Abstract
We present the results of an analytical and numerical study of the long-time behavior for certain Fermi–Pasta–Ulam (FPU) lattices viewed as perturbations of the completely integrable Toda lattice. Our main tools are the direct and inverse scattering transforms for doubly-infinite Jacobi matrices, which are well-known to linearize the Toda flow. We focus in particular on the evolution of the associated scattering data under the perturbed vs. the unperturbed equations. We find that the eigenvalues present initially in the scattering data converge to new, slightly perturbed eigenvalues under the perturbed dynamics of the lattice equation. To these eigenvalues correspond solitary waves that emerge from the solitons in the initial data. We also find that new eigenvalues emerge from the continuous spectrum as the lattice system is let to evolve under the perturbed dynamics.
Highlights
The purpose of this work is to numerically investigate the long time behavior of solutions to certain perturbations of the completely integrable Toda lattice
In the perturbed FPU lattice, the evolution equations of the scattering data become much more complicated, but we observe that the data itself continues to contain essential, readable information which describes each part of the solution in the long time limit
We numerically create solitary wave solutions of the FPU lattices, whose existence was proven in [9], and identify the scattering data corresponding to such solutions by using the scattering transform for the Toda lattice
Summary
The purpose of this work is to numerically investigate the long time behavior of solutions to certain perturbations of the completely integrable Toda lattice. It is well known that in these types of equations several different phenomena can appear over long times, among them blow-up, scattering to the free evolution, or the emergence of stable nonlinear structures, such as solitary waves and breather solutions. What we will call here solitary waves are special solutions to nonlinear PDEs or P∆Es which travel with constant speed and without changing their profile. Their existence reflects a certain balance between nonlinear and dispersive effects in a given evolution equation. We consider the classical problem of a 1-dimensional chain of particles with nearest neighbor interactions.
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