content_image/365302.jpg content_image/365301.jpg content_image/365300.jpg content_image/365299.jpg content_image/365298.jpg content_image/365297.jpg content_image/365296.jpg content_image/365295.jpg
Velarde [3]. Then a distinguished balance occurs for α, β = O ( ϵ ); so we write 

<img src='content_image/365302.jpg'>

Upon setting ˆ c = ˆ α − ˆ β , r = ϵ $^{2}$r$_{2}$ , ˆ v = 3ˆ α − 5 ˆ β , X = ϵx , τ = ϵt and T = ϵ $^{2}$t (the scalings for X and T being as in [3]), we find from (2) that 

where now 

<img src='content_image/365301.jpg'>

<img src='content_image/365300.jpg'>

The terms in M involving ˆ α and ˆ β correspond to nonlinear effects of the finite traveling-wave amplitude on the speed of the waves; see (9). 

After much algebra, the relevant (nonlinear) amplitude equations are found to be, at O ( ϵ $^{4}$) and O ( ϵ $^{5}$), 

<img src='content_image/365299.jpg'>

<img src='content_image/365298.jpg'>

<img src='content_image/365297.jpg'>

We note that in these equations the influence of dispersion arises not only through the terms involving the group velocity ˆ v , but also through the term 10 ˆ β − 3ˆ α in the equation for a$_{T}$ , in contrast to the previous case. 

To determine the stability of the traveling waves, these equations are linearized; for solutions proportional to e i LX + $^{σT}$, we find the dispersion relation 

<img src='content_image/365296.jpg'>

As in the previous section, in the limit of large L , all eigenvalues have negative real part. By contrast, in the limit of small L , if we expand σ = σ$_{1}$L + σ$_{2}$L 2 + · · · , then from (27) we find that σ$_{1}$ satisfies 

<img src='content_image/365295.jpg'>

