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

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, the fundamental as well as multi-soliton, particularly two-, three- and four-soliton, solutions are obtained for the Gardner equation. Based on these multi-soliton 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, which determine the bond length between the soliton atoms, 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 Schrödinger 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. Then, from the relative separation distance calculation we also observe that the bond length of each of the soliton molecules is not maintained during soliton molecule–molecule interaction while their structures are preserved. 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 Korteweg–de Vries and Korteweg–de Vries equations. Finally, in order to validate the existence of all the soliton structures, especially reported for the Gardner equation in the continuous limit, we simulate the corresponding FPUT chain and verified these various structures numerically and analytically as well. 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.