Extended models of non-linear waves in liquid with gas bubbles

• Non-linear waves in a liquid with gas bubbles are studied.
• Two new non-linear differential equations are derived.
• Exact solutions of the derived equations are obtained and analyzed.

In this work we generalize the models for non-linear waves in a gas–liquid mixture taking into account an interphase heat transfer, a surface tension and a weak liquid compressibility simultaneously at the derivation of the equations for non-linear waves. We also take into consideration high order terms with respect to the small parameter. Two new non-linear differential equations are derived for long weakly non-linear waves in a liquid with gas bubbles by the reductive perturbation method considering both high order terms with respect to the small parameter and the above-mentioned physical properties. One of these equations is the perturbation of the Burgers equation and corresponds to main influence of dissipation on non-linear waves propagation. The other equation is the perturbation of the Burgers–Korteweg–de Vries equation and corresponds to main influence of dispersion on non-linear waves propagation.