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Equipped with (17), we are now in a position to explore the secondary stability of weakly nonlinear spatially periodic solutions of the dispersive Nikolaevskiy equation. Such solutions correspond to plane-wave solutions of (17), which exist in the form A = Pe i( ωT + qξ $^{)}$, with P = (1 − q $^{2}$)1 / 2 and ω = q $^{2}$( b − a ) − b . To study the stability of the plane-wave solution, we write A = (1 + p ( ξ, T )) Pe i( ωT + qξ $^{)}$, which, after substitution in (17) and linearization in the perturbation p , yields 

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Then upon setting p ( ξ, T ) = R ( T )e i Lξ + S $^{∗}$( T )e − i Lξ and equating the coefficients of e i Lξ and e − i $^{Lξ}$, we have 

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Finally, with R ( T ) and S ( T ) proportional to e $^{µT}$, and expanding the growth rate in powers of the perturbation wavenumber L , we have the dispersion relation 

<img src='content_image/329901.jpg'>

If we suppose (as is generally the case) that a ̸ = b + d then it is apparent from (18) that the solution has a long-wavelength oscillatory instability whenever 

Since 1 + a ( b + d ) < 3 + a ( b + d ) + 2( b + d ) $^{2}$, we see that stability is determined by the following. If 1 + a ( b + d ) > 0 then 3 + a ( b + d ) + 2( b + d ) 2 > 0, and the plane-wave solutions are stable provided 

<img src='content_image/329900.jpg'>

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If instead 1 + a ( b + d ) < 0, then all plane waves are unstable. We note that setting a = b = d = 0 reduces (17) to a real Ginzburg–Landau equation for A , and our results reduce to the usual Eckhaus instability (with stability for q 2 < 1 / 3) [15, 16]. 

To apply this result to (2) it is necessary to indicate the regions in α , β parameter space in which the quantity 1 + a ( b + d ) is positive or negative. In Fig. 5, regions denoted by “s” indicate where 1 + a ( b + d ) > 0, so that there are some stable plane waves, as in (19); those 

