arXiv subjects

Xijun Deng

Publications and source records attributed to Xijun Deng.

4 recordsLinked to original sources

arXiv2026

Orbital Stability of Smooth Traveling Solitary Waves to the Fornberg-Whitham Equation

The Fornberg-Whitham (FW) equation was introduced by Fornberg and Whitham [Fornberg and Whitham, Phil. Trans. R. Soc. Lond. A (1978)] as a nonlocal model for unidirectional shallow water waves capable of capturing wave steepening and breaking. Despite its similarities with integrable shallow-water equations, the FW equation is not completely integrable. Nevertheless, the FW equation is part of the family of peakon-type models as it supports peaked traveling wave solutions. In this paper, we consider smooth solitary wave solutions to the FW equation. We use a variational approach to show that some are orbitally stable.

math.AP↗

arXiv2025

Stability of 2-soliton solutions for the modified Camassa-Holm equation with cubic nonlinearity

In this paper, we are concerned with the stability of 2-soliton solutions on a nonzero constant background for the modified Camassa-Holm equation with cubic nonlinearity. By employing conserved quantities in terms of the momentum variable $m$, we show that the 2-soliton, when regarded as a solution to the initial-value problem for the modified Camassa-Holm equation, is nonlinearly stable to perturbations with respect to the momentum variable in the Sobolev space $H^2$.

math.AP↗

arXiv2024

Orbital stability of smooth solitary waves for the modified Camassa-Holm equation

In this paper, we explore the orbital stability of smooth solitary wave solutions to the modified Camassa-Holm equation with cubic nonlinearity. These solutions, which exist on a nonzero constant background $k$, are unique up to translation for each permissible value of $k$ and wave speed. By leveraging the Hamiltonian nature of the modified Camassa-Holm equation and employing three conserved functionals-comprising an energy and two Casimirs, we establish orbital stability through an analysis of the Vakhitov-Kolokolov condition. This stability pertains to perturbations of the momentum variable in $H^1(\mathbb{R})$.

math.AP↗

arXiv2024

Spectral instability of peakons for the $b$-family of Novikov equations

In this paper, we are concerned with a one-parameter family of peakon equations with cubic nonlinearity parametrized by a parameter usually denoted by the letter $b$. This family is called the ``$b$-Novikov'' since it reduces to the integrable Novikov equation in the case $b=3$. By extending the corresponding linearized operator defined on functions in $H^1(\mathbb{R})$ to one defined on weaker functions on $L^2(\mathbb{R})$, we prove spectral and linear instability on $L^2(\mathbb{R})$ of peakons in the $b$-Novikov equations for any $b$. We also consider the stability on $H^1(\mathbb{R})$ and show that the peakons are spectrally or linearly stable only in the case $b=3$.

math.AP↗