### Abstract

A model of wave propagation in a slab 0≤*x*≤*L* of a *n* *o* *n* *l* *i* *n* *e* *a* *r* *r* *a* *n* *d* *o* *m* medium is considered. The index of refraction is *k*[1+ε̃(*x*,*w*‖*u* ^{ε}(*x*,*L*,*w*)‖^{2})]^{1} ^{/} ^{2}, where ε̃(*x*,α)=ε*m*(*x*)+ε^{2}[**(** *n*(*x*) +*i*δ(*x*)**)**α+θ(*x*)+*i*γ(*x*)], with ε a small parameter, *m*,*n*,δ,θ,γ suitable stochastic processes,*u* ^{ε}(*x*,*L*,*w*) the wave field, and *w*≥0 the intensity of nonlinearity. The *m* *e* *a* *n* *r* *e* *f* *l* *e* *c* *t* *e* *d* *p* *o* *w* *e* *r* is evaluated from a certain nonlinear partial differential equation satisfied by the reflection coefficient*R* ^{ε}(*L*,*w*). An infinite system of ordinary differential equations for the coefficients *R* ^{ε} _{ n }(*L*), *n*=0,1,2,..., in the expansion of *R* ^{ε}(*L*,*w*) in powers of *w*, is then derived, and the infinitesimal generator for the process [*R* _{0} *R* ^{*} _{0} *R* _{1} *R* ^{*} _{1}], *R* _{ n }≡*R* ^{0} _{ n }(ε^{2} *L*), is obtained, in the *d* *i* *f* *f* *u* *s* *i* *o* *n* *l* *i* *m* *i* *t* ε→0, *L*→∞, ε^{2} *L*=const. This allows us to compute 〈‖*R*‖^{2}〉≂〈‖*R* _{0}‖^{2}〉 +2*w* Re〈*R* _{0} *R* ^{*} _{1}〉 as a function of ε^{2} *L*. In the lossless case, δ=γ=0, there is *n* *o* correction due to the nonlinearity, in such a limit, and this remains true at least up to the order *O*(*w* ^{2}). Some effects can be observed when dissipation (δ>0, γ>0) is taken into account. Numerical results are obtained and plots are given.

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