Weakly Integrable Camassa-Holm-type Equations

In this paper,series of deformed Camassa-Holm-type equations are constructed using the Lagrangian deformation and Loop algebra splittings.By drawing into a parameter ,so that when is set to 0 the 1norm becomes the 2norm.The systematic method can lead to Camassa-Holm equation which is a completely integrable dispersive shallow water equation and obtained by using Hamiltonian methods.Depend on this,some equations in classical integrable systems will deformed,such as : the CH-NLS,CH-DNLS,CH-NLS,and CH-Hirota equations,which are weakly complete integrability.