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Lili Fan

Publications and source records attributed to Lili Fan.

17 recordsLinked to original sources
arXiv2026

Spectral stability in the modified Camassa-Holm equation

We investigate the spectral stability of small-amplitude, periodic, traveling-wave solutions of the modified Camassa-Holm equation with cubic nonlinearities. More precisely, we analyze the $L^2(\mr)$-spectrum of the associated linearized operator in a neighborhood of the origin in the spectral plane. Inspired by a recently novel method based on Kato's perturbation theory [Berti et al, Full description of Benjamin-Feir instability of Stokes waves in deep water, \textit{Invent. Math.}, 230 (2022), 651-711.], we provide a complete description of the spectrum near the origin of the linearized operator--an integro-differential operator with periodic coefficients--and thus prove that such waves are not subject to modulational instability. Moreover, a spectral analysis reveals a remarkable threshold phenomenon: such waves with wave number $k^2\leq 3$ exhibit spectral stability, while instability emerges when $k^2>3$.

math.AP↗
arXiv2026

Modulational stability of the periodic traveling wave in a local model for shallow water waves

In this paper, we investigate the modulational stability of periodic traveling waves in a local model for shallow water waves, which is an extended version of the Hunter-Saxton equation. We construct a family of small-amplitude periodic traveling waves for this local model and provide a parameterization of these waves. Using Floquet-Bloch theory, perturbation theory, and spectral analysis, we then establish the modulational stability of these background periodic traveling wave solutions. Finally, we analyze the modulational instability of another extended Hunter-Saxton equation with cubic nonlinearities, following a similar approach.

math.AP↗
arXiv2024

Efficient optimization of plasma surface high harmonic generation by an improved Bayesian strategy

Plasma surface high-order harmonics generation (SHHG) driven by intense laser pulses on plasma targets enables a high-quality extreme ultraviolet source with high pulse energy and outstanding spatiotemporal coherence. Optimizing the performance of SHHG is important for its applications in single-shot imaging and absorption spectroscopy. In this work, we demonstrate the optimization of laser-driven SHHG by an improved Bayesian strategy in conjunction with particle-in-cell simulations. A traditional Bayesian algorithm is first employed to optimize the SHHG intensity in a two-dimensional space of parameter. Then an improved Bayesian strategy, using the Latin hypercube sampling technique and a dynamic acquisition strategy, is developed to overcome the curse of dimensionality and the risk of local optima in a high-dimensional space optimization. The improved Bayesian optimization approach is efficient and robust in three-dimensionally optimizing the harmonic ellipticity, paving the way for the upcoming SHHG experiments with a considerable repetition rate.

physics.plasm-ph↗
arXiv2024

Exact solutions for nonlinear trapped lee waves in the $\beta$-plane approximation

In this paper, we construct exact solutions that character three-dimensional, nonlinear trapped lee waves propagation superimposed on longitudinal atmospheric currents in the $\beta$-plane approximation. The solutions obtained are presented in Lagrangian coordinates, and are Gerstner-like solutions. In the process, we also derive the dispersion relation and analyze the density, pressure and the vorticity qualitatively.

math.DS↗
arXiv2024

Spectral analysis of periodic $b$-KP equation under transverse perturbation

The $b$-family-Kadomtsev-Petviashvili equation ($b$-KP) is a two dimensional generalization of the $b$-family equation. In this paper, we study the spectral stability of the one-dimensional small-amplitude periodic traveling waves with respect to two-dimensional perturbations which are either co-periodic in the direction of propagation, or nonperiodic (localized or bounded). We perform a detailed spectral analysis of the linearized problem associated to the above mentioned perturbations, and derive various stability and instability criteria which depends in a delicate way on the parameter value of $b$, the transverse dispersion parameter $\sigma$, and the wave number $k$ of the longitudinal waves.

math.AP↗
arXiv2022

Rigidity of three-dimensional internal waves with constant vorticity

This paper studies the structural implications of constant vorticity for steady three-dimensional internal water waves. It is known that in many physical regimes, water waves beneath vacuum that have constant vorticity are necessarily two dimensional. The situation is more subtle for internal waves that traveling along the interface between two immiscible fluids. When the layers have the same density, there is a large class of explicit steady waves with constant vorticity that are three-dimensional in that the velocity field and pressure depend on one horizontal variable while the interface is an arbitrary function of the other. We prove the following rigidity result: every three-dimensional traveling internal wave with bounded velocity for which the vorticities in the upper and lower layers are nonzero, constant, and parallel must belong to this family. If the densities in each layer are distinct, then in fact the flow is fully two dimensional.

math.AP↗
arXiv2022

Equatorial wind-drift currents with a continuous stratification and centripetal effects in the $\beta$-plane setting

In this paper, we aim to derive an exact solution to the three-dimensional governing equations for wind-induced equatorial flows in the $\beta$-plane approximation with a depth-dependent density distribution and subjected to centripetal terms. The obtained explicit solution represents a steady purely azimuthal stratified flow with a flat surface and an impermeable flat bed that is suitable for describing the Equatorial Current. Resorting to a functional analysis, we show that the thermocline exhibits some monotonicity properties.

physics.ao-ph↗
arXiv2022

Hamiltonian model for coupled surface and internal waves over currents and uneven bottom

A Hamiltonian model for the propagation of internal water waves interacting with surface waves, a current and an uneven bottom is examined. Using the so-called Dirichlet-Neumann operators, the water wave system is expressed in the Hamiltonian form, and thus the motions of the internal waves and surface waves are determined by the Hamiltonian formulation. Choosing an appropriate scaling of the variables and employing the Hamiltonian perturbation theory from Hamiltonian formulation of the dynamics, we derive a KdV-type equation with variable coefficients depending on the bottom topography to describe the internal waves.

physics.flu-dyn↗
arXiv2022

A cylindrical coordinates approach concerning internal waves for the Antarctic Circumpolar Current

In this paper, we devise a new exact and partially explicit solution to the governing equations of geophysical fluid dynamics for an inviscid and incompressible azimuth flow with a discontinuous density distribution and subjected to forcing terms in terms of cylindrical coordinates. The obtained solution represents a steady, purely azimuthal, stratified flow with an associated free surface and an interface that is suitable for describing the Antarctic Circumpolar Current. Resorting to a functional analysis, we demonstrate that the relationship between the imposed pressure at the free surface and the resulting surface deformation is well-defined and show that the continuity of the pressure along the interface generates an equation that describes implicitly the shape of the interface. Moreover, a particular example is considered to show that the interface can be determined explicitly. Finally, we derive an infinite regularity about the interface and obtain the expected monotonicity properties between the surface pressure and its distortion.

physics.flu-dyn↗
arXiv2020

The Cauchy problem for fractional Camassa-Holm equation in Besov space

In this paper, we consider the fractional Camassa-Holm equation modelling the propagation of small-but-finite amplitude long unidirectional waves in a nonlocally and nonlinearly elastic medium. First, we establish the local well-posedness in Besov space $B^{s_0}_{2,1}$ with $s_0=2\nu-\frac 1 2$ for $\nu>\frac 3 2 $ and $s_0=\frac 5 2$ for $1<\nu\leq \frac 3 2 $. Then, with a given analytic initial data, we establish the analyticity of the solutions in both variables, globally in space and locally in time.

math.AP↗
arXiv2018

Asymptotic stability for the inflow problem of the heat-conductive ideal gas without viscosity

This paper is devoted to studying the inflow problem for an ideal polytropic model with non-viscous gas in one-dimensional half space. We showed the existence of the boundary layer in different areas. By employing the energy method, we also proved the unique global-in-time solution existed and the asymptotic stability of both the boundary layer and the superposition with the 3-rarefaction wave under some smallness conditions.

math.AP↗
arXiv2017

Asymptotic stability of viscous contact wave and rarefaction waves for the system of heat-conductive ideal gas without viscosity

This paper is concerned with the Cauchy problem of heat-conductive ideal gas without viscosity. We show that, for the non-viscous case, if the strengths of the wave patterns and the initial perturbation are suitably small, the unique global-in-time solution exists and asymptotically tends toward the corresponding the viscous contact wave or the composition of a viscous contact wave with rarefaction waves determined by the initial condition, which extended the results by Huang-Li-Matsumura[13], where they treated the viscous and heat-conductive ideal gas.

math.AP↗
arXiv2017

An exact solution for geophysical trapped waves in the presence of an underlying current

In this paper we propose an exact and explicit nonlinear solution to the governing equations which retains all the Coriolis terms. In the presence of an underlying current, we seek the trapped waves induced by these solutions in Northern and Southern hemisphere respectively, showing that the retention of the Coriolis force in the governing equations affects significantly the range of the admissible following and adverse currents.

math.AP↗
arXiv2015

Viscous Shock Wave to an Inflow Problem for Compressible Viscous Gas with Large Density Oscillations

This paper is concerned with the inflow problem for the one-dimensional compressible Navier-Stokes equations. For such a problem, F. M. Huang, A. Matsumura and X. D. Shi showed that there exists viscous shock wave solution to the inflow problem and both the boundary layer solution, the viscous shock wave, and their superposition are time-asymptotically nonlinear stable under small initial perturbation. The main purpose of this paper is to show that similar stability results still hold for a class of large initial perturbation which can allow the initial density to have large oscillations. The proofs are given by an elementary energy method and the key point is to deduce the desired uniform positive lower and upper bounds on the density.

math.AP↗
arXiv2014

Inflow Problem for the One-dimensional Compressible Navier-Stokes Equations under Large Initial Perturbation

This paper is concerned with the inflow problem for the one-dimensional compressible Navier-Stokes equations. For such a problem, Matsumura and Nishihara showed in [A. Matsumura and K. Nishihara, Large-time behaviors of solutions to an inflow problem in the half space for a one-dimensional system of compressible viscous gas. Comm. Math. Phys. 222 (2001), 449-474] that there exists boundary layer solution to the inflow problem and both the boundary layer solution, the rarefaction wave, and the superposition of boundary layer solution and rarefaction wave are nonlinear stable under small initial perturbation. The main purpose of this paper is to show that similar stability results for the boundary layer solution and the supersonic rarefaction wave still hold for a class of large initial perturbation which can allow the initial density to have large oscillation. The proofs are given by an elementary energy method and the key point is to deduce the desired lower and upper bounds on the density function.

math.AP↗
arXiv2013

Asymptotic stability of a composite wave of two viscous shock waves for a one-dimensional system of non-viscous and heat-conductive ideal gas

This paper is concerned with the asymptotic stability of a composite wave consisting of two viscous shock waves to the Cauchy problem for a one-dimensional system of heat-conductive ideal gas without viscosity. We extend the results by Huang-Matsumura \cite{Huang-Matsumura} where they treated the equation of viscous and heat-conductive ideal gas. That is, even forthe non-viscous and heat-conductive case, we show that if the strengths of the viscous shock waves and the initial perturbation are suitably small, the unique global solution in time exists and asymptotically tends toward the corresponding composite wave whose spacial shifts of two viscous shock waves are uniquely determined by the initial perturbation.

math.AP↗