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A typical example is the selffocusing of whistler waves [10,11] which will be considered in this section. We Wistler Solitons 45 confine ourselves to the ponderomotive nonlinearity. e. ν = ν(r, z). Then Eq. (2) with (4) and (8) has solutions of the form F ≡ Er − iEφ = Fm (r, z)eimφ , G ≡ Er + iEφ = Gm (r, z)eimφ , Ez = Emz eimφ , (85) where m = 0, ±1, ±2, . . Substituting (85) into (2), we arrive at the equations ∂ ω2 ∂ 2 Fm m (m−1) + Λ F − ) + ( + g)Fm = 0, (86a) + (∇ · E m m (r) ∂z 2 ∂r r c2 2 2 ∂ ∂ Gm ω m (m+1) + Λ(r) Gm − − (∇ · Em ) + 2 ( − g)Gm = 0, (86b) 2 ∂z ∂r r c 1 ∂ m ∂Emz [r(Fm + Gm )] − (Fm − Gm ) + , 2r ∂r 2r ∂z ∂ 1 ∂ (ηEmz ) + [( + g)Fm − ( − g)Gm ]} ∂z 2r ∂r m + [( + g)Fm − ( − g)Gm ] = 0, 2r ∇ · Em = (87) (88) where (n) Λ(r) = 1 ∂ ∂ n2 r − 2.

Here, σ = µ 1 + k2 λ2e /η, γ = −α1 α3 ky2 /(ηµk 6 ), β = 4kx2 1 + k 2 λ2e /(1 + 4kx2 λ2e )k 2 , and the new parameter δ = α2 (ky2 −3kx2 ) 1 + k 2 λ2e µ2 k 6 B0 /[α12 cky2 (1+k 2 λ2e − 6kx2 λ2e )], with k 2 = kx2 + ky2 and τ = t/t0 ; where t0 = ηk 2 / 1 + k 2 λ2e . A comment is in order. If we set δ = 0, which happens for ky2 = 3kx2 , then (106) to (108) reduce to the Lorenz type equations. K. Shukla and L. Stenflo A2 = − cα1 kx ky2 ηµk 6 B0 Z. (1 + k 2 λ2e − 6kx2 λ2e ) Let us now discuss the chaotic fluid behavior of electromagnetic turbulence that is governed by (106) to (108).

32)–(35). Assuming that ν ∼ µ2 , we can neglect the last term on the left hand side of Eq. I. Karpman arrive at the equations looking like (38) and (39) [Now Eq. (66)] is in fact nonlinear]. (66) and (67) describe the whistler soliton and a tunneling wave. [The density variation (63), produced by the ponderomotive force, serves as a duct, trapping the whistler wave]. Neglecting also the terms with derivatives ¯ in Eq. (67), we have G ¯ ∼ µ2 ∂ 2 F¯ . This rough estimate, however, is valid of G ξ inside the soliton, but not in the region occupied by the emitted radiation .

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A new decision method for elementary algebra by Seidenberg A.


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