $\mathbf{q}(x,t)$ to an (optimal) residual error of order $\mathcal{O}\left(t^{-3 / 4+1/(2p)}\right)$ where $2\le p<\infty$. The leading order term in this expansion is a multi-soliton whose parameters are modulated by soliton-soliton and soliton-radiation interactions. Our work strengthens and extends the earlier work regarding long-time asymptotics for solutions of the nonlinear Schr\"odinger equation with a delta potential and even initial data by Deift and Park.

nlin.SI↗

arXiv2025

Vector rogue wave patterns of the multi-component nonlinear Schrödinger equation and generalized mixed Adler--Moser polynomials

This paper investigates the asymptotic behavior of high-order vector rogue wave (RW) solutions for any multi-component nonlinear Schrödinger equation (denoted as $n$-NLSE) with multiple internal large parameters and reports some new RW patterns, including non-multiple root (NMR)-type patterns with shapes such as $ 180 $-degree sector, jellyfish-like, and thumbtack-like shapes, as well as multiple root (MR)-type patterns characterized by right double-arrow and right arrow shapes. We establish that these RW patterns are intrinsically related to the root structures of a novel class of polynomials, termed generalized mixed Adler--Moser (GMAM) polynomials, which feature multiple arbitrary free parameters. The RW patterns can be understood as straightforward expansions and slight shifts of the root structures for the GMAM polynomials to some extent. In the $(x,t)$-plane, they asymptotically converge to a first-order RW at the position corresponding to each simple root of the polynomials and to a lower-order RW at the position associated with each multiple root. Notably, the position of the lower-order RW within these patterns can be flexibly adjusted to any desired location in the $(x,t)$-plane by tuning the free parameters of the corresponding GMAM polynomials.

nlin.SI↗

arXiv2024

Stability analysis of breathers for coupled nonlinear Schrodinger equations

We investigate the spectral stability of non-degenerate vector soliton solutions and the nonlinear stability of breather solutions for the coupled nonlinear Schrodinger (CNLS) equations. The non-degenerate vector solitons are spectrally stable despite the linearized operator admits either embedded or isolated eigenvalues of negative Krein signature. The nonlinear stability of breathers is obtained by the Lyapunov method with the help of the squared eigenfunctions due to integrability of the CNLS equations.

nlin.SI↗

arXiv2024

A modified Korteweg-de Vries equation soliton gas under the nonzero background

In this paper, we consider a soliton gas of the focusing modified Korteweg-de Vries generated from the $N$-soliton solutions under the nonzero background. The spectral soliton density is chosen on the pure imaginary axis, excluding the branch cut $Σ_{c}=\left[-i, i\right]$. In the limit $N\to\infty$, we establish the Riemann-Hilbert problem of the soliton gas. Using the Deift-Zhou nonlinear steepest-descent method, this soliton gas under the nonzero background will decay to a constant background as $x\to+\infty$, while its asymptotics as $x\to-\infty$ can be expressed with a Riemann-Theta function, attached to a Riemann surface with genus-two. We also analyze the large $t$ asymptotics over the entire spatial domain, which is divided into three distinct asymptotic regions depending on the ratio $ξ=\frac{x}{t}$. Using the similar method, we provide the leading-order asymptotic behaviors for these three regions and exhibit the dynamics of large $t$ asymptotics.

nlin.SI↗

arXiv2024

Rogue wave patterns associated with Adler--Moser polynomials featuring multiple roots in the nonlinear Schrödinger equation

In this work, we analyze the asymptotic behaviors of high-order rogue wave solutions with multiple large parameters and discover novel rogue wave patterns, including claw-like, OTR-type, TTR-type, semi-modified TTR-type, and their modified patterns. A correlation is established between these rogue wave patterns and the root structures of the Adler--Moser polynomials with multiple roots. At the positions in the $(x,t)$-plane corresponding to single roots of the Adler--Moser polynomials, these high-order rogue wave patterns asymptotically approach first-order rogue waves. At the positions in the $(x,t)$-plane corresponding to multiple roots of the Adler--Moser polynomials, these rogue wave patterns asymptotically tend toward lower-order fundamental rogue waves, dispersed first-order rogue waves, or mixed structures of these rogue waves. These structures are related to the root structures of special Adler--Moser polynomials with new free parameters, such as the Yablonskii--Vorob'ev polynomial hierarchy, among others. Notably, the positions of the fundamental lower-order rogue waves or mixed structures in these rogue wave patterns can be controlled freely under specific conditions.

nlin.SI↗

arXiv2023

Elliptic-rogue waves and modulational instability in nonlinear soliton equations

We present elliptic-rogue wave solutions for integrable nonlinear soliton equations in theta functions. Unlike solutions generated on the plane wave background, these solutions depict rogue waves emerging on elliptic function backgrounds. By refining the modified squared wave function method in tandem with the Darboux-Bäcklund transformation, we establish a quantitative correspondence between elliptic-rogue waves and the modulational instability. This connection reveals that the modulational instability of elliptic function solutions triggers rational-form solutions displaying elliptic-rogue waves, whereas the modulational stability of elliptic function solutions results in the rational-form solutions exhibiting the elliptic-solitons or elliptic-breathers. Moreover, this approach enables the derivation of higher-order elliptic-rogue waves, offering a versatile framework for constructing elliptic-rogue waves and exploring modulational stability in other integrable equations.

nlin.SI↗

arXiv2023

The Riemann-Hilbert approach for the integrable fractional Fokas--Lenells equation

In this paper, we propose a new integrable fractional Fokas--Lenells equation by using the completeness of the squared eigenfunctions, dispersion relation, and inverse scattering transform. To solve this equation, we employ the Riemann-Hilbert approach. Specifically, we focus on the case of the reflectionless potential with a simple pole for the zero boundary condition. And we provide the fractional $N$-soliton solution in determinant form. Additionally, we prove the fractional one-soliton solution rigorously. Notably, we demonstrate that as $|t|\to\infty$, the fractional $N$-soliton solution can be expressed as a linear combination of $N$ fractional single-soliton solutions.

nlin.SI↗

arXiv2023

Rogue waves and their patterns for the coupled Fokas-Lenells equations

In this work, we explore the rogue wave patterns in the coupled Fokas-Lenells equation by using the Darboux transformation. We demonstrate that when one of the internal parameters is large enough, the general high-order rogue wave solutions generated at a branch point of multiplicity three can be decomposed into some first-order outer rogue waves and a lower-order inner rogue wave. Remarkably, the positions and the orders of these outer and inner rogue waves are intimately related to Okamoto polynomial hierarchies.

nlin.SI↗

arXiv2023

Inverse scattering transform for the integrable fractional derivative nonlinear Schrödinger equation

In this paper, we explore the integrable fractional derivative nonlinear Schrödinger (fDNLS) equation by using the inverse scattering transform. Firstly, we start from the recursion operator and obtain a formal fDNLS equation. Then the inverse scattering problem is formulated and solved through the matrix Riemann-Hilbert problem. Subsequently, we give the explicit form of the fDNLS equation according to the properties of squared eigenfunctions, such as squared eigenfunctions are the eigenfunctions of the recursion operator of the integrable equations. The reflectionless potential with a simple pole for the zero boundary condition is carried out explicitly by means of determinants. Finally, for the fractional one-soliton solution, we analyze the wave propagation direction and the effect of the small fractional parameter $ε$ on the wave. The fractional one-soliton solution has been verified rigorously. In addition, we also analyze the fractional rational solution obtained by taking the limit of the fractional one-soliton solution.

nlin.SI↗

arXiv2023

Stability of elliptic function solutions for the focusing modified KdV equation

We study the spectral and orbital stability of elliptic function solutions for the focusing modified Korteweg-de Vries (mKdV) equation and construct the corresponding breather solutions to exhibit the stable or unstable dynamic behavior. The elliptic function solutions of the mKdV equation and related fundamental solutions of the Lax pair are exactly represented by theta functions. Based on the `modified squared wavefunction' (MSW) method, we construct all linear independent solutions of the linearized mKdV equation and then provide a necessary and sufficient condition of the spectral stability for elliptic function solutions with respect to subharmonic perturbations. In the case of spectrum stability, the orbital stability of elliptic function solutions is established in a suitable Hilbert space. Using Darboux-Bäcklund transformation, we construct breather solutions to exhibit unstable or stable dynamic behavior. Through analyzing the asymptotic behavior, we find that the breather solution under the cn-type solution background is equivalent to the elliptic function solution adding a small perturbation as $t\to\pm\infty$.

nlin.SI↗

arXiv2022

On the large-order asymtptics of Kuznetsov-Ma breathers

We study the large-order asymptotics for the Kuznetsov-Ma breather of the nonlinear Schrödinger equation in the far-field regime. With the aid of Darboux transformation, we first derive the corresponding Riemann-Hilbert representation for the high-order Kuznetsov-Ma breathers. Under the far-field limit, there are five asymptotical regions in the space-time plane where the breathers behave differently, the genus-two region, the algebraic-decay region, and three distinct genus-zero regions. With the aid of the Deift-Zhou nonlinear steepest decent method, we give the leading order term for each region and verify the consistency between the exact solution and the asymptotic solution numerically. Compared to the previous studies about the large-order asymptotic analysis of rogue waves and solitons, we find a novel genus-two asymptotic region, which further enriches the research of large-order dynamics.

nlin.SI↗

arXiv2022

The optical rogue wave patterns in coupled defocusing systems

We systematically investigate rogue wave's spatial-temporal pattern in $N$ $(N\geq2)$-component coupled defocusing nonlinear Schrödinger equations. The fundamental rogue wave solutions are given in a unified form for both focusing and defocusing cases. We establish the quantitative correspondence between modulation instability and rogue wave patterns, which develops the previously reported inequality relation into an equation correspondence. As an example, we demonstrate phase diagrams for rogue wave patterns in a two-component coupled system, based on the complete classification of their spatial-temporal structures. The phase diagrams enable us to predict various rogue wave patterns, such as the ones with a four-petaled structure in both components. These results are meaningful for controlling the rogue wave excitations in two orthogonal polarization optical fibers.

nlin.PS↗

arXiv2022

The multi elliptic-localized solutions and their asymptotic behaviors for the mKdV equation

We mainly construct and analyze the multi elliptic-localized solutions under the background of elliptic function solutions for the focusing modified Korteweg-de Vries (mKdV) equation. Based on the Darboux-Bäcklund transformation, we provide a uniform expression for these solutions by the Jacobi theta functions. The asymptotic behaviors of multi elliptic-localized solutions are provided directly in two categories. By the consistent asymptotic expression of those solutions, we obtain that the collisions between the elliptic-breathers/solitons are elastic. Moreover, a sufficient condition of the strictly elastic collision between the solitons and breathers has been given by the symmetric analysis. In addition, as $k\rightarrow0^{+}$, the multi elliptic-localized solutions degenerate into solitons, breathers or soliton-breather solutions, which implies that we extend the solutions from the constant and vanishing backgrounds to the periodic solutions backgrounds. Moreover, we illustrate figures of the multi elliptic-localized solutions to visualize the above analysis.

nlin.SI↗

arXiv2022

Nondegenerate solitons in the integrable fractional coupled Hirota equation

In this paper, based on the nonlinear fractional equations proposed by Ablowitz, Been, and Carr in the sense of Riesz fractional derivative, we explore the fractional coupled Hirota equation and give its explicit form. Unlike the previous nonlinear fractional equations, this type of nonlinear fractional equation is integrable. Therefore, we obtain the fractional $n$-soliton solutions of the fractional coupled Hirota equation by inverse scattering transformation in the reflectionless case. In particular, we analyze the one- and two-soliton solutions of the fractional coupled Hirota equation and prove that the fractional two-soliton can also be regarded as a linear superposition of two fractional single solitons as $|t|\to\infty$. Moreover, under some special constraint, we also obtain the nondegenerate fractional soliton solutions and give a simple analysis for them.

nlin.SI↗