Modulation instability, bifurcation and chaotic behaviors for a generalized (2+1)-dimensional nonlinear wave equation in a fluid or solid

In this paper, we investigate a generalized (2+1)-dimensional nonlinear wave equation characterizing nonlinear waves in a fluid or solid. We use the Painlevé analysis to test the integrability of that equation. In order to research the modulation instability (MI) of that equation, we obtain the one-soliton solutions of that equation through the Hirota bilinear method. The propagation velocity formula and characteristic line of the one soliton are derived. Then, we perform the MI to that equation through the standard linear stability. We study the distribution of the MI gain under the parameters $K$, $\Omega$ and $R$, which are the perturbation wave numbers in $x$, $t$ directions and the initial amplitude, respectively. We find that the amplitude, bandwidth and distance to the line $\Omega =0$ or $R=0$ of the MI gain vary as the $K$ or $R$ value changes. Finally, we analyze the bifurcation behavior of the system using direction field maps and investigate its chaotic behavior under the influence of a periodic external force using phase portraits.