Zakharov-Kuznetsov equation

The Zakharov-Kuznetsov equation in 2D

u_t +(u_{xx} +u_{yy} +u^p)_x =0, \quad p = 2,3,4

governs, for example, weakly nonlinear ion-acoustic waves in a plasma comprising cold ions and hot isothermal electrons in the presence of a uniform magnetic field, appears as the amplitude equation for two-dimensional long waves on the free surface of a thin film flowing down a vertical plane with moderate values of the fluid surface tension and large viscosity.

The video shows the interaction of 2 solitons on the x-axis for 𝑝 = 2, which is elastic, as for the Korteweg-de Vries equation. The presence of radiation in the solution shows that the equation is not integrable.

The video shows the interaction of 2 solitons for 𝑝 = 2 which are not both on the x-axis, but close. The collision is again elastic.

The video shows the long time behaviour for 𝑝 = 2 of initial data given by a flattened Gaussian,

u(x,y,0) = 25\exp(-x^2-0.05y^2).

The appearance of several solitons plus radiation is a confirmation of the soliton resolution conjecture in this case.

The video shows the blow-up of initial data being 1.1 times the soliton for 𝑝 = 3.

For details and references, see 2002.07886.