Soliton molecules in Fermi-Pasta-Ulam-Tsingou lattice: Gardner equation approach

TL;DR

This paper investigates soliton molecules in the Fermi-Pasta-Ulam-Tsingou lattice by deriving the Gardner equation, obtaining multi-soliton solutions, classifying soliton molecules, analyzing their stability, and verifying results through numerical simulations.

Contribution

It introduces a Gardner equation approach to describe and classify soliton molecules in the FPUT lattice, including stability analysis and numerical verification.

Findings

Identification of table-top soliton molecules with no oscillations in coalescence

Classification into dissociated and synthetic types based on parameters

Numerical confirmation of soliton structures in the FPUT chain

Abstract

We revisit the Fermi-Pasta-Ulam-Tsingou lattice (FPUT) with quadratic and cubic nonlinear interactions in the continuous limit by deducing the Gardner equation. Through the Hirota bilinear method, multi-soliton solutions are obtained for the Gardner equation. Based on these solutions, we show the excitation of an interesting class of table-top soliton molecules in the FPUT lattice through the velocity resonance mechanism. Depending on the condition on the free parameters, we classify them as dissociated and synthetic type molecules. The main feature of the table-top soliton molecules is that they do not exhibit oscillations in the coalescence region. This property ensures that they are distinct from the soliton molecules, having retrieval force, of the nonlinear Schr\"odinger family of systems. Further, to study the stability of the soliton molecule we allow it to interact with a single (or multi) soliton(s). The asymptotic analysis shows that their structures remain constant, though the bond length varies, throughout the collision process. In addition, we consider the FPUT lattice with quadratic nonlinear interaction and FPUT lattice with cubic nonlinearity as sub-cases and point out the nature of the soliton molecules for these cases also systematically. We achieve this based on the interconnections between the solutions of the Gardner, modified K-dV and K-dV equations. Finally, we simulate the FPUT chain corresponding to the Gardner equation numerically and verify the existence of all the soliton structures associated with it. We believe that the present study can be extended to other integrable and non-integrable systems with applications in fluid dynamics, Bose-Einstein condensates, nonlinear optics, and plasma physics.