content_image/337492.jpg content_image/337493.jpg content_image/337491.jpg
When α ′ > 0, the flow equation ( k = − 1), has two solutions for b ′ $^{2}$( λ ′ ) > 1 and b ′ $^{2}$( λ ′ ) < 1. These are given as: 

<img src='content_image/337492.jpg'>

<img src='content_image/337493.jpg'>

It can be seen that b$_{0}$ = 1 corresponds to a soliton. If we consider α ′ < 0, which essentially produces an immortal solution, we can see that it expands along the flow. The conclusions mentioned in the above proposition are demionstrated in Fig.4 

<img src='content_image/337491.jpg'>

## 5. 3rd order flow on H 2 

Proposition III.5 Inclusion of the 3 rd order terms in the flow on hyperbolic space, with any choice of initial radius ( b ′ 0 ≥ 1 ) and any choice of α ′ gives rise to an immortal solution. 

