Nonlinear theory of the modulational instability at the ion-ion hybrid frequency and collapse of ion-ion hybrid waves in two-ion plasmas

We study the dynamics of two-dimensional nonlinear ion-ion hybrid waves propagating perpendicular to an external magnetic field in plasmas with two ion species. We derive nonlinear equations for the envelope of electrostatic potential at the ion-ion hybrid frequency to describe the interaction of ion-ion hybrid waves with low frequency acoustic-type disturbances. The resulting nonlinear equations also take into account the contribution of second harmonics of the ion-ion hybrid frequency. A nonlinear dispersion relation is obtained and, for a number of particular cases, the modulational instability growth rates are found. By neglecting the contribution of second harmonics, the phenomenon of collapse of ion-ion hybrid waves is predicted. It is shown that taking into account the interaction with the second harmonics results in the existence of a stable two-dimensional soliton.


I. INTRODUCTION
The presence of several species of ions is often found in both space and laboratory plasmas.In particular, space plasmas in most cases consist of several species of ions and the relative concentration of different species can vary in a fairly wide range.For example, the ionospheric and plasmaspheric plasma are composed of several species of ions [1], and in the upper ionosphere O + ions with a small addition of He + are predominant.Phenomena in multi-ion space plasmas have been intensively studied for many years [2][3][4][5].In laboratory conditions, a plasma with two ion species is of great interest primarily in relation to the ion cyclotron resonance frequency (ICRF) heating method in plasma magnetic confinement devices, where one of the most successful schemes involves minority species heating at the ion-ion hybrid resonance or at the minority cyclotron frequency [6][7][8] in H − D plasma.Recently, efficient plasma heating with the three-ion ICRH scenario with a small amount of 3 He ions in H − D mixture was suggested in Ref. [9,10].The presence of two ion species is inherent to dusty plasmas [11][12][13], where in addition to the main ion component, a second, heavy micronsize ions are present.Two-ion plasmas, although unmagnetized, also naturally arises in the inertial thermonuclear fusion experiments [14].
In a magnetized plasma consisting of electrons and two ion species with different charge-to-mass ratios, in addition to the lower-and upper-hybrid resonances, there is the so-called ion-ion hybrid resonance at the frequency ω ii defined by (1) and first introduced by Buchsbaum [15].Here, ω pα and Ω α are the plasma frequency and gyrofrequency of * Electronic address: vlashkin62@gmail.com the ions of species α = 1, 2, respectively with ω 2 pα = 4πZ 2 α e 2 n 0α /m α and Ω α = Z α eB 0 /m α c, where e is the elementary charge, n 0α , m α and Z α are the equilibrium density, the mass and charge number of the ions of species = 1, 2, respectively.Overall charge neutrality n 01 + n 02 = n 0e = n 0 is assumed, where n 0 is the equilibrium plasma density and n 0e is the equilibrium electron density.From Eq. ( 1) one can see that the ion-ion hybrid frequency ω ii lies between the gyrofrequencies Ω 1 and Ω 2 of the ions of different species.Note also that ω ii is determined only by the magnetic field B 0 and the relative population of each ion species.The presence of an additional type of ions in a magnetized plasma significantly modifies the dispersion relation and leads to the appearance of new branches of plasma oscillations that are absent in the single-ion case.The properties of such a plasma differ in many respects from the properties of single-ion plasma.Linear theory of wave propagation in plasmas with two species of ions, including an inhomogeneous plasma, has been considered in quite a few works (see, e.g., Refs.[16][17][18][19][20][21]). Parametric instabilities in a plasma with two ion species were investigated in Refs.[22][23][24][25], where the standard kinetic method for studying parametric instabilities [26,27] was used, as well as in Refs.[28,29] within the framework of fluid model.
The linear theory is valid only for sufficiently small wave amplitudes, when nonlinear effects can be neglected.Nonlinear coherent structures in plasma, in particular solitons, have been the subject of intensive theoretical study for several decades and have been experimentally observed both in laboratory and space plasmas [30][31][32].In a broad sense, a soliton is a localized structure (not necessarily one-dimensional) resulting from the balance of dispersion and nonlinearity effects.Multidimensional solitons often turn out to be unstable, and the most well-known phenomena in this case are wave collapse and wave breaking [33][34][35].Despite the obvious importance of studying nonlinear phenomena occurring in a plasma with two species of ions in the vicinity of the ionion hybrid frequency, the corresponding nonlinear theory, especially in the multidimensional case, has not been suf-ficiently developed, in contrast to the lower-hybrid (LH) and upper-hybrid (UH) resonances, for which nonlinear phenomena have been studied in more detail.In particular, in the one-dimensional case, various types of solitons, including envelope solitons, were discovered at the LH [36][37][38] and UH frequencies [39][40][41].In the multidimensional case, the phenomenon of collapse of the LH [42][43][44][45][46] and UH waves [47,48] was predicted.Taking nonlocal nonlinearity into account, stable two-dimensional UH solitons and vortex solitons were found in Ref. [49].
Note that one-dimensional solitons in a plasma with two species of ions were considered in a number of works [50][51][52][53][54][55][56][57][58], but they were, with the exception of Refs.[50,53], not the envelope solitons at the ion-ion hybrid frequency (that is, they were not ion-ion hybrid solitons), but to solitons in the frequency range of much lower or much higher the ion-ion hybrid frequency ω ii .
One-dimensional (1D) nonlinear waves near the ion-ion hybrid frequency ω ii were considered in Refs.[50,53].In both of those works, equations for the wave envelope at the ion-ion hybrid frequency ω ii were derived.In Ref. [50], an equation with nonlocal nonlinearity was obtained, however, the nonlinearity was incorrectly taken into account due to the neglect of the usual striction nonlinearity associated with the ponderomotive force, in comparison with the nonlocal nonlinearity due to the interaction with the second harmonics.In Ref. [53], the 1D nonlinear Schrödinger equation (of both focusing and defocusing types) was obtained.In that work, however, the action of the ponderomotive force of the HF field of ion-ion hybrid waves on ions was completely neglected, although the ion contribution is comparable (and sometimes exceeds) the electron contribution.Besides, the dispersion of low-frequency waves was incorrectly taken into account, so that the results obtained in Ref. [53] have a very limited area of applicability.We also note that in the one-dimensional theory there is no possibility of taking into account the vector (gyrotropic) nonlinearity, similar to the nonlinearity that occurs near the LH resonance [42,44].
The aim of this paper is to obtain two-dimensional (2D) nonlinear equations to describe the interaction of ion-ion hybrid waves with low-frequency (LF) acoustictype disturbances.In the equation for the envelope we also take into account the interaction of second harmonics at the ion-ion hybrid frequency.In the case where second harmonics can be neglected, we predict a collapse of ion-ion hybrid waves, similar to the collapse for the LH and UH waves.Taking into account the additional nonlinearity associated with the second harmonic, however, leads to a stable 2D soliton.
The paper is organized as follows.In Sec.II, we derive a set of nonlinear equations for the wave envelope and LF ion density perturbations.A nonlinear dispersion relation was obtained in Sec.III.In Sec.IV the phenomenon of collapse of ion-ion hybrid waves is predicted.A stable 2D soliton, taking into account the second harmonic, was found in Sec.V. Finally, Sec.VI concludes the paper.

II. MODEL EQUATIONS
For a cold plasma containing two ion species and immersed in a homogeneous external magnetic field B 0 = B 0 ẑ, where ẑ is the unit vector along the z-direction, the linear dispersion relation for electrostatic waves propagating normal to B 0 is given by [59] where ω pe and Ω e are the electron plasma frequency and electron-cyclotron frequency, respectively.In the high frequency range ω ≫ ω p1,2 , Ω 1,2 , where only electrons take part in the plasma motion, for the upper-hybrid frequency ω UH we get, In the intermediate frequency range Ω 1 , Ω 2 ≪ ω ≪ ω pe , Ω e , solution of Eq. ( 2) yields the frequency of the lower-hybrid resonance ω LH , Here, only ions play an active role in motion (the role of electrons is reduced to screening).Assuming ω ≪ Ω e , equation (2) can be written as In the lowest frequency range Ω 1 , Ω 2 ≪ ω p1 , ω p2 , we have and Eq. ( 5) can be simplified to On assuming that equation ( 7) yields the ion-ion hybrid frequency ω ii determined by Eq. (1).In this case, Ions of both species equally take part in the plasma motion and move in opposite phases, and it is this, compared with the cases of UH and LH resonances, that complicates the description of the behavior of electrostatic waves near the ion-ion hybrid resonance.
In this section, we derive nonlinear equations to describe the dynamics of waves near the ion-ion hybrid frequency ω ii .We consider the case of an arbitrary ratio of the ion densities n 01 and n 02 , as well as arbitrary ratios of the electron and ion temperatures.The reason is that in plasma of the Earth's ionosphere, for example, the ratios of these quantities can be, depending on the altitude, either comparable to each other, or significantly (sometimes by orders of magnitude) differ from each other in one side or another [1].

A. High-frequency disturbances
The basic equations governing the plasma dynamics are the fluid equations of motion and continuity of the ions of both species and electrons, where n α , v α , p α , e α and m α are the density, velocity, pressure, charge, and mass of the particle species α = e, i (electrons and ions), respectively.For the gas kinetic pressure we take p α = γ α n α T α , where T α is the temperature and γ α is the ratio of specific heats, and in the next we introduce the notation v T α = γ α T α /m α for the particle thermal velocity.Equations ( 10) and (11) are supplemented by the Poisson equation, Following the well-known idea dating back to the original work by Zakharov [60] of separating the slow and fast time scales and averaging over the fast time, we represent the electrostatic potential, velocities, and densities in the form where c.c. stands for the complex conjugate, v (1),( 2) α and n (1),( 2) α are assumed to vary on a timescale much more slowly than 1/ω ii .The contribution of the second harmonic of frequency ω ii is taken into account in Eqs. ( 13), ( 14) and ( 15), and we assume that the amplitudes of the first harmonic are much larger than the others so that conditions |v α , δn α are assumed to be met.Note that, as shown in Refs.[32,61,62], taking into account the second harmonic of the Langmuir frequency ω pe in a nonmagnetized plasma may halt Langmuir collapse in two or three dimensions.
The ion-ion hybrid waves have wave numbers almost normal to the external magnetic field (k z ≪ k ⊥ ), and in this paper we restrict ourselves to the 2D case of perpendicular propagation when the condition is satisfied.Substituting Eq. ( 14) into Eq.( 10) , we have where where ∇ ⊥ = (∂/∂x, ∂/∂y).Substituting Eq. ( 15) into Eq.( 11), we find where J α is the nonlinear current, As noted above, electrons do not take part in the plasma motion at the frequency of the ion-ion hybrid resonance, and ions of different species move in opposite phases, so we can write 1 + e 2 n (1) In zero order in iω ii ∂ t /(Ω 2 α − ω 2 ii ) ≪ 1, neglecting the thermal dispersion and nonlinearity, from Eq. ( 17) we obtain the perpendicular velocity and then from Eq. ( 19) the density perturbation where In the following order, taking into account the thermal dispersion and nonlin-earity, one can obtain where we have neglected non-stationary corrections ∼ ∂ t ∇n α and ∼ ∂ t F (1) in terms responsible for thermal dispersion and nonlinearity, respectively.Using Eq. ( 24) we substitute α into Eq.( 19) and get In Eq. ( 25) we substitute the zero approximation (23) in the term with ∆n α responsible for weak thermal dispersion, and then, multiplying Eq. ( 25) by 4πe α , we use Eq. ( 21).As a result, taking into account that one can finally obtain where the right hand side of Eq. ( 27) corresponds to the nonlinear terms.Equation ( 27) can be rewritten in the form where R is the dispersion length defined by In the linear approximation, taking ϕ (1) ∼ exp(ik ⊥ • r − iωt), where ω and k ⊥ the frequency and perpendicular wave vector respectively, Eq. ( 28) yields the dispersion relation of ion-ion hybrid wave, From Eqs. ( 10) and (11), for the second harmonic perturbations v α and n (2) and respectively, where v 0α and n (1) 0α are determined by Eqs. ( 22) and (23).The perturbation of the electrostatic potential at the second harmonics ϕ (2) is determined from Eq. ( 12), where we neglect the electron contribution at the frequency 2ω ii as before at ω ii .

B. low-frequency disturbances
For the LF disturbances with ω ≪ Ω 1 , Ω 2 , ions of both species are strongly magnetized and move only along the external magnetic field.Then, the LF motion is governed by the continuity equation, and the parallel momentum equation for each ion species α = 1, 2, where is the ponderomotive force (per unit ion mass) acting on the ions due to the high-frequency (HF) pressure of the ion-ion hybrid waves, and the angular brackets denote the average over the fast time.From Eqs. ( 34) and (35) we have, (37) For inertialess electrons in slow motions, one can write the force balance equation along the magnetic field, where is the ponderomotive force (per unit electron mass) acting on the electrons.From Eqs. ( 14), ( 36) and ( 39) we find, The expressions for the perpendicular ion v α,⊥ and electron v (1) e,⊥ velocities follow from Eq. ( 22) and are given by respectively.For parallel ion v α,z and electron v e,z velocities, from Eqs. ( 10) and ( 14) we have v (1)  α,z = − v (1)  e,z = ie m e ω ii ∂ϕ (1)  ∂z .
The quasineutrality condition reads Then substitution Eq. ( 49) into Eq.( 37), taking into account Eqs. ( 46) and ( 50), gives after some transformations, and where v s1 = T e /m 1 and v s2 = T e /m 2 are the ion sound speeds of species 1 and 2, respectively, and we have introduced the notation for relative ion concentration ν α = n 0α /n 0 , (α = 1, 2).When obtaining Eqs. ( 51) and ( 52), we took into account that the electron contribution to the scalar nonlinearity turns out to be smaller than the contributions of other terms by a factor of order ∼ m α /m e , and thus it can be neglected.The contributions of the electron and ion vector nonlinearities, as well as the ion scalar nonlinearity, are of the same order.In the linear approximation, assuming δn α ∼ exp(ik z z − iΩt), Eqs. ( 51) and ( 52) give the dispersion relation where k z is the parallel wave number.The linear dispersion relation (53) corresponding to two modes in a plasma with two ion species was first obtained in Ref. [21].Introducing notation q = (k, ω) and using the convolution identity (f and g are arbitrary functions) from Eqs. ( 51) and ( 52) one can write explicit expressions for the ion density perturbations in the Fourier space (taking ∼ exp(ik • r − iΩt)) as where Equations ( 51) and ( 52) describe the dynamics of LF acoustic-type disturbances (in the linear case corresponding to two branches of ion-ion sound) under the action of the ponderomotive force of the HF field of ion-ion hybrid wave.
C. neglecting second harmonics Equations ( 28), ( 31)- (33) for the HF motions along with Eqs. ( 51) and ( 52) for LF disturbances is a closed system of nonlinear equations for the HF envelope of electrostatic potential ϕ (1) and LF ion density perturbations δn 1 and δn 2 .It can be seen, however, that due to taking into account the nonlinear terms corresponding to the second harmonic of the ion-ion hybrid frequency (that is, containing ϕ (2) , v α , and n (2) α ), this system turns out to be extremely cumbersome and very difficult to analyze.In particular, taking into account second harmonics leads to terms containing F α .In addition, vector nonlinearities have, generally speaking, the same order as scalar ones.This distinguishes the case under consideration from the case of nonlinear upper-hybrid waves, where, under reasonable conditions, vector nonlinearity can be neglected, as well as from the case of lower-hybrid waves, when, on the contrary, vector nonlinearity is always dominant.However, the system of equations ( 28) and ( 31)-( 33) is greatly simplified for radially symmetric field distributions.We show below in Sec.V that taking into account the contribution of the second harmonics results in the existence of a stable 2D soliton, but for now we neglect this contribution.Then, Eq. ( 28) takes the form (here and after denoting ϕ = ϕ (1) which can be rewritten as The first term in square brackets on the right hand side of Eq. ( 59) corresponds to the scalar nonlinearity, and the second term in the form of the Poisson bracket corresponds to the so-called vector nonlinearity.The latter identically vanishes in the one-dimensional case and also for radially symmetric field distributions.

III. NONLINEAR DISPERSION RELATION
In this section we consider the linear theory of the modulational instability of a pump wave with a frequency close to the ion-ion hybrid frequency ω ii in the framework of the model equations ( 51), ( 52) and (59).We decompose the ion-ion hybrid wave into the pump wave and two sidebands, i.e.
where δ k = ω ii k 2 ⊥ R 2 /2, while the low frequency perturbations of ion plasma densities are expressed as The amplitudes of the up-shifted ϕ + and down-shifted ϕ − satellites can be calculated from Eq. ( 60).We have where D ± is the Fourier transform of the linear operator in the left hand side of Eq. ( 59) evaluated in k ± q and δ k ± Ω, and The amplitudes of the LF perturbations n1 and n2 can be found from Eqs. ( 55) and ( 56), where Ω 2 + and Ω 2 − are determined by Eq. ( 53), and where By combining Eqs. ( 63), ( 64), (68), and (69) we obtain a nonlinear dispersion relation Note, that in the case of coplanar (in the plane perpendicular to the magnetic field) wave vectors k ⊥ q ⊥ , the parametric coupling of the waves due to the vector nonlinearity is absent, while the coupling due to the scalar nonlinearity is the most effective.In general case k ⊥ ∦ q ⊥ both types of the nonlinearities yield comparable contribution, and this, taking into account that Eq. ( 73) is an equation of the sixth degree in Ω, leads to a rather complicated picture of instability.Significant simplifications are possible in a number of special cases.For example, assuming that Ω ≪ Ω − , Ω + , k ⊥ q ⊥ and k ⊥ ≫ q ⊥ , the dispersion equation ( 73) after direct calculations can be reduced to the form where v g = ω ii k ⊥ R is the group velocity of the ion-ion hybrid wave, and we have introduced the notations for the coefficients F and G, which we will use in what follows.Equation (74) predicts instability with the growth rate γ = |Im Ω|, .
(77) In the opposite case k ⊥ ≪ q ⊥ , one can get and Eq. ( 78) describes a purely growing instability with the growth rate given by Eq. (77).For example, for the upper F region/topside ionosphere (∼ 500 km), taking and Ω 1 ∼ 2 • 10 2 s −1 in accordance with Ref. [1] (subscripts 1 and 2 correspond to O + and He + , respectively, and the concentration of H + is two orders of magnitude less than the concentration of O + ), the estimate for the threshold field at q ⊥ R ∼ 0.1 is E 0 ∼ 100 mV/m.

IV. COLLAPSE OF ION-ION HYBRID WAVES
In this section, neglecting the second harmonics, we discuss the possibility of collapse of the ion-ion hybrid waves.Let us consider an important case when we can neglect the time derivatives in the LF equations ( 51) and (52).Physically, this corresponds to the balance between gas-kinetic and wave (ponderomotive) pressures.In this static approximation ("subsonic" case), from Eqs. ( 51) and ( 52) for LF perturbations of ion densities δn 1 and δn 2 one can obtain, where {ϕ (1) , ϕ (1) * } , (81) and G is determined by Eq. ( 76).The perturbation of ion densities due to the vector nonlinearity can correspond to both a density well and a hump and depends on the relative phase of ϕ and ϕ * .Introducing dimensionless variables by where F is determined by Eq. (75), and substituting Eqs.
(79) and (80) into Eq.( 60), one can obtain where we have introduced the dimensionless coefficients c 1 , c 2 and c 3 , where G 1 , G 2 and G 3 are determined in the Appendix.The nonlinear equation (85) for the envelope ψ differs significantly from the corresponding equations (in the subsonic case) for nonlinear Langmuir, lower-hybrid and upper-hybrid waves.If there is only scalar nonlinearity, that is c 1 = 0, c 2 = 0 and c 3 = 0, Eq. ( 85) is reduced to the well-known equation for nonlinear Langmuir waves [60].The case c 2 = 0 and c 3 = 0 corresponds to nonlinear upper-hybrid waves (with another dimensionless coefficient c 1 ) [49].If the scalar nonlinearity can be neglected, and also if c 2 = 0 and c 3 = 0 (again with another dimensionless coefficient c 1 ), Eq. ( 85) becomes the equation for the envelope at the lower-hybrid frequency [42][43][44][45].In all these cases, the corresponding dimensionless variables for time and space coordinates and the electrostatic potential envelope are implied.The nonlinearities corresponding to the fifth (c 2 = 0) and sixth (c 3 = 0) terms in Eq. (85) have apparently never been considered in nonlinear problems before.Equation (85) conserves the plasmon number the momentum where the momentum density is the angular momentum and the Hamiltonian Note that the second term in H is a focusing nonlinearity, while the sign of the other terms (which, as you can easily see, is real since {ψ, ψ * } purely imaginary) depend on the phase relation between ψ and ψ * and also on the signs of the coefficients c 1 , c 2 and c 3 , that is, on the ratio of ion and electron temperatures, masses of ions of different species and their relative concentrations.An essential feature of the considered model equation ( 85) is its two-dimensional nature and the cubic nonlinearity.The stationary solution of Eq. ( 85) in the form of ψ(r, t) = Ψ(r) exp(iλ 2 t) corresponds to a stationary point H for a fixed plasmon number N and resolves the variational problem that is By analogy with Langmuir, upper-hybrid [33] and lowerhybrid [32,63] waves, multiplying Eq. ( 93) by Ψ * , and then integrating over the whole D-dimension space, we get where On the other hand, one can write Next, we consider an N -preserving scaling transformation Ψ (α) = Ψ(αr) and introduce the corresponding values It is evident that from which we have, From Eqs. (94), ( 98) and (102) we then find Equation ( 103) is identical to the relationship between the Hamiltonian H and the number of plasmons N for nonlinear Langmuir waves [32,33].It can be seen that the reason for the coincidence is the same linear parts (dimensionless) and the cubic nature of the nonlinear terms.
Since for the considered 2D model we have H = 0 for stationary solutions, one can conclude that an arbitrary initial localized field distribution with H = 0 never reaches a stationary state in the course of evolution, that is, either spreads out or collapses.Hamiltonian (91) is not positively definite, despite the fact that the third term in Eq. (91), as mentioned above, may have a defocusing character.A rigorous proof of the collapse of ion-ion hybrid waves (as well as Langmuir waves in arbitrary geometry) is apparently a very difficult problem.Here we only point out, taking into account the arguments presented above, that with a negative initial Hamiltonian, the collapse of two-dimensional ion-ion hybrid waves apparently occurs.

V. STABLE 2D RADIALLY SYMMETRIC SOLITON
In the radially symmetric case, the vector nonlinearities vanish identically, and Eqs. ( 22), ( 23) and ( 31) takes the form where E = −∇ ⊥ ϕ (1) and E (2) = −∇ ⊥ ϕ (2) is the electric field at the second harmonics.Equation ( 33) can be rewritten as From Eqs. ( 32) and (107), we have Using Eq. ( 106), we eliminate v α in Eq. ( 108) and then substitute expressions for n Inserting Eq. (109) into Eq.(106) we have It can be shown that the second term in Eq. ( 32) can be neglected, and then substituting Eq. (111) into Eq.( 32) one can obtain The term corresponding to the contribution of the second harmonics on the right hand side of Eq. ( 28) can be written as ∇ ⊥ • N (2) , where In the considered radially symmetric case, we are interested here only in the radial component E r of the electric field E.Then, writing Eq. ( 60) through the electric field E with the additional term which takes into account the contribution of second harmonics, and taking its radial projection, we have for E r , 2i where ∆ r = ∂ 2 /∂r 2 + (1/r)∂/∂r is the 2D radial Laplacian, and for the considered 2D case we have used the relation (∆E) r = ∆ r E r − E r /r 2 .Further, as in the previous section, we consider the static approximation (79) and (80) for ion density perturbations, and take into account that, in the radially symmetric case, the vector nonlinearities in Eqs. ( 81) and (82) vanish identically.Next, we use the dimensionless variables defined by Eq. ( 83), and introduce the dimensionless radial electric field E through From Eqs. (79), (80), ( 113) and (114), one can finally obtain and Hamiltonian (118) An equation similar to Eq.
(116) was obtained in Ref. [64], where the influence of electron-electron nonlinearities on unstable two-dimensional and threedimensional Langmuir solitons was studied.In that work it was shown that the effective radius r ef f , defined as is bounded from below provided Q = 0, so that additional nonlinear terms proportional to Q prevent collapse (the same applies to the 3D case).Moreover, it has also been shown that the gradient norm |∂E/∂r| 2 d 2 r is bounded from above by conserved quantities Ñ and H.The authors of Ref. [64] also numerically found the 2D soliton solution and demonstrated the stability of such a soliton by direct numerical simulation of the soliton dynamics within the framework of Eq. ( 116).Thus, we can conclude that, taking into account the additional nonlinearity associated with the second harmonics of the ion-ion hybrid frequency, in the radially symmetric case there exists a stable two-dimensional ion-ion hybrid soliton.

VI. CONCLUSION
We have derived a nonlinear system of equations for the envelope of electrostatic potential at the ion-ion hybrid frequency to describe the interaction between an ion-ion hybrid waves and LF acoustic-type disturbances in a magnetized plasma with two species of ions.The resulting nonlinear equations also take into account the contribution of second harmonics of the ion-ion hybrid frequency.We have obtained a nonlinear dispersion relation predicting the modulational instability of ion-ion hybrid waves.For a number of particular cases, the modulational instability growth rates have been found.By neglecting the contribution of second harmonics, the phenomenon of collapse of ion-ion hybrid waves is predicted.It has been also shown that taking into account the interaction with the second harmonics suppresses collapse of ion-ion hybrid waves and results in the existence of a stable two-dimensional soliton.The developed theory is applicable to a wide range of theoretical and experimental problems in both space and laboratory (primarily devices with magnetic plasma confinement) plasma with two species of ions.
A number of open questions remains to be addressed: 1) We have restricted ourselves to the 2D case, when the condition ( 16) is met and the ion-ion hybrid wave propagates perpendicular to the external magnetic field.In a more general case, it is necessary to take into account an additional term of the form ∼ (k 2 z /k 2 )(m α /m e ) in the dispersion relation of the ion-ion hybrid wave Eq. (30).Then the model becomes three-dimensional and essentially anisotropic.The anisotropy of the models in the cases of upper-and lower-hybrid resonances results in the absence of a stationary point of the Hamiltonian (for a fixed number of plasmons), that is, in this case there are no three-dimensional soliton solutions (even unstable ones) [32,34].Note that the results obtained in Refs.[32,34] essentially use the cubic type of nonlinearity.A similar situation apparently occurs in the case of ion-ion hybrid resonance.
2) The model under consideration takes into account the interaction of HF ion-ion hybrid waves only with electrostatic LF disturbances which corresponds to a pertur-bation of the plasma density and neglects the interaction with nonpotential LF disturbances of the Alfvén type, which would correspond to an LF perturbation of the magnetic field (the interaction of LF Alfvén waves with HF lower-hybrid and upper-hybrid waves was considered in Refs.[46,49]).Such neglect corresponds to the smallness of the magnetic pressure in comparison with the plasma gas-kinetic pressure, and is valid under the condition v Aα ≪ v sα , where v Aα = B 0 / √ 4πn 0α m α is the Alfvén velocity of the ion species α.
3) Accounting for second harmonics is not the only reason for stopping the collapse and the existence of stable 2D solitons.In the static approximation, the Boltzmann distribution of electrons and ions leads to a saturating exponential nonlinearity.Stable multidimensional Langmuir solitons with this type of nonlinearity were obtained in Refs.[65,66].Then, apparently, the stable 2D ion-ion hybrid solitons could exist without accounting for the contribution of second harmonics.