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

Yin, Zhaoyang

Ill. J. Math. 47, No. 3, 649-666 (2003).

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.

Reviewer: Juan J. Morales-Ruiz (Barcelona)

MSC:

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

Keywords:

Cauchy problem; integrable evolution equations; Korteweg-de Vries equation; Camassa-Holm equation