Fokas-Lenells equation dark soliton and gauge equivalent spin equation

We propose the Hirota bilinearization of the Fokas–Lenells derivative nonlinear Schrödinger equation (FLE) with a non-vanishing background. In the proposed method, we have introduced an auxiliary function to transform the equation into bilinear form. The use of an auxiliary function makes the method simpler than the ones reported earlier. Using the proposed method we have obtained the dark (and bright) one soliton solution. We have also discussed the properties of the obtained soliton and mentioned the criteria upon which the nature of the soliton being dark or bright depends upon. Then we have obtained the dark (and bright) two soliton solution and discussed the respective properties and also through asymptotic analysis showed how the phase between the two individual solitons changes before and after interaction. Eventually we have proposed the scheme for obtaining N soliton solutions. The proposed method can be extended to other nonlinear equations where straightforward bilinearization is not feasible. Later, we have introduced a gauge transformation which transforms the spectral problem of FLE into a spectral problem for the Landau–Lifshitz (LL) spin system. Soliton act as an information carrier and LL system exhibits a variety of nonlinear structures so the study is worth doing.


Introduction
Fokas-Lenells equation (FLE) [1,2] is an integrable equation which describes the propagation of ultra-short pulses in an optical medium.FLE equation is a special case of a more generalized nonlinear Schrödinger equation (NLSE) which includes higher order effects.There are only a few known integrable equations of special cases, namely NLSE [3][4][5], higher order NLSE (HNLSE) [6,7], derivative NLSE (DNLSE) [8] and the fourth equation is the FLE.A common property in all the equations is the appearance of solitons.Soliton arise as a result of a balance between the nonlinear and dispersive terms of the wave equations.While the bright soliton appear as a hump on the zero background of light, the dark soliton appear as a dip on a continuous background.
The significance of FLE is due to the presence of a spatio-temporal dispersion term along with a cubic nonlinear self-steepening term which account the higher order nonlinear effects in an ultra-short optical pulse.While the study on NLSE, HNLSE and DNLSE is done extensively, the study on FLE is relatively sparse.A few of the recent notable contributions on FLE are solitary wave and elliptical solutions of FLE in presence of perturbation and modulation instability [9], combined optical solitary waves of FLE [10], dynamical behaviour of soliton solutions of FLE [11], inverse transform of FLE with non zero boundary condition [12], derivation of solitons of dimensionless FLE with perturbation term [13] and more.
The dimensionless expression for FLE [1,2] is where U is the field function that can describe the waveform of an ultra-short pulse.The suffix x and t denote the partial differentiations with respect to x and t respectively.U t is the temporal evolution of the pulse, U xx is the group velocity dispersion, U xt is the spatio-temporal dispersion, |U | 2 U is the Kerr nonlinearity effect and |U | 2 U x is the cubic nonlinear self-steepening effect of the medium.
It is interesting to note that the family of NLSE show gauge equivalence with spin system called Landau-Lifshitz equation (LLE) [27][28][29][30].FLE also belongs to this class of nonlinear Schrödinger type equations so it is worth to investigate the gauge connection of FLE with the spin system.To the best of our knowledge in the literature no such work has been reported earlier with FLE.
The structure of this manuscript is that in the following section we consider a gauge transformation that will convert eqn. 1 into a simplified form.Then using Hirota bilinearization on this simplified form, we shall derive a dark soliton solution and multi dark soliton solutions and discuss their properties and also mention the condition under which we can get bright soliton solutions.Thereafter in the third section, we propose a gauge transformation of the Lax pair for this FLE to obtain a spin system which is known as the Landau-Lifshitz spin system.Fourth section is the concluding one.

Bilinearization of FLE with vanishing background
Assuming n = 1 a 2 and m = a 1 a 2 > 0, consider the gauge transformation followed by the transformation of variables (ξ, τ ) we get the following equation here u is the field function corresponding to the new transformed system and σ = b |b| .Assuming b is positive, we can write eqn. 4 as notice that eqn. 5 is the first negative hierarchy of DNLSE.For dark soliton solution of eqn. 5 we assume a non vanishing background condition u → ρ e i (κξ+ωτ ) as ξ → ±∞.
Under this condition we expect a dark soliton solution with the bilinearization.
To write eqn. 5 in the bilinear form let us assume where g and f are two complex functions of (ξ, τ ).Consequently eqn. 5 becomes where D ξ , D τ are Hirota derivatives [6] and are defined as Notice that the last two terms in eqn.7 contains an auxiliary function s which is introduced so that the multilinear eqn.7 can be cast into two bilinear equations, namely eqns.10 and 11.Here λ is a constant to be determined by solving the bilinear eqns.9 -11.Consequently, the proposed bilinear equations in terms of g, f and s are To obtain the soliton solution, g and f are expanded with respect to an arbitrary parameter ϵ as follows and the auxiliary function s is expanded as

Dark one soliton Solution
The dark one soliton solution (1SS) is obtained by dropping terms of order greater than or equal to ϵ 3 in g, f and s.Thus from eqn. 6 the dark 1SS of eqn. 5 is 2 Let us consider the expressions for g 0 , s 0 , g 2 , s 2 and f (1) 2 are as follows 2 = K e θ+θ * (17) where θ = p ξ + Ω τ .p, Ω, K, M , T are complex parameters, ξ, ω are real constants and ρ, ρ s are positive constants.Let us consider p = p r + i p i and T = T r + i T i where p r , p i , T r and T i are real.On substituting the above equations into eqn.14, we have further putting eqns.15 -19 into eqns.9 -11 yield the following expressions where h is real and γ is complex and are represented as and the system obeys the constraint now keeping one of T r or T i fixed the other can be calculated from eqn. 29.If we fix T r then T i can be expressed as from eqn. 30 it follows that the condition |p r | ≤ κ 3 ρ 2 (1 + κρ 2 ) must satisfy to obtain 1SS.

Properties of dark one soliton Solution
In this part we shall discuss about the properties of 1SS (eqn.20) namely velocity, width inverse, the criteria upon which the nature of the one soliton obtained will be dark or bright and also discuss the amplitude.One thing to point out that there is always a background present so instead of calling 'bright soliton', an 'anti-dark soliton' term is much more accurate but we shall refer to it as 'bright soliton' for convenience.At first, let us parametrise θ as here 2p r represents the width inverse and v denotes the velocity of 1SS and is represented as eqn. 34 shows that κ can control both the sign and magnitude of v, graphically represented in figure 1a.In brief the figure 1a interprets that magnitude of v at first increases with the magnitude of κ and then decreases.And figure 1b shows that the magnitude of v decreases with p r and the sign of p r has no effect on v, in other words v is symmetric to p r .T r has no effect on v as T i is proportional to T r (from eqn.30) so T r is a common term in both numerator and denominator of eqn.34.In both the graphs v tends to zero at extreme points of κ and p r .These characteristics of v is irrespective to the nature of soliton being dark or bright.The amplitude (A) of 1SS is expressed as for positive p r , when κ and T i have different signs we get A smaller than ρ and A is bigger than ρ for the same sign of κ and T i .For the case of negative p r the criteria become vice versa.These are the criteria for the soliton to be dark and bright respectively.Figures 2a and 2b show the variation of A with respect to κ and p r respectively for dark and bright solitons.Figure 2b shows that the amplitude of the dark soliton at first decreases then increases with p r and for bright soliton the amplitude decreases monotonically with p r and in both dark and bright cases the amplitude A → ρ as p r → ± κ 3 ρ 2 (1 + κρ 2 ).One important fact to notice is that in the limit p r → 0, the dark soliton reduces to a plane wave that means algebraic dark soliton does not exist.However algebraic bright soliton exists.Figure 2b shows the same.In the same graph we see that at certain values of p r the amplitude of dark soliton reduces to zero indicating the occurrence of a black soliton.Figures 3a and 3b represent the propagation of dark and bright soliton respectively in 2D density plot.

Dark two soliton Solution
The dark two soliton solution (2SS) is obtained by dropping the terms of order greater than or equal to ϵ 5 in g, f and s.Thus from eqn. 6 the dark 2SS of eqn. 5 is Let us consider the expressions for g 2 , g 4 ,s 2 , s 4 , f 2 and f 4 in 2SS are as follows where θ j = p j ξ + Ω j τ .p j , Ω j , K j , M j , T j , K 12 , M 12 , T 12 are complex parameters (j = 1, 2).Let us consider p j = p r j + i p i j and T j = T r j + i T i j where p r j , p i j , T r j and T i j are real.On substituting the above equations into eqn.36, we have now putting eqns.37 -42 into eqns 9 -11 yield the following expressions: where h j 's are real and γ j 's are complex and are represented as and c is real expressed as the system obeys the constraints if we fix T r j then T i j can be calculated from eqn. 53 as from eqn. 54, the conditions |p r j | ≤ κ 3 ρ 2 (1 + κρ 2 ) must satisfy to obtain 2SS.

Properties of dark two soliton solution
In a similar way as we have done in section 2.1.1,to investigate the properties of 2SS we first parametrise θ j (j = 1, 2) as where v j denotes the velocity of the jth soliton and represented as and 2p r j represents the width inverse of the jth soliton of 2SS.The amplitude, A j of the jth soliton is given as The velocity and amplitude of the jth soliton obey the same characteristics as that of 1SS obey as discussed in the section 2.1.1.Interaction of two individual solitons of 2SS is demonstrated in figures 4a and 4b for a dark-dark case and bright-bright case respectively in 2D density plot.

Asymptotic analysis of 2SS
To study the amplitudes and the phase shift of 2SS we shall perform asymptotic analysis.When asymptotically apart from each other, 2SS is essentially two separated 1SS.In the limit of before interaction τ → −∞, if we fix θ 1 , this implies |e θ 1 | is finite and |e θ 2 | → ∞.This corresponds to an asymptotic form of eqn.43 which is expressed as here Eqn. 61 is nothing but 1SS (from eqn.20) with an additional phase ϕ.The arise of ϕ is due to term K 2 T 2 in eqn.60 and the expression of ϕ is given in a few steps below (in eqn.64).In the limit of after interaction τ → ∞, again we have fixed θ 1 then this will imply |e θ 1 | is finite and |e θ 2 | → 0. The corresponding asymptotic form of eqn.43 expressed as we see eqn.62 is same as eqn.20 with no additional phase.The amplitude of both the eqns.61 and 62 is same and can be expressed as where σ 1 = sign(T r 1 ).This is one of the important characteristics of a soliton that upon interaction the amplitude of the soliton remains same only a phase shift occurs.
In this case the phase shift is ϕ and its value is given by We can perform the same analysis by keeping θ 2 fixed.In the limit τ → −∞, the asymptotic form of eqn.43 becomes (65) and in the limit τ → ∞, the asymptotic form of eqn.43 becomes pair (L, M ) for eqn. 5 where Ψ is the Jost function corresponding to the field function u and (L, M ) are given by ζ is the spectral parameter and Σ and u are 2 × 2 matrices given as follow Σ = 1 0 0 -1 , u = 0 q −q * 0 from eqns.72 and 73, we can write (Ψ ξ ) τ = (Ψ τ ) ξ , this gives: eqn. 76 is the compatibility condition also called zero curvature equation.Under a local gauge transformation we write a matrix g such that: Φ is the Jost function corresponding to the spin field function S of Landau-Lifshitz (LL) system.The new Lax pair (L ′ , M ′ ) associated with Φ expressed as: and are related to (L, M ) as: where L 0 and M 0 are: the compatibility condition eqn.76 in terms of the new Lax pair (L ′ , M ′ ) takes the form: this is the compatibility condition corresponding to S. S being spin field function of LL system can be represented in terms of g as: and satisfies using eqn.81, the derivatives of S namely S ξ , S τ and also the terms SS ξ , SS τ can be written as: SS τ = − 2i ζ 0 g −1 Σug (92) again using the above four expressions in eqns.82 and 83, we get: hence, the compatibility condition eqn.86 yields: this is the gauge connected LLE for the eqn. 5.One important thing to note is that eqn.95 is consistent for the system of eqns.77, 78, 93, 94 in any matrix S of arbitrary dimension as long as eqn.88 holds.

Conclusion
We have bilinearized the Fokas-Lenells derivative nonlinear Schrödinger equation (FLDE) with a non-vanishing boundary condition.In the proposed bilinearization we have used an auxiliary function to derive explicitly the one dark soliton solution and two dark soliton solution and represent the scheme for obtaining N soliton solution.We have also discussed the criteria for the soliton to become dark or bright.Our result accepts a wide range of values for the parameters (K's, T 's) unlike the results published earlier.We have obtained a gauge equivalence between FLDE and Landau-Lifshitz (LL) spin system with the explicit construction of the new equivalent Lax pair.We believe that the derived dark (and bright) soliton solutions will be useful in optical communication and other nonlinear fields of physics where higher order effects like spatio-temporal evolution and cubic nonlinear self-steepening effects are taken into account.The gauge equivalent LL equation will be useful to study the integrability properties of the FLDE.

Figure 1 :
Figure 1: Variation of velocity (a) with respect to κ and fixed p r = 1 and 2 and (b) with respect to p r and fixed κ = ±5, ±10.In both the graphs we have fixed ρ = 2 and T r can have any value say ±1, ±2, etc. the graphs will remain the same.

Figure 2 :
Figure 2: Variation of amplitude (a) with respect to κ and fixed p r = 1 and (b) with respect to p r and fixed κ = ±5, ±10.In both the graphs we fixed ρ = 2 and T r = 6 and −6 for dark and bright soliton respectively.

Figure 3 :
Figure 3: (a) represents dark soliton with κ = −5 and T r = 6 and (b) represents bright soliton with κ = 5 and T r = −6 in density plot.In both the figures we have fixed ρ = 2.