Abstract. Consider an infinite chain of masses, each connected to its nearest neighbors by a (nonlinear) spring. This is a Fermi-Pasta-Ulam-Tsingou lattice. We prove the existence of traveling waves in the setting where the masses alternate in size. In particular we address the limit where the mass ratio tends to zero. The problem is inherently singular and we find that the traveling waves are not true solitary waves but rather "nanopterons", which is to say, waves which asymptotic at spatial infinity to very small amplitude periodic waves. Moreover, we can only find solutions when the mass ratio lies in a certain open set. The difficulties in the problem all revolve around understanding Jost solutions of a nonlocal Schrödinger operator in its semi-classical limit.Arrange infinitely many particles on a horizontal line, each attached to its nearest neighbors by a spring with a nonlinear restoring force. Constrain the motion of the particles to be within the line. This system is called a Fermi-Pasta-Ulam (FPU) or (more recently [8]) a Fermi-Pasta-Ulam-Tsingou (FPUT) lattice and it is one of the paradigmatic models for nonlinear and dispersive waves. In this article, we consider the existence of traveling waves in a diatomic (or "dimer") FPUT lattice when the ratio of the masses is nearly zero. By this we mean that the masses of the particles alternate between m 1 and m 2 along the chain andThe springs are all identical materially. The force they exert, when stretched by an amount r from their equilibrium length, is Newton's second law gives the equations of motion. After nondimensionalization, these readHere j ∈ Z and r j := y j+1 − y j . When j is odd m j = 1 and when j is even, m j = µ. In the above, y j is the nondimensional displacement from equilibrium of the jth particle. See Figure 1 for a schematic.

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 generalizes to the discrete spatial setting the results obtained in [9] 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.(1.9) with z decreasing from z 0 to 0 and with Z increasing from 0 to Z ∞ . To better understand the crucial relationship between the asymptotic phaseshift Z ∞ and the additive perturbation z, we note that the super-solution residual J = u t − u xx − g(u) is given by(1.10)Close to the interface, the term g(Φ) − g(Φ + z) ∼ −g ′ (Φ)z is negative and must be dominated by the positive termŻΦ ′ . This requires thatŻ dominate z andż. On the other hand, close to the Post-interaction regime -convergence to waveContinuous Setting In this final step of the program, the large transients generated in the interaction regime must be controlled in a frame that moves along with the unobstructed wave. As discussed above, this analysis leads naturally to a large basin nonlinear stability result for planar travelling wave solutions to the unobstructed PDE (1.2) with Ω = R 2 . For presentation purposes, we will focus our discussion here on this unobstructed special case. Indeed, the inclusion of the obstacle merely adds technical complications that do not contribute to the understanding of the differences between the continuous and discrete frameworks. 7The main task is to construct a super-solution for (1.2) with Ω = R 2 of the form( 1.15) which adds transverse effects to the Ansatz (1.9) discussed earlier. As before, the function z decreases from z 0 to 0 while Z increases from 0 to Z ∞ . Both z and Z should be thought of as small terms. By contrast, the new function θ should be allowed to be arbitrarily large at t = 0, provided that it is localized in the sense θ(·, 0) ∈ L 2 and that it decays to zero as t → ∞ uniformly in y. This function controls deformations of the wave interface in the transverse direction. We note that any localized initial perturbation from the wave can be dominated by the initial condition in (1.15) by choosing z 0 positive and as small as we wish, at the cost of a larger value for θ(·, 0) L 2 . Assuming for the moment that Z ∞ scales with z 0 , this freedom implies that we can dominate the transients caused by such a perturbation by a family of super-solutions that have arbitrarily small asymptotic phase offsets Z ∞ . A similar argument with sub-solutions then establishes the convergence to the planar wave without any asymptotic phase shift.The difference in behaviour between one and two spatial dimensions is hence caused by the extra transverse direction, al...

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 perturbations. We employ recently developed techniques involving point-wise Green's 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 study the interaction of small amplitude, long-wavelength solitary waves in the Fermi-Pasta-Ulam model with general nearest-neighbour interaction potential. We establish global-in-time existence and stability of counterpropagating solitary wave solutions. These solutions are close to the linear superposition of two solitary waves for large positive and negative values of time; for intermediate values of time these solutions describe the interaction of two counter-propagating pulses. These solutions are stable with respect to perturbations in 2 and asymptotically stable with respect to perturbations which decay exponentially at spatial ±∞.

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