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connecting a $^{′}$s or b $^{′}$s on two different strings. Let us suppose that a ′ jumps from a ′ − to a ′ + at a kink. Then the “sharpness” of a kink is defined by 

Thus the norm of the difference between a ′ − and a ′ + is | Δ a $^{′}$| $_{= 2}$√$_{ψ}$$_{. The production rate}$ of kinks is given by [13] 

<img src='content_image/354131.jpg'>

<img src='content_image/354132.jpg'>

where N ( t, ψ ) dψ denotes the number of kinks with sharpness between ψ and ψ + dψ in the volume V , ¯ Δ and γ are constants related to string networks, whose values are [13], 

<img src='content_image/354133.jpg'>

Here the subscript r ( m ) denotes the value in the radiation(matter)-dominated era. 

and we set g ( ψ ) = 0 for ψ < 0 , 1 < ψ . The correlation length of the cosmic strings ξ is given by ξ ≃ γt . 

<img src='content_image/354134.jpg'>

Produced kinks are blunted by the expansion of the Universe. The blunting rate of the kink with the sharpness ψ is given by [13] 

<img src='content_image/354135.jpg'>

where ζ is a constant which, in the radiation(matter)-dominated era, is given by ζ$_{r}$ ≃ 0 . 09 ( ζ$_{m}$ ≃ 0 . 2). 

On the other hand, the number of kinks on an infinite string decreases when it self- intercommutes since some kinks are taken away by the loop produced. The decrease rate of kinks due to this effect is given by [13] 

<img src='content_image/354136.jpg'>

where η is constant which, in the radiation(matter)-dominated era, is given by η$_{r}$ ≃ 0 . 18 ( η$_{m}$ ≃ 0 . 1). 

Taking into account these effects, the evolution of the kink number N obeys the following equation, 

<img src='content_image/354137.jpg'>

