Generalised Korteweg-de Vries equations

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

are dispersive PDEs which have applications in nonlinear waves for 𝑝 = 1 (Korteweg-de Vries equation) and 𝑝 = 2 (modified Korteweg-de Vries equation). Both equations are completely integrable. For 𝑝 > 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 𝑝 = 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 𝜀 = 0.1.

The video above shows the solution for the initial data

u_0=sech^2(x)

for 𝜀 = 0.1.

References

See 1307.0603 for references and further details.