214Weizhu Bao = � in (13.40)-(13.41). The analytic solution of the ZS (15.1)-(15.2) was derived [100] and used to test di�rent numerical methods for the ZS in [100, 28]. The solution can be written as Es (x, t; v, Emax ) = F (x �vt) exp[i(x �ut)], Ns (x, t; v, Emax ) = G(x �vt), where F (x �vt) = Emax dn(w, q), w= Emax (x �vt), (2(1 �v 2 )) G(x �vt) = q= |F (x �vt)|2 + N0 , v2 �1 , (15.8) (15.9)2 2 (Emax �Emin )Emax = v/2, L=2 v v 2N0 E 2 + Emin L = 2m, m = 1, 2, 3 , u = + �max , 2) 2 2 v v(1 �v2 2(1 �v 2 ) 2 K(q) = Emax2(1 �v 2 ) K EmaxEmin Emax,with dn(w, q) a Jacobian elliptic function, L the period of the Jacobian elliptic functions, K and K the complete elliptic integrals of the st kind satisfying K(q) = K 1 �q 2 , and N0 chosen such that Ns =1 L L 0Ns (x, t) dx = 0.16. Time-splitting spectral method for GZS In this section we present new numerical methods for the GZS (13.40), (13.41). For simplicity of notations, we shall introduce the method in one space dimension (d = 1) of the GZS with periodic boundary conditions. Generalizations to d > 1 are straightforward for tensor product grids and the results remain valid without modiations. For d = 1, the problem becomes i E + 倄x E �伪N E + |E|2 E + i E = 0, a < x 0, (16.1) 2 t N �倄x (N � |E|2 ) = 0, E(x, 0) = E(0)a < x 0,(1)(16.2) (x), (16.3) (16.4) (16.5)(x), N (x, 0) = N(x), N (x, 0) = NE(a, t) = E(b, t), N (a, t) = N (b, t),倄 E(a, t) = 倄 E(b, t), 倄 N (a, t) = 倄 N (b, t),t �0, t �0.
