Describing dynamics of nonlinear axisymmetric waves in dispersive media with new equation

• A novel nonlinear differential equation of the axisymmetric wave type was obtained.
• This equation allows simultaneously modeling both centrifugal and centripetal waves.
• It allows simulating disturbances far from the center and in its vicinity as well.
• On the basis of this new equation a number of numerical experiments were carried out.
• Evolution and interaction of bell-shaped and ring-shaped disturbances were studied.

A single nonlinear partial differential equation of the wave type for an axisymmetric case is obtained by the introduction of special auxiliary function. In contrast to cylindrical Korteweg–de Vries equation, new equation describes centrifugal and centripetal waves not only far from the center, but in its vicinity as well. With the use of this equation a number of specific problems on the evolution of the free surface disturbances are numerically solved for the cases of a horizontal bottom and a drowned concave. The research also demonstrates the difference between the results of calculations on the base of the complete equation and on the basis of the linearized equation.