where the wavenumber k = 1 + ϵq . The amplitude turns out to be given by and the speed of the wave is where c$_{p}$ is given by (4) and the second contribution to s reflects (weakly) nonlinear effects. So, regardless of the values of the dispersion parameters α and β , such spatially periodic solutions exist for r$_{2}$ > 4 q $^{2}$. We now turn to the question of the secondary stability of these solutions. In this section we first outline a numerical method for the calculation of the nonlinear traveling waves and their secondary stability, and then give the results of these computations, showing the stability boundaries of traveling waves. To calculate the secondary stability of a traveling wave solution for given values of the parameters, we first find the traveling wave solution ¯ u ( x, t ) = f ( z ), where z = x − ct . Here, c is the nonlinear wave speed, which in general is not exactly equal to the linear wave speed c$_{p}$ (4). We approximate the solution numerically using the truncated Fourier series Substitution in (2) (and calculation of the nonlinear term pseudospectrally) yields a system of nonlinear equations (solved in Matlab) for the Fourier coefficients of f ( z ), together with c , which is determined from 5 c ∫ D 0 ( f $^{′}$) $^{2}$d z = α ∫ D 0 ( f $^{′′}$) $^{2}$d z − β ∫ D 0 ( f $^{′′′}$) $^{2}$d z + ∫ D 0 f ( f $^{′}$) $^{2}$d z, (10) f ( z ) = N/ 2 ∑ − N/ 2+1 ¯ u$_{n}$ e i $^{nkz}$. a$_{0}$ = 6( r$_{2}$ − 4 q $^{2}$)1 / $^{2}$(1 + 1 $_{36}$( α − 5 β ) $^{2}$)1 / 2 (8) s = c$_{p}$ − 1 $_{6}$ϵ$^{2}$( r$_{2}$ − 4 q $^{2}$)( α − 5 β ) + o ( ϵ $^{2}$) , (9) III. SECONDARY STABILITY OF TRAVELING WAVES: NUMERICAL RE- SULTS A. Numerical method for calculating secondary stability