Modulational stability brought by cubic–quartic interactions of the nearest-neighbor in FK model subjected in a parametrized on-site potential

• Published in: Communications in nonlinear science & numerical simulation 2022-02, Vol.105, p.106088, Article 106088
• Main Authors: Mounouna, Félix Gounoko, Wamba, Etienne, Tchakoutio Nguetcho, Aurélien Serge, Bhat, Ishfaq Ahmad, Bilbault, Jean Marie
• Format: Article
• Language: English
• Subjects: Atoms & subatomic particles; Biological Physics; Chains; Direct numerical simulation; Formability; Lattice theory; Mathematical models; Modified Frenkel–Kontorova model; Modulational instability; New higher-order of interactions and nonlinearity; Nonlinear systems; Nonlinearity; Numerical analysis; Numerical simulations; Onsite; Patterns formation; Physics; Schrodinger equation; Stability; Stability analysis; Substrates
• ISSN: 1007-5704; 1878-7274
• DOI: 10.1016/j.cnsns.2021.106088

This work extends to higher-order interactions the results of Nguetcho (2017), in which we discussed only on modulational instability in one-dimensional chain made of atoms, harmonically coupled to their nearest neighbors and subjected to an external on-site potential. Here we investigate the competition between cubic–quartic nonlinearities interactions of the nearest-neighbor and substrate’s deformability, and mainly discuss its impact on the modulational instability of the system. This makes it possible to adapt the theoretical model to a real physical system such as atomic chains or DNA lattices. The governing equation, derived from the modified Frenkel–Kontorova model, is an extended nonlinear Schrödinger equation (eNLS) containing a new higher-order nonlinear term. By employing linear stability analysis, the generic properties of the MI gain spectra of the system are demonstrated. In the presence of the new quartic nonlinearity, the combinations of the system’s parameters open a large variety of gain profiles and instability domains that cannot be explored without the quartic nonlinearity. Direct numerical simulations are performed to support our analytical results, and an excellent agreement is found. •The modified Frenkel–Kontorova model with anharmonic, cubic and quartic interactions between nearest neighbor particles immersed in a parametrized on-site substrate potential is presented.•We investigate the competition between cubic–quartic nonlinearities and substrate’s deformability, and highlight its impact on the modulational instability of the system.•Various domains of gains and instabilities are provided based upon various combinations of the parameters of the system and shows that, the new quartic nonlinearity reveals the windows of instability and opens new windows of stability within the parameter domain.•We investigate the types of excitations that form within the system as a consequence of modulational instability.•Our theoretical results were validated and complemented through numerical simulations.