Velarde [3]. Then a distinguished balance occurs for α, β = O ( ϵ ); so we write 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 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 $^{5}$), 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 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 16 r$_{2}$σ 2 1 − 288 r$_{2}$q + 264 r 2 + ir$_{2}$ ˆ vσ$_{1}$ = 0 , σ 3 + 9 σ $^{2}$L2 + 24 σL 4 + 16 L 6 + 528 r 2 $_{2}$L 2 − 576 r$_{2}$qL 2 + 82 r$_{2}$σL 2 − 568 r$_{2}$L 4 + 2 r$_{2}$σ 2 − ˆ v $^{2}$σL2 − L $^{4}$ˆ v 2 + i(2 r$_{2}$ ˆ vσL + 360 r$_{2}$ ˆ βL 3 + 8ˆ vL 5 + 10ˆ vσL 3 + 2ˆ vσ $^{2}$L + 2 r$_{2}$ ˆ vL $^{3}$) = 0 . (27) ∂a ∂T = 4 ∂ $^{2}$a ∂X 2 − 24 r 1 / 2 2 ( ∂ψ ∂X ) 2 − ˆ v ∂a ∂X − 6 r 1 / 2 2 ∂f ∂X − 2 r$_{2}$a + 6 r 1 / 2 2 ( − 8 q + 22 3 r$_{2}$ + 12 ∂ 2 ∂X 2 + (10 ˆ β − 3ˆ α ) ∂ ∂X ) ∂ψ ∂X . (26) ∂f ∂T = ∂ $^{2}$f ∂X 2 − 12 r 1 / 2 2 ∂a ∂X , (25) ∂ψ ∂T = 4 ∂ $^{2}$ψ ∂X 2 − f − ˆ v ∂ψ ∂X , (24) M = (1 + ϵ $^{2}$q ) x − cτ + ϵ ( − ˆ vq + 1 $_{6}$r$_{2}$ (ˆ α − 5 ˆ β )) T + ϵψ ( X, T ) . u = ϵ (6 √ r$_{2}$ + ϵ $^{2}$a ( X, T ))e i M + c.c. + ϵ $^{3}$f ( X, T ) + · · · , α = ϵ ˆ α, β = ϵ ˆ β.