Nanopteron solutions of diatomic Fermi–Pasta–Ulam–Tsingou lattices with small mass-ratio

Abstract

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.

Nanoptera and Stokes curves in the 2-periodic Fermi–Pasta–Ulam–Tsingou equation

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Generalized solitary waves in a finite‐difference Korteweg‐de Vries equation

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The monoatomic FPU system as a limit of a diatomic FPU system

“…Numerical results on existence and non-existence of traveling solitary waves in the diatomic system (3) which are close to the traveling solitary waves of the monoatomic system (5) were reported in [13]. These numerical results inspired a number of analytical works where the authors developed the existence theory for traveling solitary waves with oscillatory tails [5,8], beyondall-order theory [16,17], and the linearized analysis of perturbations [22]. It is the purpose of this paper to give rigorous error estimates for this approximation in the context of the initial-value problem.…”