arXiv subjects

Changzheng Qu

Publications and source records attributed to Changzheng Qu.

At least 19 recordsLinked to original sources

arXiv2025

Second boundary value problem for the Hessian curvature flow

We investigate the evolution of strictly convex hypersurfaces driven by the $k$-Hessian curvature flow, subject to the second boundary condition. We first explore the translating solutions corresponding to this boundary value problem. Next, we establish the long-time existence of the flow and prove that it converges to a translating solution. To overcome the difficulty of driving boundary $C^2$ estimates, we employ an orthogonal invariance technique. Using this method, we extend the results of Schnürer-Smoczyk \cite{Schnurer2003} and Schnürer \cite{Schnurer2002} from the second boundary value problem of Gauss curvature flow to $k$-Hessian curvature flow.

math.AP↗

arXiv2025

New Revival Phenomena for Bidirectional Dispersive Hyperbolic Equations

In this paper, the dispersive revival and fractalization phenomena for bidirectional dispersive equations on a bounded interval subject to periodic boundary conditions and discontinuous initial profiles are investigated. Firstly, we study the periodic initial-boundary value problem of the linear beam equation with step function initial data, and analyze the manifestation of the revival phenomenon for the corresponding solution at rational times. Next, we extend the investigation to periodic initial-boundary value problems of more general bidirectional dispersive equations. We prove that, if the initial functions are of bounded variation, the dynamical evolution of such periodic problems depend essentially upon the large wave number asymptotics of the associated dispersion relations. Integral polynomial or asymptotically integral polynomial dispersion relations produce dispersive revival/fractalization rational/irrational dichotomies, whereas those with non-polynomial growth result in fractal profiles at all times. Finally, numerical experiments, in the concrete case of the nonlinear beam equation, are used to demonstrate how such effects persist into the nonlinear regime.

math.AP↗

arXiv2025

Equi-centro-affine extremal hypersurfaces in ellipsoid

This paper explores equi-centro-affine extremal hypersurfaces in an ellipsoid. By analyzing the evolution of invariant submanifold flows under centro-affine unimodular transformations, we derive the first and second variational formulas for the associated invariant area. Stability analysis reveals that the circles with radius $r=\sqrt{6}/3$ on $\mathbb{S}^2(1)$ are characterized as being equi-centro-affine maximal. Furthermore, we provide a detailed classification of the compact isoparametric equi-centro-affine extremal hypersurfaces on $(n+1)$-dimensional sphere, as well as the generalized closed equi-centro-affine extremal curves on $2$-dimensional sphere. These curves are shown to belong to a family of transcendental curves $\mathrm{x}_{p,q}$ ($p,q$ are two coprime positive integers satisfying that $1/2<p/q<1$ ). Additionally, we establish an equi-centro-affine version of isoperimetric inequality ${}^{ec}\hspace{-1mm}L^3\leq (4π-A)(2π-A)A$ on $\mathbb{S}^2(1)$.

math.DG↗

arXiv2024

Talbot Effect for the Manakov System on the Torus

In this paper, the Talbot effect for the multi-component linear and nonlinear systems of the dispersive evolution equations on a bounded interval subject to periodic boundary conditions and discontinuous initial profiles is investigated. Firstly, for a class of two-component linear systems satisfying the dispersive quantization conditions, we discuss the fractal solutions at irrational times. Next, the investigation to nonlinear regime is extended, we prove that, for the concrete example of the Manakov system, the solutions of the corresponding periodic initial-boundary value problem subject to initial data of bounded variation are continuous but nowhere differentiable fractal-like curve with Minkowski dimension 3/2 at irrational times. Finally, numerical experiments for the periodic initial-boundary value problem of the Manakov system, are used to justify how such effects persist into the multi-component nonlinear regime. Furthermore, it is shown in the nonlinear multi-component regime that the interplay of different components may induce subtle different qualitative profile between the jump discontinuities, especially in the case that two nonlinearly coupled components start with different initial profile.

math.AP↗

arXiv2024

An eternal hypersurface flow arising in centro-affine geometry

In this paper, the existence and uniqueness for a specific centro-affine invariant hypersurface flow in $R^{n+1}$ are studied, and the corresponding evolutionary processes in both centro-affine and Euclidean settings are explored. It turns out that the flow exhibits similar properties as the standard heat flow. In addition, the long time existence of the flow is investigated, which asserts that the hypersurface governed by the flow converges asymptotically toward an ellipsoid via systematically investigating evolutions of the centro-affine invariants. Furthermore, the classification of the eternal solutions for the flow is provided.

math.DG↗

arXiv2023

Massive Thirring Model: Inverse Scattering and Soliton Resolution

In this paper the long-time dynamics of the massive Thirring model is investigated. Firstly the nonlinear steepest descent method for Riemann-Hilbert problem is explored to obtain the soliton resolution of the solutions to the massive Thirring model whose initial data belong to some weighted-Sobolev spaces. Secondly, the asymptotic stability of multi-solitons follow as a corollary. The main difficulty in studying the massive Thirring model through inverse scattering is that the corresponding Lax pair has singularities at the origin and infinity. We overcome this difficulty by making use of two transforms that separate the singularities.

math.AP↗

arXiv2023

Solutions to the SU($\mathcal{N}$) self-dual Yang-Mills equation

In this paper we aim to derive solutions for the SU($\mathcal{N}$) self-dual Yang-Mills (SDYM) equation with arbitrary $\mathcal{N}$. A set of noncommutative relations are introduced to construct a matrix equation that can be reduced to the SDYM equation. It is shown that these relations can be generated from two different Sylvester equations, which correspond to the two Cauchy matrix schemes for the (matrix) Kadomtsev-Petviashvili hierarchy and the (matrix) Ablowitz-Kaup-Newell-Segur hierarchy, respectively. In each Cauchy matrix scheme we investigate the possible reductions that can lead to the SU$(\mathcal{N})$ SDYM equation and also analyze the physical significance of some solutions, i.e. being Hermitian, positive-definite and of determinant being one.

nlin.SI↗

arXiv2023

Orbital stability of two-component peakons

We prove that the two-component peakon solutions are orbitally stable in the energy space. The system concerned here is a two-component Novikov system, which is an integrable multicomponent extension of the integrable Novikov equation. We improve the method for the scalar peakons to the two-component case with genuine nonlinear interactions by establishing optimal inequalities for the conserved quantities involving the coupled structures. Moreover, we also establish the orbital stability for the train-profiles of these two-component peakons by using the refined analysis based on monotonicity of the local energy and an induction method.

math.AP↗

arXiv2022

Reducibility of the dispersive Camassa-Holm equation with unbounded perturbations

Considered herein is the reducibility of the quasi-periodically time dependent linear dynamical system with a diophantine frequency vector $ω\in \mathcal{O}_0 \subset \mathbb{R}^ν$. This system is derived from linearizing the dispersive Camassa-Holm equation with unbounded perturbations at a small amplitude quasi-periodic function. It is shown that there is a set $\mathcal{O}_{\infty} \subset \mathcal{O}_0$ of asymptotically full Lebesgue measure such that for any $ω\in \mathcal{O}_{\infty}$, the system can be reduced to the one with constant coefficients by a quasi-periodic linear transformation. The strategy adopted in this paper consists of two steps: (a) A reduction based on the orders of the pseudo differential operators in the system which conjugates the linearized operator to a one with constant coefficients up to a small remainder; (b) A perturbative reducibility scheme which completely diagonalizes the remainder of the previous step. The main difficulties in the reducibility we need to tackle come from the operator $J=(1-\partial_{xx})^{-1}\partial_{x}$, which induces the symplectic structure of the dispersive Camassa-Holm equation.

math.AP↗

arXiv2022

Geometric aspects of Miura transformations

The Miura transformation plays a crucial role in the study of integrable systems. There have been various extensions of the Miura transformation, which have been used to relate different kinds of integrable equations and to classify the bi-Hamiltonian structures. In this paper, we are mainly concerned with the geometric aspects of the Miura transformation. The generalized Miura transformations from the mKdV-type hierarchies to the KdV-type hierarchies are constructed under both algebraic and geometric settings. It is shown that the Miura transformations not only relate integrable curve flows in different geometries but also induce the transition between different moving frames. Other geometric formulations are also investigated.

nlin.SI↗

arXiv2021

Cauchy matrix approach to the SU(2) self-dual Yang--Mills equation

The Cauchy matrix approach is developed to solve the SU(2) self-dual Yang--Mills equation. Starting from a Sylvester matrix equation coupled with certain dispersion relation for infinite coordinates, the self-dual Yang--Mills equation under Yang's formulation is constructed. By imposing further constraints on complex independent variables, a broad class of explicit solutions are obtained.

nlin.SI↗

arXiv2020

Feature Matching and Heat Flow in Centro-Affine Geometry

In this paper, we study the differential invariants and the invariant heat flow in centro-affine geometry, proving that the latter is equivalent to the inviscid Burgers' equation. Furthermore, we apply the centro-affine invariants to develop an invariant algorithm to match features of objects appearing in images. We show that the resulting algorithm compares favorably with the widely applied scale-invariant feature transform (SIFT), speeded up robust features (SURF), and affine-SIFT (ASIFT) methods.

math.DG↗

arXiv2018

Stability of peakons for the generalized modified Camassa-Holm equation

In this paper, we study orbital stability of peakons for the generalized modified Camassa-Holm (gmCH) equation, which is a natural higher-order generalization of the modified Camassa-Holm (mCH) equation, and admits Hamiltonian form and single peakons. We first show that the single peakon is the usual weak solution of the PDEs. Some sign invariant properties and conserved densities are presented. Next, by constructing the corresponding auxiliary function $h(t,\,x)$ and establishing a delicate polynomial inequality relating to the two conserved densities with the maximal value of approximate solutions, the orbital stability of single peakon of the gmCH equation is verified. We introduce a new approach to prove the key inequality, which is different from that used for the mCH equation. This extends the result on the stability of peakons for the mCH equation (Comm. Math. Phys., 322:967-997, 2013) successfully to the higher-order case, and is helpful to understand how higher-order nonlinearities affect the dispersion dynamics.

math.AP↗

arXiv2018

Liouville correspondences between multi-component integrable hierarchies

In this paper, we establish Liouville correspondences for the integrable two-component Camassa-Holm hierarchy, the two-component Novikov (Geng-Xue) hierarchy, and the two-component dual dispersive water wave hierarchy by means of the related Liouville transformations. This extends previous results on the scalar Camassa-Holm and KdV hierarchies, and the Novikov and Sawada-Kotera hierarchies to the multi-component case.

nlin.SI↗

arXiv2017

Liouville Correspondences between Integrable Hierarchies

In this paper, we study explicit correspondences between the integrable Novikov and Sawada-Kotera hierarchies, and between the Degasperis-Procesi and Kaup-Kupershmidt hierarchies. We show how a pair of Liouville transformations between the isospectral problems of the Novikov and Sawada-Kotera equations, and the isospectral problems of the Degasperis-Procesi and Kaup-Kupershmidt equations relate the corresponding hierarchies, in both positive and negative directions, as well as their associated conservation laws. Combining these results with the Miura transformation relating the Sawada-Kotera and Kaup-Kupershmidt equations, we further construct an implicit relationship which associates the Novikov and Degasperis-Procesi equations.

nlin.SI↗

arXiv2013

Blow-up solutions and peakons to a generalized $μ$-Camassa-Holm integrable equation

Consideration here is a generalized $μ$-type integrable equation, which can be regarded as a generalization to both the $μ$-Camassa-Holm and modified $μ$-Camassa-Holm equations. It is shown that the proposed equation is formally integrable with the Lax-pair and the bi-Hamiltonian structure and its scale limit is an integrable model of hydrodynamical systems describing short capillary-gravity waves. Local well-posedness of the Cauchy problem in the suitable Sobolev space is established by the viscosity method. Existence of peaked traveling-wave solutions and formation of singularities of solutions for the equation are investigated. It is found that the equation admits a single peaked soliton and multi-peakon solutions. The effects of varying $μ$-Camassa-Holm and modified $μ$-Camassa-Holm nonlocal nonlinearities on blow-up criteria and wave breaking are illustrated in detail. Our analysis relies on the method of characteristics and conserved quantities and is proceeded with a priori differential estimates.

math.AP↗

arXiv2013

On contact symmetries of evolution equations

In this paper, we develop an algebraic approach to classifying contact symmetries of the second-order nonlinear evolution equations. Up to contact isomorphisms, all inequivalent PDEs admitting semi-simple algebras, solvable algebras of dimension $n\leq4$, and algebras having nontrivial Levi factors, belonging to the class under consideration, and corresponding contact symmetries they admitted are listed.

math-ph↗

arXiv2013

Multi-Component Integrable Systems and Invariant Curve Flows in Certain Geometries

In this paper, multi-component generalizations to the Camassa-Holm equation, the modified Camassa-Holm equation with cubic nonlinearity are introduced. Geometric formulations to the dual version of the Schrödinger equation, the complex Camassa-Holm equation and the multi-component modified Camassa-Holm equation are provided. It is shown that these equations arise from non-streching invariant curve flows respectively in the three-dimensional Euclidean geometry, the two-dimensional Möbius sphere and $n$-dimensional sphere ${\mathbb S}^n(1)$. Integrability to these systems is also studied.

nlin.SI↗