New findings for the old problem: Exact solutions for domain walls in coupled real Ginzburg-Landau equations

This work reports new exact solutions for domain-wall (DW) states produced by a system of coupled real Ginzburg-Landau (GL) equations which model patterns in thermal convection, optics, and Bose-Einstein condensates (BECs). An exact solution for symmetric DW was known for a single value of the cross-interaction coefficient, G=3 (defined so that its self-interaction counterpart is 1). Here an exact asymmetric DW is obtained for the system in which the diffusion term is absent in one component. It exists for all G>1. Also produced is an exact solution for DW in the symmetric real-GL system which includes linear coupling. In addition, an effect of a trapping potential on the DW is considered, which is relevant to the case of BEC. In a system of three GL equations, an exact solution is obtained for a composite state including a two-component DW and a localized state in the third component. Bifurcations which create two lowest composite states are identified too. Lastly, exact solutions are found for the system of real GL equations for counterpropagating waves, which represent a sink or source of the waves, as well as for a system of three equations which includes a standing localized component.