Novel soliton in dipolar BEC caused by the quantum fluctuations
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] S e p Novel soliton in dipolar BEC caused by the quantum fluctuations
Pavel A. Andreev ∗ Faculty of physics, Lomonosov Moscow State University, Moscow, Russian Federation, 119991. (Dated: September 29, 2020)Solitons in the extended hydrodynamic model of the dipolar Bose-Einstein condensate with quan-tum fluctuations are considered. This model includes the continuity equation for the scalar field ofconcentration, the Euler equation for the vector field of velocity, the pressure evolution equationfor the second rank tensor of pressure, and the evolution equation for the third rank tensor. Largeamplitude soliton solution caused by the dipolar part of quantum fluctuations is found. It appearsas the bright soliton. Hence, it is the area of compression of the number of particles. Moreover, itexists for the repulsive short-range interaction.
PACS numbers: 03.75.Hh, 03.75.Kk, 67.85.PqKeywords: quantum hydrodynamics, pressure evolution equation, extended hydrodynamics, quantum fluc-tuations, dipolar BEC.
Dipolar Bose-Einstein condensates (BECs) demon-strate the formation of quantum droplets [1], [2], [3], [4],[5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16],[17], [18], [19], [20]. Each droplet is the small cloud ofatoms, which can be considered as the soliton-like areaof increased concentration. System of droplets is thehighly nonlinear structure explained via the dipolar partof the quantum fluctuations via the generalized Gross-Pitaevskii (GP) equation including the fourth order non-linearity.The traditional GP equation contains one nonlinearterm which is proportional to the third degree of themacroscopic wave function [21]. This nonlinearity iscaused by the major part of the short-range interactioncontribution. In spite the fact that the BECs is the collec-tion of particles in the quantum state with the lowest en-ergy, a part of particles can exist in the exited states evenat the zero temperature. It happens via the interactionbeyond the mean-field approximation. This phenomenonis called the BEC depletion caused by the quantum fluc-tuations. It is studied both theoretically [22], [23], [24],[25] and experimentally [26], [27], [28]. If one considersthe quantum fluctuations caused by the short-range in-teraction within the Bogoliubov-de Gennes theory onecan find additional nonlinear term in the GP equationwhich is also caused by the short-range interaction, butit is proportional to the fourth degree of the macroscopicwave function. It contains same interaction constant asthe traditional GP equation.The dipolar BEC has long history of study over thelast 20 years [29], [30], [31], [32], [33], [34], [35], [36], [37],[38]. The nonsuperfluid fermionic dipolar gases are alsoconsidered in literature [39], [40]. Traditionally the con-densate depletion is studied in terms of Bogoliubov-deGennes theory [41], [42], [43]. It includes the depletionof the dipolar BECs. However, here we present the mi-croscopic many-particle quantum hydrodynamic theory ∗ Electronic address: [email protected] of the quantum fluctuations.Although, the dipole-dipole interaction modifies prop-erties of nonlinear structure in BECs including the widthand the amplitude of the bright and dark solitons in theBECs [44]. To the best of our knowledge, the dipole-dipole interaction brings no novel soliton formation. InRef. [44] the analytical solutions are demonstrated forthe dipole-dipole interaction of the point-like objects.Here, it is found that the dipolar part of quantum fluc-tuations causes the soliton solution.Recently, the microscopically justified quantum hy-drodynamic model describes the quantum fluctuationsvia the equations additional to the Euler and continuityequations [45], [46]. The interparticle interaction createsthe source of the third rank tensor Q αβγ , which gives thenonzero value of kinetic pressure p αβ . Both the kineticpressure and the third rank tensor Q αβγ are related to theoccupation of the excited states. Their nonzero values forthe BECs are the consequence of the quantum fluctua-tions in BECs. The pressure evolution equation containsno trace of the interaction. Hence, its value depends onthe third rank tensor Q αβγ only. The third rank tensorevolution equation contains the gradient of the concen-tration square n multiplied by the additional interactionconstant for the short-range interaction. The long-rangedipole-dipole interaction leads to the third derivative ofthe macroscopic potential of the dipole-dipole interac-tion. However, no contribution of the external field, suchas the trapping potential, is present in this equation.The description of collisions of solitons in BECs re-quires models obtained beyond the mean-field approxi-mation [47], [48], [49], [50]. In this paper we demonstratethat the described above beyond mean-field model pro-vides a novel soliton solution in dipolar BECs.Novel soliton solution in dipolar BECs is found interms of the extended quantum hydrodynamic model,where the continuity equation for the scalar field of con-centration n , the Euler equation for the vector field ofvelocity v , the pressure evolution equation for the secondrank tensor of pressure p αβ , and the evolution equationfor the third rank tensor Q αβγ are used. In this model,the quantum fluctuations are presented in the equationfor the third rank tensor evolution. This model repre-sents the microscopic motion of the quantum particlesvia the functions describing the collective dynamics [51],[52], [53], [54], [55], [56], [57], [58], which is described bythe many-particle Schrodinger equation: ı ¯ h∂ t Ψ( R, t ) = " N X i =1 (cid:18) ˆ p i m + V ext ( r i , t ) (cid:19) + 12 X i,j = i U ij + 12 µ X i,j = i − r z,ij /r ij r ij Ψ( R, t ) , (1)where m is the mass of the atom, ˆ p i = − ı ¯ h ∇ i is themomentum of i-th particle, ¯ h is the Planck constant, µ is the magnetic moment of the atom, Ψ( R, t ) is the wavefunction for the system of N quantum particles, R = { r , ..., r N } , V ext is the external potential. The short-range part of boson-boson interaction is presented viapotential U ij = ( r ij ), where r ij = | r ij | , and r ij = r i − r j .The long-range dipole-dipole interaction of align dipolesis presented by the last term of the Schrodinger equation(1).Transition to the description of the collective motion ofbosons is made via the introduction of the concentration(number density) [54], [59], [60], [61]: n = Z dR N X i =1 δ ( r − r i )Ψ ∗ ( R, t )Ψ(
R, t ) . (2)The integral in equation (2) contains the element of vol-ume in 3 N dimensional space dR = Q Ni =1 d r i .The derivation [60] shows that the concentration (2)obeys the continuity equation ∂ t n + ∇ · ( n v ) = 0 . (3)The current of particles is proportional to the momen-tum density and presented by the following equation j = Z dR N X i =1 δ ( r − r i ) ×× m i (Ψ ∗ ( R, t )ˆ p i Ψ( R, t ) + c.c. ) , (4)with c.c. is the complex conjugation. The current allowsto define the velocity vector field: v = j n .The evolution of the current (4) follows from theSchrodinger equation (1) and can be presented by theEuler equation: mn∂ t v α + mn ( v · ∇ ) v α − ¯ h m n ∇ α △√ n √ n + ∂ β T αβqf + n∂ α V ext = − gn∂ α n − n∂ α Φ d , (5) where △ = ∂ β ∂ β , and the Einstein’s rule for the summa-tion on the repeating subindex is applied.Major contribution of the short-range interaction ap-pears in the mean-field approximation [21] correspondingto the first order by the interaction radius [60]. It is pre-sented by the first term on the right-hand side of theEuler equation (5). It contains the interaction constant g expressed via the potential: g = Z d r U ( r ) . (6)The long-range of interaction is presented in the cor-relationless form corresponding to the main contributionof the interaction via the macroscopic electrostatic po-tential:Φ d = µ Z d r ′ | r − r ′ | (cid:18) − z − z ′ ) | r − r ′ | (cid:19) n ( r ′ , t ) . (7)The dipole-dipole interaction presented by potential(7) is not full dipole-dipole interaction, but the long-range asymptotics of atom-atom interaction. Potential(7) satisfies the following differential equation △ Φ d = 4 πµ ( ∂ z n − △ n ) , (8)which is the modification of the Poisson equation. Equa-tion (8) is derived for the arbitrary directions of the pairof dipoles with the further transition to the pair of aligneddipoles.The evolution of the particles current j (4) leads to theflux of momentum Π αβ which is defined as followsΠ αβ = Z dR N X i =1 δ ( r − r i ) 14 m [Ψ ∗ ( R, t )ˆ p αi ˆ p βi Ψ( R, t )+ ˆ p α ∗ i Ψ ∗ ( R, t )ˆ p βi Ψ( R, t ) + c.c. ] . (9)Equation (5) contains the momentum flux representedvia the velocity field.The continuity equation and the Euler equation arepresented via the velocity field. Transition of the gen-eral equations to this form can be made by the represen-tation of the macroscopic wave function Ψ( R, t ) via thereal functions Ψ(
R, t ) = a ( R, t ) exp( ıS ( R, t ) / ¯ h ). The realfunctions can be called the amplitude of the wave func-tion a ( R, t ), and the phase of the wave function S ( R, t ).The gradient of the wave function gives the velocity of thequantum particle: v i ( R, t ) = ∇ i S ( R, t ) /m i . The devia-tion of the velocity of quantum particle from the averagevelocity u i ( r , R, t ) = v i ( R, t ) − v ( r , t ) can be called thethermal velocity, or it is the velocity in the local comov-ing frame.Final expression for the momentum flux is obtained inthe following form:Π αβ = nv α v β + p αβ + T αβ , (10)where the tensor function p αβ in equation (10) is thekinetic pressure p αβ ( r , t ) = Z dR N X i =1 δ ( r − r i ) a ( R, t ) m i u αi u βi , (11)and the simplified form of the second rank tensor T αβ isfound as follows: T αβ = − ¯ h m (cid:18) ∂ α ∂ β n − n ( ∂ α n )( ∂ β n ) (cid:19) + T αβqf . (12)The BEC is the collection of particles in the quantumstate with the lowest energy. It corresponds to the zerotemperature T = p ββ / n . Therefore, the zero kineticpressure is the characteristic of the BEC. However, wecan consider the pressure evolution equation. Its deriva-tion is made for the arbitrary distribution of particlesover quantum states. It corresponds to the arbitrary tem-peratures. Transition to the zero temperature is madeafter derivation of the general equation.Hence, the derivation of the momentum flux evolutionequation is made by the consideration of the time deriva-tive of function Π αβ (9). Next, we introduce the velocityfield in accordance with the method shown before equa-tion (10). The final equation reduces to the equation forthe part of the quantum Bohm potential T αβ (12) causedby the quantum fluctuations T αβqf (it can be also inter-preted as the part of pressure caused by the quantumfluctuations): ∂ t T αβqf + ∂ γ ( v γ T αβqf ) + T αγqf ∂ γ v β + T βγqf ∂ γ v α + ∂ γ Q αβγqf = 0 . (13)All terms in equation (13) are proportional to T αβqf andthe flux of pressure in the comoving frame Q αβγqf . Hence,tensor Q αβγqf goes to zero together with the kinetic pres-sure at the zero temperature. Hence, equation (13) givesthe identity 0=0. However, it is the quasi-classical de-scription of the pressure evolution equation. To under-stand the full quantum picture, we need to derive theequation for the third rank tensor evolution Q αβγqf at thearbitrary temperature. Afterwords, we make the transi-tion to the zero temperature in the derived equation. Asthe result we get equation (18) presented below. Equa-tion (18) shows that ∂ t Q αβγqf = 0 at the zero tempera-ture due to the nonzero contribution of the interaction.Hence, there is the interaction related source of Q αβγqf andconsequently it gives the source of T αβqf via the pressureevolution equation. Therefore, the interaction causes theoccupation of the quantum states with nonminimal en-ergies. This description corresponds to the well-knownnature of the quantum fluctuations [23], [41], [42], [43].No interaction gives contribution in the pressure evo-lution equation (13). The form of the trapping potentialdoes not affect the pressure evolution.The evolution of the second rank tensor of pressure leads to the flux of the momentum flux: M αβγ = Z dR N X i =1 δ ( r − r i ) 18 m i (cid:20) Ψ ∗ ( R, t )ˆ p αi ˆ p βi ˆ p γi Ψ( R, t )+ˆ p α ∗ i Ψ ∗ ( R, t )ˆ p βi ˆ p γi Ψ( R, t ) + ˆ p α ∗ i ˆ p γ ∗ i Ψ ∗ ( R, t )ˆ p βi Ψ( R, t )+ ˆ p γ ∗ i Ψ ∗ ( R, t )ˆ p αi ˆ p βi Ψ( R, t ) + c.c. (cid:21) . (14)Calculations give the following representation of thethird rank tensor M αβγ via the velocity field and otherhydrodynamic functions: M αβγ = nv α v β v γ + v α p βγ + v β p αγ + v γ p αβ + Q αβγ + T αβγ , (15)where we have two new functions. One of them the quasi-classic third rank tensor in the comoving frame: Q αβγ ( r , t ) = Z dR N X i =1 δ ( r − r i ) a ( R, t ) u αi u βi u γi . (16)The quantum part of the tensor M αβγ is found: T αβγ = ¯ h m (cid:20) − n ( ∂ α ∂ β v γ + ∂ α ∂ γ v β + ∂ β ∂ γ v α )+ T βγ · v α + T αβ · v γ + T αγ · v β (cid:21) . (17)Equation for the evolution of quantum-thermal part ofthe third rank tensor is [45], [46]: ∂ t Q αβγqf + ∂ δ ( v δ Q αβγqf )+ Q αγδqf ∂ δ v β + Q βγδqf ∂ δ v α + Q αβδqf ∂ δ v γ = ¯ h m n (cid:18) g I αβγδ ∂ δ n + ∂ α ∂ β ∂ γ Φ d (cid:19) + 1 mn ( T αβqf ∂ δ T γδqf + T αγqf ∂ δ T βδqf + T βγqf ∂ δ T αδqf ) , (18)where I αβγδ = δ αβ δ γδ + δ αγ δ βδ + δ αδ δ βγ . (19)The main contribution of the short-range interactionis proportional to the second interaction constant: g = 23 Z d r U ′′ ( r ) . (20)It is obtained in the first order by the interaction radius[60].Different versions of the extended hydrodynamics forvarious physical systems are presented in Refs. [54], [59],[62], [63], [64]. Novel approaches to the development ofhydrodynamics are recently presented in Refs. [65] and[66].Consider solitons in the uniform boundless BEC withno restriction on the amplitude of soliton. Hence, wehave no external potential V ext = 0. Consider the simplified form of the hydrodynamic equa-tions giving main contribution in the soliton solution.
The continuity equation (3) requires no simplification.No simplification of the equation of field (8) is requiredtoo.We drop the traditional part of the quantum Bohmpotential in the Euler equation (5): mn∂ t v α + mn ( v · ∇ ) v α + ∂ β T αβqf = − gn∂ α n − n∂ α Φ d . (21)The pressure evolution equation simplifies to twoterms: ∂ t T αβqf + ∂ γ Q αβγqf = 0 , (22)where the quantum Bohm potential T αβqf is caused purelyby the flux Q αβγqf .Equation for the evolution of quantum-thermal part ofthe third rank tensor is: ∂ t Q αβγqf = ¯ h m n (cid:18) g I αβγδ ∂ δ n + ∂ α ∂ β ∂ γ Φ d (cid:19) , (23)where the interaction on the right-hand sides representsthe quantum fluctuations. It is assumed that the evolu-tion of the tensor Q αβγqf is mainly caused by the interac-tion.There is no independent source of interaction in thepressure evolution equation (23). Hence, the third ranktensor Q αβγqf is the single source of interaction in the pres-sure evolution equation. As it is mentioned above, wefocus on the pressure or the quantum Bohm potentialcaused by the quantum fluctuations. Hence, we considerthe interaction caused T αβqf and drop the kinematic terms(see equation (23)).We consider the one dimensional solution. We chosethe direction of wave propagation perpendicular to thedirection of titled dipoles. We seek the stationary solu-tions of the nonlinear equations. We assume the steadystate in the comoving frame. Therefore, the dependenceof the time and space coordinates is combined in the sin-gle variable ξ = x − ut . Parameter u is the constantvelocity of the nonlinear solution. Therefore, all hydro-dynamic functions depend on ξ and u . We also assumethat perturbations vanish at ξ → ±∞ . It gives the fol-lowing reduction of equations (3), (8), (21), (22), (23). Equation (38) can be integrated to obtain the ”energyintegral” in the following manner12 ( ∂ ξ n ) + m u πµ ¯ h V eff ( n ) = 0 , (24)where V eff ( n ) is the Sagdeev potential [67], [68], [69],[70], [71], [72], [73]: V eff ( n ) = 12 (cid:20)(cid:18) g − πµ − u ¯ h m g − mu n (cid:19)(cid:21) ( n − n ) . (25)Details of derivation of equations (24) and (25) are pre-sented in the Supplementary Materials. Moreover, theestimation of the area of applicability of equations (21),(22), and (23) is discussed in Supplementary Materialseither.In order the soliton to exist, the effective potential V eff ( n ) should have a local maximum in the point n = 0.Moreover, equation V eff ( n ) = 0 should have at least onereal solution n ′ = 0. This value of concentration n ′ de-termines the amplitude n ′ of the soliton as the functionof velocity u .Equation V eff = 0 can be solved analytically for n = n : n ′ = mu g − πµ − u ¯ h m g . (26)Equation (41) allows to introduce the effective dimen-sionless interaction constant G = ( n /mu )( g − πµ − h g / m u ).Let us consider the limit of the small dipole-dipole in-teraction and the small short-range part of the quantumfluctuations in compare with the mean-field of the short-range interaction presented by the Gross-Pitaevskii in-teraction constant g . Hence, we have G ≈ g/mu . Wehave soliton solution for the positive interaction constant G ∼ g > n ′ , where V ( n ′ ) = 0) decreases with theincrease of the effective interaction constant G . The in-crease of the velocity of soliton u decreases constant G and, consequently, decreases the amplitude. The increase FIG. 1: The Sagdeev potential as the function of the dimen-sionless deviation ∆ = ( n − n ) /n of the concentration n fromthe equilibrium value n (the concentration at the infinity n = n ( x = ±∞ )) is demonstrated for the value of ∆ below 1in accordance with the area of applicability of the simplifiedhydrodynamic equations in accordance with Supplementarymaterials. The Sagdeev potential depends on one parameter G , which is the combination of the interaction constants in-cluding the effective interaction constant for the dipole-dipoleinteraction g d = 4 πµ . The figure shows that there is the ”po-tential gap” in the area of positive deviations ∆. It means thatthere is a bright soliton (the area of increased concentration).FIG. 2: The dimensionless soliton amplitude value for con-centration n ′ /n as the function of the effective interactionconstant is demonstrated. This dependence is given analyt-ically by equation (26). Here, the decrease of amplitude n ′ corresponds to the decrease of area of negative potential inFig. 1. Decrease of G gives the monotonic increase of ampli-tude n ′ . Hence, the area of applicability of obtained solutioncan be broken. Therefore, this figure shows the restrictionson the effective interaction constant G . of the soliton velocity u diminish the role of the short-range interaction part of the quantum fluctuations in theeffective interaction constant.The traditional bright and dark solitons in neutralatomic BEC are caused by the nonlinearity created bythe short-range interaction in the Gross-Pitaevskii ap-proximation. There are different generalizations of theGross-Pitaevkii model include effects beyond the mean-field approximation. An example of the beyond mean-field model of BEC has been derived in this paper. Thequantum fluctuations have been included here via theextended hydrodynamic model which includes the conti-nuity equation for the scalar field of concentration, theEuler equation for the vector field of velocity, the pressureevolution equation for the second rank tensor of pressure,and the evolution equation for the third rank tensor.The dipolar part of quantum fluctuations is presentedby the term proportional to the third derivative of theelectrostatic potential. For one dimensional perturba-tions in the single fluid species the potential is propor-tional to the variation of the concentration from the equi-librium value.Hence, the high derivatives of the concentration ap-pears in term presenting the dipolar part of quantumfluctuations.We acknowledge that the work is supported by theRussian Foundation for Basic Research (grant no. 20-02-00476). [1] H. Kadau, M. Schmitt, M. Wenzel, C. Wink, T. Maier,I. Ferrier-Barbut, T. Pfau, Nature , 194 (2016).[2] I. Ferrier-Barbut, H. Kadau, M. 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We consider the one dimensional solution. We chosethe direction of wave propagation perpendicular to thedirection of titled dipoles. We seek stationary solutionsof the nonlinear equations. We assume the steady statein the comoving frame. Therefore, the dependence ofthe time and space coordinates is combined in the singlevariable ξ = x − ut . Parameter u is the constant velocityof the nonlinear solution. Therefore, all hydrodynamicfunctions depend on ξ and u . We also assume that per-turbations vanish at ξ → ±∞ . It gives the followingreduction of equations (3), (8), (21), (22), (23).The reduced continuity equation is − u∂ ξ n + ∂ ξ ( nv x ) = 0 , (27)where the time derivatives ∂ t is replaced by − u∂ ξ in ac-cordance with the variable ξ introduced for the stationarysolution.The reduced Euler equation is − umn∂ ξ v x + mnv x ∂ ξ v x + ∂ ξ T xxqf = − gn∂ ξ n − n∂ ξ Φ d . (28)The Poisson equation (8) transforms to ∂ ξ Φ d = − πµ ∂ ξ n. (29)The reduced pressure evolution equation is − u∂ ξ T xxqf + ∂ ξ Q xxxqf = 0 . (30)Equation for the evolution of quantum-thermal part ofthe third rank tensor is: − u∂ ξ Q xxxqf = ¯ h m n (cid:18) g ∂ ξ n + ∂ ξ Φ d (cid:19) , (31)where we have the source for the dispersion of the nonlin-ear wave presented by the third derivative of the potentialΦ d (7).The neglecting of the quantum Bohm potential (12)in the Euler equation corresponds to the following re-striction for the velocities of the soliton propagation u ≪ µ n . B. Derivation of the soliton solution
The continuity equation (27) can be integrated n ( v x − u ) = − un , (32) where we use the boundary conditions with nonzero con-centration at infinity n ( ξ → ±∞ ) = n , and zero valueof velocity at infinity v x ( ξ → ±∞ ) = 0.The Euler equation (28) can not be integrated at thisstage due to the third term, which is caused by the quan-tum fluctuations, which is proportional to ∂ ξ T xxqf n . It willbe considered later after the analysis of solution for T xxqf .The derivative of the second rank tensor T xxqf is pre-sented via the derivative of the third rank tensor Q xxxqf by equation (30). Equation (30) can be integrated, butit is not necessary since we need the derivative of thesecond rank tensor ∂ ξ T xxqf for the Euler equation (28).The derivative of the third rank tensor ∂ ξ Q xxxqf is foundfrom equation (31). The extra multiplier n in front ofthe third derivative of the potential Φ d is canceled inexpression ∂ ξ T xxqf n , so the final form of the Euler equationcan be integrated.Equation (29) can be integrated twice. After first in-tegration including the zero derivatives of the concentra-tion and the potential of electric field at infinity we have ∂ ξ Φ d = − πµ ∂ ξ n. (33)Next, we make the second integration to find the expres-sion of the potential:Φ d = − πµ ( n − n ) . (34)The final expression for the derivative of the secondrank tensor is following: ∂ ξ T xxqf = − u ¯ h m n (cid:18) g ∂ ξ n + ∂ ξ Φ d (cid:19) . (35)It gives the following form of the Euler equation: m ( − u ) ∂ ξ v x − u ¯ h m (cid:18) g ∂ ξ n − πµ ∂ ξ n (cid:19) + mv x ∂ ξ v x = − g∂ ξ n + 4 πµ ∂ ξ n, (36)where the potential of electric field Φ d is expressed viathe concentration in accordance with equation (33).We integrate Euler equation (36) m ( − u ) v x + 12 mv x − u ¯ h m (cid:18) g ( n − n ) − πµ ∂ ξ n (cid:19) + ( g − πµ )( n − n ) = 0 . (37)The third and fourth terms can be combined togethersince they are proportional to n − n . Hence, the quantumfluctuations caused by the short-range interaction givesthe variation of the Gross-Pitaevskii interaction constant g . The dipolar part of the quantum fluctuations playsthe crucial contribution in the formation of novel soli-ton. Equation (37) becomes the differential equation due FIG. 3: The Sagdeev potential as the function of the dimen-sionless deviation ∆ = ( n − n ) /n of the concentration n from the equilibrium value n is demonstrated for the rela-tively large values of the effective interaction constant G . Dis-appearance of the area of negative potential is demonstratedat the increase of the parameter G . to the dipolar part of the quantum fluctuations. If wedrop the dipolar part of the quantum fluctuations we ob-tain the constant value of concentration. It means thatno soliton exist in this limit.Using the integral of the continuity equation (32) weexpress the velocity via the concentration. Hence, equa-tion (37) becomes the equation relatively one function: ∂ ξ n + m u πµ ¯ h f ( n ) = 0 , (38)where f ( n ) = − mu n − n n − u ¯ h m g ( n − n )+ ( g − πµ )( n − n ) . (39)Equation (38) can be integrated to obtain the ”energyintegral” in the following manner12 ( ∂ ξ n ) + m u πµ ¯ h V eff ( n ) = 0 , (40)where V eff ( n ) is the Sagdeev potential V eff ( n ) = − mu ( n − n ) n + 12 (cid:18) g − πµ − u ¯ h m g (cid:19) ( n − n ) . (41) C. Justification of the simplified equations
Consider full set of equations (3), (5), (13), (18) forthe one dimensional case to estimate the contribution ofthe dropped terms: ∂ t n + ∂ x ( nv x ) = 0 , (42) FIG. 4: The Sagdeev potential as the function of the dimen-sionless deviation ∆ = ( n − n ) /n of the concentration n fromthe equilibrium value n is demonstrated for the relativelylarge values of ∆. Here, the amplitude of soliton becomeslarge enough to brake regime of existence of the simplifiedhydrodynamic equations. mn∂ t v x + mnv x ∂ x v x + gn∂ x n + n∂ x Φ d + ∂ x T xxqf = ¯ h m n∂ x ∂ x √ n √ n , (43) ∂ t T xxqf + ∂ x Q xxxqf = − v x ∂ x T xxqf − T xxqf ∂ x v x , (44)and ∂ t Q xxxqf − ¯ h m n (cid:18) g ∂ x n + ∂ x Φ d (cid:19) = − v x ∂ x Q xxxqf − Q xxxqf ∂ x v x + 3 mn T xxqf ∂ x T xxqf , (45)where the left-hand side in equations (43), (44), (45) con-tains the terms used in the simplified regime. The right-hand side gives the terms dropped earlier for their furtherestimation.Equation (40) shows existence of the soliton solutionfor the concentration. Equations (27)-(35) allows to ex-press other hydrodynamic functions via the concentra-tion: ˜ v x = u (cid:18) − n n (cid:19) . (46)Transition to the frame comoving with the soliton wechange ∂ t on − u∂ ξ and ∂ x = ∂ ξ . Equation (31) canbe integrated and we obtain corresponding simplified ex-pression of tensor Q xxx : u ˜ T xxqf = ˜ Q xxxqf = 3¯ h m u g n + π ¯ h µ m u n∂ ξ n − π ¯ h µ m u ( ∂ ξ n ) , (47)where equation (30) is used to get solution for T xxqf in thesimplifies regime. Next, we use this solution to integratethe right-hand side of equation (45).The right-hand side of equation (45) contains thefollowing combination of the hydrodynamic functions˜ v x ∂ x ˜ Q xxxqf + 4 ˜ Q xxxqf ∂ x ˜ v x . Using solution (47) this combi-nation can be found as the function of the concentration˜ v x ∂ x ˜ Q xxxqf +4 ˜ Q xxxqf ∂ x ˜ v x = − ¯ h m ( n − n )[3 g ∂ ξ n − πµ ∂ ξ n ] − h m g n ( ∂ ξ n ) n + 4 πµ ¯ h n m ( ∂ ξ n ) n ∂ ξ n − πµ ¯ h n m ( ∂ ξ n ) n . (48)Here, we obtain the generalized expression for thederivative of the third rank tensor via the concentration ∂ ξ Q xxxqf = − u ( ¯ h m n (cid:18) g ∂ ξ n − πµ ∂ ξ n (cid:19) + ¯ h m ( n − n )[3 g ∂ ξ n − πµ ∂ ξ n ] + 3¯ h m g n ( ∂ ξ n ) − πµ ¯ h n m ( ∂ ξ n ) n ∂ ξ n + 2 πµ ¯ h n m ( ∂ ξ n ) n ) , (49)where the last term of equation (45) mn T xxqf, ∂ x T xxqf, isdropped since it is proportional to ¯ h (see equation (47)).Next, we use equation (49) to find the right-hand sideof equation (45) and we use (47) to find the right-handside of equation (45) ∂ ξ T xxqf = − u ( ¯ h m n (cid:18) g ∂ ξ n − πµ ∂ ξ n (cid:19) + ¯ h m ( n − n )[3 g ∂ ξ n − πµ ∂ ξ n ] + 15¯ h m g n ( ∂ ξ n ) − πµ ¯ h n m ( ∂ ξ n ) n ∂ ξ n + 7 πµ ¯ h n m ( ∂ ξ n ) n ) , (50)where the right-hand side is given by˜ v x ∂ x ˜ T xxqf + 3 ˜ T xxqf ∂ x ˜ v x = (cid:18) − n n (cid:19) ∂ x ˜ Q xxxqf + 3 n ˜ Q xxxqf ∂ x nn = − ¯ h m u ( n − n )[3 g ∂ ξ n − πµ ∂ ξ n ]+3 n ∂ ξ nn (cid:20) h g m u n + π ¯ h µ m u n∂ ξ n − π ¯ h µ m u ( ∂ ξ n ) (cid:21) . (51)Expressions (46) and (50) are substituted in the Eulerequation (43) to get equation for the concentration n . Itgives the generalization of equation (36) − umn∂ ξ v x + mnv x ∂ ξ v x + gn∂ ξ n + n∂ ξ Φ d = ¯ h m n∂ ξ ∂ ξ √ n √ n + 1 u ( ¯ h m n (cid:18) g ∂ ξ n − πµ ∂ ξ n (cid:19) + ¯ h m ( n − n )[3 g ∂ ξ n − πµ ∂ ξ n ] + 15¯ h m g n ( ∂ ξ n ) − πµ ¯ h n m ( ∂ ξ n ) n ∂ ξ n + 7 πµ ¯ h n m ( ∂ ξ n ) n ) . (52)The second term on the right-hand side causes the soli-ton obtained in this paper. The third term has similarstructure, but ( n − n ) is placed instead of n . Therefore,the third term can be dropped if n ≈ n . Consequently,parameter ∆ ≪≪