II. DYNAMICS OF COSMIC STRINGS III. DISTRIBUTION FUNCTION OF KINKS The dynamics of a cosmic string, whose width can be neglected, is described by the Nambu-Goto action, where X = X ( ζ ) is embedding of the world sheet on the background metric. If the back- ground space-time is Minkowski one, we can select the coordinate system ( ζ $^{0}$, ζ$^{1}$) = ( τ, σ ) which satisfies the gauge conditions The time scale of a GW burst is much shorter than the Hubble expansion, hence we consider an individual burst event on the Minkowskian background. The general solution of the equation of motion derived from the action (1) is where u = σ + t, v = σ − t . We call a ( u ) ( b ( v )) the left (right)-moving mode. Then, Eq. (2) can be rewritten in terms of a ( u ) and b ( v ), Kinks can be defined as discontinuities of a ′ or b $^{′}$. They are produced when two infinite strings collide and reconnect because a ′ and b ′ on the new infinite string are created by S = − µ ∫ d $^{2}$ζ √ − det( γ$_{ab}$ ) . (1) T $^{µν}$( x ) = µ ∫ d $^{2}$ζ √ − det( γ$_{ab}$ ) γ $^{ab}$xµ $_{,a}$x ν $_{,b}$δ $^{4}$( x − X ( ζ )) , (2) τ = t (physical time) , x$_{,τ}$ · x$_{,σ}$ = 0 , x 2 ,τ + x 2 ,σ = 0 . (3) x µ = 1 2 ( a $^{µ}$( u ) + b $^{µ}$( v )) , a ′ $^{2}$( u ) = b ′ $^{2}$( v ) = 1 (4) T $^{µν}$( k ) = µ 4 ( I µ $_{+}$( k ) I ν $_{−}$( k ) + I µ − ( k ) I ν $_{+}$( k )) , (5) I µ $_{+}$( k ) = ∫ dua ′ $^{µ}$( u ) e ik · a ( u ) / $^{2}$, I µ − ( k ) = ∫ dvb ′ $^{µ}$( v ) e ik · b ( v ) / $^{2}$, (6) where T $^{µν}$( k ) is the Fourier transform of the T $^{µν}$( x ), i.e. T $^{µν}$( k ) = ∫ d $^{4}$xT $^{µν}$( x ) e ik · $^{x}$. 4