Periodic and homoclinic travelling waves in infinite lattices

✍ Scribed by Percy D. Makita

• Publisher: Elsevier Science
• Year: 2011
• Tongue: English
• Weight: 277 KB
• Volume: 74
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/j.na.2010.11.011

✦ Synopsis

Consider an infinite chain of particles subjected to a potential f , where nearest neighbours are connected by nonlinear oscillators. The nonlinear coupling between particles is given by a potential V . The dynamics of the system is described by the infinite system of second order differential equations

We investigate the existence of travelling wave solutions. Two kinds of such solutions are studied: periodic and homoclinic ones. On one hand, we prove under some growth conditions on f and V , the existence of non-constant periodic solutions of any given period T > 0, and speed c > c 0 , where the constant c 0 depends on f ′′ (0) and V ′′ (0). On the other hand, under very similar conditions, we establish the existence of non-trivial homoclinic solutions, of any given speed c > c 0 , emanating from the origin. Moreover, we prove that these homoclinics decay exponentially at infinity. Each homoclinic is obtained as a limit of periodic solutions when the period goes to infinity.