An implicit difference scheme for the long-time evolution of localized solutions of a generalized Boussinesq system
For the longitudinal displacement and the transversal one in the one- dimensional model of an atomic chain the authors obtain a system of differential-difference equations. As the continuum limit of this system they get a generalized Boussinesq-like equation coupled to a linear wave equation. Under some auxiliary assumptions and appropriate boundary conditions both discrete and continuous problems are treated numerically by a fully implicit difference scheme employing Newton's quasilinearization. The scheme is stable for a wide range of time steps and initial conditions and as a seven-diagonal it turns out to be well prepared for solving. Next, a decoupled case, i.e. the sole Boussinesq-like equation is investigated numerically for various initial conditions and for different values of time increment. Two distinct classes of solutions appear then. The first one consists of oscillatory pulses whose envelopes are localized waves. The second one consists of solutions whose shapes are either heteroclinic or homoclinic. They exhibit many interesting features. After a suitable rescaling of the pure Boussinesq-like equation the authors numerically confirm the existence of self-similar (in some sense) solutions. Finally, the coupled system is carefully considered by calculations for many combinations of initial conditions and parameters describing the system. Results of the paper seem to be important for metallurgists.
scientific article; zbMATH DE number 740608