Volume 36 • Number 4 • 2013   • Existence of Homoclinic Travelling Waves in Infinite Lattices Zhisu Liu, Shangjiang Guo and Ziheng Zhang Abstract. By using critical point theory, we investigate the existence of homoclinic travelling waves in an one-dimensional infinite lattice with nearest-neighbor interactions and a on-site potential (density) $f$. The system is described by the infinite system of second-order differential equations: $$\ddot{q}_{j}+f'(q_{j}(t))=V'(q_{j+1}(t)-q_{j}(t))-V'(q_{j}(t)-q_{j-1}(t)), \quad t\in\mathbb{R}, \ j\in\mathbb{Z},$$ where $f,V\in C^{1}(\mathbb{R},\mathbb{R})$. We establish three new criteria ensuring the existence of non-trivial homoclinic travelling wave solutions, for any given speed $c$ bigger (or smaller) than some constant depending on $f$ and $V$. Relevant results in the literatures are extended. 2010 Mathematics Subject Classification: 37K60, 34C25, 34C37 Full text: PDF