T\overline{T} Deformations of nonrelativistic models
TTCDMATH 21-03
T T deformations of non-relativistic models
Chantelle Esper and Sergey Frolov †‡ School of Mathematics and Hamilton Mathematics Institute,Trinity College, Dublin 2, Ireland
Abstract
The light-cone gauge approach to
T T deformed models is used to derive the
T T de-formed matrix nonlinear Schr¨odinger equation, the Landau–Lifshitz equation, and theGardner equation. Properties of one-soliton solutions of the
T T deformed nonlinearSchr¨odinger and Korteweg–de Vries equations are discussed in detail. The NLS solitonexhibits the recently discussed phenomenon of widening/narrowing width of particlesunder the
T T deformation. However, whether the soliton’s size is increasing or decreas-ing depends not only on the sign of the deformation parameter but also on soliton andpotential parameters. The
T T deformed KdV equation admits a one-parameter familyof one-soliton solutions in addition to the usual velocity parameter. The extra parametermodifies the properties of the soliton, in particular, it appears in the dispersion relation. † Correspondent fellow at Steklov Mathematical Institute, Moscow. ‡ email: [email protected], [email protected] a r X i v : . [ h e p - t h ] F e b ontents T T deformed models 4
T T deformed sigma model . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2
T T deformed matrix nonlinear Schr¨odinger model . . . . . . . . . . . . . 62.3
T T deformed Landau–Lifshitz model . . . . . . . . . . . . . . . . . . . . 82.4
T T deformed Gardner equation . . . . . . . . . . . . . . . . . . . . . . . 12
T T deformed NLS soliton . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2
T T deformed KdV soliton . . . . . . . . . . . . . . . . . . . . . . . . . . 19
The irrelevant
T T deformation of two-dimensional Lorentz invariant models introducedin [1] has many interesting properties. In particular, if a seed model is integrable thenthe
T T deformed model is also integrable at least at the classical level [2, 3]. Assumingthe
T T operator is well-defined at the quantum level, the factorisation of two-pointcorrelation functions at large separation and a CFT limit at short distances, one can showthat the spectrum of a
T T deformed model is governed by an inhomogeneous inviscidBurgers equation. If the spectrum depends regularly on the deformation parameter thenit is completely fixed by the spectrum of the seed model [1]. The Burgers equation canbe used to derive the CDD factor which relates the S-matrices of the deformed and seedmodels [2]. The same CDD factor appears in the world-sheet S-matrix of the light-conegauge-fixed AdS × S string sigma model [4] and in the study of effective bosonic stringtheory in flat space [5]. It also describes the world-sheet scattering of light-cone stringson AdS backgrounds without RR fields [6, 7, 8]. Its relation to the T T deformationwas pointed out in [9]. For many other aspects of
T T deformed models see the lecturenotes [10].There are various connections of
T T deformed relativistic models to two-dimensionalgravity. A
T T deformed S-matrix and the partition function can be obtained by couplinga seed model to the flat space Jackiw-Teitelboim (JT) gravity and its generalisations[11, 12, 13]. This leads to the interpretation of the
T T deformation as a nonlocalfield dependent change of space-time coordinates of the seed model [14]. The partition2unction of a deformed model can also be derived by coupling a seed model to a randomgeometry [15]. The action of a
T T deformed model can be obtained by interpretingit [16, 17] as the action of a non-critical string sigma model in a parameter dependentuniform light-cone gauge introduced in [18]. Most of the
T T deformed Lorentz invariantactions [19, 3, 20, 21, 16, 22, 23, 24, 25] derived by using other methods are particularcases of the
T T deformed action for a very general system of any number of bosons andfermions with an arbitrary potential which was derived in [17] by using the light-conegauge approach. In fact, for
T T deformations with the canonical stress-energy tensorthis action is universal and can be applied to any model.The
T T deformation of non-Lorentz invariant models is also very interesting to studyeven at the classical level. Many non-relativistic models, for example the nonlinearSchr¨odinger (NLS) equation, the Landau-Lifshitz (LL) equation and the Gardner equa-tion which is a combination of the Korteweg–de Vries (KdV) and the modified KdV(mKdV) equation, play important roles in describing various phenomena in nonlinearoptics, hydrodynamics, plasma physics and condensed matter physics. Some aspects ofnon-Lorentz invariant
T T deformed models have been studied in [28, 29, 30, 31, 32, 33].The light-cone gauge approach to
T T deformed models works equally well for rel-ativistic and non-relativistic models. In particular, as was mentioned in [17], it couldbe used to derive the
T T deformed action for the chiral SYK model and the matrixnonlinear Schr¨odinger model.In this paper we derive the
T T deformed actions for the matrix NLS equation, theLL equation and the Gardner equation by using the light-cone gauge approach. Theresulting actions are written in the first-order form and depend on auxiliary fields. Forthe deformed matrix NLS and LL models, the auxiliary fields satisfy algebraic equationsof motion and can be eliminated leading to Nambu-Goto type actions. The
T T deformedGardner model is more involved because the auxiliary fields appear in the deformedaction together with their space derivatives, and it is unlikely that there exists a localdeformed action depending only on the physical field. Moreover, the Gardner field whichappears in the Gardner equation is not the physical field of the Gardner model actionbut one of the auxiliary fields.We then find one-soliton solutions of the deformed NLS and KdV models. Thedeformed NLS soliton clearly exhibits the general phenomenon of widening/narrowingthe width of particles under the
T T deformation recently discussed in [29]. However,in the nonrelativistic case whether the soliton’s size is increasing or decreasing dependsnot only on the sign of the deformation parameter but also on soliton and potentialparameters. As to the
T T deformed KdV soliton, we find a one-parameter family ofsolutions where the extra parameter is related to the time dependence of the physicalfield at space infinities. If one fixes the dependence, then the extra parameter can beinterpreted as the parameter of the deformation by the time component of the conservedcurrent due to the invariance of the
T T deformed Gardner model under constant shiftsof its physical field. The parameter modifies the properties of the soliton, in particular,it appears in the dispersion relation. All these solutions reduce to the usual KdV solitononce one takes the
T T deformation parameter to 0.3he
T T deformed action for the (non-matrix) NLS model has been also found in[31, 32, 33] by using different and substantially more complicated methods than thelight-cone gauge one, and some deformed soliton solutions have been analysed in [32].The paper is organised as follows. In section 2 we first review the universal
T T deformed action derived in [17] and introduce our notations. Then in section 2.1, as awarm-up, we obtain the well-known
T T deformed Lagrangian of a sigma-model of scalarfields with arbitrary potential and B -field. In section 2.2-2.4 we get the T T deformedactions for the matrix NLS, the LL and the Gardner models. In section 2.3 we also showhow the deformed NLS and sine-Gordon models can be obtained from the deformedLL model by taking appropriate limits generalising the well-known results for the seedmodels [34]. In section 3.1 we discuss a one-soliton solution of the
T T deformed NLSequation with the potential which in addition to the usual quartic term also includes thedensity of particles. This term is unimportant for the undeformed NLS model becauseit can be removed by a time dependent U(1) transformation of the fields. The
T T deformed model and its solutions however depend on it in a nontrivial way. In section3.2 we consider a one-parameter family of one-soliton solutions of the
T T deformedKdV equation which is the simplest case of the Gardner equation. In Conclusions wesummarise the results obtained and discuss numerous open problems. Technical detailscan be found in several Appendices.
T T deformed models
All models we are going to discuss in this paper are
T T deformations of a seed modeldescribed by the following action S = Z d x d t L , L = P ta (Ψ) ∂ t Ψ a + P xa (Ψ) ∂ x Ψ a − V(Ψ) . (2.1)Here Ψ a , a = 1 , . . . , n are bosonic and fermionic fields which can be real or complex.If a field is complex then the set (Ψ a ) also includes its complex conjugate field. P ta , P xa and V are chosen so that the action (2.1) is real and Grassmann even but otherwisethey are arbitrary functions of the fields Ψ a . The seed action is written in the first-orderformalism with respect to both time and space, and as a result many of the fields arenon-dynamical. If each Ψ a belongs to a Lorentz group representation and P ta , P xa belongto the conjugate representation, and V is a Lorentz scalar then the seed model is Lorentzinvariant.The light-cone gauge approach to T T deformed models developed in [17] then leadsto the following deformed Lagrangian L = K tt + K xx − V + α (K tt K xx − K tx K xt )1 + α V = L − α (cid:15) γρ (cid:15) µν K µγ K νρ α V , (2.2)where K tγ ≡ P ta ∂ γ Ψ a , K xγ ≡ P xa ∂ γ Ψ a , γ = t, x , (2.3)4nd the skew-symmetric Levi-Civita symbol is defined by (cid:15) = (cid:15) tx = 1 = (cid:15) xt = (cid:15) . Toget (2.2) from the Lagrangian (3.53) in [17] one should make the following replacementsin (3.53): Ψ ± → Ψ, Ψ a K + ab → − i P tb , Ψ a K − ab → − i P xb , ∂ + → ∂ t , ∂ − → ∂ x .The canonical stress-energy tensor of the deformed model can be easily calculated T µν = ∂ L ∂∂ µ Ψ a ∂ ν Ψ a − δ µν L (2.4) T tt = − K xx + V1 + α V , T xt = K xt α V , T tx = K tx α V , T xx = − K tt + V1 + α V , (2.5)and used to check that the deformed Lagrangian (2.2) satisfies the flow equation ∂ L ∂α = T tt T xx − T tx T xt (2.6)Since any seed model can be written in the form (2.1), the T T deformed Lagrangian(2.2) is universal. However, in a non-relativistic case the seed Lagrangian (2.1) mayalso include total derivative terms which do not change the equations of motion of theseed model but they do change the canonical stress-energy tensor and as a result theLagrangian and the equations of motion of the deformed model may depend on the totalderivative terms. This dependence does not seem to be spurious, and we do not thinkthat it can be undone by a field redefinition.
T T deformed sigma model
As a warm-up, in this subsection we discuss the well-known deformation of a sigma-model of n scalar fields described by the Lagrangian L = 12 η αβ ∂ α X i ∂ β X j G ij ( X ) + 12 (cid:15) αβ ∂ α X i ∂ β X j B ij ( X ) − U ( X ) , (2.7)where η αβ = diag(1 , − (cid:15) = (cid:15) tx = 1 = (cid:15) xt , and U is an arbitrary potential.To bring the Lagrangian to the form (2.1), we introduce the momentum vectors P αi = ∂ L ∂∂ α X i = (cid:16) η αβ G ij + (cid:15) αβ B ij (cid:17) ∂ β X j . (2.8)The component P ti is the momentum conjugate to X i .Solving these equations for ∂ α X µ , one finds ∂ α X i = (cid:16) η αβ e G ij + (cid:15) αβ e B ij (cid:17) P βj , (2.9)where e G ij and e B ij satisfy G ij e G jk + B ij e B jk = δ ki , G ij e B jk + B ij e G jk = 0 , (2.10)5hich can be solved as e G ij (cid:16) G jk − B jl G lm B mk (cid:17) = δ ik , e B ij = − e G ik B kl G lj = − G ik B kl e G lj . (2.11)Note that e G is symmetric and e B is anti-symmetric.It is then straightforward to rewrite L in the first-order formalism L = P γi ∂ γ X i − (cid:16) η γρ e G ij + (cid:15) γρ e B ij (cid:17) P γi P ρj − U . (2.12)It is the form of L we need. The set (Ψ a ) consists of X i , and P γi , andK tt = P ti ∂ t X i , K xx = P xi ∂ x X i , K tx = P ti ∂ x X i , K xt = P xi ∂ t X i ,V = 12 (cid:16) η γρ e G ij + (cid:15) γρ e B ij (cid:17) P γi P ρj + U . (2.13)Thus, the
T T deformed Lagrangian of the sigma model is L = P γi ∂ γ X i − (cid:16) η γρ e G ij + (cid:15) γρ e B ij (cid:17) P γi P ρj − U − α (cid:15) γρ (cid:15) µν P µi ∂ γ X i P νj ∂ ρ X j α (cid:16) η γρ e G ij + (cid:15) γρ e B ij (cid:17) P γi P ρj + αU . (2.14)One can get rid of the auxiliary fields P γi by using their equations of motion and, choosinga proper solution of the resulting quadratic equation on L , one gets the well-knownanswer L ph = − α + 12 ˜ α + 12 ˜ α q α ( ˙ X − X ) − α ( ˙ X X − ( ˙ XX ) ) + ˙ X i X j B ij , (2.15)where˙ X ≡ G ij ˙ X i ˙ X j , X ≡ G ij X i X j , ˙ XX ≡ G ij ˙ X i X j , ˜ α = α (1 + αU ) . (2.16)It is worth stressing that the Lagrangian (2.14) describes both the perturbative andnon-perturbative in α solutions of the quadratic equation on L . T T deformed matrix nonlinear Schr¨odinger model
The Lagrangian of the matrix nonlinear Schr¨odinger model is L = i ψ ˙ ψ − ˙¯ ψψ ) − ¯ ψ ψ − U , U = κ ¯ ψψ ¯ ψψ − µ ¯ ψψ . (2.17) To find L ph which depends only on the physical fields X i it is not necessary to solve the equationsof motion for P αi . Since L depends just on K γρ and V it is sufficient to know only them to find L ph . Thiscan be done by expressing V in terms of L and K γρ , and substituting it into the equations of motion for P αi . This leads to simple linear equations for K γρ which can be easily solved. The consistency conditionof the solution with the expression for V in terms of L and K γρ leads to a quadratic equation for L withcoefficients which depend only on the physical fields. ψ = ( ψ ai ) , ¯ ψ = ψ † = ( ψ ∗ ia ) , a = 1 , . . . , n , i = 1 , . . . , m (2.18)are complex n × m and m × n matrices hermitian conjugate to each other. Then, thetrace is implied in (2.17), i.e.¯ ψ ˙ ψ ≡ ˙ ψ ∗ ia ψ ai , ¯ ψψ ¯ ψψ ≡ ψ ∗ ia ψ aj ψ ∗ jb ψ bi . (2.19)To bring the Lagrangian into the desired form we introduce two auxiliary matriceshermitian conjugate to each other A = ( A ai ) , ¯ A = A † = ( A ∗ ia ) , a = 1 , . . . , n , i = 1 , . . . , m , (2.20)and rewrite (2.17) as L = i ψ ˙ ψ − ˙¯ ψψ ) − ¯ Aψ − ¯ ψ A + ¯ AA − U (2.21)Thus, the set (Ψ a ) consists of ψ, ¯ ψ, A, ¯ A , andK tt = i ψ ˙ ψ − ˙¯ ψψ ) , K xx = − ¯ Aψ − ¯ ψ A , K tx = i ψψ − ¯ ψ ψ ) , K xt = − ¯ A ˙ ψ − ˙¯ ψA , V = U − ¯ AA , (2.22)where the trace is implied.The
T T deformed Lagrangian of the matrix nonlinear Schr¨odinger model, therefore,is L = K tt − ¯ Aψ − ¯ ψ A + ¯ AA − U − α (cid:16) K tt ( ¯ Aψ + ¯ ψ A ) − K tx ( ¯ A ˙ ψ + ˙¯ ψA ) (cid:17) − α ( ¯ AA − U ) . (2.23)Eliminating the auxiliary fields A, ¯ A by using their equations of motion and, choosingthe regular in α solution of the resulting quadratic equation on L , one gets L ph = − α + 1 + αK tt + √ Λ2 ˜ α , ˜ α = α (1 + αU ) , Λ = (1 + αK tt ) (1 − α ¯ ψ ψ ) + 4 a ˜ α (1 + αK tt ) K tx ( ˙¯ ψψ + ¯ ψ ˙ ψ ) − α ˜ α ( K tx ) ˙¯ ψ ˙ ψ (2.24)where in the expression for Λ the trace is implied.It is clear that the deformation drastically modifies the Poisson structure of themodel, and developing a Hamiltonian formulation requires dealing with an intricatesystem of second-class constraints. The same seems to be valid for any non-relativisticmodel. 7 .3 T T deformed Landau–Lifshitz model
We mostly follow the notations in [34].The Landau-Lifshitz equation is ∂S i ∂t = 1 R (cid:15) ijk S j ∂ S k ∂x + (cid:15) ijk S j J k S k , (2.25)where S i = R , and we sum over repeated indices even if there are 3 of them.The fields S i have the Poisson structure { S i ( x ) , S j ( y ) } = − η (cid:15) ijk S k ( x ) δ ( x − y ) , (2.26)and the LL equation follows from the Hamiltonian H = Z dx η (cid:18) R (cid:18) ∂S k ∂x (cid:19) − J k S k + J R (cid:19) . (2.27)where the constant J R guaranties the vanishing of the Hamiltonian density in therapidly decreasing case where we impose the conditions S k ( ±∞ ) = δ k R . By rescaling S k and x, y one can set R = 1 and η = 1. We prefer to keep these two parameters tosimplify taking the limits to the NLS and sine-Gordon models.To find the T T deformed LL model we need its Lagrangian description. To this endwe multiply (2.25) by (cid:15) lmi S m , and, changing the indices, get (cid:15) ijk S j ∂S k ∂t = − (cid:16) δ ij − R S i S j (cid:17)(cid:18) ∂ S j ∂x + R J j S j (cid:19) , S i = R . (2.28)These equations can be derived from the following Lagrangian L = Z dx Z ∞ dr ηR (cid:15) ijk S i ∂S j ∂r ∂S k ∂t + Z dx η (cid:18) − R (cid:18) ∂S k ∂x (cid:19) + J k S k − J R (cid:19) , (2.29)where S k are subject to the sphere constraint S k = R . In the first term S k dependon an extra radial coordinate r , and satisfy the conditions S k ( x, t, r ) | r =0 = S k ( x, t ), S k ( x, t, r ) | r = ∞ = δ k R . This is a WZNW type term, and its variation is δ Z dx Z ∞ dr ηR (cid:15) ijk S i ∂S j ∂r ∂S k ∂t = − Z dx (cid:15) ijk ηR S i δS j ∂S k ∂t , (2.30)where the variation δS k is tangent to the sphere, i.e. it obeys the constraint δS k S k = 0.Because of this, any products V k δS k have to be replaced with V k ( δ km − S k S m R ) δS k . Itproduces all the terms on the r.h.s. of the equations of motion (2.28). Introducing anycoordinates φ a , a = 1 , S k = R , one can bring the WZNW term to thetotal derivative form1 ηR (cid:15) ijk S i ∂S j ∂r ∂S k ∂t = − ∂∂r (cid:16) P a ∂φ a ∂t (cid:17) + ∂∂t (cid:16) P a ∂φ a ∂r (cid:17) , (2.31)8here P a satisfies the condition P a ( x, t, ∞ ) = 0 to ensure the absence of the contributionfrom the first term at r = ∞ . We will always drop the total time derivative term, inte-grate the remaining term over r and, as a result, use the following Lagrangian (density)for the T T deformation L = P k ˙ S k − ηR (cid:18) ∂S k ∂x (cid:19) + 12 η ( J k S k − J R ) − U add ( S k ) , (2.32)where P k are such that P k ˙ S k = P a ˙ φ a , and U add is an additional potential term which canbe an arbitrary function of S k . We will choose it later so that the T T deformed NLSmodel could be obtained as a special limit of the
T T deformed LL model.In particular, in spherical coordinates S = cos φ sin θ , S = sin φ sin θ , S = cos θ , (2.33)the WZNW term takes the form P k ˙ S k = 1 ηR (cos θ −
1) ˙ φ = − ηR sin θ φ . (2.34)Now, introducing an auxiliary vector A i , the LL model Lagrangian can be written as L = P k ˙ S k + A k S k + ηR A k + 12 η ( J k S k − J R ) − U add (2.35)We see that the set (Ψ a ) consists of S k , A k , andK tt = P k ˙ S k , K tx = P k S k , K xx = A k S k , K xt = A k ˙ S k ,V = − ηR A k + U , U = − η ( J k S k − J R ) + U add . (2.36)Thus, the T T deformed Lagrangian of the LL model is L = P k ˙ S k + A k S k + ηR A k + η ( J k S k − J R ) − U add + αP k A l ( ˙ S k S l − S k ˙ S l )1 − αηR A k − α η ( J k S k − J R ) + αU add . (2.37)One can get rid of the auxiliary fields A k by using their equations of motion and, choosinga proper solution of the resulting quadratic equation on L , one gets L ph = − α + 1 + αK tt + √ Λ2 ˜ α , ˜ α = α (1 + αU ) , Λ = (1 + αK tt ) (1 − αηR S k ) + 4 aηR ˜ α (1 + αK tt ) K tx S k ˙ S k − α ˜ αηR ( K tx ) ˙ S k . (2.38)The similarity of this Lagrangian with (2.24) for the NLS model is obvious, and notaccidental. It is well-known that the NLS model can be obtained from the LL model934]. Since the NLS model has a U(1) symmetry we need to set J = J = J . Then,the LL model also has the symmetry and S is proportional to the density of the U(1)current, and it can be added to the LL Lagrangian while preserving the integrability ofthe model. Thus, the potential U we are going to use is U = 12 η ( J − J )( R − S ) + ν ( R − S ) , (2.39)where ν is any constant.Next, we use the spherical coordinates (2.33), and getK tt = P k ˙ S k = − ηR sin θ φ , K tx = P k S k = − ηR sin θ φ , ηR S k = 1 ηR ( θ + sin θφ ) , ηR ˙ S k = 1 ηR ( ˙ θ + sin θ ˙ φ ) , ηR S k ˙ S k = 1 ηR ( θ ˙ θ + sin θφ ˙ φ ) , U = 12 η ( J − J ) sin θ + ν (1 − cos θ ) . (2.40)Now, we set R = 1, and rescale the angle θ as θ = √ η ρ . (2.41)We want to take the limit η → tt → − ρ ˙ φ , K tx → − ρ φ , ηR S k → ρ + ρ φ ) , ηR ˙ S k →
2( ˙ ρ + ρ ˙ φ ) , ηR S k ˙ S k → ρ ˙ ρ + ρ φ ˙ φ ) . (2.42)To make contact with the NLS model, we introduce ψ, ¯ ψ as ψ = ρ e iφ , ¯ ψ = ρ e − iφ , (2.43)and find − ρ ˙ φ = i ψ ˙ ψ − ˙¯ ψψ ) , − ρ φ = i ψψ − ¯ ψ ψ ) , ρ + ρ φ ) = 2 ¯ ψ ψ ,
2( ˙ ρ + ρ ˙ φ ) = 2 ˙¯ ψ ˙ ψ , ρ ˙ ρ + ρ φ ˙ φ ) = ¯ ψ ˙ ψ + ˙¯ ψψ . (2.44)This is exactly what we have in (2.24), and the only question remaining is what happenswith the potential U in the limit. Expanding the potential in powers of ρ , one gets U = ( J − J + ην ) ρ − η ( η ν − J + 4 J ) ρ + O ( ρ ) . (2.45)Now, to reproduce the NLS model potential we impose the conditions J − J + ην = − µ , − η ( η ν − J + 4 J ) = κ , (2.46)10nd get J = J + µ − κη , ν = 2 κη − µ η . (2.47)It is then easy to check that in the limit η → U → κρ − µρ = κ ( ¯ ψψ ) − µ ¯ ψψ , (2.48)which is indeed the NLS model potential.The sine-Gordon model is also a limiting case of the LL model. To get the SG modelwe set U add = 0, and parametrise S k as [34] S = − βπ , S = s R − β π βφ , S = s R − β π βφ , (2.49)where β is a new constant, and π and φ are the fields parametrising S k . We then getK tt = P k ˙ S k = β η π ˙ φ , K tx = P k S k = β η π φ , ηR S k = 1 ηR β (cid:16) φ ( β π − R ) + 16 R π (cid:17) R − β π , ηR ˙ S k = 1 ηR β (cid:16) ˙ φ ( β π − R ) + 16 R ˙ π (cid:17) R − β π , ηR S k ˙ S k = 1 ηR β (cid:16) φ ˙ φ ( β π − R ) + 16 R π ˙ π (cid:17) R − β π ,U = − β π ( J (cos( βφ ) − − J (cos( βφ ) + 1) + 2 J ) + 4 ( J − J ) R (cos( βφ ) − η . (2.50)Now, we choose η = β , take the limit R → ∞ , and getK tt → π ˙ φ , K tx → π φ , ηR S k → φ , ηR ˙ S k → ˙ φ , ηR S k ˙ S k → φ ˙ φ . (2.51)Finally, we choose J k as [34] J = J + 1 , J = J + 1 + m R , (2.52)and in the limit R → ∞ get UU = 12 π + m β (1 − cos βφ ) . (2.53)Thus, in this limit we get the T T deformation of a model with the seed Lagrangian L = π ˙ φ − φ − π − m β (1 − cos βφ ) , (2.54)which is indeed the SG model Lagrangian.11 .4 T T deformed Gardner equation
The Gardner equation is a combined KdV-mKdV equation˙ u + µ u + 6 g uu − h u u + u = 0 , (2.55)where g, h and µ are constants. If u satisfies periodic boundary conditions then µ canbe removed by a constant shift of uu → u − c , hc + gc − µ , (2.56)which also changes g . For decreasing boundary conditions such a shift is obviouslyforbidden. The Gardner equation is the continuity equation for the current J t = u , J x = µ u + 3 g u − h u + u , (2.57)and if the charge Q = R dx u exists then it is conserved. In what follows we only considerthe case where Q exists.The Gardner equation can be derived from the action S = Z d x d t L , L = κ ( − ˙ φφ − µφ − gφ + hφ + φ ) , (2.58)where the field φ satisfies the boundary conditions φ ( t, ∞ ) − φ ( t, −∞ ) = Q φ = const , (2.59) κ is any constant, and u is related to φ as u = φ . (2.60)Obviously, in the undeformed case Q φ = Q . The equation of motion for φ is invariantunder a shift of φ by any function of time. By using this invariance one may require φ ( t, ±∞ ) to be constant. However, as we will see, in the deformed case this invarianceis broken, and different time dependence of φ ( t, ∞ ) leads to different solutions.To write the Lagrangian (2.58) in the form (2.1), we first introduce an auxiliary field A satisfying the equation of motion A = φ , and cast L into the form L = κ ( − ˙ φφ − µφ − gφ + hφ + 2 Aφ − A ) . (2.61)Then, we introduce auxiliary fields for φ and ˙ φ u = − κ ∂ L ∂ ˙ φ = φ , B = − κ ∂ L ∂φ = ˙ φ + 2 µφ + 6 gφ − hφ + 2 A , (2.62)and get the desired form of the Lagrangian L = κ ( − u ˙ φ − Bφ + 2 A u + u B − µ u − g u + h u − A ) . (2.63)12learly, the auxiliary field u is the Gardner field u , and the existence of the conservedcurrent (2.57) is the consequence of the invariance of L under constant shifts of φ .We see that the set (Ψ a ) consists of φ, u , B, A , andK tt = − κ u ˙ φ , K xx = − κ Bφ + 2 κ A u , K tx = − κ u φ , K xt = − κ B ˙ φ + 2 κ A ˙ u ,V = − κ ( u B − µ u − g u + h u − A ) . (2.64)Therefore, the T T deformed Lagrangian of the Gardner model is L = κ − u ˙ φ − Bφ + 2 A u + u B − µ u − g u + h u − A − ακ A u ( u ˙ φ − ˙ u φ )1 − ακ ( u B − µ u − g u + h u − A ) , (2.65)where the field φ satisfies the same boundary conditions (2.59) as in the undeformedcase. The undeformed Lagrangian (2.63) changes under the transformation φ → φ + f ( t ) , B → B + dfdt , (2.66)by a derivative term L → L − κ ∂∂x (cid:16) dfdt φ (cid:17) . (2.67)The T T deformed Lagrangian (2.65), however, transforms in a nontrivial way, and there-fore the time dependence of φ at x = ±∞ changes physical properties of the T T deformedGardner model.In the undeformed model the auxiliary field u coincides with the Gardner field u . Itis therefore reasonable to use the same identification in the T T deformed Lagrangian(2.63). One might try to use the fact that the Gardner equation is the continuityequation, and to identify φ or J t = − κ ∂ L ∂ ˙ φ with u . Both φ and J t are time componentsof conserved currents and coincide with u in the undeformed case. Our analysis of theone-soliton solution of the T T deformed KdV equation indicates that the auxiliary field u is a better choice.It is doubtful that one can get rid of all the auxiliary fields because the Lagrangiandepends on derivatives of u . In what follows without loss of generality we set κ = 1. T T deformed NLS soliton
In this subsection we discuss a one-soliton solution of the
T T deformed NLS model. Letus first recall some properties of the seed model. Its Lagrangian is given by (2.21) where ψ, ¯ ψ (and A, ¯ A ) are complex fields conjugate to each other. The Lagrangian is invariantunder the Galilean transformations x → x − v t , t → t , ψ → e i v t − i v x ψ , A → e i v t − i v x ( A − i v ψ ) (3.1)13hich implies the usual nonrelativistic dispersion relation for a one-soliton solution, andallows one to recover a full solution from a soliton at rest. It is also invariant underthe U(1) transformations ψ → e iζ ψ , A → e iζ A , and the finite density term µ ¯ ψψ isproportional to the time component of the conserved U(1) current. It can therefore beremoved by the following time-dependent U(1) transformation ψ → e − iµ t ψ , A → e − iµ t A . (3.2)Thus, in the rapidly decreasing case the finite density term plays no essential role in theundeformed NLS model.The one-soliton solution we are going to deform exists for κ <
0, and to simplify theformulae below we introduce a new coupling constant g > κ as κ = − g , (3.3)Then, the one-soliton solution is given by ψ = ug (cid:16) u ( x − vt ) (cid:17) e iφ , φ = v x − vt ) + t (cid:16) u + v + 4 µ (cid:17) , A = ψ , (3.4)where v is the velocity of the soliton, and u > Q , the momentum P and the energy E of the soliton are Q = Z ∞−∞ dx ¯ ψψ = 4 ug ,P = − Z ∞−∞ dx T tx = 2 u vg = m v , m = 2 ug = Q ,E = Z ∞−∞ dx T tt = uv g − u g − uµg = P m − g m − µ Q , (3.5)and up to a constant the dispersion relation is indeed nonrelativistic, and the U(1) chargeis twice the mass of the soliton.To find a T T deformation of the soliton (3.4), we begin with the
T T deformedLagrangian (2.23) which for the NLS model simplifies to L = i ( ¯ ψ ˙ ψ − ˙¯ ψψ ) − ¯ Aψ − ¯ ψ A + ¯ AA − U + α i ( ¯ Aψ + ¯ ψA )( ˙¯ ψψ − ¯ ψ ˙ ψ )1 − α ( ¯ AA − U ) , (3.6) U = − g ψψ ) − µ ¯ ψψ . (3.7)It is clear from the Lagrangian (3.6) that the U(1) transformation (3.2) does not removethe µ -dependent terms, and therefore, T T deformed soliton properties depend on it.14t is convenient to introduce the polar coordinates for ψ and redefine the auxiliaryfields as follows ψ = ρ e iφ , ¯ ψ = ρ e − iφ , A = ρ A e iφ , ¯ A = ¯ ρ A e − iφ , (3.8)because the U(1) symmetry is realised just by shifts of φ , and the Lagrangian dependsonly on the derivatives of φ . Clearly, ρ is the amplitude and φ is the phase of the soliton.In terms of the fields the Lagrangian (3.6) takes the form L = − ρ ˙ φ − ( ρ A + ¯ ρ A ) ρ + i ( ρ A − ¯ ρ A ) ρ φ + ¯ ρ A ρ A + g ρ + µ ρ − αρ ( ρ A + ¯ ρ A )( ˙ ρφ − ρ ˙ φ )1 − α ( ¯ ρ A ρ A + g ρ + µ ρ ) , (3.9)where ρ A and ¯ ρ A are complex conjugate to each other.The deformed one-soliton solution can be derived by explicitly solving the equationsof motion by using the following ansatz ρ ( t, x ) = ρ ( x − vt ) , ρ A ( t, x ) = ρ A ( x − vt ) , ρ ( ±∞ ) = 0 ,φ = ω t + ϕ ( x − vt ) , ω = u + v µ . (3.10)The phase φ of the soliton is at most the sum of a linear function of x, t which we canchoose without loss of generality to be the same as in the undeformed case, and of afunction of x − vt due to the restricted dependence of the other fields.The derivation is sketched in appendix A, and the solution can be expressed in termsof ρ as follows ρ = ± ρ √ u − g ρ αρ ( − g ρ + u − v − µ ) , ρ A = 12 ρ (cid:18) iv ± q u − g ρ (cid:19) ,x − vt = x ± − (cid:18) u √ u − g ρ (cid:19) u ∓ α √ u − g ρ ( u + 3 v + 12 µ + 2 g ρ )6 g ,φ = 12 v ( x − vt ) + 14 t (cid:16) u + v + 4 µ (cid:17) ± αv ( u − g ρ ) / g . (3.11)Since the phase φ and the auxiliary field ρ A are smooth functions of x and t if theamplitude ρ is, we discuss only the properties of ρ . Unlike the undeformed soliton, theamplitude has a nontrivial dependence on the chemical potential µ . However, it entersthe amplitude only through the combination v + 4 µ . Without loss of generality we canset t = 0 and x = 0. Clearly, the maximum of ρ ( x ) is equal to u/g , and it is at x = 0.From the equation for ρ we see that ρ is a single-valued function of x only if ρ = ∞ forall x which leads to the condition4 + αρ (cid:16) − g ρ + u − v − µ (cid:17) = 0 for 0 ≤ ρ ≤ ug . (3.12) These variables are also useful for analysing the JT -type deformations [35] of the NLS model.
15o analyse (3.12) it is convenient to introduce a new parameter W = u − v − µ . (3.13)Then, the roots of the equation ρ = ∞ are given by ρ ± = W ± q g α + W g , ρ (cid:12)(cid:12)(cid:12) ρ = ρ ± = ∞ . (3.14)A simple analysis shows that the roots ρ ± are outside the interval (0 , u/g ) ifI. W ∈ R and − g W < α ≤ ⇒ complex ρ ± II. W = u − v − µ < α ≤ − g W ⇒ ρ − ≤ ρ < W − u = − u − v − µ > g u − u W < α ≤ − g W ⇒ u g < ρ − ≤ ρ IV. W − u = − u − v − µ < < α < g u − u W ⇒ ρ − < < u g < ρ V. W − u = − u − v − µ > α > ⇒ ρ − < < u g < ρ (3.15)Introducing the following two critical values of αα − ≡ − g ( u − v − µ ) < , α + ≡ g u ( u + v + 4 µ ) , (3.16)we can combine these regions as followsA. − ∞ < u − v − µ < − ∞ < α < α + , α + > < u − v − µ < u and α − < α < α + , α + > u < u − v − µ < u and α + < α − < α < ∞ D. 4 u < u − v − µ < ∞ and α + < α < ∞ , α + < α − < v > u − µ which imposes a lower bound on v if u > µ .If µ ≥ u , v but C and D are never satisfied.The condition D is satisfied if v < − u − µ which imposes an upper bound on v if3 u < − µ . If µ < ρ ( x ) is an evenfunction of x , and the differential equation for ρ allows one to replace the integrationof any expression over x with the integration over ρ . The U(1) charge, energy andmomentum of the soliton are easily found, appear to be unchanged by the deformation,16 = α = α - / α = α - α = α - ρ ( x ) α = α = α - / α = α - α = α - ρ ( x ) α = α = α - / α = α - α = α - ρ ( x ) Figure 1: Left: Case B, µ = 0, α − = −
32, displaying formation of shockwave solution fornegative α . Centre: Boundary case of B and C, µ = − / α − = −
8, example of competingshockwave and narrowing behaviours creating a double-loop solution. Right: Case C, µ = − . α − = − . α − , after which it forms a loop. and are given by (3.5). The shape of the soliton obviously changes, and, in particular,we can define its size by using the full-width-half-maximum F W HM = − α √ u ( u + 2 v + 8 µ )4 g + 4 log (cid:16) √ (cid:17) u . (3.18)The soliton clearly exhibits the general phenomenon of widening/narrowing the widthof particles under the T T deformation [29]. However, whether the size is increasing ordecreasing depends not only on the sign of α but also on the sign of s ≡ u + 2 v + 8 µ .Obviously, it is positive for all values of u and v only if µ ≥
0. It is also positive ifthe soliton parameters satisfy condition A but it is negative for conditions C or D. Forparameters satisfying condition B one can have both positive and negative s if µ isnegative. The visually distinct solutions are demonstrated in figure 1. Further plots forall cases are shown in figures 6 and 7 in Appendix B, which display the same behavioursas in case C. Since the amplitude depends only on v + 4 µ we set v = 0 without lossof generality when plotting solutions. We set g = 1 , u = 1, so that the graphs areparametrised by µ . If the soliton base widens (or remains constant if u + 2 v + 8 µ = 0)as the magnitude of α increases then the peak flattens as in figures 6 and 7. Let us alsomention that as one can see from (3.18) the heavier and speedier the soliton is the widerit is. That is very different from the undeformed case where the width is independentof speed and decreases with mass increasing.Let us now assume that u, v, µ satisfy one of the conditions (3.17) but α is at aboundary of its allowed values, i.e. it takes one of the critical values α ± . Then, a shock-wave singularity develops, and away from the critical values the solution ρ ( x ) becomes amulti-valued function of x . In this case at least one of the roots ρ ± is inside the interval(0 , u/g ). Regions where only one root exists form loops as in figures 1, 7 and 6, due to x ( ρ ) (given explicitly in (3.20)) becoming negative. This happens if u, v, µ satisfy eitherconditions A and B with α > α + > α < α + <
0. Where both roots17xists the solution is either a bell shape which happens if u, v, µ satisfy condition B with α < α − <
0, or it is a double loop shape, both shown in figure 1. The double loop shapehappens if u, v, µ satisfy condition C with α + < α < α − <
0. The conditions for theappearance of these solutions are summarised belowLoop: ( u + v + 4 µ > , α > α + > , ρ − < < ρ + < ug u + v + 4 µ < , α < α + < , < ρ − < ug < ρ + Bell: 0 < u + v + 4 µ < u , α < α − < − u < u + v + 4 µ < , α + < α < α − < T T deformation. We attempt to fix this by redefining the amplitudefunction as a piecewise smooth curve by exploiting the translational invariance of x − vt . ρ ( x ) ρ ( x ) ρ ( x ) Figure 2: Demonstration of the gluing procedure on the loop (Left), bell (Centre) and double-loop (Right) soliton solutions, indicating the points where ρ becomes singular. We set t = 0, x = 0, choose the upper sign in the solution (3.11), and introduce thefunction x ( ρ ) = 2 coth − (cid:18) u √ u − g ρ (cid:19) u − α √ u − g ρ ( u + 3 v + 12 µ + 2 g ρ )6 g , ≤ ρ ≤ ug . (3.20)In terms of x ( ρ ) the piece-wise smooth solutions can be written asLoop: x + L ( ρ ) = − x ( ρ ) θ ( ρ − ρ + ) + (cid:16) x ( ρ ) − x + (cid:17) θ ( ρ + − ρ ) x − L ( ρ ) = − x ( ρ ) θ ( ρ − ρ − ) + (cid:16) x ( ρ ) − x − (cid:17) θ ( ρ − − ρ )Bell:Double Loop: ) x B ( ρ ) = x ( ρ ) θ ( ρ − ρ + ) + (cid:16) x + − x ( ρ ) (cid:17) θ ( ρ + − ρ ) θ ( ρ − ρ − )+ (cid:16) x ( ρ ) − x + − x − (cid:17) θ ( ρ − − ρ ) (3.21)18here θ is the Heaviside function and x ± = x ( ρ ± ). Each of these functions is a positivedecreasing function of ρ with a continuous first derivative. The soliton profile ρ ( x ) isan even function of x given for x ≥ x but it is an integrablesingularity. Since they depend on ρ and ρ , the energy, momentum and charge are givenby the same expressions (3.5). The three forbidden solution types are reconstructed intovalid amplitudes in figure 2. Note that all these new solutions increase in width as α increases in magnitude. Whether such a gluing procedure is legitimate remains to beseen but there are examples of models with singular solitons, see e.g. [36].Let us finally mention that the inverse function x ( ρ ) can also be derived through adynamical coordinate transformation as described in [14], and used in [32] to find the T T deformed one-soliton solution for the case µ = 0. T T deformed KdV soliton
In this subsection we discuss a one-soliton solution of the
T T deformed KdV equationwhich corresponds to the g = 1 , h = 0 case of the Gardner equation˙ u + µ u + 6 uu + u = 0 . (3.22)The constant µ is usually set to 0 but we prefer to keep it so that for µ < u = 2 w cosh (cid:16) w ( x − vt ) (cid:17) , w = 12 √ v − µ > ,φ = 2 w tanh ( w ( x − vt )) + f ( t ) , (3.23)where f ( t ) is any function of t . As was discussed in the previous section, in the unde-formed case the soliton properties are independent of f ( t ). In particular, the charge Q ,momentum P and energy E of the soliton are Q = Z ∞−∞ dx u = 4 w ,P = Z ∞−∞ dx u = 163 w ,E = Z ∞−∞ dx ( µ u + 2 u − u ) = 163 µw + 645 w = µP + 35 (cid:18) (cid:19) / P / . (3.24)A funny property of the soliton is that its momentum is always positive even if thevelocity v is negative which requires µ to be negative too. This is counter-intuitive andfor v < P and E which is equivalentto setting κ = − α in the T T deformed Lagrangian (2.65). Then, for small P the dispersion relation would19e approximately the one for a massless relativistic particle. In what follows to have auniform description we will continue using κ = 1 for all values of v .The T T deformed soliton solution depends on the function f ( t ) in a nontrivial way,and we only consider the simplest case f ( t ) = b t where b is an arbitrary constant.In fact, redefining φ as φ → φ + bt , we find that the T T deformed Lagrangian (2.65)transforms as
L → L − b J t , and therefore b can be interpreted as the parameter of thedeformation by the time component of the conserved current due to the invariance of(2.65) under constant shifts of φ .In this case all auxiliary fields are only functions of x − vt , and the T T deformedsolution can be found by using the equations of motion and the ansatz φ = φ ( x − v t ) + b t , u = u ( x − v t ) , A = A ( x − v t ) , B = B ( x − v t ) . (3.25)The full derivation is described in Appendix C. We find that u rather than φ is thenatural field to express our results in terms of. We define ˜ w = v − µ − α b = w − α b to simplify the following expressions. The solution can be written as a set of equationsexpressing u , φ , A and B in terms of uu = ± u √ w − u α u (4 u − w − αb ) , φ = u − α b u α u (4 u − w − αb ) ,B = (cid:16) µ + 4 ˜ w (cid:17) u + b , A = ± u √ w − u , ˜ w = w − α b . (3.26)For the solutions to be real, ˜ w >
0, or equivalently, v > µ + α b . For fixed v, µ, b thiscondition imposes an upper bound on allowed values of α : µ − vb > α . Note also that for α < w = v − µ < b causes the deformed quantities of energy, momentum and thedispersion relation to be dependent on both α and b . E = 1615 ˜ w (cid:16)
12 ˜ w + 5( µ − αb ) (cid:17) , P = 163 ˜ w ,E ( P ) = P (cid:16) µ − αb (cid:17) + 35 (cid:18) (cid:19) / P / . (3.27)The appearance of α in the dispersion relation is due to the fact that the T T deformedKdV model is intrinsically nonlocal and sensitive to the boundary behaviour of φ .Furthermore, the parameter b causes the previously identical conserved charges of J t and φ to become independent Q = Z dx J t = 4 ˜ w (cid:18) w αb (cid:19) , Q φ = Z dxφ = 4 ˜ w (cid:18) −
43 ˜ w αb (cid:19) (3.28)We also find that b defines a flow equation for a deformation under the current J t ∂ L ∂b = − u ( αb u + 1)1 − α u ( − u + αb + 8 ˜ w ) = −J t . (3.29)20ntegrating the equation for u in (3.26), we find the inverse expression for u x − vt = x ± arctanh (cid:16) √ w − u w (cid:17) ˜ w ∓ √ α √ w − u (cid:16) (cid:16) w − u (cid:17) (cid:16) u + 4 ˜ w (cid:17) + 5 αb (cid:16) u + 4 ˜ w (cid:17)(cid:17) , (3.30)which displays both shockwave and looping solutions as in the NLS case. With t = 0and x = 0 the maximum of u ( x ) occurs at x = 0 for u (0) = 2 ˜ w . The full-widthhalf-maximum of the soliton is F W HM = 2arcoth (cid:16) √ (cid:17) ˜ w − √ α ˜ w (cid:16) αb + 28 ˜ w (cid:17) , (3.31)and for positive α it decreases.The derivative u becomes singular when the denominator in the equation for u in(3.26) vanishes d ( u ) ≡ α u (cid:16) u − w − αb ) + 1 = 0 , < u < w . (3.32)In much the same way as the NLS case, restricting the roots of this expression to lieinside the range of u will generate the conditions for the solution to become multi-valued.A detailed analysis of the equation (3.32) can be found in Appendix C where it isshown that at least one root of the equation d ( u ) = 0 lies inside the interval (0 , w ) if4 ˜ w = v − µ − αb = 4 w − αb > ( b = 0 , α < α − < b = 0 , w > b , α (2)+ < α < α (3)+ < w b Bell or Double Loop: ( b = 0 , α > w b = 0 , w > b , < α (1)+ < α < α (2)+ (3.33)Here the critical values of α are given by α − = − q w + 2 | b | − w b , α (2)+ = 2 w − q w − | b | b , α (3)+ = 2 w + q w − | b | b , (3.34)and α (1)+ is the positive root smaller than w b of the following equation1 − α w − αb ! = 0 . (3.35)As one can see from (3.33), the soliton solution is single-valued for α (3)+ < α < α max = 4 w b , w b > , (3.36)21t is interesting that this region is nonperturbative in α .The complex evolution of the solution for α > α ( i )+ are real is shown inFigure 3. For large α the dominating factor is the α dependence in ˜ w which enables theexistence of the nonperturbative regular solutions for α > α (3)+ . These regular solutionsare shown in Figure 4, along with the negative α behaviour. The solution profiles for b = 0 are shown in Figure 8 in Appendix C. α = α = α +( ) / α = α +( ) u ( x ) α = α +( ) α = α α = α α = α +( ) u ( x ) α = α +( ) α = α α = α α = α +( ) u ( x ) Figure 3: Evolution of KdV soliton solutions for w = 1 , b = 1 , α >
0, transitioning betweendifferent types of multi-valued solutions. Left: Width is decreasing with increasing α , α (1)+ ≈ .
23. Centre: Formation of double-loop solution for α (1)+ < α < α (2)+ , with a singular solutionat α = α (2)+ ≈ .
59. The intermediate values are equally spaced, α = α (1)+ + α (2)+ , α = α (1)+ +2 α (2)+ . Right: Amplitude decreasing, transitioning to singular peak at α = α (3)+ ≈ . α = α (2)+ + α (3)+ , α = α (2)+ +2 α (3)+ . α = α +( ) α = α = ( x ) α = α = α = ( x ) α = α = α - / α = α - α = α - u ( x ) Figure 4: Left: Continuation of evolution from figure 3, displaying single-valued solution for α > α (3)+ ≈ .
41. The extreme flattening of the solution in the limit α → w → w = 1 , b = 3, the soliton remains regular for all 0 < α < /
9, after which itceases to exist in a similar fashion. Right: w = 1 , b = 1 , α < α − ≈ − .
4. Solution widens,but with nonzero b develops into a loop solution. = α = α c / α = α c α = α c ϕ ( x ) α = α =- α =- ϕ ( x ) Figure 5: Plots of the soliton φ ( x − vt ) with w = 1. Note shock-wave behaviour is onlydisplayed for α > α c = 27 / Let us also mention that for b = 0 the T T deformed soliton solution can be easilyfound by using the dynamical coordinate transformation [14]. We denote the coordinatesof the undeformed soliton (3.23) with f ( t ) = 0 by τ , σ , and its stress-energy tensor by T γδ , and computing it on the soliton solution, we get T τ τ = 4 w sech ( w ( σ − vτ )) (cid:16) µ − w + 8 w sech ( w ( σ − vτ )) (cid:17) , T στ = v T τ τ , T τ σ = − w sech ( w ( σ − vτ )) , T σσ = v T τ σ . (3.37)The dynamical coordinate transformation is given by dt = (1 + α T σσ ) dτ − α T τ σ dσ = dτ − α T τ σ d ( σ − vτ ) ,dx = (1 + α T τ τ ) dσ − α T στ dτ = dσ + α T τ τ d ( σ − vτ ) ,d ( x − vt ) = (cid:16) α ( T τ τ + v T τ σ ) (cid:17) d ( σ − vτ ) . (3.38)Integrating this relation we find that the deformed inverse relation is x − vt = σ − vτ + 32 αw
15 tanh ( w ( σ − τ v )) (cid:16) ( w ( σ − τ v )) − (cid:17) ,x − vt = 1 w arctanh φ w ! + αφ (cid:16) φ − w (cid:17) . (3.39)Note that φ ( x − vt ) ∈ ( − w, w ) and so the α -dependent term has a fixed sign for allvalues of w . The deformed behaviour of the soliton is fixed by the sign of α . By requiringthe roots of d ( x − vt ) dφ to be real and within the range of φ we find that the critical value ofthe deformed parameter is α c = w . For α > α c the soliton becomes multi-valued asit transitions into a shock-wave solution. For all α < α c the soliton exists and becomeswider as α → −∞ . These behaviours are shown in figure 5, and they are consistentwith [29]. It is easy to check that this solution agrees with (3.30) for b = 0. One can23lso see that φ exhibits a physical shock wave formation. Since φ develops singularitiesas α approaches α c , it cannot be identified with the T T deformed KdV field.
In this paper we have applied the light-cone gauge approach to the
T T deformation tonon-Lorentz invariant models. We have seen that the deformation drastically modifiesthe Poisson structure of all the models we considered, and developing a Hamiltonianformulation requires dealing with an intricate system of second-class constraints. Thisactually makes
T T deformed non-relativistic models more complicated than the rela-tivistic ones where the Hamiltonian formulation is straightforward.We have found one-soliton solutions of the deformed NLS and KdV models, and dis-cussed their properties. The width of the solitons appears to depend on the deformationparameter according to the general phenomenon of widening/narrowing the width ofparticles under the
T T deformation [29]. However, whether soliton’s size is increasingor decreasing depends not only on the sign of the deformation parameter but also onthe potential and soliton parameters. In the NLS case this more complicated behaviouris caused by the addition of the time component of the conserved U(1) current to theseed model. After the
T T deformation this cannot be undone by a time dependent U(1)transformation (3.2).For any values of the parameters of the solitons there is at least one critical value α cr at which solitons begin to exhibit the shock-wave behaviour. We proposed that forvalues of α beyond α cr a soliton solution may be constructed by gluing together the twobranches of the soliton solution at the points where the first derivative of the solitonfield diverges. Despite the divergency, the soliton energy and momentum are finite, andthe dispersion relation is defined for all values of α . A natural expectation is that theglued soliton is unstable, and it would be interesting to check it.The T T deformed KdV equation admits at least a one-parameter family of one-soliton solutions. The extra parameter b can be introduced explicitly in the T T deformedLagrangian by shifting the field φ by bt , and requiring that φ asymptotes to constantsat space infinities. Then, b can be interpreted as the parameter of the deformation bythe time component of the conserved current due to the invariance of the T T deformedGardner model under constant shifts of φ . Since the parameter b modifies the propertiesof the soliton, in particular, it appears in the dispersion relation, such an interpretationis probably the right one. It is however unclear to us why one has to impose constantspace asymptotes on φ . If b does not vanish then there is an upper bound on α , andapproaching the bound the soliton’s amplitude decreases and finally vanishes. Choosingproperly other parameters of the soliton, one can make the bound negative. Thus, theparameter b allows one to construct solutions which do not exist in the seed model.There are many open questions to be addressed, and below we list some of them.We have only discussed the deformed models on a line. In this case the requirementof finiteness of the energy singles out the perturbative in α branch of the deformed24agrangian depending only on the physical fields. It would be interesting to put themodels on a circle and look for solutions nonperturbative in α with energy divergent inthe limit α →
0. In fact, these solutions may exist even for Lorentz invariant models, e.g.for the
T T deformed sigma model described by the Lagrangian (2.14). If they do existthen it would imply that the spectrum of
T T deformed relativistic models previouslydiscussed is incomplete and must be supplemented by a nonperturbative part.The seed models we have considered are integrable, and it is believed that their
T T deformations are integrable too. The first step in proving the integrability would befinding Lax pairs for the deformed models. Lax pairs of several models including theNLS model were recently found in [37] by using the dynamical coordinate transformation[14]. Their results agree with the previously known Lax pairs of the sine-Gordon andLiouville models [21, 38]. It should be possible to apply the method of [37] to thematrix NLS model and the LL model. It would be interesting to see if their method canbe generalised to include models of the Gardner type where auxiliary fields cannot beeliminated and one has to deal with them.As has been mentioned above, understanding the Poisson structure and developing aHamiltonian formulation of the deformed models is important and probably very hard.Given a Lax pair (
V, U ) and a Hamiltonian formulation of the NLS model, onecan calculate the Poisson bracket between U ’s, and see how the r -matrix structure ismodified, and whether it can be quantised.If a seed model possesses an additional conserved U (1) current J then one can con-sider J T deformations [35] which have properties similar to the
T T deformation. TheNLS model is one of the simplest nonrelativistic models with the U (1) symmetry, and itwould be interesting to analyse the properties of the model deformed by J T operators.Some steps in this direction have been made in [31, 32]. The light-cone gauge approachto the
T T deformation of relativistic sigma models can be readily generalised to includethe
J T deformations and deformations by operators linear in conserved currents [39].It should be possible to consider in the same framework nonrelativistic models. As waspointed out in [40], since the
J T deformations break Lorentz invariance the deformationsby operators linear in conserved currents are necessary to derive flow equations for thespectrum. In fact, for nonrelativistic models it seems necessary to include the lineardeformations even to derive the flow equations for the
T T deformation.The
T T deformation of nonrelativistic models is defined with the help of the La-grangian flow ∂ α L = − T T . This modifies the Poisson structure of a seed model, andmakes it difficult to derive flow equations for the spectrum. It would be interesting tosee whether one can define the deformation as the Hamiltonian flow ∂ α H = T T whichpreserves the Poisson structure of a seed model. This can be done for a
T T deformedmassive Dirac fermion [19] but for a bosonic model the Hamiltonian might appear to benonlocal in space. 25
Deformed NLS soliton solution
We start from the Lagrangian expressed in polar coordinates as in equation 3.9 andderive the equations of motion. Then we apply the ansatz as described in equation 3.10.Furthermore we decompose ρ A into real and imaginary components as ρ A = X + iY .In addition to the equations of motion for ( ρ, ρ A , ¯ ρ A ) we have the following simplifiedequations from the continuity of the stress tensor and the fact that φ is a cyclic variable − v ∂ L ∂ ˙ φ + ∂ L ∂φ = c , − vT tt + T xt = c , − vT tx + T xx = c . (A.1)Applying the boundary conditions of ρ ( ±∞ ) = 0 to each of the equations of motion andcontinuity equations yields X ( ±∞ ) = Y ( ±∞ ) = 0 = c = c = c . (A.2)Solving the equations of motion for φ yields a simple relation for Y . Applying this tothe continuity equations yields the relation for X Y = 12 vρ , X = − ρ (cid:16) g ρ + 4 µ + v − ω (cid:17) . (A.3)The two continuity equations for the stress tensor become dependent at this stage. Fromthe real part of the equations of motion for ρ A we find X = − ρ (cid:16) αg ρ + αv ρ + 4 αµρ − αωρ − (cid:17) . (A.4)By substituting this into the continuity equation we find the first-order differential equa-tion in ρ ρ = ± ρ √ u − g ρ − αρ (2 g ρ + 4 µ − u + v ) . (A.5)Where we redefine the arbitrary parameter ω = (4 µ + u + v ). Then, if one considersthe imaginary part of the equations of motion for ρ A without substituting this newrelation, we can find φ in terms of ρ, ρ ϕ = 14 v (cid:16) αρ (cid:16) αρ (cid:16) g ρ + 4 µ − u + v (cid:17) − (cid:17) + 2 (cid:17) . (A.6)Recalling that the integration variable is x − vt , can trivially integrate the constant termto get ϕ ( α = 0) = v/ x − vt ) and can use a change of coordinates d ( x − vt ) = dρ ( ρ ) − .Then we find the expression for φ in terms of ρφ = t (cid:16) µ + u + v (cid:17) + v x − vt ) ± αv ( u − g ρ ) / g . (A.7)The auxiliary fields are ρ A = 12 ρ (cid:18) iv ± q u − g ρ (cid:19) , ¯ ρ A = 12 ρ (cid:18) − iv ± q u − g ρ (cid:19) . (A.8)26he stress-energy tensor becomes T xt = − ρ ( − g ρ − µ + u + v ) αρ (2 g ρ + 4 µ − u + v ) − vT tt ,T xx = 2 v ρ αρ (2 g ρ + 4 µ − u + v ) − vT tx . (A.9) B Additional Graphs of NLS deformed soliton α = α = α + α = α + ρ ( x ) α = α =- α =- ρ ( x ) α = α = α + α = α + ρ ( x ) Figure 6: Left & Centre: Case A, µ = 1, α + = 4 /
5, displaying loop formation for α > α + > α <
0. Right: Case B, µ = 0, α + = 4, loop solution appears for α >
0, thisis the only case with a finite region of valid α . α = α = α = ρ ( x ) α = α = α + / α = α + α = α + ρ ( x ) α = α = α = ρ ( x ) Figure 7: Left: Case C, µ = − . α − = − . α >
0. Centre & Right: Case D, µ = − α + = − /
39. Loop formation for α < α + < α >
0. Note the varying rate of soliton widening between the two cases. Deformed KdV soliton solution
The starting deformed Lagrangian is given by L = − A (cid:16) A + 2 u (cid:16) α u ˙ φ − (cid:17) − α u ˙ u φ (cid:17) + B ( u − φ ) − u (cid:16) u (2 u + µ ) + ˙ φ (cid:17) αA + α u ( u (2 u + µ ) − B ) + 1 (C.1)In addition to the equations of motion for each of the fields, we use the simplifiedcontinuity equations for the stress tensor and the equation of motion for φ − v J t + J x = c = v u − BαA + α u ( u (2 u + µ ) − B ) + 1 ,T σσ − vT τσ = c = A + b u − B u + µ u + 2 u αA − αB u + αµ u + 2 α u + 1 , (C.2)where we have applied the ansatz given by 3.25.From the equations of motion for φ , we find an expression for B , which we substituteinto the stress tensor continuity equation to solve for A B = αc ( A + u (2 u + µ )) − v u + c αc u − ,A = u ( − u (2 u ( αc −
1) + αbc + αc µ − αc v − µ + v ) + b + c ) − c αc − . (C.3)Removing the A in the solution for B , we then apply this to the equation of motionfor A . We can then solve for A and then create another equation by requiring the twosolutions for A be consistent B = c − u ( αbc − αc v + v ) αc − ,A = u ( α u (2 u (2 u ( αc −
1) + αbc + αc µ − αc v − µ + v ) − b − c ) + 2 αc − αc − αc φ − . (C.4)At this stage we aim to fix the constants c , c by evaluating the expressions as x − vt →∞ . Initially we only have that φ → φ and the consistency equation for the A solutionsare nontrivial. However, the set of solutions for which these equations hold each require u = 0 and hence u = 0 at infinity. With the new boundary conditions, we find that c = 0. Applying the solutions and boundary conditions for the equation of motion for u then sets c = − b . We find u in terms of u , φ by solving the consistency equation forthe A solutions, and applying this to the equation of motion for B we then find the lastrelation for φ in terms of u . u = ± u ( αbφ − q ( v − αb − µ − u )( α u (2 u (2 u + αb + µ − v ) − b ) + 1) , φ = u − αb u α u (4 u + αb + 2 µ − v ) + 1 . (C.5)28ow we have expressed all the fields in terms of u and have a first-order differentialequation for said field. The T T flow equation holds on shell, and the solutions hold inthe undeformed limits α → b →
0. We can define the variable ˜ w = v − µ + b α = w + b α to simplify the expressions, where w was used in the undeformed descriptionof the soliton. We can use a change of variables from dx → ( u ) − d u to perform spatialintegration. The stress tensor on-shell is given by T σσ = u ( αb + µ + 4 ˜ w ) α u ( − u + αb + 8 ˜ w ) − vT τσ ,T στ = ( µ + 4 ˜ w ) u ( − u − µ + 4 ˜ w ) + b α u ( − u + αb + 8 ˜ w ) − vT ττ − b . (C.6)And the conserved current from the equations of motion is J t = u ( αb u + 1)1 − α u ( − u + αb + 8 ˜ w ) . (C.7)Let us now analyse the equations(3.32) which we repeat here for convenience d ( u ) ≡ α u (cid:16) u − w − αb ) + 1 = 0 , < u < w , (C.8)and determine the values of the parameters for which the solution becomes multi-valued.Calculating the values of d ( u ) at the boundaries of the allowed values of u , one finds d (0) = 1 , d (2 ˜ w ) = 1 − α b ˜ w = 1 − α b (4 w − α b ) ≤ . (C.9)Then, we find the first and second derivatives of d ( u ), and its extremal points d ( u ) = 2 α u (cid:16) u + αb − w (cid:17) , d ( u ) = 2 α (cid:16) u + αb − w (cid:17) , u ex1 = 0 , d (0) = − α (cid:16) w − αb (cid:17) , u ex2 = 8 w − αb , d ( u ex2 ) = 2 α (cid:16) w − αb (cid:17) (C.10)Let us now fix v, µ, b and find for which values of α the equation (C.8) has solutions inthe interval (0 , w ).We begin with the simplest case b = 0. Then, ˜ w = w and b = 0 : d (0) = d (2 ˜ w ) = 1 , u ex2 = 4 w < w , d ( u ex2 ) = 16 αw . (C.11)Thus, the second extremal point is always inside the interval (0 , w ), and if α < , w ). If α >
0, then itis a minimum, and b = 0 : d ( u ex2 ) = 1 − αw . (C.12)29 = α = α cr / α = α cr α = α cr u ( x ) α = α =- α =- ( x ) α = α =- α =- ( x ) Figure 8: KdV soliton solutions for w = 1 , b = 0. Double-loop solution forms only for α >α c = 27 / α <
0, solution remains single-valued and increases in width. Rightmostplot examines peak of α <
It is clear now that for b = 0 (C.8) has two roots the interval (0 , w ) for α > α cr = w ,and the solution is first of a bell shape and then of a double loop shape as on the leftpicture of Figure 8.If b = 0 there may be critical values of α for both signs.Let us first consider the α < w = v − µ >
0, then 8 w − αb > w > α < u ex1 = 0 is a minimum, and the second extremalpoint is a maximum. Therefore, one can get a root of (C.8) which is inside the interval(0 , w ) only if d (2 ˜ w ) becomes negative. Solving the equation d (2 ˜ w ) = 0 with α < αα − = − q w + 2 | b | − w b , (C.13)and for α < α − the solution is of a loop shape, see the right plot of Figure 4.Then, if w < α < w b <
0, and for w b < α < w b the first extremal point u ex1 = 0 is a maximum, and the second extremal point u ex2 < α < w b , u ex1 becomes a minimum, and u ex2 > w > d (2 ˜ w ) to be negative which again happens at α = α − givenby (C.13). Depending on values of w and b , α − may be greater or less than 8 w /b .Let us now consider the α > α > u ex1 = 0 isa maximum, and therefore the second extremal point is a minimum, and b = 0 : d ( u ex2 ) = 1 − α w − αb ! . (C.14)As for the b = 0 case, critical values are given by roots of the equation d ( u ex2 ) = 0. It iseasy to see that d ( u ex2 ) as a function of α has the only minimum at α min = 2 w b ⇒ d ( u ex2 ) = 1 − w b . (C.15)30n fact, α min is also the minimum of d (2 ˜ w ) and u ex2 = 2 ˜ w = w for α = α min .Thus, if w b < α >
0, and if w b > d ( u ex2 ) = 0, and the critical value α (1)+ is the positiveroot which is smaller than α min . For values of α slightly greater than α (1)+ there are tworoots of the equation (C.8) for values of u for which u = ∞ , and therefore the solutionis first of a bell shape and then of a double loop shape as for b = 0 case. The two rootsare inside the interval (0 , w ) until α becomes equal to α (2)+ = 2 w − q w − | b | b . (C.16)At α = α (2)+ one gets d (2 ˜ w ) = 0 and therefore the larger root is equal to 2 ˜ w . Increasing α more moves the larger root away from the interval (0 , w ), and the solution is of aloop shape. Finally, the smaller root leaves the interval (0 , w ) at α (3)+ = 2 w + q w − | b | b , (C.17)because d (2 ˜ w ) is again equal to 0 for α = α (3)+ . Thus, for α (3)+ < α < α max = 4 w b , w b > , (C.18)the solution is regular again.The discussion above is summarised in eq.(3.33). References [1] A. B. Zamolodchikov, “Expectation value of composite field T anti-T in two-dimensional quantum field theory,” hep-th/0401146.[2] F. A. Smirnov and A. B. Zamolodchikov, “On space of integrable quantumfield theories,” Nucl. Phys. B (2017) 363 doi:10.1016/j.nuclphysb.2016.12.014[arXiv:1608.05499 [hep-th]].[3] A. Cavagli`a, S. Negro, I. M. Sz´ecs´enyi and R. Tateo, “ T ¯ T -deformed 2D Quan-tum Field Theories,” JHEP (2016) 112 doi:10.1007/JHEP10(2016)112[arXiv:1608.05534 [hep-th]].[4] G. Arutyunov, S. Frolov and M. Zamaklar, “The Zamolodchikov-Faddeev alge-bra for AdS(5) x S**5 superstring,” JHEP (2007), 002 doi:10.1088/1126-6708/2007/04/002 [arXiv:hep-th/0612229 [hep-th]].[5] S. Dubovsky, R. Flauger and V. Gorbenko, “Solving the Simplest Theoryof Quantum Gravity,” JHEP (2012) 133 doi:10.1007/JHEP09(2012)133[arXiv:1205.6805 [hep-th]]. 316] M. Baggio and A. Sfondrini, “Strings on NS-NS Backgrounds as Integrable Defor-mations,” Phys. Rev. D (2018) no.2, 021902 doi:10.1103/PhysRevD.98.021902[arXiv:1804.01998 [hep-th]].[7] A. Dei and A. Sfondrini, “Integrable spin chain for stringy Wess-Zumino-Wittenmodels,” JHEP (2018) 109 doi:10.1007/JHEP07(2018)109 [arXiv:1806.00422[hep-th]].[8] A. Dei and A. Sfondrini, “Integrable S matrix, mirror TBA and spectrumfor the stringy AdS x S x S x S WZW model,” JHEP (2019) 072doi:10.1007/JHEP02(2019)072 [arXiv:1812.08195 [hep-th]].[9] M. Caselle, D. Fioravanti, F. Gliozzi and R. Tateo, “Quantisation of the ef-fective string with TBA,” JHEP (2013) 071 doi:10.1007/JHEP07(2013)071[arXiv:1305.1278 [hep-th]].[10] Y. Jiang, “Lectures on solvable irrelevant deformations of 2d quantum field theory,”[arXiv:1904.13376 [hep-th]].[11] S. Dubovsky, V. Gorbenko and M. Mirbabayi, “Asymptotic fragility, near AdS holography and T T ,” JHEP (2017) 136 doi:10.1007/JHEP09(2017)136[arXiv:1706.06604 [hep-th]].[12] S. Dubovsky, V. Gorbenko and G. Hern´andez-Chifflet, “
T T partition functionfrom topological gravity,” JHEP (2018) 158 doi:10.1007/JHEP09(2018)158[arXiv:1805.07386 [hep-th]].[13] A. J. Tolley, “
T T deformations, massive gravity and non-critical strings,” JHEP (2020), 050 doi:10.1007/JHEP06(2020)050 [arXiv:1911.06142 [hep-th]].[14] R. Conti, S. Negro and R. Tateo, “The TT perturbation and its geometric interpre-tation,” JHEP (2019) 085 doi:10.1007/JHEP02(2019)085 [arXiv:1809.09593[hep-th]].[15] J. Cardy, “The T T deformation of quantum field theory as random geometry,”JHEP (2018), 186 doi:10.1007/JHEP10(2018)186 [arXiv:1801.06895 [hep-th]].[16] M. Baggio, A. Sfondrini, G. Tartaglino-Mazzucchelli and H. Walsh, “On T T defor-mations and supersymmetry,” JHEP (2019), 063 doi:10.1007/JHEP06(2019)063[arXiv:1811.00533 [hep-th]].[17] S. Frolov, “ T T
Deformation and the Light-Cone Gauge,” Proc. Steklov Inst. Math. (2020), 107-126 doi:10.1134/S0081543820030098 [arXiv:1905.07946 [hep-th]].[18] G. Arutyunov, S. Frolov and M. Zamaklar, “Finite-size effects from giant magnons,”Nucl. Phys. B (2007) 1, hep-th/0606126.3219] L. F. Alday, G. Arutyunov and S. Frolov, “New integrable system of 2dim fermionsfrom strings on AdS,” JHEP (2006) 078, hep-th/0508140.[20] G. Bonelli, N. Doroud and M. Zhu, “ T ¯ T -deformations in closed form,” JHEP (2018) 149 doi:10.1007/JHEP06(2018)149 [arXiv:1804.10967 [hep-th]].[21] R. Conti, L. Iannella, S. Negro and R. Tateo, “Generalised Born-Infeldmodels, Lax operators and the TT perturbation,” JHEP (2018) 007doi:10.1007/JHEP11(2018)007 [arXiv:1806.11515 [hep-th]].[22] H. Jiang, A. Sfondrini and G. Tartaglino-Mazzucchelli, “ T ¯ T deformations with N = (0 ,
2) supersymmetry,” arXiv:1904.04760 [hep-th].[23] C. K. Chang, C. Ferko and S. Sethi, “Supersymmetry and
T T deformations,” JHEP (2019) 131 doi:10.1007/JHEP04(2019)131 [arXiv:1811.01895 [hep-th]].[24] C. K. Chang, C. Ferko, S. Sethi, A. Sfondrini and G. Tartaglino-Mazzucchelli, “ T ¯ T Flows and (2,2) Supersymmetry,” arXiv:1906.00467 [hep-th].[25] E. A. Coleman, J. Aguilera-Damia, D. Z. Freedman and R. M. Soni, “ T ¯ T -DeformedActions and (1,1) Supersymmetry,” arXiv:1906.05439 [hep-th].[26] S. Chakrabarti, D. Gupta, A. Manna and M. Raman, “Irrelevant Deformations ofChiral Bosons,” [arXiv:2011.06352 [hep-th]].[27] H. Ouyang and H. Shu, “ T ¯ T deformation of chiral bosons and Chern–Simons AdS gravity,” Eur. Phys. J. C (2020) no.12, 1155 doi:10.1140/epjc/s10052-020-08738-6 [arXiv:2006.10514 [hep-th]].[28] J. Cardy, “ T T deformations of non-Lorentz invariant field theories,”[arXiv:1809.07849 [hep-th]].[29] J. Cardy and B. Doyon, “