Breather modes of fully two dimensional triangular lattice. #

Reem Almarashi, Jonathan Wattis, Rachel Nicks

11:30 Wednesday in 4Q04.

Part of the Nonlinear dynamics and applications session.

Abstract #

 We consider a triangular lattice  of particles, each connected to their six nearest neighbours by nonlinear springs and subject to an additional onsite potential.  We use small amplitude asymptotic analysis to describe the motion of nodes within the plane of the lattice $u_{m,n}(t),v_{m,n}(t)$. Thus, we have a fully two dimensional (2D) Hamiltonian $H(u_{m,n},v_{m,n})$. We illustrate suitable form for the onsite potential $V_0(u_{m,n},v_{m,n})$.    

The system’s linear frequency spectrum is composed of two branches. An asymptotic reduction of this Klein-Gordon system using $u_{m,n} \sim \varepsilon e^{ikm+i \sqrt{3}ln} F\left( \varepsilon (m-Ut),\varepsilon (\sqrt{3}n-Vt) , \varepsilon^2 t \right), v_{m,n} \sim \varepsilon e^{ikm+i \sqrt{3}ln} P\left( \varepsilon (m-Ut),\varepsilon (\sqrt{3}n-Vt) , \varepsilon^2 t \right) $ yields a 2D Nonlinear Schrödinger (NLS) equation of the form $ i \Omega F_{T}= D_{M} |F|^2F+D_{Z}\left( F_{ZZ}+\mathcal{E} F_{\xi\xi} \right) $. Wavenumbers $(k,l)$ that satisfy the ellipticity constraint $\mathcal{E}(k,l)>0 $, and the focusing condition $D_{M}D_{Z}>0$, are necessary for localised solitary wave solutions exist.