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Chungen Liu

Publications and source records attributed to Chungen Liu.

29 records · Page 2Linked to original sources
arXiv2011

Multiple brake orbits on compact convex symmetric reversible hypersurfaces in $\R^{2n}$

In this paper, we prove that there exist at least $[\frac{n+1}{2}]+1$ geometrically distinct brake orbits on every $C^2$ compact convex symmetric hypersurface $\Sg$ in $\R^{2n}$ for $n\ge 2$ satisfying the reversible condition $N\Sg=\Sg$ with $N=\diag (-I_n,I_n)$. As a consequence, we show that there exist at least $[\frac{n+1}{2}]+1$ geometrically distinct brake orbits in every bounded convex symmetric domain in $\R^{n}$ with $n\ge 2$ which gives a positive answer to the Seifert conjecture of 1948 in the symmetric case for $n=3$. As an application, for $n=4$ and 5, we prove that if there are exactly $n$ geometrically distinct closed characteristics on $\Sg$, then all of them are symmetric brake orbits after suitable time translation.

math.DS↗
arXiv2011

Iteration theory of $L$-index and Multiplicity of brake orbits

In this paper, we first establish the Bott-type iteration formulas and some abstract precise iteration formulas of the Maslov-type index theory associated with a Lagrangian subspace for symplectic paths. As an application, we prove that there exist at least $[\frac{n}{2}]+1$ geometrically distinct brake orbits on every $C^2$ compact convex symmetric hypersurface $Σ$ in $\mathbb{R}^{2n}$ satisfying the reversible condition $NΣ=Σ$, furthermore, if all brake orbits on this hypersurface are nondegenerate, then there are at least $n$ geometrically distinct brake orbits on it. As a consequence, we show that there exist at least $[\frac{n}{2}]+1$ geometrically distinct brake orbits in every bounded convex symmetric domain in $\mathbb{R}^{n}$, furthermore, if all brake orbits in this domain are nondegenerate, then there are at least $n$ geometrically distinct brake orbits in it. In the symmetric case, we give a positive answer to the Seifert conjecture of 1948 under a generic condition.

math.SG↗
arXiv2011

Existence and multiplicity of Homoclinic solutions for the second order Hamiltonian systems

In this paper we study the existence and multiplicity of homoclinic solutions for the second order Hamiltonian system $\ddot{u}-L(t)u(t)+W_u(t,u)=0$, $\forall t\in\mathbb{R}$, by means of the minmax arguments in the critical point theory, where $L(t)$ is unnecessary uniformly positively definite for all $t\in \mathbb{R}$ and $W_u(t, u)$ sastisfies the asymptotically linear condition.

math.DS↗
arXiv2010

Existence of Nontrivial Solutions for p-Laplacian Equations in {R}^{N}

In this paper, we consider a p-Laplacian equation in {R}^{N}with sign-changing potential and subcritical p-superlinear nonlinearity. By using the cohomological linking method for cones developed by Degiovanni and Lancelotti in 2007, an existence result is obtained. We also give a result on the existence of periodic solutions for one-dimensional $p$-Laplacian equations which can be proved by the same method.

math.AP↗
arXiv2010

Symmetrical Symplectic Capacity with Applications

In this paper, we first introduce the concept of symmetrical symplectic capacity for symmetrical symplectic manifolds, and by using this symmetrical symplectic capacity theory we prove that there exists at least one symmetric closed characteristic (brake orbit and $S$-invariant brake orbit are two examples) on prescribed symmetric energy surface which has a compact neighborhood with finite symmetrical symplectic capacity.

math.SG↗
arXiv2010

Brake subharmonic solutions of first order Hamiltonian systems

In this paper, we mainly use the Galerkin approximation method and the iteration inequalities of the $L$-Maslov type index theory to study the properties of brake subharmonic solutions for the first order non-autonomous Hamiltonian systems. We prove that when the positive integers $j$ and $k$ satisfies the certain conditions, there exists a $jT$-periodic nonconstant brake solution $z_{j}$ such that $z_{j}$ and $z_{kj}$ are distinct.

math.DS↗
arXiv2009

Minimal period estimates for brake orbits of nonlinear symmetric Hamiltonian systems

In this paper, we consider the minimal period estimates for brake orbits of nonlinear symmetric Hamiltonian systems. We prove that if the Hamiltonian function $H\in C^2(\Bbb R^{2n}, \Bbb R)$ is super-quadratic and convex, for every number $τ>0$, there exists at least one $τ$-periodic brake orbit $(τ,x)$ with minimal period $τ$ or $τ/2$ provided $H(Nx)=H(x)$.

math.DS↗
arXiv2008

Block Density Matrix Renormalization Group with Effective Interactions

Based on the contractor renormalization group (CORE) method and the density matrix renormalization group (DMRG) method, a new computational scheme, which is called the block density matrix renormalization group with effective interactions (BDMRG-EI), is proposed to deal with the numerical computation of quantum correlated systems. Different from the convential CORE method in the ways of calculating the blocks and the fragments, where the DMRG method instead of the exact diagonalization is employed in BDMRG-EI, DMRG-EI makes the calculations of larger blocks and fragments applicable. Integrating DMRG's advantage of high accuracy and CORE's advantage of low computational costs, BDMRG-EI can be widely used for the theoretical calculations of the ground state and low-lying excited states of large systems with simple or complicated connectivity. Test calculations on a 240 site one-dimensional chain and a double-layer polyacene oligomer containing 48 hexagons demonstrate the efficiency and potentiality of the method.

cond-mat.str-el↗