Analytic black branes in Lifshitz-like backgrounds and thermalization

Using black brane solutions in 5d Lifshitz-like backgrounds with arbitrary dynamical exponent $\nu$, we construct the Vaidya geometry, asymptoting to the Lifshitz-like spacetime, which represents a thin shell infalling at the speed of light. We apply the new Lifshitz-Vaidya background to study the thermalization process of the quark-gluon plasma via the thin shell approach previously successfully used in several backgrounds. We find that the thermalization depends on the chosen direction because of the spatial anisotropy. The plasma thermalizes thus faster in the transversal direction than in the longitudinal one. To probe the system described by the Lifshitz-like backgrounds, we also calculate the holographic entanglement entropy for the subsystems delineated along both transversal and longitudinal directions. We show that the entropy has some universality in the behavior for both subsystems. At the same time, we find that certain characteristics strongly depend on the critical exponent $\nu$.


Introduction
The gravity/gauge duality provide an alternative tool for understanding dynamics of the strong coupling system, where standard methods lacks. One such system is the quark-gluon plasma (QGP), which can be produced in heavy-ion collisions and represents a strongly coupled fluid with a small viscosity [1]. The QGP goes through several stages of evolution. It is believed that the QGP is created after a very short time after the collision τ therm ≈ few 0.1 fm/c and the holographic approach, in particular, is aimed to describe this and a short nearest period of evolution [2,3]. There are indications that in this time the QGP is anisotropic. Since at time scales of τ ≈ few 0.1 fm/c it is in thermal equilibrium, one can try to apply anisotropic holographic hydrodynamics to describe its isotropization. The anisotropic stage of the QGP takes place for 0.1 fm/c τ 0.3 − 2 fm/c [4] and can be studied also holographically [5,6].
Through the gauge/gravity duality, the thermalization of the field theory in the boundary corresponds to the process of black hole formation in the bulk. According to the holographic dictionary, the scenario of a heavy-ion collision can be represented as a shock wave collision in which trapped surface is formed [7]- [16]. After the collision the shocks slowly decay, leaving the plasma described by hydrodynamics in the middle. The creation of the black hole is also described by the Vaidya metric of an infalling shell with a horizon corresponding to the location of the trapped surface [17]- [21].
By now both standard existing models and the holographic approach with AdS backgrounds, as well as its conformally equivalent deformations for bulk geometries, have failed to reproduce the particle multiplicity at high energies. However, if one performs the holographic estimations of multiplicities in Lifshitz-like spacetimes [22,23,25], one can fit the experimental data for certain values of the critical exponents [24]. In this paper, we consider following the 5-dimensional Lifshitz-like metric: The choice of the geometry (1.1) is motivated by studies of the anisotropic phase of the QGP. As it is known, the QGP in the 4d gauge theory can be characterized by the energymomentum tensor T µν = diag(ε, p L , p T , p T ), with the particle momenta p 2 L < p 2 T at early times of the QGP formation. To reproduce this anisotropy from the gravity side, one of the possible backgrounds is the Lifshitz-like metric (1.1). It has been shown in [24] that for the wall-on-wall collision in the 5d Lifshitz-like background with the critical exponent ν = 4, the dependence of multiplicity on the energy is desirable, i.e. behaves as E 1/3 .
Another possible implementation of the 5d Lifshitz-like spacetime is which differs from (1.1) by the anisotropic scaling taking place only for one spatial direction. The embedding of this background and its non-zero temperature generalization into supergravity IIB was done in [25] for ν = 3/2. Solutions interpolating between Lifshitz-like (1.2) and AdS geometries were intensively studied in [26]- [32] within the context of applications to the anisotropic QGP. However, the results for multiplicities calculated using the background (1.2) in [24] do not fit the experimental data unlike the case of the metric (1.1).
Since after the shock wave collision the trapped surface argument supports black hole formation, it is natural to construct the corresponding Vaidya-type solution. In the present paper we start from the generalization of (1.1) to the non-zero temperature case for an arbitrary critical exponent. Further, we construct a Vaidya-type geometry asymptoting to the Lifshitz-like solution to model a gravitation collapse in order to study the holographic thermalization. The Vaidya metric with Lifshitz scaling was used for the examination of the holographic thermalization in [33,34]. There, it has been shown that for the metric with anisotropy between time and spatial directions the propagation of thermalization represents a similar "horizon" behavior as that seen in the AdS case. The Vaidya metric was also generalized to the Lifshitz spacetimes with a hyperscaling violating factor [35][36][37]. As another application of our solutions, we consider the time evolution of the holographic entanglement entropy during the process of thermalization. The behavior of the entanglement entropy modeling the thermalization and "quench" processes is a subject of intensive studies during last years, see [36,[38][39][40][41][42] and refs. therein. For Lifshitz metrics the time evolution of the entanglement entropy turns out to have a linear regime [33]. In this work, we examine the influence of spatial anisotropy on the behavior of the entanglement entropy.
The paper is organized as follows. Sect. 2 is devoted to constructing the exact solutions which asymptotes to the Lifshitz-like metric (1.1). In Sect. 2.1 we present the 5d black brane background. In Sect. 2.2 we generalize it to the Vaidya type solution, which describes a thin shell collapsing to a black hole in the Lifshitz-like background. In Sect. 3 we numerically calculate thermalization times using our Lifshtiz-Vaidya type solution. Sect. 4 is devoted to studies of the holographic entanglement entropy at equilibrium as well as its out-ofequilibrium behavior. We conclude in Sect. 5 with a discussion of our results. Appendix A collects some technical details used for constructing analytic solutions. In Appendix B we present some details concerning numerical solutions to EOM related the functional of the entanglement entropy.

Black branes in Lifshitz-like backgrounds
In [24] we considered a collision of two domain walls in the five-dimensional Lifshitz-like background where ν is the critical exponent. Note that (2.1) is equivalent to (1.1) via the change of coordinate z =r −ν and the rescaling (t, x, y 1 , y 2 ) → ν −1 (t, x, y 1 , y 2 ).
In [24,25] the metric ansatz (2.1) was considered for a 5d model governed by the action where m 0 and Λ are constant and the 3-form H (3) and the 2-form B (2) are related by However, it seems difficult to find an analytic black brane (hole) solution for the model (2.2) due the dependence (2.3) for the gauge fields.
In this paper we consider another bulk theory, possessing the metric (2.1) as a solution to Einstein equations, with the following action where the 2-form F (2) is the gauge field with φ is the dilaton scalar field, λ is a dilatonic coupling constant and Λ is the cosmological constant 1 . The model (2.4) can be considered as a truncated supergravity IIA in the style of [25]. Another possible underlying theory is the 5d SO(6) gauged supergravity [44]. The Einstein equations of motion can be written as The scalar field equation is We then select the following anzatz for the dilaton and Maxwell fields: where µ and q are two constants. One can see that this ansatz has some features. Firstly, the dilaton has the linear dependence in the radial coordinate (2.11). Black hole solutions with a linear dilaton in the supergravity context were discussed in [43]. At the same time the similar ansatz for the gauge fields (2.12) emerges to support AdS 2 × R 3 , AdS 3 × R 2 , AdS 2 × R 2 solutions and their non-zero temperature analogues of gauged supergravity in [44]- [45]. The 6-dimensional Lif 4 × R 2 background with a constant two-form field was found in [46]. The model (2.4) with the fields given by (2.11)-(2.12) can be generalized to the nonzero temperature case without changing the field ansatz. The metric of the black brane solution reads with the blackening function given by (2.14) For the particular case ν = 4, the metric (2.13)-(2.14) along with the ansatz (2.11)-(2.12) solves the field equations (2.6)-(2.8) provided that the constants take the following values: If the dilaton is constant and the Maxwell field vanishes, the metric (2.13) with ν = 1 turns out to be the black brane solution in the AdS background: (2.17) or in terms of variabler This corresponds to r → 0 or the UV limit.

The Vaidya-Lifshitz geometry
To study the thermalization process we need to use the infalling shell approach based on the Vaidya solution [47]. First, we introduce the coordinate z = e −νr , which, after the rescaling (t, x, y 1 , y 2 ) → ν −1 (t, x, y 1 , y 2 ), allows one to rewrite the metric (2.13) in the form , (2.20) with the blackening function To write down the Vaidya-Lifshitz solution, one should consider the ingoing null geodesics and introduce the Eddington-Finkelstein coordinate system (v, x, y 1 , y 2 , z) via Owing to (2.23) we can represent (2.20) in the following form where M is a constant and θ(v) is the Heaviside function. One can also consider a smooth function m(v) and get, for instance,

The thermalization process
In [24] we have shown that there is a trapped surface, which forms in the collision of two shock waves in the background (2.1), controlled by boundary points z a and z b , with z a < z b . This trapped surface defines the location of the horizon for (2.24)-(2.25).
Calculations of the thermalization time t therm at the scale is based on finding geodesics with endpoints located at the distance for a bulk particle. Then, the thermalization time t therm is the time when this geodesic covered by the shell (2.24)-(2.25).
The general case for the Lagrangian of the pointlike probe has the form where τ is a parameter. Here we have two possibilities for the choice of τ with respect to the transverse directions.

Thermalization along the longitudinal direction
Consider the first case taking τ = x, which can be interpreted as a longitudinal direction. Now we obtain where we define The integrals of motion corresponding to (3.2) are From the relations (3.4) and (3.5) we get (3.6) The turning point z * can be found from the equation For simplicity, we put I = 0 and we get from (3.7) For the distance between the ends of the geodesic and the thermalization time one gets . (3.10) Note that here we assume that the turning point lies above the horizon, i.e. z h > z * . The behavior of the thermalization time as a function of the distances for (3.9)-(3.10) is represented in Fig.1.A. We see that the thermalization time behaves linearly with . The results match to those for modified AdS models from [34] and coincide for all values of the dynamical exponent ν.

Thermalization along the transversal direction
Now turn to the second case when τ = y 1 , that we interpret as the thermalization along a transversal direction. From (3.1) we have where we put The integrals of motion corresponding to (3.11) read (3.14) From (3.13) and (3.14) we get that the turning point z * is defined from For I = 0 this equation is simplified to give and we also getż From (3.17) one gets the relation between the ends of geodesic and the thermalization time .
Here we remove the regularization since ν > 0. The dependence (3.19) on (3.18) is given in Fig. 1.B. We see that the thermalization time in the transversal direction depends on the anizotropic parameter ν. In particular, for ν = 2 the thermalization process is more then twice faster as compared to the longitudinal direction. By increasing ν we make the thermalization in the transversal direction faster. We also see that for larger values of ν the dependence on the mass m becomes more essential.

Entanglement entropy
In this section we explore the evolution of entanglement entropy in the context of the holographic prescription. We perform calculations using both black brane (2.13)-(2.14) and Vaidya-Lifshitz time dependent backgrounds (2.24)-(2.25). The entanglement entropy can be useful to probe correlations in the background measuring an entanglement of a quantum system. If the system is divided into two spatially disjoint parts A and B, the entanglement entropy S(A) gives an estimation of the amount of information loss corresponding to the restriction of an A. It seems not to be simple to calculate the entanglement entropy from the strongly coupled system side. However, one can compute its holographic dual using the suggestion from works [48]- [50]. The holographic formula for the entanglement entropy of a subsystem A where A is the area of the minimal three-dimensional surface whose boundary coincides with the boundary of the region A. The area of the surface is defined by the relation where From (4.4) we see that the entanglement entropy depends on the direction along which the subsystem is delineated. There are two possible cases for the subsystems both for the black brane and the thin shell we have we study and compare each other.

Entanglement entropy in a time-independent background
To begin with, we compute the entanglement entropy for the black hole (2.20)-(2.21).
Here we present the results for the two subsystems cut out both along longitudinal and transversal directions.

Subsystem delineated along the longitudinal direction
First, consider the subsystem A cut out along x-direction, say, the belt is located as We assume that the minimal area surface is invariant under the y 1 and y 2 planar directions and the embedding function is the function of only one coordinate, z = z(x). Thus, the three-dimensional minimal surface is defined by Taking into account (2.20), one has where it is supposed that = d dx . The integral of motion corresponding to the system with the Lagrangian L reads The function z = z(x) that minimizes the surface area is then given by the equation of motion where the turning point z * is related with C as z The length scale l x can be found from We can remove from the upper limit in (4.10) under the assumption that the turning point z * is above the horizon. Indeed, if the function f is given by (2.21) with the horizon defined by the integrand in (4.10) near z = z * can be represented as thus, we have the integrable singularity for z * < z h . However, for z * = z h one obtains (4.13) which leads to the logarithmic singularity. For calculations of the entropy in the black hole background we assume that the turning point is below the horizon, while for the case of the shell we present the results for the case when the horizon is crossed.
Substituting (4.9) into (4.7) and coming to the integration with respect to z-variable, one has (4.14) In (4.14) we remove assuming m = 1, but keep UV regularization z 0 . The latter expression can be represented in terms of the dimensionless variable w = z/z * as The renormalized functional for the minimal surface reads (4.16) Taking into account (4.12) one can also rewrite (4.10) in the w-variable for z * = z h . One can see that the relation for the entanglement entropy is proportional to the area of the boundary ∂A = L y 1 L y 2 which is in agreement to the area law.
The behavior of the area (4.16) is presented in Fig.2 A. To get the dependence of the entanglement entropy of the length for small values of one can consider the massless case. We see that for m = 0 the integrals (4.16) and (4.17) can be calculated explicitly. By analogy with [25] one gets as a function of (4.17) in the 5d Lifshitz black brane background (2.20)-(2.21) for ν = 2, 3, 4 (the upper gray, middle green and lower brown curves, respectively). From numerical calculations we see that for large To keep the correct dimension we have to assume It should also be noted, that from Fig.2.A the dependence on the mass of the black brane for the intermediate is rather small. The physical meaning of estimation (4.19) is that our surface for large becomes like a smothered parallelepiped almost touching to the horizon.

Subsystem delineated along the transversal direction
Another possible subsystem A can be divided along the y 1 -direction (which is equivalent to dividing it along y 2 ). It is also assumed that z = z(y 1 ) and (4.21) The three-dimensional minimal surface bordering on ∂A has the form (4.22) This dynamical system has the following integral of motion The corresponding equation of motion reads where z * is related with C as z The length scale l y 1 can be defined in the following way (4.25) We note that for the lower limit in (4.25) one can take z 0 = 0. At the same time, we can remove for the upper limit of (4.25) for z * < z h , by the same reason as above in (4.10).
Owing to (4.24) the relation (4.22) in terms of the dimensionless w-variable takes the form (4.26) The renormalized functional for the minimal surface (4.26) reads Numerical results for the entanglement entropy density (4.27) for different values of ν are shown in Fig. 2 for large . We see the dependence on the mass in Fig.2.B for large . Note that the functions γ L (m) (4.19) and γ T (m) (4.29) are different.

Entanglement entropy in a time-dependent background
Now we come to studies of the evolution of entanglement entropy in the Lifshitz-Vaidya background (2.24)-(2.25), describing the infalling shell. As before we will consider subsystems delineated along both transversal and longitudinal directions.

Subsystem delineated along the longitudinal direction
We once again start from the consideration of a subsystem A extending along x-direction, assuming that the minimal surface area is parameterized by Taking into account (4.5), the volume functional corresponding to the minimal threedimensional surface is given by (4.33) Here we suppose that ≡ d dx . Substituting (4.32) in (4.1) we get the expression for the holographic entanglement entropy.
The Lagrangian L in (4.32) has the integral of motion given by where we denote which coincide with those from [51] for ν = 1. Taking into account (4.36) the equations of motion (4.37)-(4.38) can be rewritten as Here we assume that the function f has the form (2.27). We can solve these equation numerically using the following initial conditions We are interested in solutions which reach z = 0 at some point x s , similar boundary conditions have been proposed for example in [52,53] . The point x s is, in fact, a singular point of the solutions. Solutions to eqs. (4.37)-(4.38) obeying (4.40)-(4.41) are presented in Fig.3. We can observe that there are two types of such solutions that have a z * below and above the horizon. It is useful to study a domain of the initial data, where these solutions can exist, see Appendix B. It is also instructive to see the behavior of the quantity f v + z . An assumption, that the function f does not depend on v, yields to the fact that f v + z is some conserved quantity. At the same time for f defined by (2.27) it changes that we observe on Fig.15. However, it can also be seen that at the end of the curve z = z(x) at x = x U V , this quantity does not vary significantly and admits the approximation ∂ v f = 0.
Owing to (4.36) the minimal three-dimensional surface (4.32) can be represented as (4.42) Coming to the z-variable one can rewrite (4.42) in the following form with a(z) defined by (4.44) In (4.44) the RHS is taken with the negative sign since z < 0 for the solutions of our interest.
To calculate the entanglement entropy we have to study the behavior of the integrand a in (4.42). For z ∼ 0 we expect the following behaviour It is also convenient to introduce the quantity b(z), defined by We study the behaviour of function b(z) on the solution to eqs.(4.37)-(4.38) for ν = 2 and different masses is shown in Fig.16 (Appendix B). We see that b(z) → C = 0 for any value of mass, and therefore, we have a(z) ∼ C/z 1+2/ν . From Fig.16 one can see that for z * = 1 we have C = 1. Hence, the UV divergence is similar to the shell free case and one can perform the similar renormalization Returning to the variable x we obtain the finite contribution to the entanglement entropy of the shell Now we can define the quantity ∆A Shell−LV It should be noted that the holographic entanglement entropy for the Lifshitz-Vaidya background depends on two parameters, z * and v * , whereas for the pure Lifshitz case it depends only on z * . One takes z * in the second term in such a way that it gives the same distance as in the first term. From (4.17) and for f = 1 one gets Taking into account that where a 2,ren = −0.5991, a 3,ren = −1.03468, and a 4,ren = −1.49367, we can explicitly write down ∆A Shell−LV . Thus, we have the difference between the entropy in the current time and thermal entropy. We observe the kink in the evolution which was considered for Lifshitz (ν = 2) and AdS (ν = 1) backgrounds in [33] and [51], respectively. From Figs. 5, 6 we see that the entanglement entropy increases almost linearly with time. We note that after the saturation point had been reached the entropy flattens out. It should also be mentioned that the saturation is faster for small values of and is almost independent on the anisotropic parameter ν.

Subsystem delineated along the transversal direction
Now we turn to the case when a subsystem A is delineated along y 1 (y 2 )-direction. Parameterizing the minimal surface are by v = v(y 1 ), z = z(y 1 ) with (4.21), we have where it is supposed that ≡ d dy 1 . The integral of motion corresponding to the system with Lagrangian L (4.57) is with (4.60) we get the conserved quantity The EOM corresponding to (4.57) can be presented in the form Taking into account (4.62) the equations of motion (4.63)-(4.65) can be re written as Here we assume that the function f is given by (2.27) as in the previous section and we solve (4.63)-(4.65) with the same boundary conditions We consider once again only solutions that reach z = 0 at some point x s , that is in fact a singular point of the solutions. In Fig.7 we plot solutions to (4.63)-(4.65) with (4.67)-(4.68). We also present domains of the initial data plane, where these solutions can exist in Fig.21-23, see Appendix B. We show the position of the singular point corresponding to the solution with given z * and varying v * . We present more details about solutions to eqs. One can rewrite (4.57) in the following way On the solution z the functional (4.69) can be presented as As in the previous section we derive the factor b(z) thus b(z) → 1 (see, Appendix B). In this case the UV behaviour is the same as in the vacuum case and we can represent the answer to (4.69) in the following form (4.73) The finite contribution to the holographic entanglement entropy can be represented in the following way (4.74) The renormalized entanglement entropy (4.74) as a funciton of is presented in Fig.8. From Fig.2 B and Fig.8 D one can see the entanglement entropy in time-dependent background has the similar behavior as for the static case. For small we observe the dependence of the entropy on ν, which vanishes for large , where the entropy has linear behavior. In three left panels of Fig.11 we present the renormalized entanglement entropy (4.74) as a function of and t. Now as above let us define ∆A Shell−LV Taking into account that for A LV reg we have Now we get a ν,ren . (4.80) For ν = 2, 3, 4 one can write down explicitly We present the time evolution of the entanglement entropy (4.74) for different values of the critical exponent ν in Fig.9 and Fig.10 . In Figs.9 we show the difference between the entropy in the shell background and the value of the entropy in the Lifshitz vacuum. The evolution in time of the quantity which represents the difference between the entropy in the current time and thermal entropy is demonstrated in Figs. 10. For each value of we observe that the entropy grows linearly at small times. Then it approaches saturation and we see a kink in the dependence on time, which is much sharper for greater values . We note that the evolution of the entanglement entropy has more essential dependence on the anisotropic parameter ν comparing to the case when the subsystem cut out along the longitudinal direction.
Three right panels in Fig.11 also demonstrate the behavior of the entanglement entropy with subtracted vacuum values as a function of and t for different ν.  Figure 10. The time dependence of the holographic entanglement entropy A ren for the Lifshitz-Vaidya metric after the corresponding subtraction of the state when the black brane has already been formed (t = 2.5) at fixed l = 1, 1.4, 1.9, 2.2, 2.5, 2.8 for a subsystem delineated along the transversal direction, ν = 2, 3, 4 (A,B,C, respectively). In D we plot the time dependence of A ren − A ren | t=2.5 at l = 2 for ν = 2, 3, 4 (from bottom to top, respectively).

Conclusions
In this paper, we have investigated the holographic thermalization process of the quarkgluon plasma in anisotropic backgrounds. For this purpose, we have used an analytic black brane solution which asymptotes to the Lifshitz-like spacetime with arbitrary critical exponent. We also have built the corresponding Lifshitz-Vaidya solution, which metric interpolates between the vacuum Lifshitz-like and the black brane geometries. This background has been used to describe the thermalization process as well to model the "quench" process. Let us note that 4-dimensional Lifshitz spacetimes with black hole are widely used in AdS/CMT models [54][55][56].
We have considered thermalization processes both in the transversal and longitudinal directions, which differ by the contribution of the anisotropic exponent. The thermalization along the longitudinal direction turned to have the linear regime similar to that in modified AdS models. At the same time, in the transversal direction the thermalization process is much faster and behaves linearly only for large distances. It should also be noted that the thermalization along the longitudinal direction is independent of the value of the dynamical exponent, while results obtained for the transversal direction strongly depend on the anisotropic parameter and are more sensitive to the value of mass.
Holographic entanglement entropy have also been studied for the subsystems delineated along both transversal and longitudinal directions. For a subsystem cutting out along the longitudinal direction in the black brane background, we have found that the dependence of the entropy on the critical exponent for small distances was absent and appeared for larger values of . In the transversal direction we have observed that the entropy depends on ν at small distances and approaches a linear behavior which is the same for all ν. Thus, for both subsystems at large , the entanglement entropy comes to a linear regime, which, depending on the chosen direction, depends or does not on the critical exponent.
The regime is similar to the one found for the Lifshitz metrics in [33], which, however, is independent on ν. This is related to that the anisotropy between the spatial coordinates is absent in the Lifshitz backgrounds considered in [33] unlike the Lifshitz-like metrics suggested in [22,25].
The most interesting results concern the holographic entanglement entropy in the Vaidya-Lifshitz solution that we constructed. Here we again studied subsystems divided along two possible directions. The common feature of the time evolution of the entropy for both subsystems is the kink observed already for small distances. The entropy increases linearly in time until it approaches the saturation point. We found that the form of the kink is sharper for large values of . The dependence on the critical exponent looks similar to this one in the black brane background. Since the subsystems differ by the contribution of the critical exponent, the rate of approaching saturation and the saturation value of the entanglement entropy were seen to be different for each case.
It would be interesting to study other non-local operators, like two-point correlation functions and Wilson loops operators, in the backgrounds considered in this paper and compare their velocity bounds as well as estimate with experimental data. We shall address these problems in our future work [27].
preparation of this work.

A.1 The LHS of the Einstein Equations
Without any black brane, the metric reads The non-zero components of the Ricci tensor are The scalar curvature is The metric for a black brane solution that asymptotes to the Lifshitz background (A.2) is given by The geometry (A.6)-(A.7) is supported by e λφ = µe 4r , F (2) = 1 2 q dy 1 ∧ dy 2 . (A.8) The non-zero components of the Ricci tensor of the metric (A.6) are R 00 = e 2νr f (r) 2(ν 2 + ν)f (r) + (2ν + 1) ∂f (r) ∂r + 1 2 The generalization of (A.6)-(A.7) to the Vaidya background reads ds 2 = −e 2νr f (v, r)dv 2 + 2e νr dvdr + e 2νr dx 2 + e 2r dy 2 1 + dy 2 2 , (A.14) where The solution (A.14)-(A.15) is supported by the fields (A.8), plus the infalling shell of null dust, whose energy-momentum tensor is The non-zero components of the Ricci tensor of the metric (A.14) are leads to a solution for the constants λ, µ, q and Λ and the component T s 00 of the shell energy-momentum tensor. Thus, one finds that the ansatz (A.8) for the fields is valid. For ν = 4, the solution is formed by the values (2.15) for the constants λ, µ, q and Λ, as well as by the following expression of the shell energy-momentum: In Fig.12 and Fig.13 we show the position of the singular point (x-axis) of the solution with given z * (y-axis) and varying v * . From Fig.12 we see that for the given value of z * ≤ 1 varying v * , say from v * = v 1 to v * = v 2 , we get different positions of l s lying between l s (v 1 ) and l s (v 2 ). It is interesting to note that for small z * the position of the singular point is not considerably depends on value of v * . For z * → 1 this dependence is more significant.
We also see that one given value of l s corresponds to different values of z * and v * .  Figure 13. The same as in Fig.12 for ν = 2, 3, 4 (blue,brown,green,respectively), A: z * ≤ 1, B: z * > 1.
The position of the singular point for different values of critical exponent ν is presented in Fig.13.
In Fig.14 we present the contour plots for the boundary time and z(l sing ) as functions of z * and v * . The values of the initial conditions taken from regions of white colour yields solutions to eqs. (4.37)-(4.38), which do not obey the boundary constraints.
It is also interesting to find the behaviour of the function f (2.27) as a function of position on the constructed solutions to eqs. (4.37)-(4.38). In Fig.15.A we present the behavior of f (x) near to 1 in the region of the singular point.
In Fig.16 we check the asymptotic behaviour of b(z) defined by (4.46) on the solution z(x) to equations (4.37)-(4.38) for ν = 2. For these solutions z * is taken to be 1. We see that for x → , i.e. near the end of the profile, b(z) → 1.     is conserved on the solution, we can say the solution can be approximated by the static solution in this region. We also present the dependence of the "quasi" conserved quantity Q on x for ν = 2 in Fig.19 .
In Fig.20 we check the asymptotic behaviour of b(z) defined by (4.72). We present the dependence of l sing on the turning point z * and v * in Fig.21 and Fig.22. Here we again observe that one can get different positions of l s in the range from l s (v 1 ) to l s (v 2 ) varying v * and fixing z * .
In the three left panels of Fig.23 we show contour plots for the boundary time depending on the initial conditions z * , v * for ν = 2, 3, 4. In the three right panels of Fig.23 we present contour plots for z(l sing ) as a function of initial conditions z * , v * for ν = 2, 3, 4. As in the previous case, regions of white colour correspond to the irrelevant initial conditions.