Nonlinear dissipative wave trains in a system of self-propelled particles

The paper addresses the existence of modulated nonlinear periodic wave trains in a system of self-propelled particles (SPPs). The reductive perturbation method reduces the model hydrodynamics equations to a one-dimensional (1D) complex Ginzburg-Landau (CGL) equation. The modulational instability (MI) phenomenon is studied, where an expression for the instability growth rate is proposed. The latter is used to discuss regions of parameters where trains of solitonic waves are likely to be obtained. This is highly influenced by the values of the variances of Gaussian noise in self-diffusion and binary collision. Solutions for the CGL equations are also studied via the Porubov technique, using a combination of Jacobi and Weierstrass elliptic functions. Wave propagation in the self-propelled particles flock includes modulated nonlinear wave trains, nonlinear spatially localized periodic patterns, and continuous waves.