Multi-branch Ermakov Equation

The study on Ermakov system has received much attention in the last two decades. Ermakov systems was nothing but a time-dependent harmonic oscillator coupled to Pinney's equation. A central feature of Ermakov systems is their property of always possessing a first integral, commonly known as the L-R-R invariants. The L-R-R invariant of Ermakov systems plays a central role in the study of Ermakov systems. For instance, it can be used to construct a nonlinear superposition law and liberalize system.The aim of this letter is to extend the two-component Ermakov systems to multi-component Ermakov systems. It is shown that certain three-component Ermakov sys-tems and four-component Ermakov systems admit L-R-R invariants and nonlinear superposition law.In this thesis, the study of multi-component Ermakov systems well be mainly discussed.In partⅠ, the purpose of this part is to study the Lie point symmetry and their invariant of Pinney equation and two-component Ermakov systems. In partⅡ, the aim of this letter is to extend the two-component Ermakov systems to multi-component Ermakov systems. Therefore we extend the L-R-R invariant form two-component to multi-component.The L-R-R invariant of two-component Ermakov systems can be used to con-struct nonlinear superposition laws. Therefore, in partⅢ, it is interest to discuss the superposition laws for Ermakov systems.In partⅣ, Geometric integrability of multi-component Camassa-Holm and Hunter-Saxton systems. The multi-component Hunter-Saxton andμ-Camassa-Holm system are introduced. It is shown that the multi-component Camassa-Holm, Hunter- Saxton andμ-Camassa-Holm system are geometrically integrable, namely they de-scribe posedo-spherical surfaces. As a consequence, their infinite number of conser-vation laws can be directly constructed.