content_image/371149.jpg content_image/371148.jpg content_image/371147.jpg content_image/371145.jpg content_image/371146.jpg content_image/371143.jpg content_image/371144.jpg content_image/371142.jpg
Kudryashov and Migita [10] found stable periodic waves in their numerical simulations of the dispersive PDE (2), in this regime. 

We begin by introducing the weakly nonlinear expansion 

<img src='content_image/371149.jpg'>

with r given by (6). Then substitution in (2) and consideration of successive orders in ϵ leads to the following. 

At O ( ϵ ), we find that 

<img src='content_image/371148.jpg'>

where c$_{0}$ = α − β , and where the amplitude A varies slowly in space and in time, in principle depending on the slow variables 

<img src='content_image/371147.jpg'>

A consideration of the terms proportional to e i( x − c 0 t ) at O ( ϵ $^{2}$) then shows that in fact A = A ( ξ, T ), where 

is a coordinate moving at the group velocity of the waves. Then solving the problem at this order in ϵ yields 2 

Here f is a slow varying function of X , τ and T , chosen to appear at this order to balance forcing terms appearing at the next order in ϵ . 

<img src='content_image/371145.jpg'>

<img src='content_image/371146.jpg'>

At O ( ϵ $^{3}$), we find, from the respective consideration of the terms in (2) proportional to e i( x − c 0 t ) and e 0i( x − c 0 t $^{)}$, the amplitude equations 

Since A = A ( ξ, T ), the second amplitude equation suggests taking f = f ( ξ, T ), in which case (14) becomes 2 

<img src='content_image/371143.jpg'>

<img src='content_image/371144.jpg'>

<img src='content_image/371142.jpg'>

