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The case λ < 0 has been the object of discussions (Rubenchik and Zakharov 1986), but a similar conclusion holds (Rypdal and Rasmussen 1989 and references therein). A discussion is presented by Kuznetsov, Rubenchik, and Zakharov (1986). In optical systems with negligible time dispersion (where t holds for the coordinate along the direction of propagation), the transverse instability can be arrested by spatially modifying the refractive index so that it provides waveguiding in one transverse direction, while self-trapping occurs in the perpendicular one.

The Sobolev spaces H s are also considered for fractional powers s. In the entire space, it is convenient to define Sobolev norms in terms of the Fourier transform u of the function u by |u|2H s = (1 + |k|2 )s |u(k)|2 dk. In some instances, we shall write the variable as an index or its domain in parentheses to avoid confusion. For example, Lpx1 = Lp (R). Also, Lq ([0, T ), Lr (Rd )) denotes the space of functions u in Lq ([0, T )) in the t-variable with values in Lr (Rd ). 1) ψ(x, 0) = ϕ(x). 1 The Linear Schr¨ odinger Operator 45 defines a unitary transformation group in L2 .

4 Variance Identity 39 Note that this “collapse” does not necessarily corresponds to a singularity for the primitive wave equation, but only to a breakdown of the modulational analysis leading to the (cubic when σ = 1) NLS equation. 35) which remains globally well-posed. Regularization were proposed by retaining higher order saturating terms in the expansion, corresponding to higher order nonlinearity or to the breaking of the paraxial approximation (see Chapter 9 and also Xin (1998) for a detailed modeling of the saturation in the context of the Sine–Gordon equation).

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