Approximate Analytic Solutions to Coupled Nonlinear Dirac Equations
We consider the coupled nonlinear Dirac equations (NLDE’s) in 1+1 dimensions with scalar-scalar self interactions as well as vector-vector interactions of the form Writing the two components of the assumed solitary wave solution of these equation in the form , , and assuming that have the same functional form they had when =0, which is an approximation consistent with the conservation laws, we then find approximate analytic solutions for which are valid for small values of and . In the nonrelativistic limit we show that both of these coupled models go over to the same coupled nonlinear Schrödinger equation for which we obtain two exact pulse solutions vanishing at .
pacs:05.45.Yv, 03.70.+k, 11.25.Kc
The nonlinear Dirac (NLD) equation in dimensions iva () has a long history and has emerged as a useful model in many physical systems such as extended particles fin (); ffk (); hei (), the gap solitons in nonlinear optics bar (), light solitons in waveguide arrays and experimental realization of an optical analog for relativistic quantum mechanics lon (); dre (); tra (), Bose-Einstein condensates in honeycomb optical lattices had (), phenomenological models of quantum chromodynamics fil (), as well as matter influencing the evolution of the universe in cosmology sah (). Further, the multi-component BEC order parameter has an exact spinor structure and serves as the bosonic analog to the relativistic electrons in graphene. To maintain the Lorentz invariance of the NLD equation, the self interaction Lagrangian is built using the bilinear covariants. Of special interest are scalar bilinear covariant and vector bilinear covariant which have particularly attracted a lot of attention.
Classical solutions of nonlinear field equations have a long history as a model of extended particles sol (). In 1970, Soler proposed that the self-interacting 4-Fermi theory was an interesting model for extended fermions. Later, Strauss and Vasquez str () were able to study the stability of this model under dilatation and found the domain of stability for the Soler solutions. Solitary waves in the 1+1 dimensional nonlinear Dirac equation have been studied lee (); nog () in the past in case the nonlinearity parameter , i.e. the massive Gross-Neveu gro () (with , i.e. just one localized fermion) and the massive Thirring thi () models. In those studies it was found that these equations have solitary wave solutions for both scalar-scalar (S-S) and vector-vector (V-V) interactions. The interaction between solitary waves of different initial charge was studied in detail for the S-S case in the work of Alvarez and Carreras alv () by Lorentz boosting the static solutions and allowing them to scatter.
In a previous paper coo () we extended the work of these preceding authors to the case where the nonlinearity was taken to an arbitrary power for both the scalar-scalar and vector vector couplings and were able to find solitary wave solutions for an arbitrary nonlinearity parameter . In this paper we will extend the previous models in a new direction by looking for solitary wave solutions to the problem of two coupled NLDE’s and considering the scalar-scalar coupling as well as the vector-vector coupling between the two fields. Our strategy is to write the components of the two Dirac equations for solitary waves as and then assume that the conservation law for linear momentum is satisfied independently for . This assumption is equivalent to saying that have the same functional form they had when =0. Once one makes that assumption we obtain an analytic expression for which we then show approximately solves the differential equation for . The one situation which restricts the validity of this solution occurs in the scalar-scalar interaction case when one of the solitary wave solutions (when ) is of a double humped variety. In that case the solution is valid only when the dimensionless coupling constants and are . Otherwise the approximate analytic solutions we have found seem to be numerically accurate in both the scalar-scalar as well as the vector-vector coupled NLD equation as long as the two dimensionless constants are .
Ii scalar-scalar interactions
We are interested in solitary wave solutions of the coupled nonlinear Dirac equations (NLDEs) given by
We can eliminate one of the coupling constants by rescaling the fields, that is if we let , , so that there are two independent dimensionless coupling constants
as we will discover later. The field equations can be derived from the Lagrangian
We notice the Lagrangian is symmetric under the interchange and .
We next choose the following representation of the matrices:
where the are the usual Pauli spin matrices.
In the rest frame we assume that the two components of the solutions can be written as
In the absence of interactions (), the solutions are of two types coo (). When then the solutions are single humped as they are always in the case of vector-vector interactions discussed below. However for the case the solutions are double humped and in that regime if the solutions when are of two different types, then we will find the approximate solutions we obtain are only valid for very small . In component form these two coupled NLDEs can be written as
These are symmetric under the interchange and . These four equations can also be written if we let as:
We can rewrite these equations in terms of the two dimensionless coupling constants by scaling , .
ii.1 Conservation Laws
We have that energy and momentum are conserved, namely
where the energy-momentum tensor is defined as
and is given by Eq. (II). From total momentum conservation, we find, just like for the single field NLDE, that for a solution that vanishes at we have
which becomes using our ansatz
One also has that energy is conserved. The energy density is given by
with the total energy being conserved
The other conserved quantities are charges and defined by
ii.2 Approximate Solution
We will obtain our approximate analytic solution by assuming that each of the two terms in Eq. (15) is identically zero. Then we obtain
whose solutions are the same as when , namely
where is the value of when , and for . Letting we have that the values of various trigonometric functions of , valid when are given by:
We see from Eqs. (II.2) and (II.2) that the solution for after rescaling depends on the two dimensionless coupling constants . Our solutions for were found using the differential equation for . So to see the accuracy of this solution we need to check how well the Eqs. (II) are satisfied. We will find that this depends on whether one of the solutions is double humped, since then its derivative near will be opposite that of the single humped one and then the left hand side can have behavior different from the right hand side in various scenarios near . First, let us look at the case when both are single humped. We can always after rescaling take . For simplicity we will also choose , and are then left with which for for illustrative purposes we will usually choose . For the choice , , , , the left hand side of Eq. (II) is shown in blue and the right hand side in yellow in Fig. 1.
Here if we take the ratio we would find that this was always less than over the entire range. For this choice of parameters is much more modified by the interaction than . This is shown in Figs. 2 and 3.
The problematic case is when one of the is double humped. In that case the derivatives of determined from our approximation can be positive, whereas the right hand side can still be negative in our approximation for a range of small and . An example of this is given for the values: , , , , in Fig. 4
In this case if we reduce to be , then we again get good agreement between the left and right hand sides of the equation for . This is shown in Fig. 5. When , is slightly modified from its value when . However, the double humped solution is barely modified by the interaction as seen in Figs. 6 and 7.
Iii Vector-Vector Interactions
The coupled nonlinear Dirac equations (NLDEs) with vector-vector interactions are given by
We notice that as in the scalar-scalar case, the Lagrangian in this case is also symmetric under the interchange , and . Again using the representation as given by Eq. (II), we have the equations for the components of the two coupled NLDEs which can be written as
These are symmetric under the interchange , and . These four equations can also be written if we let as:
This reduces, when to the Eqs. (14) in Chang et al. Cha ().
iii.1 Conservation Laws
We again have energy-momentum conservation governed by Eqs. (10) and (11) but where is now given by Eq. (30). From total momentum conservation, we find, just like for the scalar case, that for a solution that vanishes at , and are again given by Eqs. (12) and (13) respectively, but where is as given by Eq. (30). Multiplying Eq. (28) on the left by and Eq. (29) on the left by and adding those two equations and then using Eq. (13) to eliminate the interaction terms of , we as in the scalar case, again obtain Eq. (14). On using the ansatz (II) we then again obtain
iii.2 Approximate Solution
We will obtain our approximate analytic solution by assuming that each of the two terms in Eq. (34) is identically zero. Then we obtain
whose solutions are given by Eq. (20). We can again solve for and and obtain
Since in the absence of interactions we have
we can rewrite Eq. (III.2) as
So we see that we need both and for this approximation to make sense. Now let us see to what extent we violate Eq. III. We have, letting , the approximate expression for , valid when , given by Eq. (20).
Now unlike the scalar-scalar case, the solutions are single humped and so typical values of the parameters give generic results.
Setting and and , , , we find that the relative error on comparing the lhs and rhs of Eq. (III), i.e , is less than , (see Fig. 8 ) for , At the same time, is changed quite a bit from its uncoupled value when we choose these values of the parameters as seen in Fig. 9. The effect is not as dramatic for for these values as seen in Fig. 10.
Iv Nonrelativistic Limit
In our previous paper coo (), we had started with NLD equations and using Moore’s decoupling method moo () we had obtained the nonrelativistic limit of our NLD equations in both the scalar and the vector coupling cases. In this section we essentially follow the same decoupling method to obtain the nonrelativistic limit of coupled NLD equations in both scalar and vector coupling cases.
and the Moore’s decoupling method as well as essentially following the steps given in our previous paper coo (), we find that in both the scalar-scalar and the vector-vector cases we get the coupled NLS equations
under the assumption that and .
Let us now look for exact solutions of the coupled Eqs. (43) and (44) under the assumption that both and vanish in the limit . It turns out that there are two such solutions and we discuss these one by one.
iv.1 Solution I
iv.2 Solution II
It is thus worth noting that while the first solution is valid for any values of , the second solution is only valid when .
In this paper we have introduced and initiated discussion about coupled NLD equations with both scalar-scalar and vector-vector interactions. In particular, we have given the first (approximate) analytic solitary wave solution to two coupled NLDEs for both scalar-scalar interactions and vector-vector interactions. These solutions are relevant in nonlinear optics bar () as well as for light solitons in waveguide arrays lon (); dre (); tra () among other applications in BECs and cosmology. Further,we have shown using the Moore’s decoupling method that in the nonrelativisticlimit, NLDEs with both scalar-scalar and vector-vector interactions reduce tothe same coupled nonlinear Schrödinger equation (NLSE). We have obtained two exact pulse solutions to these coupled NLSE. Using the results found in coo (), one can extend these solutions to the case where the scalar-scalar as well as vector-vector interactions are taken to an arbitrary (nonlinearity) power . We hope to address this issue as well as the question of stability of the solutions found here in the near future.
Acknowledgements.This work was performed in part under the auspices of the U.S. Department of Energy. F.C. would like to thank the Santa Fe Institute and the Center for Nonlinear Studies, Los Alamos National Laboratory, for its hospitality. A.K. is grateful to Indian National Science Academy (INSA) for awarding him INSA Senior Scientist position at Savitribai Phule Pune University, Pune, India.
- (1) D.D. Ivanenko, Zhurn. Experim. Teoret. Fiz. 8, 260 (1938).
- (2) R. Finke;stein, R. Lelevier and M. Ruderman, Phys. Rev. 83, 326 (1951).
- (3) R. J. Finkelstein, C. Fronsdal, and P. Kaus, Phys. Rev. 103, 1571 (1956).
- (4) W. Heisenberg, Rev. Mod. Phys. 29, 269 (1957).
- (5) I.V. Barashenkov, D.E. Pelinovsky, and E.V. Zemlyanaya, Phys. Rev. Lett. 80, 5117 (1998).
- (6) S. Longhi, Opt. Lett. 35, 235 (2010).
- (7) F. Dreisow, M. Heinrich, R. Keil, A. Tünnermann, S. Nolte, S. Longhi, and A. Szameit, Phys. Rev. Lett. 105, 143902 (2010).
- (8) T.X. Tran, S. Longhi, and F. Biancalana, Ann. Phys. 340, 179 (2014).
- (9) L.H. Haddad and L.D. Carr, Physica D 238, 1413 (2009); arXiv:1006.3893.
- (10) F. Fillion-Gourdeau, H. Herrmann, M. Mendoza, S. Palpacelli, and M. Succi, Phys. Rev. Lett. 111 160602 (2013).
- (11) B. Saha, Int. J. Theor. Phys. 51, 1811 (2012).
- (12) M. Soler, Phys. Rev. D 1, 2766 (1970).
- (13) W. Strauss and L. Vazquez, Phys. Rev. D 34, 641 (1986).
- (14) S.Y. Lee, T. K. Kuo, and A Gavrielides, Phys. Rev. D 12, 2249 (1975).
- (15) Y. Nogami and F. M. Toyama, Phys. Rev. A 45, 5258 (1992).
- (16) D. J. Gross and A. Neveu, Phys. Rev. D 10, 3235 (1974).
- (17) W. Thirring, Ann. Phys. 3, 91 (1958).
- (18) A. Alvarez and B. Carreras, Phys. Lett. 86A, 327 (1981).
- (19) F. Cooper, A.Khare, B. Mihaila, and A. Saxena, Phys. Rev. E 82, 036604 (2010).
- (20) S-J. Chang, S.D. Ellis, and B.W. Lee, Phys. Rev. D 11, 3572 (1975).
- (21) T.C. Scott, R.A. Morre, G.J. Fee, M.B. Monagan, and E.R. Vrscay, J. Comp. Phys. 87, 366 (1990).