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On the Cauchy problem for an integrable equation with peakon solutions. (English) Zbl 1061.35142

The non-linear family of partial differential equations,

u t +c 0 u x +γu xxx -α 2 u txx =(c 1 u 2 +c 2 u x 2 +c 3 uu xx ) x ,

contains the Korteweg-de Vries and the Camassa-Holm equations as particular cases. These two equations are considered “integrable”, because for some boundary conditions they can be solved using linear methods. Another differential equation in this family with similar “integrability” properties is

u t -u txx +4uu x =3u x u xx +uu xxx ·

The paper under review studies the Cauchy problem for the above equation.

MSC:
35Q58Other completely integrable PDE (MSC2000)
37K40Soliton theory, asymptotic behavior of solutions
35G25Initial value problems for nonlinear higher-order PDE
35L05Wave equation (hyperbolic PDE)