Effects of nonlinear coupling parameters on the formation of intrinsic localized modes in a quantum 1D mixed Klein–Gordon/Fermi–Pasta–Ulam chain

We analyze the effects of the nonlinear coupling parameters $k_4$ and ${\alpha }_4$ on the formation of intrinsic localized modes (ILMs) in a one-dimensional quantum mixed Klein–Gordon/Fermi–Pasta–Ulam (KG/FPU) chain. Our results indicate that the frequency amplitude is more significantly affected by increasing the values of the nonlinear parameter ${\alpha }_4$ compared to the parameter $k_4$. Additionally, we demonstrate that the mixed system admits breather solutions known as ILMs, with energy exhibiting symmetrical properties. Numerical simulations conducted to support our analytical findings reveal that the $k_4$ parameter in the KG potential leads to minimal scattering of phonons, while the ${\alpha }_4$ parameter in the FPU potential results in strong phonon scattering phenomena in the mixed KG/FPU system. Furthermore, we confirm the existence of ILMs through a linear stability analysis, deriving the criteria for the appearance of modulational instability (MI). Our findings show that the shape of MI regions and the instability growth rate are significantly influenced by increases in the nonlinear parameter ${\alpha }_4$. The theoretical predictions have been validated through numerical tests using the discrete spatial Fourier method, demonstrating full agreement with our analytical results.