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) wi...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.展开更多
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 equati...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.展开更多
Difference equations or discrete systems are mathematical models of various fields such as physics, chemistry, biology, and economics and have been subjects of extensive study of both pure mathematicians and applied m...Difference equations or discrete systems are mathematical models of various fields such as physics, chemistry, biology, and economics and have been subjects of extensive study of both pure mathematicians and applied mathematicians. Through its interaction with modern integrable systems, the theory of difference equations is enriched greatly and has been undergoing a rapid development. SIDE-10, the tenth of a series of biennial conferences devoted to Symmetries and Integrability of Difference Equations and related topics, was held during 10-16 June, 2012 at Ningbo, China. It was sponsored and supported by the National Natural Science Foundation of China, Ningbo Association of Science and Technology, Ningbo University, Academy of Mathematics and Systems Science of Chinese Academy of Sciences, China University of Mining and Technology (Beijing), Tsinghua University, and Shanghai University. The conference attracted over 100 participants from more than a dozen of countries. During the conference, 44 contributed talks were arranged and the topics covered by the meeting include展开更多
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 multi-component extension of the integrable ...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 multi-component 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.展开更多
We study a two-component Novikov system,which is integrable and can be viewed as a twocomponent generalization of the Novikov equation with cubic nonlinearity.The primary goal of this paper is to understand how multi-...We study a two-component Novikov system,which is integrable and can be viewed as a twocomponent generalization of the Novikov equation with cubic nonlinearity.The 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 solitons.We 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 system.Furthermore,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 demonstrated.In 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.展开更多
In this paper,the invariant geometric flows for hypersurfaces in centro-affine geometry are explored.We first present evolution equations of the centro-affine invariants corresponding to the geometric flows.Based on t...In this paper,the invariant geometric flows for hypersurfaces in centro-affine geometry are explored.We first present evolution equations of the centro-affine invariants corresponding to the geometric flows.Based on these fundamental evolution equations,we show that the centro-affine heat flow for hypersurfaces is equivalent to a system of ordinary differential equations,which can be solved explicitly.Finally,the centro-affine invariant normal flows for hypersurfaces are investigated,and two specific flows are provided to illustrate the behaviour of the flows.展开更多
基金Project supported by the National Natural Science Foundation of China for Distinguished Young Scholars (No.10925104)the National Natural Science Foundation of China (No.11001240)+1 种基金the Doctoral Program Foundation of the Ministry of Education of China (No.20106101110008)the Zhejiang Provincial Natural Science Foundation of China (Nos.Y6090359,Y6090383)
文摘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.
基金supported by the National Natural Science Foundation of China(Nos.11631107,11471174)。
文摘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.
文摘Difference equations or discrete systems are mathematical models of various fields such as physics, chemistry, biology, and economics and have been subjects of extensive study of both pure mathematicians and applied mathematicians. Through its interaction with modern integrable systems, the theory of difference equations is enriched greatly and has been undergoing a rapid development. SIDE-10, the tenth of a series of biennial conferences devoted to Symmetries and Integrability of Difference Equations and related topics, was held during 10-16 June, 2012 at Ningbo, China. It was sponsored and supported by the National Natural Science Foundation of China, Ningbo Association of Science and Technology, Ningbo University, Academy of Mathematics and Systems Science of Chinese Academy of Sciences, China University of Mining and Technology (Beijing), Tsinghua University, and Shanghai University. The conference attracted over 100 participants from more than a dozen of countries. During the conference, 44 contributed talks were arranged and the topics covered by the meeting include
基金National Natural Science Foundation of China(Grants Nos.12271424 and 11871395)National Natural Science Foundation of China(Grants Nos.11971251,11631007 and 12111530003)。
文摘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 multi-component 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.
基金supported by National Natural Science Foundation of China(Grant Nos.11631007,11471174 and 11471259)。
文摘We study a two-component Novikov system,which is integrable and can be viewed as a twocomponent generalization of the Novikov equation with cubic nonlinearity.The 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 solitons.We 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 system.Furthermore,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 demonstrated.In 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.
基金This work was supported by National Natural Science Foundation of China(Grant Nos.11631007 and 11971251).
文摘In this paper,the invariant geometric flows for hypersurfaces in centro-affine geometry are explored.We first present evolution equations of the centro-affine invariants corresponding to the geometric flows.Based on these fundamental evolution equations,we show that the centro-affine heat flow for hypersurfaces is equivalent to a system of ordinary differential equations,which can be solved explicitly.Finally,the centro-affine invariant normal flows for hypersurfaces are investigated,and two specific flows are provided to illustrate the behaviour of the flows.