Related papers: On Hamiltonian stable Lagrangian tori in complex h…

On a two-dimensional flat torus, the Laplacian eigenfunctions can be expressed explicitly in terms of sinusoidal functions. For a rectangular or square torus, it is known that every first eigenstate is orbitally stable up to translation…

The aim of this paper is to introduce a class of Hamiltonian autonomous systems in dimension 4 which are completely integrable and their dynamics is described in all details. They have an equilibrium point which is stable for some rare…

We show that multisoliton solutions to the Benjamin--Ono equation are uniformly orbitally stable in $H^s(\mathbb{R})$ for every $-\tfrac12<s\leq \frac12$. This improves the regularity required for stability up to the sharp well-posedness…

The problem of linear instability of a nonlinear traveling wave in a canonical Hamiltonian system with translational symmetry subject to superharmonic perturbations is discussed. It is shown that exchange of stability occurs when energy is…

We prove that a monotone Lagrangian torus in $S^2\times S^2$ which suitably sits in a symplectic fibration with two sections in its complement is Hamiltonian isotopic to the Clifford torus.

In this paper we give an upper bound of the first eigenvalue of the Laplace operator on a complete stable minimal hypersurface $M$ in the hyperbolic space which has finite $L^2$-norm of the second fundamental form on $M$. We provide some…

In lines 8-11 of \cite[pp. 2977]{Lu} we wrote: "For integer $m\ge 3$, if $M$ is $C^m$-smooth and $C^{m-1}$-smooth $L:\R\times TM\to\R$ satisfies the assumptions (L1)-(L3), then the functional ${\cal L}_\tau$ is $C^2$-smooth, bounded below,…

Isorotating ${\mathbb C}P^2$ Q-solitons in $2+1$ dimensions were studied. Hamiltonian formalism as a more physically meaningful yet fairly demanding approach was adopted during the investigation, which helped to exclude unobservable…

We derive the precise stability criterion for smooth solitary waves in the b-family of Camassa-Holm equations. The smooth solitary waves exist on the constant background. In the integrable cases b = 2 and b = 3, we show analytically that…

In this paper, we will prove a very general result of stability for perturbations of linear integrable Hamiltonian systems, and we will construct an example of instability showing that both our result and our example are optimal. Moreover,…

Gravitational stability of torsion and inflaton field in a four-dimensional spacetime de Sitter solution in scalar-tensor cosmology where Cartan torsion propagates is investigated in detail. Inflaton and torsion evolution equations are…

For a family of minimal helicoids H_a in the hyperbolic 3-space, there exists a constant a_c=2.17966 such that the following statements are true: (1) H_a is a globally stable minimal surface if 0<=a<=a_c, and (2) H_a is an unstable minimal…

We show that relativistic quantum electrodynamics in the Coulomb gauge satisfies the following bound, which establishes stability: let $H(\Lambda,V)$ denote the Hamiltonian of $QED_{1+3}$ on the three-dimensional torus of volume $V$ and…

We give a necessary and sufficient condition for strong stability of low dimensional Hamiltonian systems, in terms of the iterates of a closed orbit and the Conley-Zehnder index. Applications to Mathieu equation and stable harmonic…

We consider the classical momentum- or velocity-dependent two-dimensional Hamiltonian given by $$\mathcal H_N = p_1^2 + p_2^2 +\sum_{n=1}^N \gamma_n(q_1 p_1 + q_2 p_2)^n ,$$ where $q_i$ and $p_i$ are generic canonical variables, $\gamma_n$…

We study the linear stability of regular $n$-gon rotating equilibria in the $n$-body problem with logarithm interaction. In the presence of a central mass $M$, linear stability is insured if $M$ is bounded below and above by constants…

We prove the orbital stability of soliton solutions for 2D Maxwell--Lorentz system with extended charged particle. The solitons corresponds to the uniform motion and rotation of the particle. We reduce the corresponding Hamilton system by…

We consider Hamiltonian diffeomorphisms of the Euclidean space, generated by compactly supported time-dependent perturbations of hyperbolic quadratic forms. We prove that, under some natural assumptions, such a diffeomorphism must have…

We prove that higher-derivative and genuinely nonlocal Lagrangian systems can be Lyapunov-stable even when their Hamiltonians lack a lower bound. Explicit free and coupled Pais-Uhlenbeck oscillators, together with a genuine nonlocal model,…

We construct a family of Lagrangian tori $\Theta^n_s$ $\subset$ $(\mathbb{C}P^1)^n$, $s \in (0,1)$, where $\Theta^n_{1/2} = \Theta^n$, is the monotone twist Lagrangian torus described by Chekanov-Schlenk. We show that for $n = 2m$ and $s…