content_image/334142.jpg content_image/334143.jpg content_image/334141.jpg content_image/334140.jpg
where the wavenumber k = 1 + ϵq . The amplitude turns out to be given by 

and the speed of the wave is 

<img src='content_image/334142.jpg'>

<img src='content_image/334143.jpg'>

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. 

## III. SECONDARY STABILITY OF TRAVELING WAVES: NUMERICAL RE- SULTS 

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. 

## A. Numerical method for calculating secondary stability 

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 

<img src='content_image/334141.jpg'>

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 

<img src='content_image/334140.jpg'>

