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). 2πIH 0 0 Z ∞  2 (2l + 1)tanh ξ2 I (l) = f 2(ξ) − − dξ; p1 l 1 2 3 Variational analysis 0 (cosh ξ) ξ Z ∞ (l) = 4 2 ; (l) = (l) + (l); In the framework of the variational approach (see Anderson, Ip3 ξ tanh ξfl (ξ)dξ Ip2 Ip1 Ip3 1983) the GNSE (3) is formulated as a reduced variational 0 R 1 Z ∞ problem for functional Ldζ I (l) = ξf 6(ξ)dξ. (23) k (l) 2 l Z (2πIH ) 0 δ Ldζ = 0, (17) Stationary points of the system Eq. (21) coincide with the extrema of the Hamiltonian functional Eq. (22): where L = R ld2r, l is the Lagrangian density. In cylindrical coordinates (r, θ, ζ ) the latter may be written as: ∂H ∂H = 0; = 0. (24) ∂β ∂µ  ∗  " 2 2# µ=µ0, β=β0 µ=µ0, β=β0 i ∂ψ ∗ ∂ψ ∂ψ 1 ∂ψ l = ψ − ψ + D + 2 ∂ζ ∂ζ ∂r r2 ∂θ These points determine the waveguide parameters µ0 and β0 for a given number of quanta N. Introducing the notation 2 1 ∂  ∂ψ  1 ∂2ψ B K −P r + − |ψ|4 − |ψ|6 .(18) (l)  (l) (l)  (l) (l)2 2 2 DI I − I 3P I − I r ∂r ∂r r ∂θ 2 3 d p1 p3 2 p1 p3 N1 = ; N = , (25) | | (l) (l) 2 (l) (l) Here we will use a trial function in the form: B Ib Ip3 4Ip3 KIk √ µ N   ψ = fl(µr) exp iγ ln cosh µr + ilθ + iϕ(ζ ) ,(19)  (l)2 (l) q (l) 4 DI KI  2 πI dp k N1 2 H σ = = , 2 (26)  (l) (l) N2 where µ = µ(ζ ), γ = γ (ζ ), and the real function fl(ξ) 3 BIb PIp3 determines the envelope profile: where |l| ξ (l) (l) (l) (l)
fl(ξ) = . (20) I − I 2I /I − 1 cosh(ξ) (l) = p1 p3 d1 d2 Idp , I (l) − I (l) This function describes soliton-like (at l = 0) or vortex- p1 p3 like (at l 6= 0) nonlinear waveguides. The nonlinear phase one obtains for the waveguides with β = 0 the following dependence on r-coordinate for this function is fixed to be relation connecting µ2 and N: ∼ iγ ln cosh µr, as it was in the case of 1D exact chirped |B|I (l) soliton solution found by Davydova and Zaliznyak (2000). (l) + b 2 DId 2 N 2 After performing a Ritz optimization procedure, the waveg- µo = < µo max, (27) KI (l) µ β = γ µ (l) + k 2 uide parameters and are obtained from the canon- 2(P Ip1 3 N ) ical Hamiltonian set of equations where dβ ∂H =   , v (l) (l) dη ∂µ DI (l) u 3(BI )2PI 2 d  u b p3  dµ ∂H µo max = 1 + t1 + . (28) = − , (21) 4PI (l)  4(DI (l))2KI (l)  dη ∂β p1 d1 k 128 Yu. A. Zaliznyak et al.: New types of stable nonlinear whistler waveguides

One can also find stationary solutions of Eq. (24) with β0 6= 0 which correspond to electromagnetic beams with curved N wave fronts. In particular for l = 0 one has C O

DI (0) 1 − N/N µ2 = d 1 , (29) S US c  (0) (0) 2 − 1 − (N/N2) P Ip1 Ip3

N2 DI (0) DI (0) 2 d d O µc(N = 0) = > > P (I (0) − I (0)) PI (0) C p1 p1 p2 S N1 DI (0) d = µ2(N = 0). (30) US (0) o 2PIp1 US O Some stationary solutions are stable under the variation of 2 2 m two parameters, β and µ, when they give maximum or min- m d/(2p ) d/p cO max d/(p -p ) imum to the Hamiltonian and they are unstable when they 1 2 1 3 correspond to the Hamiltonian saddle point. It gives a stabil- (A) ity criterion in the framework of our variational approach in the form: N O 2 2 2 2 ! C ∂ H ∂ H ∂ H h = − β=β , µ=µ > 0. (31) ∂β2 ∂µ2 ∂β∂µ 0 0 S US 2 N The qualitative dependencies of N on µ0 for waveguides 1 with β0 = 0, l = 0 (marked “O”) and for β0 6= 0, l = 0

(marked “C”) are plotted in the Fig. 1. The dashed line in- N2 dicates the region where β2 < 0 and thus where waveguides C O with β 6= 0 cannot exist. Stable and unstable branches are US marked in this figure by “S” and “US”, respectively. It is seen S that the variational approach with a trial function Eq. (19) predicts a bistability phenomenon for stationary waveguide O propagation states: the coexistence of two waveguides with US the same width but with different numbers of quanta, see 2 2 Fig. 1a in the region (d/p2, µ ). The conclusion of bista- m o max d/(2p ) d/p d/(p1 -p3 ) bility also will be confirmed by the direct numerical solution 1 2 of the stationary and nonstationary equations in the next sec- (B) tion. 2 = (l) = (l) Fig. 1. N versus µ , d DId , pσ PIσ (a): for the case when N < N (σ < 1).; (b): for the case when N > N (σ > 1). Numerical modeling 1 2 1 2

In order to perform a numerical modeling of GNSE waveg- uides, it is more convenient to use it in a fully 3D form in The Eq. (33) was integrated numerically using the relax- Cartesian coordinates (ξ, η, ζ ). This equation has no pecu- ation process in the Fourier space which is a generalization liarity at the point r = 0. After the standart rescalings, this of a well-known stabilizing multiplier method, described in equation takes a form Petviashvili and Pokhotelov (1989). To check the structure’s stability, the evolutionary Eq. (32) was also integrated nu- ∂ψ 2 2 4 merically by a standart split step Fourier method (see Taha i + 1⊥ψ + 1⊥ψ − ψ |ψ| + κψ |ψ| = 0, (32) ∂ζ and Ablowitz, 1984). Simulation results are presented in a form of energy dis- where κ = KD2/P B2. Without loss of generality we shall persion diagrams (EDD) – dependencies N(λ). EDD for assume below that D = P = 1,B = −1 and vary pa- soliton, like (l = 0) and vortex, like (l = 1) nonlinear struc- rameter K. Stationary (along OZ) solutions have a form tures with β = 0 for different values of κ are plotted in ψ(ξ, η, ζ ) = ψ(ξ, η) exp(iλζ ) and obey the partial differ- 0 Fig. 2a and b. ential equation: It is seen that soliton like and vortex-like waveguides exist 2 2 4 −λψ + 1⊥ψ + 1⊥ψ − ψ |ψ| + κψ |ψ| = 0. (33) inside the restricted (both in N and in λ) area and the size Yu. A. Zaliznyak et al.: New types of stable nonlinear whistler waveguides 129

N 100 Y 90 K=0.1 3.0 80 K=0.2 K=0.3 2.5 70

60 2.0

50 1.5 40 1.0 30 0.5 20

10 0.0 0 l -0.5 r -2.0 -1.8 -1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 51015 (A) Y (A) 300 N 3 K=0.1 250 K=0.2

200 2

150

1 100

50 0 r 0 51015 -2,0 -1,5 -1,0 -0,5 l (B) (B)

Fig. 2. An energy dispersion diagram for stationary soliton, like Fig. 3. Stable stationary radially, symmetric soliton, like (l = 0, (l = 0, case (a)) and vortex, like (l = 1, case (b)) solutions of case (a), λ = −1.89) and vortex, like (l = 1, case (b), λ = −2.10) GNSE. Numerical result. Dashed lines indicate N(λ) dependencies solutions of GNSE with D = P = −B = 1,K = 0.1. predicted by the variational approach. Here D = P = −B = 1.

Vakhitov criterion which takes place in the case of P = 0. of their existence domain decreases when K increases. At When the parameter K exceeds some threshold value Kcr , every solitary or vortex branch there exists a range of λ – localized solutions of Eq. (33) disappear. values, where our numerical simulations confirm the varia- It was also confirmed numerically that soliton parameter λ tional conclusions about the bistability of stationary waveg- is bounded from below: for every K there exists some mini- uide propagation states in the sense of the coexistence of two mum value of λ = λmin (see Fig. 4). states with the same λ but with different energies (numbers of quanta) and spatial scales. Examples of bistable soliton like and vortex like states are presented in Figs. 3a and b. A numerical verification of stationary waveguide stabil- Conclusions ity has shown that all waveguides found above with l = 0 and l = 1 are stable even with respect to sufficiently high The propagation of whistler wave beams along the magnetic symmetric and asymmetric perturbations of the initial state. field lines with frequencies near the half electron gyrofre- Their parameters (width, amplitude, etc.) oscillate nonlin- quency (ω ≈ ωBe/2) in the ionosphere is described by the early in the vicinity of the stationary point. A conclusion single nonlinear Schrodinger¨ equation for the parallel elec- about the waveguide stability does not depend on the sign of tric field component Eq. (3). This equation includes a term 2 ∂N/∂λ derivative in contrast to the well-known Kolokolov- ∼ P 1⊥Ez in its linear part. 130 Yu. A. Zaliznyak et al.: New types of stable nonlinear whistler waveguides

l be applied. min Direct numerical integration of stationary (with respect to -0.5 Z) (Eq. 8) and nonstationary (Eq. 3) equations showed that there really exists soliton, like (with zero topological charge) -1.0 and vortex, like (first azimuthal mode l = 1) stable non- -1.5 linear waveguides. Simulations found exactly the bistability Solitons l=0 regions for waveguides (see Fig. 2). All stationary solutions Vortices l=1 -2.0 were shown to be stable even with respect to sufficiently high -2.5 symmetric and asymmetric perturbations of the initial state. The developed theory can be helpful for explaining the -3.0 regular structures formed during ionospheric heating experi- -3.5 ments (Bharuthram et al., 1992), e.g. for altitude h = 300 km that corresponds to maximum of the F-layer of the iono- -4.0 6 3 ◦ 6 sphere: ne ≈ 2 · 10 cm, Te ≈ 10 K, ωBe ≈ 8 · 10 −1 ≈ · −2 · ≈ · −5 ≈ − · 3 -4.5 c . D 2.5 10 1, P 1.6 10 , B 8 10 , K 2 0.0 0.2 0.4 0.6 0.8 K ≈ 0.8ν2 and κ = KD ≈ 10−7ν212  1, where e PB2 e  1 = ωBe
2 − > 1 2ω 1 0. Thus typical space scales√ Fig. 4. The minimum possible value of λ (where soliton, like l = 2 of the structures depend on 1: L⊥[cm]' 6 · 10 / 1, 0 and vortex, like l = 1 waveguides exist) versus K. Numerical L [cm]' 6 · 102/12. We think that there is a possibility result. Here D = P = −B = 1. || to observe such structures in ionosphere.

In the considered case D > 0, P > 0, B < 0, K > 0, References the Hamiltonian of GNSE (3) is bounded from below and above for every N, which indicates that there exists at least Anderson, D.: Variational approach to nonlinear pulse propagation one stable solitary solution, in the Lyapunov sense, which in otptical fibers, Physical Review, A 27, 3135–3145, 1983. corresponds to the Hamiltonian’s exact extremum. Bharuthram, R., Shukla, P. K., Stenflo, L., Nekrasov, A. K., and The sum of all linear terms in the virial relation for the Pokhotelov, O. A.: Generation of coherent structures caused by waveguide’s effective width acts in a defocusing manner, ionospheric heating, IEEE Trans. Plasma Sciense, 20, 6, 803– which indicates that any linear wave packet described by a 809, 1992. linear GNSE (3) always spreads out. At the same time the Davydova, T. A. and Zaliznyak, Yu. A.: Chirped Solitons Near nonlinear part of the virial relation includes focusing, as well Plasma Resonances in the Magnetized Dlasmas, Physica Scripta, as defocusing terms. It may result in a coexistence of several 69, 476–484, 2000. Gyrevich, A. V. and Shvarzburg, A. B.: Nonlinear theory of stationary nonlinear structures with different spatial scales in radiowave propagation in ionosphere (in Russian), Moskow, the framework of Eq. (3). Nauka, 1973. A variational approach with the trial function Eq. (19) pre- Petviashvili, V. I. and Pokhotelov, O. A.: Solitary waves in plasma dicts a bistability phenomenon for the stationary waveguide and atmosphere (in Russian), Moscow, Energoatomizdat, 1989. propagation states: the coexistence of two stable solutions Taha, M. J. and Ablowitz, M. J.: Analytical and Numerical As- with different energies (numbers of quanta N) and spatial pects of Certain Nonlinear Evolution Equations. II. Numerical, scales for the same value of the nonlinear shift of wave num- Nonlinear Schrodinger¨ Equation, J. Comput. Phys., 55, 203–230, ber λ. It also predicts that the sign of the derivative ∂N/∂λ 1984. may change within the region of stability, thus a commonly Zharova, N. A. and Sergeev, A. M.: On the stationary self-influence used Vakhitov-Kolokolov criterion (which works for GNSE of whistler waves Fizika Plazmy, 15, 1175–1179, 1989. with P = 0) in the considered case P > 0, K > 0 cannot