A remark about the justification of the nonlinear Schr¨odingerequation in quadratic spatially periodic media Carsten Blank, Martina Chirilus Bruckner, Christopher Chong, Vincent Lescarret, Guido Schneider, and Hannes Uecker October 1, 2007 Abstract We prove the validity of a technical assumption necessary in a proof of the validity of the nonlinear Schr¨odinger equation as envelope equation in quadratic spatially periodic media. The dynamics of the envelopes of spatially and temporarily oscillating wave packets advancing in dispersive spatially periodic media can be approximated by solutions of a Nonlinear Schr¨odinger equation. In [1] the semilinear wave equation ∂ 2 t u(x, t)= χ 1 (x)∂ 2 x u(x, t) − χ 2 (x)u(x, t) − χ 3 (x)u ϑ (x, t) (1) has been considered as a model problem for this, where x ∈ R,t ∈ R,u(x, t) ∈ R,ϑ = 2 or ϑ = 3, and χ j (x)= χ j (x +2π) for j =1, 2, 3. Under a number of technical assumptions in [1] a proof for the validity of the Nonlinear Schr¨odinger equation as an amplitude equation has been given. In the present paper we explain that the technical assumption (7) in [1] for the much more advanced quadratic case ϑ = 2 is always satisfied if χ 1 ∈ C 2 per . The linearized problem ∂ 2 t u(x, t)= χ 1 (x)∂ 2 x u(x, t) − χ 2 (x)u(x, t) (2) is solved by the Bloch waves u(x, t)= f n (ℓ, x)e iℓx e ±iωn(ℓ)t where n ∈ N,ℓ ∈ (−1/2, 1/2], with ω n (ℓ) ∈ R satisfying ω n+1 (ℓ) ≥ ω n (ℓ), and f n (x, ℓ) satisfying f n (ℓ, x)= f n (ℓ, x +2π) and f n (ℓ, x)= f n (ℓ +1,x)e ix . Slow modulations in time and space of such a Bloch mode (indexed with n 0 ) are described by the ansatz u(x, t)= εA(ε(x + c g t),ε 2 t)f n 0 (ℓ 0 ,x)e iℓ 0 x e iωn 0 (ℓ 0 )t + cc + h.o.t., (3) where cc means complex conjugate, h.o.t. means terms of order ε 2 and higher, 0 < ε ≪ 1 is a small parameter, c g = ∂ ℓ ω n 0 (ℓ 0 ) is the negative group velocity, and where 1