arXiv subjects

Qin Xing

Publications and source records attributed to Qin Xing.

5 recordsLinked to original sources

arXiv2026

Instability of the Standing Pulse in Skew-Gradient Systems and Its Application to FitzHugh-Nagumo Type Systems

Classical results from Sturm-Liouville theory establish that the Morse index of a one-dimensional Sturm-Liouville operator defined on $\mathbb{R}$ is equal to the number of its associated conjugate points. Recent advancements by Beck et al.~\cite{BCJLM18} have extended these results to higher-dimensional Sturm-Liouville operators on $\mathbb{R}$, utilizing the Maslov index to characterize the spectral stability of nonlinear waves in multi-component systems. In this paper, we extend this framework further to non-self-adjoint settings by investigating skew-gradient reaction-diffusion systems. By utilizing the Maslov index and spectral flow, we derive an instability criterion for standing pulses. This approach bridges the gap between variational methods and the stability index in systems where the standard self-adjoint structure is absent. As a primary application, we apply our results to FitzHugh-Nagumo type systems, where the reaction terms for both the activator and inhibitor exhibit intrinsic nonlinearities. This provides a robust topological method to account for the influence of nonlinear inhibition on pulse stability in the non-self-adjoint regime.

math.AP↗

arXiv2025

Morse index theorem for heteroclinic, homoclinic and halfclinic orbits of Lagrangian systems

The purpose of this paper is to prove a new, more general version of the Morse index theorem for heteroclinic, homoclinic, and half-clinic solutions in general Lagrangian systems. In the final section, we compute the Morse index for specific heteroclinic and half-clinic solutions in classical mechanical models such as the mathematical pendulum, the Nagumo equation, and a four-dimensional competition-diffusion system.

math.DS↗

arXiv2025

Morse Index Theorem for Sturm-Liouville Operators on the Real Line

The classical Morse index theorem establishes a fundamental connection between the Morse index-the number of negative eigenvalues that characterize key spectral properties of linear self-adjoint differential operators-and the count of corresponding conjugate points. In this paper, we extend these foundational results to the Sturm-Liouville operator on $\mathbb{R}$. In particular, for autonomous Lagrangian systems, we employ a geometric argument to derive a lower bound for the Morse index. As concrete applications, we establish a criterion for detecting instability in traveling waves within gradient reaction-diffusion systems.

math.DS↗

arXiv2020

Index theory for traveling waves in reaction diffusion systems with skew gradient structure

A unified geometric approach for the stability analysis of traveling pulse solutions for reaction-diffusion equations with skew-gradient structure has been established in a previous paper [9], but essentially no results have been found in the case of traveling front solutions. In this work, we will bridge this gap. For such cases, a Maslov index of the traveling wave is well-defined, and we will show how it can be used to provide the spectral information of the waves. As an application, we use the same index providing the exact number of unstable eigenvalues of the traveling front solutions of FitzHugh-Nagumo equation.

math.AP↗

arXiv2019

Morse index and stability of the planar N-vortex problem

This paper concerns the investigation of the stability properties of relative equilibria which are rigidly rotating vortex configurations sometimes called vortex crystals, in the N-vortex problem. Such a configurations can be characterized as critical point of the Hamiltonian function restricted on the constant angular impulse hypersurface in the phase space (topologically a pseudo-sphere whose coefficients are the circulation strengths of the vortices). Relative equilibria are generated by the circle action on the so-called shape pseudo-sphere (which generalize the standard shape sphere appearing in the study of the N-body problem). Inspired by the planar gravitational $N$-body problem, and after a geometrical and dynamical discussion, we investigate the relation intertwining the stability of relative equilibria and the inertia indices of the central configurations generating such equilibria. In the last section we apply our main results to some symmetric three and four vortices relative equilibria.

math.DS↗