Generalised Korteweg-de Vries equations

u_t+u^pu_x+ \epsilon^2 u_{xxx}=0, \quad p=1,2,โฆ

are dispersive PDEs which have applications in nonlinear waves for ๐ = 1 (Korteweg-de Vries equation) and ๐ = 2 (modified Korteweg-de Vries equation). Both equations are completely integrable. For *๐* > 3, solutions to the equation can blow up in finite time which provides an interesting model to study the loss of regularity of a solution: for ๐ = 3 it is known that sufficiently rapidly decreasing initial data with a mass larger than the mass of the soliton become infinite in finite time (Martel, Merle, Raphael).

The video above shows the solution for the initial data

u_0=sech^2(x)

for ๐ = 0.1.

The video above shows the solution for the initial data

u_0=sech^2(x)

for **๐** = 0.1.

### References

See 1307.0603 for references and further details.