Nonlinear dynamics of damped DNA systems with long-range interactions

•A long-range version of the Peyrard-Bishop DNA model is investigated.•The model takes into account Stokes and long-range hydrodynamical damping forces.•The complex Ginzburg-Landau equation is obtained through the discrete difference operator technique.•The breather solution solitons are obtained . We investigate the nonlinear dynamics of a damped Peyrard-Bishop DNA model taking into account long-range interactions with distance dependence |l|−s on the elastic coupling constant between different DNA base pairs. Considering both Stokes and long-range hydrodynamical damping forces, we use the discrete difference operator technique and show in the short wavelength modes that the lattice equation can be governed by the complex Ginzburg-Landau equation. We found analytically that the technique leads to the correct expression for the breather soliton parameters. We found that the viscosity makes the amplitude of the breather to damp out. We compare the approximate analytic results with numerical simulations for the value s=3 (dipole-dipole interactions).