Existence and multiplicity of wave trains in 2D lattices

CDSNS Colloquium
Monday, August 10, 2015 - 11:00
1 hour (actually 50 minutes)
Skiles 005
College of Mathematics and Econometrics, Hunan University
We study the existence and branching patterns of wave trains in a two-dimensional lattice with linear and nonlinear coupling between nearest particles and a nonlinear substrate potential. The wave train equation of the corresponding discrete nonlinear equation is formulated as an advanced-delay differential equation which is reduced by a Lyapunov-Schmidt reduction to a finite-dimensional bifurcation equation with certain symmetries and an inherited Hamiltonian structure. By means of invariant theory and singularity theory, we obtain the small amplitude solutions in the Hamiltonian system near equilibria in non-resonance and $p:q$ resonance, respectively. We show the impact of the direction $\theta$ of propagation and obtain the existence and branching patterns of wave trains in a one-dimensional lattice by investigating the existence of travelling waves of the original two-dimensional lattice in the direction $\theta$ of propagation satisfying $\tan\theta$ is rational