obvious that it will be a convenient one to be used when a higher accuracy in numerical schemes is required. The possibility that it may also help to control (or at least delay) the numerical instabilities found in previous works (see for instance [9, 33] and references therein) will be the subject of further work. Let us remind again that, as we already pointed out in the introduction, these results are formal since the higher order derivatives of the field are not under control. As was indicated in the introduction, a pseudo-spectral spatial discretization scheme has been recently introduced [14–16, 36]. In this section we show the relation existing between the present analysis and the indicated pseudo-spectral spatial discretization scheme. The pseudo-spectral discretization procedure starts by Fourier expanding the field h ( x, t ) with and Eq. (3) can be rewritten as Thus Eq. (2) reads since ξ ( x, t ) is also assumed to be L –periodic as a function of x . A sufficient condition is that In this context “discretize” means to consider only N Fourier modes, including k = 0. If M ≡ ( N − 1) \ 2, then 11 IV. PSEUDO-SPECTRAL DISCRETIZATION h ( x, t ) = ∞ ∑ k = −∞ ̂ h$_{k}$ ( t ) exp ( i 2 π L kx ) , (32) ̂ h$_{k}$ ( t ) = 1 L ∫ L 0 d x h ( x, t ) exp ( − i 2 π L kx ) , (33) φ ( x, t ) = exp   λ 2 ν ∞ ∑ k = −∞ ̂ h$_{k}$ ( t ) exp ( i 2 π L kx )   . (34) ∞ ∑ k = −∞ exp ( i 2 π L kx )   $^{˙}$ ̂ h$_{k}$ ( t ) + ( 2 π L ) 2 k ̂ h$_{k}$ ( t )   νk + λ 2 ∞ ∑ k ′ = −∞ k ′ ̂ h$_{k}$ ′ ( t ) exp ( i 2 π L k $^{′}$x )   − ε ̂ ξ$_{k}$ ( t )    = 0 , (35) ˙ ̂ h$_{k}$ ( t ) = − ( 2 π L ) 2 k ̂ h$_{k}$ ( t )   νk + λ 2 ∞ ∑ k ′ = −∞ k ′ ̂ h$_{k}$ ′ ( t ) exp ( i 2 π L k $^{′}$x )   + ε ̂ ξ$_{k}$ ( t ) . (36) ˙ ̂ h$_{k}$ ( t ) = − ( 2 π L ) 2 k ̂ h$_{k}$ ( t )   νk + λ 2 M ∑ k ′ = M +1 − N k ′ ̂ h$_{k}$ ′ ( t ) exp ( i 2 π L k $^{′}$x )   + ε ̂ ξ$_{k}$ ( t ) . (37)