Exact particular solutions of Klein–Gordon equation and their application in fluid mechanics

The behavior of a rapid wave (the precursor) spreading in a liquid with gas bubbles has been studied. For its description, the nonlinear Klein–Gordon equation with dissipative components was modeled. Its exact partial solutions were constructed, describing the displacement of solitons (solitary waves), both at a subsonic speed (known earlier) and at a supersonic speed. Record of dissipation (viscosity) leads to solutions that describe the structures of shock waves in the examined mixture. The obtained analytic solutions correctly reflect the process of dissemination of waves observed in the experiment.