Investigation of bright and dark solitons in α, β-Fermi Pasta Ulam lattice

Abstract

We consider the Hamiltonian of α, β-Fermi Pasta Ulam lattice and explore the Hamilton–Jacobi formalism to obtain the discrete equation of motion. By using the continuum limit approximations and incorporating some normalized parameters, the extended Korteweg–de Vries equation is obtained, with solutions that elucidate on the Fermi Pasta Ulam paradox. We further derive the nonlinear Schrödinger amplitude equation from the extended Korteweg–de Vries equation, by exploring the reductive perturbative technique. The dispersion and nonlinear coefficients of this amplitude equation are functions of the α and β parameters, with the β parameter playing a crucial role in the modulational instability analysis of the system. For β greater than or equal to zero, no modulational instability is observed and only dark solitons are identified in the lattice. However for β less than zero, bright solitons are traced in the lattice for some large values of the wavenumber. Results of numerical simulations of both the Korteweg–de Vries and nonlinear Schrödinger amplitude equations with periodic boundary conditions clearly show that the bright solitons conserve their amplitude and shape after collisions.