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

The non-linear family of partial differential equations, \[ u_t+c_0u_x+\gamma u_{xxx}-\alpha^2 u_{txx}= (c_1u^2+c_2u_x^2+c_3uu_{xx})_x, \] 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 \[ u_t-u_{txx}+4uu_x= 3u_xu_{xx}+uu_{xxx}. \] The paper under review studies the Cauchy problem for the above equation.


35Q58 Other completely integrable PDE (MSC2000)
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
35G25 Initial value problems for nonlinear higher-order PDEs
35L05 Wave equation