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 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 where c$_{0}$ = α − β , and where the amplitude A varies slowly in space and in time, in principle depending on the slow variables 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 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 . 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 9 − v ∂f ∂ξ = − ∂ | A | 2 ∂ξ , ∂A ∂T = ( r$_{2}$ − 1 − i( α − 5 β ) / 6 36 + ( α 5 β ) 2 | A | 2 ) A + (4 + i(3 α − 10 β )) ∂ $^{2}$A ∂ξ 2 − i f A, (13) ∂f ∂τ = − ∂ | A | 2 ∂ξ . (14) u$_{2}$ = − i A 2 36(1 + i( α − 5 β ) / 6) e 2i( x − c 0 t ) + c.c. + f. ξ = X − (3 α − 5 β ) τ ≡ X − vτ X = ϵx, τ = ϵt, T = ϵ $^{2}$t. u$_{1}$ = A e i( x − c 0 t ) + c.c. , u = ϵu$_{1}$ + ϵ $^{2}$u$_{2}$ + ϵ $^{3}$u$_{3}$ + · · · , (12)