arXiv subjects

Shuguan Ji

Publications and source records attributed to Shuguan Ji.

At least 19 recordsLinked to original sources

arXiv2026

Blowup analysis of a Camassa-Holm type equation with time-varying dissipation

This paper is concerned with the local well-posedness, wave breaking, blow-up rate for a Camassa-Holm type equation with time-dependent weak dissipation. Firstly, we obtain the local well-posedness of solutions by using Kato's theory. Secondly, by using energy estimates, characteristic methods, and comparison principles, we derive two blowup criteria involving both pointwise gradient conditions and mixed amplitude-gradient conditions, and prove the blowup rate is universally $-2$. Our results extend wave breaking analysis to physically relevant variable dissipation regimes.

math.AP↗

arXiv2025

Construction of Exceptional Points in Time-Modulated High-Contrast Elastic Media

Spatially periodic elastic metamaterials, comprising hard inclusions within a soft matrix in $d$-dimensional space ($d\geq 2$), exhibit a rich spectrum of physical phenomena. This paper investigates such a model and presents the following contributions. First, we analyze the system's subwavelength quasi-frequencies under static conditions, establishing their functional dependence on the high-contrast parameter. This analysis enables the determination of the subwavelength quasi-frequency range. Second, for time-modulated structures, we derive a system of ordinary differential equations (ODEs) within the subwavelength regime. We demonstrate that this ODE system accurately captures the quasi-frequency behavior of the original elastic system. Finally, leveraging the derived ODEs and Floquet's theorem, we construct concrete examples of first-order asymptotic exceptional points (EPs) in three dimensions.

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arXiv2025

Analysis on a generalized two-component Novikov system

In this paper, we study the Cauchy problem for a generalized two-component Novikov system with weak dissipation. We first establish the local well-posedness of solutions by using the Kato's theorem. Then we give the necessary and sufficient condition for the occurrence of wave breaking in a finite time. Finally, we investigate the persistence properties of strong solutions in the weighted $L^{p}(\mathbb{R})$ spaces for a large class of moderate weights.

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arXiv2025

Averaging method for quasi-periodic response solutions

In this paper, we present an averaging method for obtaining quasi-periodic response solutions in perturbed, real analytic, quasi-periodic systems with Diophantine frequency vectors. Under the assumptions that the averaged system possesses a non-degenerate equilibrium and that the eigenvalues of its linearized matrix are pairwise distinct, we show that the original system admits a quasi-periodic response solution for parameters in a Cantorian set. The proof relies on KAM techniques. It is worth mentioning that our results do not require the equilibrium to be hyperbolic, meaning that the eigenvalues of the linearized matrix of the averaged system may be purely imaginary. Furthermore, the proposed averaging method is applicable to second-order systems, and a higher-order averaging framework is also established.

math.DS↗

arXiv2024

Melnikov's persistence for completely degenerate Hamiltonian systems

In this paper, we study the Melnikov's persistence for completely degenerate Hamiltonian systems with the following Hamiltonian \begin{equation*} H(x,y,u,v)=h(y)+g(u,v)+\varepsilon P(x,y,u,v),~~~(x,y,u,v)\in \mathbb{T}^n\times{G}\times \mathbb{R}^d\times \mathbb{R}^d, \end{equation*} where $n\geq2$ and $d\geq1$ are positive integers, $G\subset\mathbb{R}^n$, $g=o(|u|^2+|v|^2)$ admits complete degeneracy and certain transversality, and $\varepsilon P$ is the small perturbation. This is a try in studying lower-dimensional invariant tori in the normal complete degeneracy. Under Rüssmann-like non-degenerate condition and transversality condition, we apply the homotopy invariance of topological degree to remove the first order terms about $u$ and $v$ and employ the quasi-linear KAM iterative procedure to derive the persistence of lower-dimensional invariant tori.

math.DS↗

arXiv2024

Breather solutions for a radially symmetric curl-curl wave equation with double power nonlinearity

This paper is concerned with breather solutions of a radially symmetric curl-curl wave equation with double power nonlinearity. By considering the solutions with a special form, we obtain a family of ordinary differential equations (ODEs) parameterized by the radial variable. Then we characterize periodic behaviors and analyze the joint effects of the double power nonlinear terms on the minimal period and the maximal amplitude. Under certain conditions, we construct a breather solution for the original curl-curl wave equation and find such a solution which can generate a continuum of phase-shifted breathers.

math.AP↗

arXiv2023

Lower dimensional invariant tori for multi-scale Hamiltonian systems

The ``Fundamental Theorem" given by Arnold in [2] asserts the persistence of full dimensional invariant tori for 2-scale Hamiltonian systems. However, persistence in multi-scale systems is much more complicated and difficult. In this paper, we explore the persistence of lower dimensional invariant tori for multi-scale Hamiltonian systems, which play an important role in dynamics of resonant Hamiltonian systems. Moreover, using the corresponding results we give a quasi-periodic Poincaré Theorem for multi-scale Hamiltonian systems, i.e., at least $2^{m_0}$ families resonant tori survive small perturbations, where the integer $m_0$ is the multiplicity of resonance.

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arXiv2023

Positive ground state solutions for generalized quasilinear Schrödinger equations with critical growth

This paper concerns the existence of positive ground state solutions for generalized quasilinear Schrödinger equations in $\mathbb{R}^N$ with critical growth which arise from plasma physics, as well as high-power ultrashort laser in matter. By applying a variable replacement, the quasilinear problem reduces to a semilinear problem which the associated functional is well defined in the Sobolev space $H^1(\mathbb{R}^N)$. We use the method of Nehari manifold for the modified equation, establish the minimax characterization, then obtain each Palais-Smale sequence of the associated energy functional is bounded. By combining Lions's concentration-compactness lemma together with some classical arguments developed by Brézis and Nirenberg \cite{bn}, we establish that the bounded Palais-Smale sequence has a nonvanishing behaviour. Finally, we obtain the existence of a positive ground state solution under some appropriate assumptions. Our results extend and generalize some known results.

math.AP↗

arXiv2023

Global conservative weak solutions for a class of nonlinear dispersive wave equations beyond wave breaking

In this paper, we study the global conservative weak solutions for a class of nonlinear dispersive wave equations after wave breaking. We first transform the equations into an equivalent semi-linear system by introducing new variables. We then establish the global existence of solutions for the semi-linear system by using the standard theory of ordinary differential equations. Finally, returning to the original variables, we obtain the global conservative weak solutions for the original equations.

math.AP↗

arXiv2023

Blow-up and decay for a class of variable coefficient wave equation with nonlinear damping and logarithmic source

In this paper, we consider the long time behavior for the solution of a class of variable coefficient wave equation with nonlinear damping and logarithmic source. The existence and uniqueness of local weak solution can be obtained by using the Galerkin method and contraction mapping principle. However, the long time behavior of the solution is usually complicated and it depends on the balance mechanism between the damping and source terms. When the damping exponent $(p+1)$ (see assumption (H3)) is greater than the source term exponent $(q-1)$ (see equation (1.1)), namely, $p+2>q$, we obtain the global existence and accurate decay rates of the energy for the weak solutions with any initial data. Moreover, whether the weak solution exists globally or blows up in finite time, it is closely related to the initial data. In the framework of modified potential well theory, we construct the stable and unstable sets (see (2.8)) for the initial data. For the initial data belonging to the stable set, we prove that the weak solution exists globally and has similar decay rates as the previous results. For $p+2<q$ and the initial data belonging to the unstable set, we prove that the weak solution blows up in finite time for a little special damping $g(u_{t})=|u_{t}|^{p}u_{t}$.

math.AP↗

arXiv2023

Stability for Time-domain Elastic Wave Equations

This paper is concerned with the inverse scattering problem involving the time-domain elastic wave equations in a bounded $d$-dimensional domain. First, an explicit reconstruction formula for the density is established by means of the Dirichlet-to-Neumann operator. The reconstruction is mainly based on the modified boundary control method and complex geometric optics solutions for the elastic wave. Next, the stable observability is obtained by a Carleman estimate. Finally, the stability for the density is presented by the connect operator.

math.AP↗

arXiv2021

Periodic solutions of a semilinear variable coefficient wave equation under asymptotic nonresonance conditions

We consider the periodic solutions of a semilinear variable coefficient wave equation arising from the forced vibrations of a nonhomogeneous string and the propagation of seismic waves in nonisotropic media. The variable coefficient characterizes the inhomogeneity of media and its presence usually leads to the destruction of the compactness of the inverse of linear wave operator with periodic-Dirichlet boundary conditions on its range. In the pioneering work of Barbu and Pavel (1997), it gives the existence and regularity of periodic solution for Lipschitz, nonresonant and monotone nonlinearity under the assumption $η_u>0$ (see Sect. 2 for its definition) on the coefficient $u(x)$ and leaves the case $η_u=0$ as an open problem. In this paper, by developing the invariant subspace method and using the complete reduction technique and Leray-Schauder theory, we obtain the existence of periodic solutions for such a problem when the nonlinear term satisfies the asymptotic nonresonance conditions. Our result not only does not need any requirements on the coefficient except for the natural positivity assumption (i.e., $u(x)>0$), but also does not need the monotonicity assumption on the nonlinearity. In particular, when the nonlinear term is an odd function and satisfies the global nonresonance conditions, there is only one (trivial) solution to this problem on the invariant subspace.

math.AP↗

arXiv2020

Global Stability of Traveling Wave Fronts for a Population Dynamics Model with Quiescent Stage and Delay

This paper is concerned with the globally exponential stability of traveling wave fronts for a class of population dynamics model with quiescent stage and delay. First, we establish the comparison principle of solutions for the population dynamics model. Then, by the weighted energy method combining comparison principle, the globally exponential stability of traveling wave fronts of the population dynamics model under the quasi-monotonicity conditions is established.

math.DS↗

arXiv2020

Periodic solutions of a semilinear Euler-Bernoulli beam equation with variable coefficients

This paper is devoted to the study of periodic solutions for a semilinear Euler-Bernoulli beam equation with variable coefficients. Such mathematical model may be described the infinitesimal, free, undamped in-plane bending vibrations of a thin straight elastic beam. When the frequency $ω=\frac{2π}{T}$ is rational, some properties of the beam operator with variable coefficients are investigated. We obtain the existence of periodic solutions when the nonlinear term is monotone and bounded.

math.DS↗

arXiv2018

Infinitely many periodic solutions for a semilinear wave equation with $x$-dependent coefficients

This paper is devoted to the study of periodic (in time) solutions to an one-dimensional semilinear wave equation with $x$-dependent coefficients under various homogeneous boundary conditions. Such a model arises from the forced vibrations of a nonhomogeneous string and propagation of seismic waves in nonisotropic media. By combining variational methods with an approximation argument, we prove that there exist infinitely many periodic solutions whenever the period is a rational multiple of the length of the spatial interval. The proof is essentially based on the spectral properties of the wave operator with $x$-dependent coefficients.

math.DS↗

arXiv2018

Existence of multiple periodic solutions to a semilinear wave equation with $x$-dependent coefficients

This paper is concerned with the periodic (in time) solutions to an one-dimensional semilinear wave equation with $x$-dependent coefficient. Such a model arises from the forced vibrations of a nonhomogeneous string and propagation of seismic waves in nonisotropic media. By combining variational methods with saddle point reduction technique, we obtain the existence of at least three periodic solutions whenever the period is a rational multiple of the length of the spatial interval. Our method is based on a delicate analysis for the asymptotic character of the spectrum of the wave operator with $x$-dependent coefficients, and the spectral properties play an essential role in the proof.

math.DS↗

arXiv2017

Existence of multiple periodic solutions for a semilinear wave equation in an $n$-dimensional ball

This paper is devoted to the study of periodic solutions for a radially symmetric semilinear wave equation in an $n$-dimensional ball. By combining the variational methods and saddle point reduction technique, we prove there exist at least three periodic solutions for arbitrary space dimension $n$. The structure of the spectrum of the linearized problem plays an essential role in the proof, and the construction of a suitable working space is devised to overcome the restriction of space dimension.

math.DS↗

arXiv2016

Flux ratios and channel structures

We investigate Ussing's unidirectional fluxes and flux ratios of charged tracers motivated particularly by the insightful proposal of Hodgkin and Keynes on a relation between flux ratios and channel structure. Our study is based on analysis of quasi-one-dimensional Poisson-Nernst-Planck type models for ionic flows through membrane channels. This class of models includes the Poisson equation that determines the electrical potential from the charges present and is in that sense consistent. Ussing's flux ratios generally depend on all physical parameters involved in ionic flows, particularly, on bulk conditions and channel structures. Certain setups of ion channel experiments result in flux ratios that are universal in the sense that their values depend on bulk conditions but not on channel structures; other setups lead to flux ratios that are specific in the sense that their values depend on channel structures too. Universal flux ratios could serve some purposes better than specific flux ratios in some circumstances and worse in other circumstances. We focus on two treatments of tracer flux measurements that serve as estimators of important properties of ion channels. The first estimator determines the flux of the main ion species from measurements of the flux of its tracer. Our analysis suggests a better experimental design so that the flux ratio of the tracer flux and the main ion flux is universal. The second treatment of tracer fluxes concerns ratios of fluxes and experimental setups that try to determine some properties of channel structure. We analyze the two widely used experimental designs of estimating flux ratios and show that the most widely used method depends on the spatial distribution of permanent charge so this flux ratio is specific and thus allows estimation of (some of) the properties of that permanent charge, even with ideal ionic solutions. ...

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