Quasi-integrability in the modified defocusing non-linear Schr\"odinger model and dark solitons

The concept of quasi-integrability has been examined in the context of deformations of the defocusing non-linear Schr\"odinger model (NLS). Our results show that the quasi-integrability concept, recently discussed in the context of deformations of the sine-Gordon, Bullough-Dodd and focusing NLS models, holds for the modified defocusing NLS model with dark soliton solutions and it exhibits the new feature of an infinite sequence of alternating conserved and asymptotically conserved charges. For the special case of two dark soliton solutions, where the field components are eigenstates of a space-reflection symmetry, the first four and the sequence of even order charges are exactly conserved in the scattering process of the solitons. Such results are obtained through analytical and numerical methods, and employ adaptations of algebraic techniques used in integrable field theories. We perform extensive numerical simulations and consider the scattering of dark solitons for the cubic-quintic NLS model with potential $V =\eta I^2 - \frac{\epsilon}{6} I^3 $ and the saturable type potential satisfying $V'[I] =2 \eta I - \frac{\epsilon I^q}{1+ I^q},\,q \in \IZ_{+}$, with a deformation parameter $\epsilon \in \IR$ and $I=|\psi|^2$. The issue of the renormalization of the charges and anomalies, and their (quasi)conservation laws are properly addressed. The saturable NLS supports elastic scattering of two soliton solutions for a wide range of values of $\{\eta, \epsilon, n\}$. Our results may find potential applications in several areas of non-linear science, such as the Bose-Einstein condensation.


Introduction
The theory of soliton collisions in the integrable models is well known in the current literature. In particular, the solitons emerge with their velocities and shapes completely unaltered after collision between them, the only noticed outcome being their phase shifts. However, certain non-linear field theory models with relevant physical applications possess soliton-like solutions and it is difficult to know a priori if they are in fact true solitons. The so-called quasi-integrability concept has recently been put forward in the context of certain deformations of the integrable sine-Gordon (SG), Bullough-Dodd (BD) and focusing non-linear Schrödinger (NLS) models [1,2]. The idea is that many non-integrable theories posses solitary wave solutions that behave similar to solitons, i.e. the scattering of such solitons basically preserve their shapes and velocities.
So, certain deformations of the SG, BD and focusing NLS models were considered as quasi-integrable theories possessing infinite number of charges that are asymptotically conserved. This means that during the collision of two soliton-like solutions such charges do vary in time and sometimes significantly, however their values remain the same in the past (before the collision) and in the future (after the collision). The deformations of the topological SG and BD models studied in [1] furnish kink-type solitary waves presenting soliton-like behaviour. Whereas, the deformed focusing NLS model considered in [2] supports bright solitary waves presenting soliton-like behaviour. The both numerical and analytical techniques were used to describe the concept of quasi-integrability.
Regarding the analytical calculations of solitary wave collisions in deformed NLS models some results have been obtained for the cases of small perturbations of the integrable NLS model [3,4,5]. For nearly integrable models it has been implemented a perturbation theory allied to the the inverse scattering transform (IST) method [6]. These methods have been applied mainly to the focusing NLS solitons that decay at infinity, i.e. the bright solitons. On the other hand, the integrable defocusing NLS model supports dark solitons which are intensity dips sitting on a continuous wave background (cwb) with a phase change across their intensity minimum. In order to apply to the dark soliton perturbation and the non-vanishing boundary conditions (nvbc), inherent to this type of solution, various improvements have been made to the calculations and methods [7,8,9] based on the so-called complete set of squared Jost solution (eigenstates of the linearised NLS operator). The implementation of direct methods, however, consider small perturbations around the integrable NLS model [5,10]. Moreover, in [5] it has been argued that the squared Jost functions associated with the dark soliton might be an insufficient basis, so rendering problematic the issue of the completeness of the basis for the solution space of a deformed NLS equation. The solitary wave collisions of certain deformed NLS models, beyond small perturbations, have been considered in the recent literature; e.g. in [11] it has been computed the spatial shifts of fast dark-dark soliton collisions and in [12] some properties of slow dark soliton collisions have been studied.
It would be interesting to understand the issue of quasi-integrability in the other variant of the NLS model, i.e. the deformed defocusing NLS model, since such theories also appear in many areas of non-linear science, condensed matter physics, plasma physics and, in particular, in the study of Bose-Einstein condensates. In this context, our aim is to predict the results of solitary wave collisions and test the quasiintegrability concept in deformed defocusing NLS models with nvbc. However, the non-vanishing boundary condition inherent to the dark solitons, which may change when the deformations are present, introduces some complications when applying the techniques developed for bright solitons in order to study the quasiintegrability concept. In the analytic treatments the nvbc introduces the issue of the renormalization of the charges and anomalies in the expressions of the quasi-conservation laws which must be properly addressed. This is closely related to the fact that the dark soliton solution is a composite object, i.e. it incorporates the continuous wave background (cwb) plus the dark soliton itself. In addition, to simulate the time dependence of field configurations for computing dark solitons properties the choice of the efficient and accurate numerical methods, the so-called time-splitting cosine pseudo-spectral finite difference (TSCP) and the time-splitting finite difference with transformation (TSFD-T) methods [13] will be made in order to control the highly oscillatory phase background. In fact, these methods allowed us to improve in several orders of magnitude the accuracy in the computation of the charges and anomalies presented in [2].
The deformed focusing NLS with bright soliton solutions and the structures responsible for the phenomenon of quasi-integrability have been discussed in [2]. In this context it has been shown that this model possesses an infinite number of asymptotically conserved charges. An explanation found so far for this behaviour of the charges is that some special soliton type solutions are eigenstates of a space-time parity transformation. On the other hand, the deformed defocusing NLS with dark soliton solutions also presents the above remarkable properties, however in this case we can say even more. As we will show, there are some soliton like solutions which present a special space-reflection symmetry for any time, so this property implies the exact conservation of the sequence of the even order charges, which in the absence of this special symmetry would be conserved only asymptotically, as in the models studied in [1,2]. Here we will show the connection between the space-reflection parity and the exactly conserved charges which involves an interplay between the space-reflection parity and internal transformations in the affine Kac-Moody algebra underlying the anomalous Lax equation. However, it seems to be that such parity property is a sufficient but not a necessary condition in order to have the sequence of exactly conserved charges. In fact, as we will show by numerical simulations, there are certain soliton like configurations without this symmetry which also exhibit such conserved charges. So, these properties constitute the distinguishing new features associated to the deformed defocusing NLS with dark soliton solutions, as compared to the previous quasi-integrable models [1,2].
The paper is organized as follows. In the next section we discuss the deformed defocusing NLS model with non-vanishing boundary conditions. In section 3, we discuss the concept of quasi-integrability for deformed defocusing NLS, based on an anomalous zero curvature condition. In 3.1 and 3.2 we discuss the relationships between the space-time parity and asymptotically conserved charges, and the space-reflection parity and the exactly conserved charges, respectively. In section 4 we perform the renormalizations of the charges and anomalies relevant for dark soliton solutions. In sections 5.1 and 5.2 we discuss the space-time and spacereflection symmetries of dark soliton solutions of the integrable NLS model, respectively. In 6 we show the vanishing of the anomalies associated to solitary waves. In 6.1, by using general symmetry considerations, we show that the anomalies β (n) r vanish when evaluated on a general solitary wave of the deformed NLS model. In 6.2 we show, by direct computation, the vanishing of the first non-trivial anomaly β (4) r when evaluated on a general solitary wave of the deformed NLS. In section 7 we present the results of our numerical simulations which allowed us to compute and study various properties of dark soliton scattering of the full equations of motion for two deformations of the NLS model and several values of the deformation parameter ǫ. In 7.1 we present the results of the simulations of the first deformation with V = η|ψ| 4 − ǫ 6 |ψ| 6 , and in 7.2 the second deformation with V ′ [I] = 2ηI − ǫ I q 1+I q . In 8 we present some conclusions and discussions. The appendix A presents the eqs. of motion in the R and ϕ parametrizations, as well as some useful identities and B presents relevant expressions corresponding to the gauge transformation.

Deformations of defocusing NLS
We will consider non-relativistic models of a complex scalar field in (1 + 1)−dimensions with Lagrangian density given by whereψ stands for the complex conjugation of ψ and V : IR + → IR . The Euler-Lagrange equation following from (2.1) becomes The model ( [14,15]. In [16,14] the bright solitary waves of the cubic-quintic focusing NLS have been regarded as quasi-solitons presenting partially inelastic collisions in certain region of parameter space. Among the models with saturable non-linearities [6], the case V [I] = 1 2 ρ s (I + ρ 2 s I+ρs ) also exhibits analytical dark solitons [17]. The deformed NLS model with V ′ [I] = 2ηI − ǫ I q 1+I q will deserve a careful consideration below. It passes the Painlevé test for arbitrary positive integers q ∈ Z Z + and ǫ = 1 [18]. However, its Lax pair formulation and analytical solutions for a general set {η, ǫ, q}, to the best of our knowledge, are not known in the literature.
The qualitative properties of travelling waves of the NLS model for general non-linearities and nonvanishing boundary conditions have been studied in [19]. Exact analytical dark-soliton solutions of the deformed NLS model with arbitrary potential V are not available, and one can resort to numerical simulations to obtain such solutions. However, dark-soliton solutions can be presented in an implicit form as a curve in a complex plane (for the complex variable Z = A + iB, where ψ ≡ (A + iB)e (kx+wt) ) (see [6] and references therein) or in a small-amplitude approximation where the deformed NLS model is reduced to the Korteweg-de Vries equation [20].
Among the possible deformations of the NLS model the case V = 2η 2+ǫ |ψ| 2 2+ǫ , ǫ ∈ IR, for η < 0 has recently been considered in [2] in order to study the concept of quasi-integrability for bright soliton collisions.
An analytical solution with vanishing boundary condition (bright soliton) for this potential is well known in the literature. However, an analytical dark soliton solution, as far as we know, is not known in the literature.
In this paper we will study analytically and numerically some deformations of the NLS model of the type (2.2). The first deformation we will consider in our study is defined by the non-integrable cubic-quintic NLS dy 2 ). This deformation possesses analytical solitary waves as we will present below.
The second deformation considers a saturable non-linearity and it is defined by the equation We do not know any analytical solutions for a general set of parameters of this model, however dark solitary wave solutions will be obtained numerically.
The both deformations reproduce the integrable defocusing NLS model in the limit ǫ → 0.

Quasi-integrability of deformed NLS
Next we undertake an analytical study of the properties of the deformed NLS model (2.2), in order to do this we will use the well known techniques from the integrable field theories. We follow the developments and notations put forward in [1,2] on quasi-integrability. Then, one considers an anomalous zero curvature representation of the deformed NLS model (2.2) with the connection given by where the above Lax potentials are based on a sl(2) loop algebra with commutation relations One can verify that the curvature of the connection (3.1) is given by (3.5) Note that when the equation of motion (2.2) and its complex conjugate, i.e −i∂ tψ + ∂ 2 xψ −ψ δV δ|ψ| 2 = 0, are satisfied the terms proportional to the Lie algebra generators T 0 ± vanish. In addition, the quantity X vanishes for the usual non-linear Schrödinger potential Then, the curvature vanishes for the NLS model making this theory an integrable field theory. For the generalizations of this theory, i.e. deformations of the potential (3.6), the above curvature does not vanish and the model is regarded as non-integrable [2]. Next, we use some algebraic techniques borrowed from the theory of integrable models in order to analyse the equation (3.4) corresponding to a model with nonvanishing anomaly X (3.5), or equivalently F xt in (3.3). Let us denote Substituting the parametrization (3.7) into (2.2) one gets the system of eqs. of motion We specialize the parametrizations for the case of the defocusing NLS, i.e. the case η > 0. In addition, define a new basis of the sl(2) loop algebra as The connection in the new basis is obtained through the gauge transformation So, inserting the connection (3.1) into the above expression one gets So, considering the fields which satisfy the equation (2.2) and its complex conjugate, i.e. the terms in the directions of T 0 ± vanish in (3.4), the curvature becomes (3.14) Next, in order to construct the quasi-conservation laws we perform a further gauge transformation The parameters ζ of the transformation will be determined below so that the a x component of the new connection lies in the abelian sub-algebra generated by the elements b n . Under the gauge transformation (3.15) the curvature (3.14) transforms as The sl(2) loop algebra is furnished with an integer gradation associated to the grading operator d defined . . .
As usual we will use the generator b 1 as a semi simple element in the sense that its adjoint action decomposes the sl(2) loop algebra G into its kernel and image as follows such that Ker and Im have no elements in common. According to the commutation relations in (3.11) one has that the b n 's form a basis of Ker, and F n i , i = 1, 2, a basis of Im. Next, one can make all the components of a x in the subspace Im to vanish by choosing conveniently the parameters ζ . This can be done recursively, and so on. So, the a x component get rotated onto the subspace Ker Notice that in this process it has not been used neither the equation of motion (2.2) nor its complex conjugate.
The components a (3,−n) x will depend on the real fields R and ∂ x ϕ, and their x-derivatives, but not on the potential V . In the Appendix B we provide the first components of a x .
Next, the component a t of the connection does not get rotated onto the subspace Ker even when the equation of motion (2.2) and its complex conjugate are used. Then, one has (3.23) Since a x lies in Ker one has that the term [a t , a x ] in (3.17) has components only in Im. Let us denote then, we find that In the appendix C we present the first quantities α (3,−n) .
Next, let us discuss the nvbc associated to dark soliton solutions. These solutions present a difficulty in the sense that the field ψ does not vanish at x → ±∞. In general, the behaviour of the solutions are such that where θ 0 is a constant phase. The connection (3.1) is not suitable to construct the charges because However, making the gauge transformation (3.12) and the reparametrization (3.7) the new connection A µ depends upon the modulus R and its derivatives, and only on the derivatives of the phase ϕ. Therefore, one has that for dark solitons In addition the abelianized potential a µ in (3.15) also satisfy that condition, i.e.
because the group element also satisfies in the appendix B and one notes that g depends on R and its derivatives and only on the derivatives of the phase ϕ.
Then, in the defocusing NLS case with dark soliton solutions one has that the a t component of the connection satisfies a non-vanishing boundary condition such that a then from (3.25) we have the anomalous conservation laws Thus, the non-vanishing of the quantity X given in (3.14) and the anomalies β n above imply the nonconservation of the charges. Therefore, the charges and anomalies in (3.31) are valid for the deformed NLS model with nvbc and dark soliton solutions.

Space-time parity and asymptotically conserved charges
Next, let us discuss a special symmetry which plays an important role in the study of quasi-integrable theories. For certain solutions of the theory (2.1) the charges Q (n) satisfy the so-called mirror symmetry.
This symmetry is realized on a special subset of solutions of the deformed NLS model (2.2). For every solution belonging to this subset one can find a point (x ρ , t ρ ) in space-time and define a parity transformation around this point as such that the fields R and ϕ transform as In order to realize this symmetry at the level of the anomalous zero-curvature equation let us combine the above parity transformation with the order two Z Z 2 automorphism of the sl(2) loop algebra given by For more details of this demonstration we refer to [2].
Next, let us note that the X in the integrand of the anomaly β n in (3.31) is a x−derivative of a functional of R. Since R is assumed to satisfy (3.33) one has that the time-integrated anomaly vanishes where the integration is meant to be performed on any rectangle with center in (x ρ , t ρ ), and the parameters A remarkable consequence of the vanishing of the integrated anomaly is that the charges (3.31) satisfy a mirror time-symmetry around the point t ρ This relationship shows that the scattering of two-soliton solutions may present an infinite number of charges which are asymptotically conserved. For some very special two-bright soliton configurations in the context of focusing deformed NLS it has been verified that the charges are asymptotically conserved, i.e. even though they vary in time during collision their values in the far past and the far future are the same [2]. As we will show below, for the case of defocusing NLS it is a remarkable fact that for any two-dark soliton configuration it is possible to find a point in space-time (x ρ , t ρ ), such that the fields R and ϕ always satisfy the space-time parity symmetry (3.32)-(3.33) without any additional restriction on the soliton parameters.
Next we will summarize the main steps in the construction of solutions in perturbation theory, for a detailed account see [2]. One can choose as a zero order solution a solution of the NLS equation, such that it satisfies (3.33), i.e. it exhibits only the pair R + 0 , ϕ − 0 and nothing of the pair R − 0 , ϕ + 0 . So, let us expand the fields in powers of the deformation parameter ǫ Then the equations of motion for the components R + 1 , ϕ − 1 of the first order solution satisfy non-homogeneous equations and so they can never vanish. However, the pair R − 1 , ϕ + 1 satisfy homogeneous equations and so they can vanish. In addition, if the pair of components without definite parity ( 3. Then, one has an infinite set of asymptotically conserved charges, i.e. as we will discuss in this work.

Space-reflection symmetry and conserved charges
Next, we consider some special solutions which exhibit a space-reflection symmetry such that the fields R and ϕ transform as As we will see below, in the special case of 2-dark solitons moving in opposite directions and equal velocities, such that they undergo a head-on collision, we have a space-reflection symmetry. The implication of this additional symmetry of the fields under space-reflection (for any shifted time), i.e. R and ϕ being even fields, on the behaviour of the quantities α (3,−n) deserves a further analysis. We will show that some α (3,−n) 's possess an even parity under P x for any shifted time. Since X is odd the anomaly β n in (3.31) will vanish identically providing an exact conservation equation for the charges dQ (n) dt = 0, n = 0, 2, 4, ....
Therefore, these type of charges will not vary during collision, implying their exact conservation in the whole process of scattering.
In order to realize the symmetry (3.48)-(3.49) at the level of the anomalous zero curvature equations (3.14) and (3.17) we introduce a transformation Σ in the internal space along with the space-reflection symmetry P x . So, let us consider another order two Z Z 2 automorphism Σ of the sl(2) loop algebra given by Actually, each term of A x in (3.13) possesses this property. Next, we analyse the transformation property of the new connection a x in (3.19). Thus, one can have The term A (0) x in the third eq. of (3.19) as defined in (3.20) is odd under Ω, then it follows from the third equation in (3.19) that The F (−n) have been selected to rotate a x into the kernel of the adjoint action of b 1 and the r.h.s. of (3.54) is in the image of the adjoint action, so for consistency the both sides of (3.54) must vanish Therefore, F (−1) is even under Ω. Taking into account this result we see from the fourth equation in (3.19) that Then, following similar arguments we conclude that and so that F (−2) is even under Ω as well. Furthermore, from the fifth equation in (3.19) one has that , and following similar arguments as above we find that By repeating the same arguments as above one can show that all F (−n) are even under Ω. This property can be used to show that the group element g which enter in the gauge transformation (3.15) is even, i.e.
Since A x and ∂ x are odd under Ω, and since g is even one can show that a x in (3.15) is odd under Ω. In addition, since the F (−n) are even under Ω and the generators satisfy (3.50), it follows from (3.16) that , a = 1, 2. One can verify these results by inspecting the explicit expressions for the fields ζ −n a provided in the appendix B.
In order to find the parity of the factor α (3,−n) in the integrand of the anomaly β n in (3.31) let us consider some properties of the sl(2) loop algebra. The Killing form is given by It is realized by Tr(⋆) = 1 2πi dλ λ tr(⋆), where tr is the usual finite matrix trace, and T n i = λT i , i = 3, ±. So, from (3.24) one has that where the invariance property of the Killing form under Σ has been used, and the fact that the b n transform as in (3.50). Then, taking into account (3.59) one has that and so one can see that the α (3,−n) 's with n = 0, 2, 4, ... are even under P x . On the other hand, since we have assumed that R is even under P x and that X given in (3.14) is a x−derivative of a functional of R, one has that X is odd under P x , i.e. P x (X) = −X and so that Therefore, the anomalous conservation laws (3.31) imply Consequently, the even order charges are exactly conserved. Notice that it holds for any potential V which depends only on the modulus |ψ|. The only requirement is that the field components satisfy (3.49).
Next let us analyse the solutions which admit the system of equations (3.8)-(3.9) such that the field components R, ϕ satisfy the space-reflection symmetry (3.48)-(3.49). We perform this construction in perturbation theory around solutions of the NLS model, so we expand the equations of motion and the solutions into power series in ǫ, as The deformed potential and its gradient can been expanded as An interesting solution is the one satisfying R = R + , ϕ = ϕ + , i.e. the odd components vanish R − = 0, ϕ − = 0. We will show that this property is maintained in all orders of perturbation theory. So, substituting Then, the solution of (3.70)-(3.71) with vanishing odd components will satisfy Now, assuming (R 0 , ϕ 0 ) = R + 0 , ϕ + 0 one has that the first order terms O(ǫ) of the even and odd parts of the eqs. of motion (3.8)-(3.9) satisfy Some comments are in order here. First, the pair of fields R + 1 and ϕ + 1 satisfies the linear system of equations (3.73) and (3.75) with variable coefficients, similarly the pair R − 1 and ϕ − 1 satisfies the linear system of equations (3.74) and (3.76). The couple of first order fields (R + 1 , ϕ + 1 ) and (R − 1 , ϕ − 1 ) in the eqs. above form two decoupled system of equations, respectively. Second, the equations of motion for the even components R + 1 , ϕ + 1 , (3.73) and (3.75), satisfy a non-homogeneous coupled system of equations (the non-homogeneous term being the last term in (3.75 ) and so they can never vanish. However, the odd part components R − 1 , ϕ − 1 , (3.74) and (3.76), satisfy a homogeneous system of equations and so they can vanish. Third, if the pair of components without definite parity (R 1 , ϕ 1 ) is a solution, so is the pair Then, one can always choose a first order solution which is even under the space-reflection parity, i.e.
Choosing the zeroth (3.72) and first order (3.77) solutions one has that the second order terms O(ǫ 2 ) of (3.8)-(3.9) splits under P x as For this choice one can show that the second order solution has similar properties, i.e. the components satisfy a non-homogeneous system of equations, (3.78) and (3.80), and the odd components satisfy homogeneous equations, (3.79) and (3.81). So, we can always choose the second order Then one can choose again the second order solution to be even, and this process repeats in all orders O(ǫ n ).
By repeating this procedure, order by order, one can construct a perturbative solution which satisfies (3.48)-

Renormalized charges and anomalies
The difficulty with dark solitons is the fact that the charges and anomalies (3.31) diverge when evaluated on the dark-solitons. However, the divergence comes from the vacuum behavior of the solution at x → ±∞, i.e. the cwb solution given in (3.26). So, one can renormalize them by substructing the vacuum contribution (which is infinite). Using the notation ψ = √ Re iϕ/2 , we consider the following boundary condition The nvbc (4.1) is suitable for soliton solutions with the continuous wave background such that R = |ψ 0 | and ϕ/2 = kx + wt + x 0 .
The vacuum solution is given in (3.26) and we have So, we define the renormalized charges and anomalies and they satisfy Let us evaluate the first charges and anomalies in (3.31).
The anomaly does not get renormalized and it vanishes The charge Q So, it can be shown that the anomaly β 2 vanishes and This charge is related to the space translation symmetry of the model and using the expression (3.7) it can be written as Since the solution ψ incorporates the cwb solutions along with the dark soliton itself, it is a composite object. The momentum associated to the dark soliton itself must be renormalized even for k = 0, and it is provided by [6] where we have used the eq. of motion for R given in the appendix. Therefore (3.31) implies We then define the fake charge as in (4.16) and using the relationship |∂ x ψ| 2 = 1 4 R(∂ x ϕ) 2 + (∂ x R) 2 /R one can define the charge as and so the renormalized charge becomes So, the renormalized Hamiltonian of the usual NLS model (V = η|ψ| 4 ) can be written as Therefore, the NLS Hamiltonian can be written as a linear combination H N LS = Q r . Notice that the expressions (4.15) and (4.25) evaluated for a NLS 1-dark soliton solution will provide a standard relationship ∂H ∂P = v as in classical mechanics, where v is the velocity of the soliton. This indicates that the dark soliton can be considered as an effective particle [6].

The Q
(4) r charge: the first asymptotically-conserved charge Therefore we have that the renormalized charge and anomaly are Note that β with and This is the first non-trivial non-vanishing anomaly associated to an asymptotically conserved charge for solutions satisfying the symmetry (3.33). Moreover, one can have vanishing anomaly β r for solutions satisfying the symmetry (3.49). In addition, we will show below that this anomaly vanishes for a general solitary wave solution of the deformed NLS.
Discarding a total derivative associated to the first term in (4.32) (i.e. the term (V ′′ − 2η) 2η∂ x R 3 must vanish once integrated in the whole line taking into account the b.c. (4.1)) the anomaly β (−4) r can be rewritten in a form amenable to numerical calculations. Then, discarding the 'surface' term, one is left with the expression For later purposes we defined the integrand function γ(x, t). The time integrated anomaly becomes One notices that in the limit ǫ → 0 the anomaly β (4) r vanishes identically since V ′′ [R] → 2η in this limit.
In addition, this anomaly vanishes when |ψ| = |ψ 0 |, i.e. when evaluated on the cwb. We will compute numerically the anomaly β (4) r for certain dark soliton configurations in the both type of deformed NLS described above.

Space-time symmetries of defocusing NLS and dark solitons
Next we discuss the both space-time and space-reflection symmetries in the general one and two dark soliton solutions of the integrable defocusing NLS model. So, consider the defocusing NLS equation Here we consider the Hirota method and construct the one and two dark solitons. Let us introduce the Hirota tau functions as [21] where h(x, t) = |ψ 0 | exp [i(kx + wt + x 0 )] and |ψ 0 |, k, w, x 0 are real constants. The tau function G is a complex function, whereas F is real. Notice that for F = G = 1 in (5.2) one has the solution which is a continuous wave background (cwb) solution of (5.1) provided that w = −k 2 − 2η|ψ 0 | 2 .
Substituting the expression (5.2) into (5.1) one gets The 1-dark soliton solution is given by where y is a complex variable. The functions θ,θ are defined as where the bar stands for complex conjugation. The parameters satisfy the relationships Putting these tau functions into the NLS function ψ in (5.2) and making use of the identity 1+y exp φ 1+exp φ = This solution possesses three arbitrary real parameters, say |ψ 0 |, k and the phase associated to y. In fact, from the relationships (5.9) one can get the remaining ones w, p in terms of the set {|ψ 0 |, k, y}. From the third relationship in (5.9) one gets the condition (assume k > 0, p = p R + i p I ) The intensity function |ψ| moves at the velocity v = −2p I , which is the velocity of the dark soliton. The dark soliton approaches constant amplitude |ψ 0 | as |x| → ±∞. As x varies from −∞ to +∞ the soliton acquires a 2δ phase, where this quantity is the phase of the parameter y y = e 2iδ . (5.12) We restrict −π/2 < δ < π/2. Moreover, at the center of the soliton, i.e. for θ(x, t) +θ(x, t) + ρ ≡ 0, one has that the intensity becomes |ψ| center = |ψ 0 | cos δ.
where v s is the sound speed. Therefore the speed of the dark soliton must satisfy The 2-dark soliton solution is given by where r is a real constant, and y j (j = 1, 2) are constant phases. The functions θ j ,θ j are defined as where the bar stands for complex conjugation. The parameters satisfy the following relationships For later purposes let us record the next relationship between the soliton parameters in the above 2-dark soliton solution where v j = −2p jI , a j = 2p jR , p j ≡ p jR + ip jI , j = 1, 2. The v j are the soliton velocities and the a j are the parameters associated to the width (∼ 1 |aj| ) of each soliton.

Space-time parity transformation
Let us verify the parity properties of the above solutions under the transformation
The last expression can be rewritten as Notice that the parameters x ρ , t ρ and Λ 1 , Λ 2 are real.
The parity transformation (5.22) can be written in terms of z ± as P : (z + , z − ) → (−z + , −z − ) . The phase φ of N in (5.29) can be written as Therefore it can directly be verified that the squared absolute value R = |ψ| 2 and phase ϕ = 2(kx + wt + It must be emphasized that the above parity symmetry is satisfied for any set of values {δ 1 , δ 2 } characterizing the 'darkness' of each one-dark soliton component. In fact, there is no further restriction on the parameter space of this two-dark soliton solution in order to satisfy the parity property. Therefore, for this type of solutions it is expected that the integrated anomalies to vanish and the associated charges to be asymptotically conserved. This behaviour is in contradistinction from the one described in the two-bright analytical soliton solution of the focusing NLS model, in which only a subset of the general solution satisfies that symmetry (i.e. the solutions such that the 2-soliton parameter c satisfies c = n π 2 , n ∈ Z Z in the eq. The general form of the one and two dark soliton solutions as presented in (5.23) and (5.27), respectively, have been derived in [21,22]. In the first reference a hybrid of the dressing transformation and tau function methods has been used, whereas in the second one they have been derived by the KP-hierarchy reduction method. The ansatz (5.2) for the general N-soliton solution in terms of the tau functions G and F follows the construction presented in [21].

Space-reflection parity transformation
It is interesting to see the behaviour of the fields R and φ for the two-soliton solutions possessing a spacereflection symmetric amplitude R(− x, t) = R( x, t) and phase φ(− x, t) = φ( x, t) for any given shifted time t. In fact, when the continuous wave background is at rest, i.e. k = 0, one has the parameter relationships , which satisfy the eq. (5.21) for two-soliton solutions. These parameters correspond to two dark solitons moving in opposite directions with equal velocities such that they undergo a head-on collision (see Fig. 1). So, for this choice of parameters, from eqs. (5.28)-(5.33) and (5.36) we have In Fig. 1 we plot the functions R and φ for three successive times. Note that they are symmetric under the reflection x → − x for any value of the variable t = t. There is another solution with similar behaviour for the set of parameters {k = 0, Therefore, it is clear that the above solution satisfies This type of solution will be associated to the even order exactly conserved charges as discussed in section 3.2. In our numerical simulations we will verify the properties for this kind of solutions of deformed NLS models, as well as for configurations without this type of symmetry. We will consider translationally invariant solutions of (2.2) of the form

Solitary waves and anomalies
satisfying the following non-vanishing boundary condition (nvbc) The nvbc (6.2) is suitable for wave solutions in the form of localized 'dark' pulses created on the continuous wave background (cwb) at rest is the phase shift experienced by the cwb with intensity |ψ 0 | 2 , in the non-linear medium with intensity |ψ| 2 and varying according to the law − ∂V [|ψ| 2 ] ∂|ψ| 2 . Moreover, we will consider a family of equations such that small oscillations around |ψ| = |ψ 0 | of (2.2) defines a hyperbolic system with sound Then the dark solitons of the model (2.2) must propagate with the velocity v such that |v| < v s .
is also a solution. So, in the co-moving coordinate of the solitary wave one can write the solitary wave solutions Φ 2 ( x) and Θ( x) as dependents only on the variable x. As an example, we can rewrite the solitary wave in (6.5)-(6.6) as Then, the space-time transformation (3.32) applied to this form of solution furnishes the next symmetry So, for solutions satisfying this type of space-time symmetry one has that the factor α (3,−n) in the anomaly β n in (3.31) is even according to (3.38). The first expressions of α (3,−n) provided in the appendix B exhibit this property. Therefore, the associated charges Q (n) satisfy a mirror type symmetry (3.40) and an asymptotically conservation law (3.41).
Moreover, in this case we can also explore the space-reflection symmetry of the anomalies which can be useful in order to estimate their values, by using general symmetry considerations. So, note that the time dependence of the solution ψ(x, t) in (6.1) for functions of the form Φ 2 ( x) and Θ( x) appears only in the cwb phase (6.3). In addition, the components α (3,−n) are polynomials in R( x) and ∂ x ϕ( x), and their x−derivatives (see the first five expressions in the appendix B). So, the space-time parity symmetry can be used in order to see the properties of these expressions. In fact, the parities of R and ∂ x ϕ of this special soliton solution can be obtained by taking t = 0 in this frame. Therefore, using the result (3.38) one has that the α (3,−n) are even, whereas X is odd under the above space-reflection, then one has that the anomalies β (n) vanish for this type of solitary wave. Therefore, in the co-moving frame one has dQ (n) dt = 0, n = 0, 1, 2, 3, ...
In fact, the integrand of these expressions are some polynomials containing the functions R(z) and ∂ x ϕ(z), and their x−derivatives. Next, we can rewrite the integration (6.19) by making a coordinate transformation x → z, t → t. Furthermore, one can relabel z → x, and the outcome would be an equivalent integral expression to the one in the co-moving frame we have already analized above. Therefore, also for the travelling waves the anomalies vanish and the associated charges will be conserved. Rmin (...)dR, where the limits of integration R min and |ψ 0 | 2 characterize the minimum intensity dip of the dark soliton and the cwb amplitude, respectively. Let us make a coordinate transformation x → z, t → t and rewrite (4.33) as
Moreover, using (A.5) one has that the term inside parenthesis can be written as . Therefore, all the terms in the integrand above, except the last one, can be rewritten in the form K[R]R ′ , which can be written as a total derivative in z, then this 'surface term' does not contribute to the anomaly for the boundary condition (4.1) with k = 0. Regarding the last term, we can rewrite the integration in field space as follows

Simulations
Firstly, we have solved numerically the ordinary NLS model (5.1) without deformation (ǫ = 0). We develop this procedure for a general initial condition with a cwb, for the one-dark and two-dark soliton solutions, respectively. Then, it is convenient to rewrite the dark soliton (5.10) as where α ≡ 2p R , v ≡ −2p I . Notice that the dark soliton (7.1) is characterized by the width 1/α and the velocity v. The background density becomes |ψ 0 | = (α 2 + v 2 ) 1/2 / 4|η|.
In order to study the interaction of 2-dark solitons we will choose the initial data (t = 0) as (consider The two solitons are initially centered at ±x 0 (x 0 > 0), the soliton centered initially at −x 0 (t = 0) moves to the right with velocity we can take two one dark solitons some distance apart as the initial condition for our numerical simulations.
For the second deformed model (2.4) ǫ = 0, q ∈ Z Z + we do not know any analytical expression for its solitary wave; however one can generate numerically a solitary wave, as we will explain below. We will study the dark soliton collisions numerically for the both deformed NLS models and compare the outcomes with the properties summarized above for the integrable NLS model.

First deformation:
The collision of dark solitons in the cubic-quintic NLS equation (2.3) can been simulated numerically by considering the initial condition ψ(x) defined as where two solitary wave solutions (6.5)-(6.6) of the cubic-quintic NLS model have been located some distance apart. The function ψ is continuous at x = 0, i.e. it has continuous absolute value |ψ| = Φ(x 0 ) and phase equal to Θ(x 0 ). The parameters {ξ 1 , ξ 2 , r, k ± } are defined in (6.5)-(6.6), so the ± signs correspond to the relevant sign of the deformation parameter ǫ. The two solitons are initially centered at ±x 0 (x 0 > 0), the soliton centered initially at −x 0 (t = 0) moves to the right with velocity v > 0, whereas the soliton initially (t = 0) centered at x 0 travels to the left with velocity (−v). Notice that the direction of motion of the soliton is related to the sign of the phase slope in (7.3). In addition, we will consider initially well-separated solitons, i.e. the parameter 2x 0 is chosen to be several times the width of the solitons (∼ 1 k ± ) and 2x 0 < 2L. So, the initial condition uses two of its static analytic solitary waves which are stitched together at the middle point, and then we allow the scattering of them, absorbing the radiation at the edges of the grid. It amounts to maintain the nvbc at the edges of the grid for each time step of the numerical simulation. For solitons colliding with different velocities (different amplitudes) we must modify conveniently this initial condition.    r in (4.30). Therefore, the charges must be exactly conserved, within numerical accuracy. So, our numerical simulations suggest that the space-reflection P x symmetry is a sufficient but not a necessary condition for the vanishing of the anomaly β                         Notice that the charge Q (4) in (4.30)-(4.31) is conserved to within an accuracy of ≈ 10 −11 in the second modified NLS model even for deformation parameters of the order of |ǫ| ≈ 0.1, whereas in the first modification the CQNLS model this error is of the order of ≈ 10 −5 for |ǫ| ≈ 0.01. In our extensive simulations we noticed that in general the vanishing of the anomalies occur with small numerical errors provided that the parameter value ǫ is small and the solitons are fast. Then, we are quite confident of our simulations in the second modified NLS model in which the anomalies vanish within an error of less than 10 −10 for |ǫ| < 1.5 and n ≥ 2 in the whole process of the collisions of two fast and/or slow solitons. The main conclusion that we can make from our simulations for several two-soliton collisions is that the charge Q (4) is not only asymptotically conserved, but exactly conserved during the whole collision process, within the numerical accuracy.
The results of our simulations of the second deformation of the NLS model (2.4) and the fact that it possesses a submodel (with ǫ = 1, q ∈ Z Z + ) which passes the Painlevé test [18] suggest that it might be an exactly integrable model for some set of the parameters {|ψ 0 |, η, ǫ, q}. To the best of our knowledge its Lax pair formulation and analytic soliton solutions are not known in the literature and the search for them are worth attempting. However, the existence of two-solitons is not indicative of integrability. In fact, some non-linear models present elastic scattering of two solitons but they do not posses three-soliton solutions. Nevertheless, according to the ref. [25] the above deformed NLS model (2.4) may be regarded as a partially integrable model in the sense that it possesses two-soliton solutions, even though it has been obtained numerically. Moreover, the existence of three-soliton solutions in partially integrable models is quite demanding and it is very rare. In fact, in the Hirota's bilinear approach the existence of three-solitons is a strict integrability criterion [25]. So, it would be interesting to tackle this problem in the lines discussed above for some deformations of the NLS model, since the presence of three-soliton combinations of any type may indicate integrability of the model.

Discussions and some conclusions
In this paper we have studied some properties of the deformed non-linear Schrödinger model defined by the eq. of motion (2.2), with the potential involving a deformation parameter ǫ and the non-vanishing boundary condition (3.26). We have shown that such deformed NLS theories possess some sectors where the solitary wave solutions exhibit properties very similar to solitons in exactly integrable field theories.
They possess an infinite number of exactly conserved quantities for one-soliton type solutions, i.e. for dark solitary waves travelling with a constant speed. For solutions possessing a special space-time parity symmetry, using analytical methods borrowed from integrable theories, it can be shown that those quantities are asymptotically conserved for two-soliton type solutions. This means that these quantities can vary in time during the collision process of two one-solitons but return, in the distant future (after the collision), to the values they had in the distant past (before the collision). Furthermore, for solutions possessing the space-time parity symmetry and in addition, a special space-reflection parity symmetry we can say even more. Using an algebraic technique which combines the space-reflection and an order two Z Z 2 automorphism of the affine sl (2)  We have confirmed the predictions of our analytical calculations through an efficient and accurate numerical method, the so-called time-splitting cosine pseudo-spectral finite difference method, which is appropriate for the case in which the cwb is at rest. The numerical code was devised for two important deformations of the NLS. The first deformation is related to the cubic-quintic non-integrable NLS model and its numerical simulation considers as an initial condition two of its solitary waves located some distance apart. The second deformation considers a saturable type potential and its solitary wave is generated numerically by taking as a seed solution the analytic solitary wave of the cubic-quintic NLS model.
For the above two deformations we have computed the first non-trivial anomaly β (4) of the Q r charge's non-conservation law. We have verified that this anomaly vanishes, and consequently the exact conservation of the charge Q Then, we may argue that the parity property (3.48)-(3.49) is not the cause of the exact conservation of the charges, but according to our analytical calculations it is a sufficient condition for these phenomena to happen. Further research work is necessary to settle such questions which involve the non-linear dynamics of the scattering. So, the symmetries involved in the quasi-integrability phenomenon deserve further investigation and they may have relevant applications in many areas of non-linear sciences.