A non-linear vibration of the stretch of hydrogen bonds has been studied theoretically. The authors have used the potential V(r)=Ar²-Br³, where A and B are constants, to describe approximately the H-bond energy. They have obtained analytical solutions of the motion equation for a particle moving in this potential field. The theoretical results have been applied to some realistic physics systems, including water molecules, the DNA double helix and the alpha -helix protein. It is shown that the calculated vibration frequencies are in agreement with experimental data observed by the Raman scattering and other techniques. The asymmetry of the H-bond energy is also discussed.