content_image/348651.jpg content_image/348662.jpg content_image/348661.jpg
<img src='content_image/348651.jpg'>

## 3. 4th order flow on S 2 

Proposition III.3 Inclusion of the 4 th order terms in the flow on a canonical two-sphere, we observe the following behavior: 

• α ′ > 0 and any choice of a ′ 0 gives rise to ancient solution. 

• α ′ < 0 , a ′ 0 = 1 . 3048 : soliton. For a ′ 0 ̸ = 1 . 3048 we find, a ′ ∞ =const 

• α ′ < 0; a ′ 0 < 1 . 3048 : immortal solution and . a ′ 0 > 1 . 3048 : eternal solution. 

The solution of the flow equation with the fourth order term included is found to be, 

<img src='content_image/348662.jpg'>

where the constants are: 

<img src='content_image/348661.jpg'>

