Asymptotic solutions of the problem on standing capillary-gravity waves in a fluid at near-critical modes

We construct asymptotic solutions of the nonlinear 2D problem on standing capillary-gravity waves at the interface between two fluids of different densities in the form of expansions in powers of the wave amplitude, which is assumed to be small. The solutions of the problem and nonlinear corrections to the wave frequency are obtained in Lagrange variables up to fourth-order infinitesimals. The solutions are studied in the problem parameter range where the second-order frequency corrections are zero (the "critical mode") or nearly zero ("near-critical modes"). We show that the amplitude-frequency curves in near-critical modes have two branches. A detailed analysis of the problem on waves on the free surface of a homogeneous fluid of finite depth as well as of the problem on internal waves on the interface between two semi-infinite fluid layers of different densities is carried out. The amplitude-frequency curves and wave profiles are presented graphically. The results are compared with laboratory experiments for gravity waves