Statistical solution and Liouville type theorem for coupled Schrödinger-Boussinesq equations on infinite lattices

<p style='text-indent:20px;'>In this article, we are concerned with statistical solutions for the nonautonomous coupled Schrödinger-Boussinesq equations on infinite lattices. Firstly, we verify the existence of a pullback-<inline-formula><tex-math id="M1">\begin{document}$ {\mathcal{D}} $\end{document}</tex-math></inline-formula> attractor and establish the existence of a unique family of invariant Borel probability measures carried by the pullback-<inline-formula><tex-math id="M2">\begin{document}$ {\mathcal{D}} $\end{document}</tex-math></inline-formula> attractor for this lattice system. Then, it will be shown that the family of invariant Borel probability measures is a statistical solution and satisfies a Liouville type theorem. Finally, we illustrate that the invariant property of the statistical solution is indeed a particular case of the Liouville type theorem.</p>

Inhomogeneous Folding Modes in Infinite Lattices of Rigid Triangulated Miura-ori

Infinite two-dimensional tessellations of triangulated Miura-ori with rigid panels are known to exhibit only homogeneous modes of folding, thereby limiting their usefulness in engineering applications. In this work, we show that the corresponding one-dimensional lattices are less restricted and can exhibit inhomogeneous folding modes of deformation. We demonstrate this by looking at the modes in the null space of Bloch-reduced compatibility matrix in a nodal-displacement-based formulation, that is typically employed in the context of origami structural analysis. We compute the deformation modes that vary non-uniformly across the lattice depending on their wavelength, and identify the minimal number of modes that can represent such deformations. We then present a more efficient formulation based on folding-angles to study the deformation modes of infinite one-dimensional rigid triangulated origami lattices. We derive the degrees of freedom of the tessellations in terms of the minimal number of folding-angles that are required to capture the periodic inhomogeneous deformations of the infinite lattices. Within this formulation, we provide the framework to analytically derive the stiffness matrix of the lattice. Finally, we verify the new formulation by comparing the results with the bar-and-hinge model that is based on nodal-displacements. The observations from our work could have implications for the use of rigid panel origami lattices as acoustic metamaterials.

Existence and approximation of attractors for nonlinear coupled lattice wave equations

<p style='text-indent:20px;'>This paper is concerned with the asymptotic behavior of solutions to a class of nonlinear coupled discrete wave equations defined on the whole integer set. We first establish the well-posedness of the systems in <inline-formula><tex-math id="M1">\begin{document}$ E: = \ell^2\times\ell^2\times\ell^2\times\ell^2 $\end{document}</tex-math></inline-formula>. We then prove that the solution semigroup has a unique global attractor in <inline-formula><tex-math id="M2">\begin{document}$ E $\end{document}</tex-math></inline-formula>. We finally prove that this attractor can be approximated in terms of upper semicontinuity of <inline-formula><tex-math id="M3">\begin{document}$ E $\end{document}</tex-math></inline-formula> by a finite-dimensional global attractor of a <inline-formula><tex-math id="M4">\begin{document}$ 2(2n+1) $\end{document}</tex-math></inline-formula>-dimensional truncation system as <inline-formula><tex-math id="M5">\begin{document}$ n $\end{document}</tex-math></inline-formula> goes to infinity. The idea of uniform tail-estimates developed by Wang (Phys. D, 128 (1999) 41-52) is employed to prove the asymptotic compactness of the solution semigroups in order to overcome the lack of compactness in infinite lattices.</p>

Ground states for infinite lattices with nearest neighbor interaction

Sun and Ma (J. Differ. Equ. 255:2534–2563, 2013) proved the existence of a nonzero T-periodic solution for a class of one-dimensional lattice dynamical systems, $$\begin{aligned} \ddot{q_{i}}=\varPhi _{i-1}'(q_{i-1}-q_{i})- \varPhi _{i}'(q_{i}-q_{i+1}),\quad i\in \mathbb{Z}, \end{aligned}$$ q i ¨ = Φ i − 1 ′ ( q i − 1 − q i ) − Φ i ′ ( q i − q i + 1 ) , i ∈ Z , where $q_{i}$ q i denotes the co-ordinate of the ith particle and $\varPhi _{i}$ Φ i denotes the potential of the interaction between the ith and the $(i+1)$ ( i + 1 ) th particle. We extend their results to the case of the least energy of nonzero T-periodic solution under general conditions. Of particular interest is a new and quite general approach. To the best of our knowledge, there is no result for the ground states for one-dimensional lattice dynamical systems.

“Eggs in egg cartons”: co-crystallization to embed molecular cages into crystalline lattices

Discrete coordination cages were connected into the infinite lattices via shape-complementary co-crystallization with networked coordination hosts in the “eggs-in-an-egg-carton” styles.

Multi-bump solutions of -Δn = K(x)u (n+2)/(n-2) on lattices in ℝ n

We consider the following semilinear elliptic equation with critical exponent: Δ u = K(x) u^{(n+2)/(n-2)} , u > 0 in \mathbb{R}^{n} , where {n\geq 3} , {K>0} is periodic in ( x_{1} ,…, x_{k} ) with 1 \leq k < (n-2)/2. Under some natural conditions on K near a critical point, we prove the existence of multi-bump solutions where the centers of bumps can be placed in some lattices in {\mathbb{R}^{k}} , including infinite lattices. We also show that for k \geq (n-2)/2, no such solutions exist.