arXiv subjects
Publications and source records attributed to Yuqi Ruan.
In this paper, by considering two non-isospectral problems with matrices chosen on the color Lie algebra $\mathfrak{sp}_{1}(6)$, we construct (1+1)-dimensional and (2+1)-dimensional super integrable systems on $\mathfrak{sp}_{1}(6)$. Moreover, based on the supertrace identity, their super Hamiltonian structures are also constructed.
In this paper, we investigate two non-isospectral problems on the loop algebra of the Lie superalgebra osp(1,6), and construct two super-integrable systems and their super Hamiltonian structure using the supertrace identity. The resulting super-integrable system can be reduced to the super-AKNS hierarchy under certain conditions. By reconsidering a new (2 + 1)-dimensional non-isospectral problem with spectral matrices satisfying these conditions, we obtain a (2 + 1)-dimensional generalization of the super-AKNS hierarchy.
In this paper, we consider nonisospectral problems of two distinct dimensions on the loop algebra of the symplectic Lie algebra $\mathfrak{sp}(6)$, and construct two integrable systems. Furthermore, we derive their Hamiltonian structures using the Tu scheme. Additionally, we construct an integrable hierarchy on the generalized Lie algebra $\mathfrak{Gsp}(6)$ and establish its Hamiltonian structure as well.
In this paper, by selecting appropriate spectral matrices within the loop algebra of symplectic Lie algebra sp(6), we construct two distinct classes of integrable soliton hierarchies. Then, by employing the Tu scheme and trace identity, we derive the Hamiltonian structures of the aforementioned two classes of integrable systems. From these two classes of integrable soliton hierarchies, we select one particular hierarchy and employ the Kronecker product to construct an integrable coupling system.
We consider the evolutionary Hamilton-Jacobi equation \begin{align*} w_t(x,t)+H(x,Dw(x,t),w(x,t))=0, \quad(x,t)\in M\times [0,+\infty), \end{align*} where $M$ is a compact manifold, $H:T^*M\times R\to R$, $H=H(x,p,u)$ satisfies Tonelli conditions in $p$ and the Lipschitz condition in $u$. This work mainly concerns with the Lyapunov stability (including asymptotic stability, and instability) and uniqueness of stationary viscosity solutions of the equation. A criterion for stability and a criterion for instability are given. We do not utilize auxiliary functions and thus our method is different from the classical Lyapunov's direct method. We also prove several uniqueness results for stationary viscosity solutions. The Hamiltonian $H$ has no concrete form and it may be non-monotonic in the argument $u$, where the situation is more complicated than the monotonic case. Several simple but nontrivial examples are provided, including the following equation on the unit circle \[ w_t(x,t)+\frac{1}{2}w^2_x(x,t)-a\cdot w_x(x,t)+(\sin x+b)\cdot w(x,t)=0,\quad x\in \mathbf{S}, \] where $a$, $b\in R$ are parameters. We analyze the stability, and instability of the stationary solution $w=0$ when parameters vary, and show that $w=0$ is the unique stationary solution when $a=0$, $b>1$ and $a\neq0$, $b\geqslant 1$. The sign of the integral of $\frac{\partial H}{\partial u}$ with respect to the Mather measure of the contact Hamiltonian system generated by $H$ plays an essential role in the proofs of aforementioned results. For this reason, we first develop the Mather and weak KAM theories for contact Hamiltonian systems in this non-monotonic setting. A decomposition theorem of the Ma\~n\'e set is the main result of this part.