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On the Cauchy problem and peakons of a two-component Novikov system

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Science China Mathematics 2020年第10期63卷 1965-1996页

作者： changzheng qu Ying Fu School of Mathematics and Statistics Ningbo UniversityNingbo 315211China School of Mathematics Northwest UniversityXi’an 710127China

We study a two-component Novikov system,which is integrable and can be viewed as a twocomponent generalization of the Novikov equation with cubic *** primary goal of this paper is to understand how multi-component equations,nonlinear dispersive terms and other nonlinear terms affect the dispersive dynamics and the structure of the peaked *** establish the local well-posedness of the Cauchy problem in Besov spaces B^s/p,r with 1 p,r+∞,s>max{1+1/p,3/2}and Sobolev spaces H^s(R)with s>3/2,and the method is based on the estimates for transport equations and new invariant properties of the ***,the blow-up and wave-breaking phenomena of solutions to the Cauchy problem are studied.A blow-up criterion on solutions of the Cauchy problem is *** addition,we show that this system admits single-peaked solitons and multi-peaked solitons on the whole line,and the single-peaked solitons on the circle,which are the weak solutions in both senses of the usual weak form and the weak Lax-pair form of the system.

关键词： two-component Novikov system Hamiltonian structure Camassa-Holm type equation wellposedness peaked soliton

Nonlocal Symmetries of the Camassa-Holm Type Equations

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Chinese Annals of Mathematics,Series B 2020年第3期41卷 407-418页

作者： Lu ZHAO changzheng qu Center for Nonlinear Studies and Department of Mathematics Ningbo UniversityNingbo 315211ZhejiangChina

A class of nonlocal symmetries of the Camassa-Holm type equations with bi-Hamiltonian structures, including the Camassa-Holm equation, the modified Camassa-Holm equation, Novikov equation and Degasperis-Procesi equation, is studied. The nonlocal symmetries are derived by looking for the kernels of the recursion operators and their inverse operators of these equations. To find the kernels of the recursion operators, the authors adapt the known factorization results for the recursion operators of the KdV, modified KdV, Sawada-Kotera and Kaup-Kupershmidt hierarchies, and the explicit Liouville correspondences between the KdV and Camassa-Holm hierarchies, the modified KdV and modified Camassa-Holm hierarchies, the Novikov and Sawada-Kotera hierarchies, as well as the Degasperis-Procesi and Kaup-Kupershmidt hierarchies.

关键词： Nonlocal symmetry Recursion operator Camassa-Holm equation Modified Camassa-Holm equation Novikov equation Degasperis-Procesi equation Liouville correspondence

Symplectic invariants for curves and integrable systems in similarity symplectic geometry

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Science China Mathematics 2015年第7期58卷 1415-1432页

作者： LI YanYan qu changzheng College of Mathematics and Information Science Xianyang Normal University Department of Mathematics Ningbo University

In this paper, similarity symplectic geometry for curves is proposed and studied. Explicit expressions of the symplectic invariants, Frenet frame and Prenet formulae for curves in similarity symplectic geometry are obtained by using the equivariant moving frame method. The relationships between the Euclidean symplectic invariants, Frenet frame, Frenet formulae and the similarity symplectic invariants, Frenet frame, Frenet formulae for curves are established. Invariant curve flows in four-dimensional similarity symplectic geometry are also studied. It is shown that certain intrinsic invariant curve flows in four-dimensional similarity symplectic geometry are related to the integrable Burgers and matrix Burgers equations.

关键词： similarity symplectic geometry integrable system symplectic invariant moving frame method matrix Burgers equation

Invariant subspaces and conditional Lie-Bcklund symmetries of inhomogeneous nonlinear difusion equations

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Science China Mathematics 2013年第11期56卷 2187-2203页

作者： qu changzheng JI LiNa Center for Nonlinear Studies Ningbo University Department of Information and Computational Science Henan Agricultural University

The inhomogeneous nonlinear diffusion equation is studied by invariant subspace and condi- tional Lie=Bgcklund symmetry methods. It is shown that the equations admit a class of invariant subspaces governed by the nonlinear ordinary differential equations, which is equivalent to a kind of higher=order conditional Lie-B^icklund symmetries of the equations. As a consequence, a number of new solutions to the inhomogeneous nonlinear diffusion equations are constructed explicitly or reduced to solving finite-dimensional dynamical sys- tems.

关键词： inhomogeneous nonlinear diffusion equation invariant subspace conditional Lie-B/icklund sym-metry functionally generalized separable solution

Symmetry and regularity of solutions to a system with three-component integral equations

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Science China Mathematics 2012年第10期55卷 1991-2004页

作者： qu changzheng DOU JingBo Department of Mathematics Ningbo UniversityNingbo 315211China School of Statistics Xi'an University of Finance and EconomicsXi'an 710100China

Consider the system with three-component integral equations{u（x）：fRn│x - y│α-nw（y）^v（y）^qdy,v（x） =fRn│x-y│^α-nu（y）^pw（y）^rdy, w（x） =fRn│x - y│α-nv（y）qu（y）Pdy,where 0 〈 a 〈 n, n is a positive constant, p, q and r satisfy some suitable conditions. It is shown that every positive regular solution （u（x）, v（x）, w（x）） is radially symmetric and monotonic about some point by developing the moving plane method in an integral form. In addition, the regularity of the solutions is also proved by the contraction mapping principle. The conformal invariant property of the system is also investigated.

关键词： system of integral equations symmetry regularity conformal invariance

Orbital stability of two-component peakons

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Science China Mathematics 2023年第7期66卷 1395-1428页

作者： Cheng He Xiaochuan Liu changzheng qu School of Mathematics and Statistics Ningbo UniversityNingbo 315211China School of Mathematics and Statistics Xi'an Jiaotong UniversityXi'an 710049China

We prove that the two-component peakon solutions are orbitally stable in the energy *** system concerned here is a two-component Novikov system,which is an integrable multi-component extension of the integrable Novikov *** 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 ***,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.

关键词： Novikov equation two-component Novikov system peakons orbital stability conservation law Camassa-Holm equation

Symmetry and nonexistence of positive solutions to an integral system with weighted functions

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Science China Mathematics 2011年第4期54卷 753-768页

作者： DOU JingBo qu changzheng HAN YaZhou Center for Nonlinear Studies Northwest UniversityXi'an 710069China School of Statistics Xi'an University of Finance and EconomicsXi'an 710100China Department of Mathematics College of ScienceChina Jiliang UniversityHangzhou 310018China

Consider the system of integral equations with weighted functions in Rn,{u(x) =∫Rn|x-y|α-nQ(y)v(y)qdy1,v(x)=∫Rn|x-y|α-nK(y)u(y)pdy,where 0 < α < n,1/(p+1) + 1/(q+1)≥(n-α)/n1,α/(n-α) < p1q < ∞1,Q(x) and K(x) satisfy some suitable *** is shown that every positive regular solution(u(x)1,v(x)) is symmetric about some plane by developing the moving plane method in an integral ***,regularity of the solution is ***,the nonexistence of positive solutions to the system in the case 0 < p1q <(n+α)/(n-α) is also discussed.

关键词： Hardy-Littlewood-Sobolev inequality system of integral equations symmetry regularity conformally invariant property

Maximal Dimension of Invariant Subspaces to Systems of Nonlinear Evolution Equations

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Chinese Annals of Mathematics,Series B 2012年第2期33卷 161-178页

作者： Shoufeng SHEN changzheng qu Yongyang JIN Lina JI Department of Applied Mathematics Zhejiang University of TechnologyHangzhou 310023China Department of Mathematics Ningbo UniversityNingbo 315211ZhejiangChina Department of Information and Computational Science Henan Agricultural UniversityZhengzhou 450002China

In this paper, the dimension of invariant subspaces admitted by nonlinear sys- tems is estimated under certain conditions. It is shown that if the two-component nonlinear vector differential operator F = （F1, F2） with orders {k1, k2} （k1≥ k2） preserves the invariant subspace Wn1^1× Wn2^2 （n1 ≥ n2）, then n1 - n2 ≤ k2, n1 ≤2（k1 ＋ k2）＋ 1, where Wnq^q is the space generated by solutions of a linear ordinary differential equation of order nq （q = 1, 2）. Several examples including the （1＋1）-dimensional diffusion system and Ito＇s type, Drinfel＇d-Sokolov-Wilson＇s type and Whitham-Broer-Kaup＇s type equations are presented to illustrate the result. Furthermore, the estimate of dimension for m-component nonlinear systems is also given.

关键词： Invariant subspace Nonlinear PDEs Exact solution Symmetry Dynamical system

Lie group classification of the N-th-order nonlinear evolution equations

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Science China Mathematics 2011年第12期54卷 2553-2572页

作者： SHEN ShouFeng qu changzheng HUANG Qing JIN YongYang Department of Applied Mathematics Zhejiang University of Technology Hangzhou 310023 China Department of Mathematics Ningbo University Ningbo 315211 China Department of Mathematics Northwest University Xi＇an 710069 China

In this paper, Lie group classification to the N-th-order nonlinear evolution equation Ut ： UNx ＋ F（x, t, u, ux, . . . , U（N-1）x）is performed. It is shown that there are three, nine, forty-four and sixty-one inequivalent equations admitting one-, two-, three- and four-dimensionM solvable Lie algebras, respectively. We also prove that there are no semisimple Lie group 50（3） as the symmetry group of the equation, and only two realizations oral（2, R） are admitted by the equation. The resulting invariant equations contain both the well-known equations and a variety of new ones.

关键词： group classification Lie algebra nonlinear evolution equation