Nonlinear Processes in Geophysics (2002) 9: 125–130 Nonlinear Processes in Geophysics c European Geophysical Society 2002 New types of stable nonlinear whistler waveguides Yu. A. Zaliznyak, T. A. Davydova, and A. I. Yakimenko Plasma Theory Department, Institute for Nuclear Research, Nauki Ave. 47, Kiev 03680, Ukraine Received: 7 June 2001 – Accepted: 28 August 2001 = 2 2 − 2 = − 2 2 − Abstract. The stationary self-focusing of whistler waves where g (ω/ωBe)ωpe/(ω ωBe), ε⊥ ωpe/(ω with frequencies near half of the electron-cyclotron fre- 2 = − 2 2 ωBe), ε|| ωpe/ω . Taking into account the nonlinear quency in the ionospheric plasma is considered in the frequency shift (see Gyrevich and Shvarzburg (1973)) framework of a two-dimensional generalized nonlinear s Schrodinger¨ equation including fourth-order dispersion ef- ω2 + ν2 4ν2 E 2 ω = − e 1 + e z − 1 (2) fects and nonlinearity saturation. New types of soliton- nl 2 2 2 2νe ω + νe Ep like (with zero topological charge) and vortex-like nonlinear waveguides are found, and their stability confirmed both an- 4 and expanding Eq. (2) accurately to ∼ Ez/Ep , where alytically and numerically. p 2 2 2 ω ≈ ωBe/2, Ep = 3mT (ω + νe )δ/e , δ ≈ 2m/M, νe is the frequency of electron collisions, n0 is the density of electrons, T is the electron temperature, we obtain the Gen- 1 Introduction eralized Nonlinear Schrodinger¨ Equation (GNSE): ∂ψ 2 2 4 i + D1⊥ψ + P 1 ψ + Bψ |ψ| + Kψ |ψ| = 0, (3) Plane whistler waves which propagate along the magnetic ∂ζ ⊥ = field B (0, 0,Bz) and obey the dispersion relation 2 2 2 2 2 2 2 where 1√⊥ = ∂ /∂ξ + ∂ /∂η , ψ = Ez/F0, δ ≈ 2m/M, k c /ωpe = ω/(ωBe − ω), are fully electromagnetic right- F0 = 12πT n0δ, ζ = zω/c, ξ = xω/c, η = yω/c, hand polarized waves having k⊥ = 0, E = 0. However, z ω ≈ ω /2, and the coefficients of equation are: D = when the wave beam is localized in the transverse plane, Be √ (1/(4u2) − 1)/(2 v), P = 1/(8v3/2), B = −v3/2, K = or when the direction of propagation deviates from the OZ 5/2 2 2 2 2 v ν /ω , u = ω/ωBe, v = ω /ω . Equation (3) general- axis, these properties disappear. Together with nonzero k⊥ e P e izes Eq. (1) of Zharova and Sergeev (1989) to the case of 2D whistler waves gain elliptic polarization (which can be as- geometry and takes into account the nonlinearity saturation sumed as a presence of the left-hand polarized part of the effects. Gyrevich and Shvarzburg (1973) and Bharuthram wave field) and the electrostatic wave component (E 6= 0). z et al. (1992) pointed out that the main nonlinear effect in Here we consider propagation of beams of whistler waves the ionosphere is the plasma extrusion from the HF field re- with frequencies near half of the electron-cyclotron fre- gion due to heating and a pressure increase. Near the point quency (0 < ω /(2ω) − 1 1) strictly along the magnetic Be ω ≈ ω /2 the thresholds of modulational instabilities for a field in the ionosphere. Using a hydrodynamic approxima- Be whistler wave decreases dramatically, and one must account tion for cold plasma one can find that the electrostatic com- for the next terms in the nonlinearity expansion which are of ponent E (x, y) of the electrical field of the whistler wave in z the same order as the linear terms. It leads to the appear- a linear case is governed by the following equation ance of cubic-quintic saturable nonlinearity in the GNSE (3). ( " 2 2 2 # In this paper we consider the case u ≤ 1/2, which means ε|| ω ω g 12 + −k2 − k2 + (ε + ε ) − 1 ω ≤ ω /2 and the signs of GNSE coefficients are the fol- ⊥ z z 2 || ⊥ 2 ⊥ Be ε⊥ c c ε⊥ lowing: D > 0, P > 0, B < 0, K > 0. We are interested 2 2 ! 4 to find the conditions of the whistler wave propagation inside ε|| ω ω + k2 − ε − g2 E = , stationary waveguides, or channels, formed due to nonlinear x 2 ⊥ 4 z 0 (1) ε⊥ c c self-interaction and localization in the plane perpendicular to direction of propagation: OZ axis. Wave intensity profiles Correspondence to: Yu. A. Zaliznyak ([email protected]) across these waveguides does not depend on z. 126 Yu. A. Zaliznyak et al.: New types of stable nonlinear whistler waveguides 2 General properties of nonlinear whistler waveguides the Hamiltonian functional is always positive. Using integral inequalities Any wave packet localized in the plane perpendicular to z- direction which evolves along the z-direction according to Z Z 1/2 |∇ψ|2d2r < N 1/2 |1ψ|2d2r , (12) Eq. (3) has conserved quantities: number of quanta: ZZ 1/2 2 2 Z Z N = |ψ| d r, (4) |ψ|4d2r < N 1/2 |ψ|6d2r , (13) momentum: we find the following estimate for the Hamiltonian: ZZ i ∗ ∗ 2 I = − ψ ∇ψ − ψ∇ψ d r, (5) Z 1/2 2 H < DN 1/2 |1ψ|2d2r angular momentum: Z Z 1/2 2 2 |B| 1/2 6 2 i ZZ −P |1ψ| d r + N |ψ| d r M = − ψ∗ [r × ∇ψ] − ψ r × ∇ψ∗ d2r, (6) 2 2 " # K Z N D2 3 B2 − |ψ|6d2r < + . (14) Hamiltonian: 3 4 P 4 K ZZ 2 2 H = D |∇ψ| − P |1⊥ψ| − It indicates that for a fixed value of the number of quanta N, B K a Hamiltonian functional is bounded from below and above. |ψ|4 − |ψ|6 d2r. (7) This guarantees that for any N there exists at least one stable 2 3 solitary solution which corresponds to Hamiltonian’s abso- Stationary (along the OZ axis) waveguides in the framework lute extremum. of Eq. (3) have the form ψ(r) exp(iλζ ), where the complex Using integral inequalities Eqs. (12) and (13) and the iden- function ψ(r) obeys the partial differential equation tity Eq. (9) it is easy to show that the waveguide parameter λ (the nonlinear shift of wave number) is bounded from below 2 2 4 −λψ + D1⊥ψ + P 1⊥ψ + Bψ |ψ| + Kψ |ψ| = 0. (8) and to estimate a range of accessible λ – values: ∗ Multiplying Eq. (8) by ψ and integrating over space coordi- D2 B2 D2 nates in the perpendicular plane, an integral relation − − < λ < − . (15) 4P 4K 4P ZZ ZZ 2 2 2 2 λN = P |1⊥ψ| d r − D |∇ψ| d r From the virial relation for the waveguide’s effective width ZZ ZZ Z 2π Z ∞ + | |4 2 + | |6 2 2 = −1 2 | |2 B ψ d r K ψ d r (9) reff N r ψ rdrdθ, 0 0 is obtained. Multiplying Eq. (8) by r2dψ∗/dr, integrating namely, from and adding the complex conjugate, another integral identity d2r2 Z ∞ Z 2π n h is found: N eff = D2 |∇ψ|2 − DP |1ψ|2 2 8 4 ZZ dζ 0 0 2 2 λN = −P |1⊥ψ| d r+ i BD +4P 2 |∇1ψ|2 − |ψ|4 B ZZ K ZZ 2 |ψ|4 d2r + |ψ|6 d2r. (10) KD 2 3 − |ψ|6 − 3BP r |∇ψ|2 ∇ |ψ|2 3 One restriction on the parameter λ: λ < −D2/4P easily fol- 2 −P Kr ∇ |ψ|2 6 |ψ|2 |∇ψ|2 + ∇ |ψ|2 rdrdθ,(16) lows from the asymptotic behavior |r| → ∞ of any localized solution of Eq. (8). After excluding λ from Eqs. (9)–(10) one finds a simplified expression for a Hamiltonian of solitary one finds that: (i) when P = K = 0 and BD < 0, GNSE (which means decaying at infinity) solutions: (3) has no localized solutions at all; any wave packet moving along the z-axis spreads out in a radial direction; (ii) in the ZZ ZZ 2 2 K 6 2 case of P = 0 = K = 0, BD > 0 the virial relation gives H = P |1⊥ψ| d r + |ψ| d r 3 2 2 D ZZ B ZZ d reff = |∇ψ|2 d2r − |ψ|4 d2r. (11) N = 8DH, 2 4 dς 2 From Eq. (11) it follows immediately that for D > 0, P > and predicts the collapsing of any wave packet having 0, B < 0, K > 0 (which is the case under consideration) DH < 0; (iii) the sum of all linear terms in the virial relation Yu. A. Zaliznyak et al.: New types of stable nonlinear whistler waveguides 127 = 1 R z 2 are defocusing. This follows from the integral inequality where η N 0 µ dζ , and the Hamiltonian Z ∞ Z ∞ 2 dψ d1r ψ " (l) |1 ψ| rdr = − rdr ≤ N (l) 2 (l) 2 BIb 2 r dr dr H = D(I µ + I β ) − Nµ 0 0 (l) d1 d2 2 2π IH Z ∞ 2 !1/2 Z ∞ 2 !1/2 dψ d1r ψ (l) rdr rdr . KIk (l) (l) (l) i dr dr − N 2µ4 −P (I µ4 + I µ2β2 + I β4) . (22) 0 0 3 p1 p2 p3 Thus, any linear wave packet described by the linear GNSE (l) Integrals Iσ are defined by the choice of trial function: will always spread out. When D > 0, B < 0, P > 0, K > 0 the virial relation includes focusing (proportional to KD and Z ∞ ∂f (ξ)2 (l) = l ; to BP ) as well as defocusing (proportional to BD and PK) Id1 ξ dξ 0 ∂ξ nonlinear terms. Different linear and nonlinear terms come Z ∞ (l) = 2 2 ; into play at different spatial scales and on different values Id2 ξfl (ξ) tanh (ξ)dξ of field amplitude; thus it is natural to expect that several 0 1 Z ∞ Z ∞ stationary nonlinear structures may coexist in the framework I (l) = ξf 4(ξ)dξ; I (l) = ξf 2(ξ)dξ; b (l) l H l of GNSE (3).