In the exponent in Eq. 3.7.6, integrating over r is carried out not from r to ro, as in Eq. 3.7.5, but from r to ^ since we consider here that a limiting of the solar wind is provided automatically by its non-linear interaction with CR. Therefore in the nonlinear theory, in contrast to the linear theory, it is not necessary to introduce any assumptions about the dimension ro of the region of solar wind propagation. This fact is one of the essential advances of non-linear theory as compared to the linear theory.

In the papers Dorman and Dorman (1968a,b), Dorman and Babayan (1975), Babayan et al. (1976), a similar equation with some simplifying assumptions was transformed to a non-linear differential equation of the second order, for which it appeared to be possible to obtain the analytical solution only for the regions not very distant from the Sun, i.e. for the regions where the effect of non-linear interaction is small.

In the work Babayan and Dorman (1977a) Eq. 3.7.6 was solved numerically by iteration method. The expression

was taken as no, where y = 1.5 and 2, and Ekmin = 0.1, 0.01 and 0.001 GeV. The calculations were also carried out for the spectrum described by Eq. 3.7.7 but over rigidity R with the index y + 1 = 2.5. The coefficient a was determined from the condition that a density of kinetic energy of CR WCR is 1 eV/cm3 at Ek,min = 0.1

GeV. A dependence of A on R was taken as in Dorman and Dorman (1968 a,b) and on a distance r, according to power law A ^ rß with the power index ß = 0 and 1, i.e.,

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