arXiv subjects

Jian-Wen Zhang

Publications and source records attributed to Jian-Wen Zhang.

4 recordsLinked to original sources

arXiv2026

Optical dispersive shock waves of initial pulses in optical fibers with high-order dispersion and quintic nonlinearity effects

This paper probes the dispersive shock waves (DSWs) theory in nonlinear optical systems through Whitham modulation theory for the high-order Chen-Lee-Liu (HOCLL) equation. We systematically derived the one-phase periodic solutions and the corresponding Whitham equations. For all feasible initial discontinuous conditions, we delineate a classification of wave structures during the evolutionary process by virtue of the initial distribution of Riemann invariants, including both convex and non-convex cases. In this classification, we find that the evolved wave structures in non-convex cases incorporate an additional and more complex region-contact DSW region, compared with those in convex cases. So we analyze the propagation behavior of this region. Additionally, we also take into account the problem that the initial wave at the instant of wave breaking approximated by the cubic root function, and elaborately analyze the propagation of photons in optical fibers under the effects of high-order dispersion and quintic nonlinearity.

math-ph↗

arXiv2025

Dynamics of the semi-discrete Gardner equation under two types of non-vanishing boundary conditions: heteropolar solitons and kinks

In this work, we will use inverse scattering transform to study the semi-discrete Gardner equation under two types of non-vanishing boundary conditions, and investigate two interesting nonlinear waves in the presence of discrete spectrum, namely heteropolar solitons and kinks. When $u_n\rightarrow -\frac{a}{2b}$ as $n\rightarrow \pm \infty$, this is a symmetric boundary condition, for which the heteropolar solitons, i.e., two kinds of single soliton solutions with different polarities will be obtained. If considering two sets of discrete eigenvalues, there will be two types of soliton collisions, head-on and overtaking collision, depending on the position of discrete spectrum. Interestingly, the energy gathered at the moment of collision with different polarities, producing the so-called rogue wave phenomenon with a large amplitude more than twice the background, and its generation mechanism is briefly analyzed. When $u_n\rightarrow \frac{c_{\pm}\sqrt{ a^2+4b }-a}{2b}$ as $n\rightarrow \pm \infty$, the kink, i.e., the undercompressive dispersive shock wave, will be obtained under the specific step-like boundary condition.

math-ph↗

arXiv2024

Inverse scattering transform for the discrete nonlocal PT symmetric nonlinear Schrödinger equation with nonzero boundary conditions

In this paper, the inverse scattering transform for the integrable discrete nonlocal PT symmetric nonlinear Schrödinger equation with nonzero boundary conditions is presented. According to the two different signs of symmetry reduction and two different values of the phase difference between plus and minus infinity, we discuss four cases with significant differences about analytical regions, symmetry, asymptotic behavior and the presence or absence of discrete eigenvalues, namely, the existence or absence of soliton solutions. Therefore, in all cases, we study the direct scattering and inverse scattering problem, separately. The Riemann-Hilbert problem is constructed and solved as well as the reconstruction formula of potential is derived, respectively. Finally, combining the time evolution, we provide the dark, bright, dark-bright soliton solutions on the nonzero background in difference cases under the reflectionless condition.

math-ph↗

arXiv2024

The semi-discrete complex modified Korteweg-de Vries equation with zero and non-zero boundary conditions: Riemann-Hilbert approach and N-soliton solutions

We focus on the semi-discrete complex modified Korteweg-de Vries (DcmKdV) equation in this paper. The direct and inverse scattering theory is developed with zero and non-zero boundary conditions (BCs) of the potential. For direct problem, the properties of the eigenfunctions and the scattering matrix, including analyticity, asymptotics and symmetries, are investigated, which facilitates the establishment of the Riemann-Hilbert (RH) problems. By solving the RH problems in the inverse problem part, the reconstruction potential formulas are obtained, which allows us to derive the N-soliton solutions in the reflectionless case. Meanwhile, the trace formulas are derived by means of studying the corresponding RH problems. Furthermore, the dynamic characteristics of the 1-soliton and 2-soliton with zero and non-zero boundary are demonstrated by graphical simulation.

nlin.SI↗