We consider scalar lattice differential equations posed on square lattices in two space dimensions. Under certain natural conditions we show that wave-like solutions exist when obstacles (characterized by holes) are present in the lattice. Our work g
eneralizes to the discrete spatial setting the results obtained in a paper of Berestycki, Hamel and Matano for the propagation of waves around obstacles in continuous spatial domains. The analysis hinges upon the development of sub and super-solutions for a class of discrete bistable reaction-diffusion problems and on a generalization of a classical result due to Aronson and Weinberger that concerns the spreading of localized disturbances.
The Backlund Transform, first developed in the context of differential geometry, has been classically used to obtain multi-soliton states in completely integrable infinite dimensional dynamical systems. It has recently been used to study the stabilit
y of these special solutions. We offer here a dynamical perspective on the Backlund Transform, prove an abstract orbital stability theorem, and demonstrate its utility by applying it to the sine-Gordon equation and the Toda lattice.
We consider general reaction diffusion systems posed on rectangular lattices in two or more spatial dimensions. We show that travelling wave solutions to such systems that propagate in rational directions are nonlinearly stable under small perturbati
ons. We employ recently developed techniques involving point-wise Greens functions estimates for functional differential equations of mixed type (MFDEs), allowing our results to be applied even in situations where comparison principles are not available.
We prove that multi-soliton solutions of the Toda lattice are both linearly and nonlinearly stable. Our proof uses neither the inverse spectral method nor the Lax pair of the model but instead studies the linearization of the Backlund} transformation
which links the ($m-1$)-soliton solution to the $m$-soliton solution. We use this to construct a conjugation between the Toda flow linearized about an $m$-solition solution and the Toda flow linearized about the zero solution, whose stability properties can be determined by explicit calculation.
By combining results of Mizumachi on the stability of solitons for the Toda lattice with a simple rescaling and a careful control of the KdV limit we give a simple proof that small amplitude, long-wavelength solitary waves in the Fermi-Pasta-Ulam (FP
U) model are linearly stable and hence by the results of Friesecke and Pego that they are also nonlinearly, asymptotically stable.
We study traveling waves for reaction diffusion equations on the spatially discrete domain $Z^2$. The phenomenon of crystallographic pinning occurs when traveling waves become pinned in certain directions despite moving with non-zero wave speed in ne
arby directions. Mallet-Paret has shown that crystallographic pinning occurs for all rational directions, so long as the nonlinearity is close to the sawtooth. In this paper we show that crystallographic pinning holds in the horizontal and vertical directions for bistable nonlinearities which satisfy a specific computable generic condition. The proof is based on dynamical systems. In particular, it relies on an examination of the heteroclinic chains which occur as singular limits of wave profiles on the boundary of the pinning region.
We prove the existence of asymptotic two-soliton states in the Fermi-Pasta-Ulam model with general interaction potential. That is, we exhibit solutions whose difference in $ell^2$ from the linear superposition of two solitary waves goes to zero as time goes to infinity.
We study the interaction of small amplitude, long wavelength solitary waves in the Fermi-Pasta-Ulam model with general nearest-neighbor interaction potential. We establish global-in-time existence and stability of counter-propagating solitary wave so
lutions. These solutions are close to the linear superposition of two solitary waves for large positive and negative values of time; for intemediate values of time these solutions describe the interaction of two counterpropagating pulses. These solutions are stable with respect to perturbations in $ell^2$ and asymptotically stable with respect to perturbations which decay exponentially at spatial $pm infty$.}
