Modulation instability, periodic anomalous wave recurrence, and blow up in the Ablowitz–Ladik lattices

The Ablowitz–Ladik equations, hereafter called AL+ and AL− , are distinguished integrable discretizations of respectively the focusing and defocusing nonlinear Schrödinger (NLS) equations. In this paper we first study the modulation instability of the homogeneous background solutions of AL± in the periodic setting, showing in particular that the background solution of AL− is unstable under a monochromatic perturbation of any wave number if the amplitude of the background is greater than 1, unlike its continuous limit, the defocusing NLS. Then we use Darboux transformations to construct the exact periodic solutions of AL± describing such instabilities, in the case of one and two unstable modes, and we show that the solutions of AL− are always singular on curves of spacetime, if they live on a background of sufficiently large amplitude, and we construct a different continuous limit describing this regime: a NLS equation with a nonlinear and weak dispersion. At last, using matched asymptotic expansion techniques, we describe in terms of elementary functions how a generic periodic perturbation of the background solution (i) evolves according to AL+ into a recurrence of the above exact solutions, in the case of one and two unstable modes, and (ii) evolves according to AL− into a singularity in finite time if the amplitude of the background is greater than 1. The quantitative agreement between the analytic formulas of this paper and numerical experiments is perfect.

The AL equations (1) characterize [36] the quantum correlation function of the XY-model of spins [40].If η = 1, it is relevant in the study of anharmonic lattices [62]; it is gauge equivalent to an integrable discretization of the Heisenberg spin chain [35], and appears in the description of a lossless nonlinear electric lattice (η = 1) [43].At last, if η = 1, the AL hierarchy describes the integrable motions of a discrete curve on the sphere [15].
It is well-known that the homogeneous background solutions of the NLS equations (3) a exp(2 i η |a| 2 τ ), a complex constant parameter, (8) is unstable under the perturbation of waves with sufficiently large wave length in the focusing case η = 1 [9,8,69,71], and always stable in the defocusing case η = −1, and the modulation instability (MI) of the focusing case is the main cause for the formation of anomalous (rogue) waves (AWs) [26,17,51,28,29,50,71].Since (8) is also the exact homogeneous solution of the AL equations (replacing τ by t), it is natural to investigate their linear instability properties under monochromatic perturbations with respect to the AL dynamics, and study how such instability develops into the full nonlinear regime.
We remark that, as in the NLS case, the AL equations have the elementary gauge symmetry (if u is solution, also ũn = u n exp(iρ), ρ ∈ R is solution); then a could be chosen to be positive without loss of generality (but we shall not do it here).Unlike the NLS case, for which, if v(ξ, τ ) is a solution, also ṽ(ξ, τ ) = b v(bξ, b 2 τ ), b ∈ R is solution), the AL equations do not possess any obvious scaling symmetry.It follows that a in (8) cannot be rescaled away as in the NLS case.Therefore one expects that, unlike the NLS case, the amplitude a of the background (8) play a crucial role in its stability properties under perturbation.
The Cauchy problem of the periodic AWs of the focusing NLS equation (3) has been solved in [20,21], to leading order and in terms of elementary functions, for generic periodic initial perturbations of the unstable background: in the case of a finite number N of unstable modes, using a suitable adaptation of the finite-gap (FG) method.In the simplest case of a single unstable mode (N = 1), the above finite gap solution provides the analytic and quantitative description of an ideal Fermi-Pasta-Ulam-Tsingou (FPUT) recurrence [18] without thermalization, of periodic NLS AWs over the unstable background (8), described, to leading order, by the well-known Akhmediev breather (AB) solution of focusing NLS for the arbitrary real parameters ã, ρ, k, X, T , but with different parameters at each appearance [20].See also [22] for an alternative and effective approach to the study of the AW recurrence in the case of a single unstable mode, based on matched asymptotic expansions; see [23] for a finite-gap model describing the numerical instabilities of the AB and [24] for the analytic study of the linear, nonlinear, and orbital instabilities of the AB within the NLS dynamics; see [25] for the analytic study of the phase resonances in the AW recurrence; see [56] and [13] for the analytic study of the FPUT AW recurrence in other NLS type models: respectively the PT-symmetric NLS equation [3] and the massive Thirring model [63,44].The AB, describing the nonlinear instability of a single mode, and its Nmode generalization were first derived respectively in [7] and [39].The NLS recurrence of AWs in the periodic setting has been investigated in several numerical and real experiments, see, f.i., [66,67,30,46,54], and qualitatively studied in the past via a 3-wave approximation of NLS [34,64].
In addition, a perturbation theory describing analytically how the FPUT recurrence of AWs is modified by the presence of a perturbation of NLS has been recently introduced in [10], in the simplest case of a small linear loss or gain, giving a theoretical explanation of previous interesting real and numerical experiments [30,60].This theory has been applied in [11] to the complex Ginzburg-Landau [48] and Lugiato-Lefever [41] models, treated as perturbations of NLS (see also [12]).
These results suggest two interesting problems.i) the construction of the analytic and quantitative description of the dynamics of periodic AWs of the AL lattices; ii) the understanding of the effect of a perturbation of the AL lattices on such a dynamics.
The solution of the first problem is the main goal of this paper; the solution of second problem is contained in the paper [14].
We remark that the terminology "focusing" and "defocusing" NLS equations should not be exported to their AL discretizations, since the background solution of the AL equation reducing to the defocusing NLS is unstable under any monochromatic perturbation if |a| > 1, and such an instability leads generically to blow up in space-time.Therefore we prefer to call hereafter the AL equations ( 1) with η = ±1 as AL ± equations, instead of using the terminology "focusing and defocusing AL equations" often present in the literature.
The paper is organized as follows.In §2 we investigate the linear stability properties of the background (8), extending results already present in the literature [6,52], and showing that, unlike the NLS case, the background of AL − is unstable under any monochromatic perturbation if |a| > 1.In §3 we present the exact solutions of AL ± describing the instability of one and two unstable modes, and we show that: i) the solutions of AL + are always regular, but with an amplitude, relative to the background, growing as |a| 2 and as |a| 4 respectively in the case of one and two unstable modes; ii) the solutions of AL − develop singularities in closed curves of spacetime, and when these curves intersect a line x = n 0 ∈ Z (the generic case), the solution blows up at finite time in the site n = n 0 .In §4 we use the matched asymptotic expansion approach developed in [22] to solve to leading order the periodic Cauchy problem of the AWs, i) describing in terms of elementary functions the associated AW recurrence of one and two unstable modes (in this second case for non generic initial perturbations) in the AL + model; ii) showing analytically how a smooth perturbation blows up at finite time in the AL − model.The Appendix is dedicated to the construction of the exact solutions studied in §3 using the Darboux transformations (DTs) of AL ± [19].
To the best of our knowledge, known results concerning AWs of the AL ± models prior to our work are the following.The exact solution of AL + over the background, corresponding to a spectral parameter in general position, containing as limiting cases the discrete analogues of the Akhmediev breather (24), of the Kuznetsov-Ma [33,42] and Peregrine [53] solutions were first constructed in [47] using the Hirota method [27].The amplitude growth of (24) for large |a| was investigated in [6], together with the linear instability properties of the background solution of AL + .Numerical experiments for the Cauchy problem of AWs for AL + are reported in [58].The linear instability properties of the background solution of the AL − model were investigated in [52], where Peregrine type solutions of any order of the AL ± models were constructed using the Hirota method, observing that they are singular in the AL − case.

Linear stability properties of the background in the AL case
To study the (linear) stability properties of the background solutions (8) in the AL ± dynamics, we seek solutions of (1) in the form obtaining, at O( ), the linearized AL ± equations for ξ n : If the perturbation is a monochromatic wave: equation ( 12) reduces to the system of ODEs whose solution reads: where and ν, µ are two arbitrary complex parameters.From now on we fix the following constraint on the wave number since the negative values are covered by the second exponential in (11), and the growth rate ( 16) depends on κ through cos κ.
The growth rate ( 16) implies the following stability features of the background solutions (8) of the AL ± .The case η = −1.Equation AL − reduces to the defocusing NLS in the continuous limit, for which the background ( 8) is stable under a perturbation of any wave number.The stability properties of the AL − background are much richer [52]; indeed we distinguish three sub-cases: • |a| > 1 ⇒ σ > 0 ⇒ exponential growth and linear instability ∀ κ (see Figure 1); S 0 = 0 with linear growth, stability otherwise, ∀ κ; Since σ depends on κ through cos κ, these stability properties are 2πperiodically extended to the whole real κ axis, with basic period (−π, π).In the unstable case |a| > 1, there are two subcases: i) 1 < |a| < √ 2, then σ(k) has its absolute minimum at κ = 0 with σ(0) = 0, and absolute maxima at , then σ(k) has its absolute minimum at κ = 0 with σ(0) = 0, relative minima at κ = ±π, with σ(±π) = 4 |a| 2 − 1, and absolute maxima 1).The case η = 1.The AL + equation reduces to the focusing NLS (3) in the continuous limit, for which the background ( 8) is unstable for monochromatic perturbations of wave number k such that |k| < 2|a|, and the parameter a can be rescaled away due to the scaling and trivial gauge symmetries of NLS.Also in this case the stability properties of the AL background are richer than those of the NLS background, since now the amplitude |a| cannot be rescaled away, and is involved in the following nontrivial way.Define κ a as [6] κ (see Figure 2); then The growth rate has maxima at ±κ M , with κ M = arccos and σ(±κ M ) = 2|a| 2 .The instability curve is similar to that of focusing NLS, except for the 2π periodicity (see Figures 2).
As before, these stability properties are 2π-periodically extended to the whole real κ axis.Summarizing the results of this section, we have the following "instability regions" of the background (8): where κ a is the smallest positive branch of arccos.They were found in [6] for η = 1, and in [52] for η = −1.
As we shall see in the following, in all the cases discussed above in which the background ( 8) is unstable, we find it convenient to introduce the parameter φ defined by We remark that φ is real in both unstable cases (19) (it is therefore an angle) and, in terms of it, the growth rate σ (16) takes the same simple form as in the NLS case [20,21,22].

Exact periodic AW solutions of the AL equations
Since the background solution ( 8) is linearly unstable under monochromatic perturbations in the cases (19), it is important to describe how the corresponding exponential growth evolves into the nonlinear stage of MI described by the full nonlinear model.In this section we present the exact periodic solutions of AL ± describing the nonlinear instability of a single nonlinear mode and of two interacting nonlinear modes.The construction of these solutions using the DTs of the AL equations [19] is presented in the Appendix.

The case of a single unstable mode
The instability of a single nonlinear mode K 1 of AL ± is described by the solution: where K 1 is the wave number and σ(K 1 ), defined in (16), is the growth rate of the linearized theory in the unstable cases (19), the angle θ 1 is defined as in ( 20) and X 1 , T 1 and ρ are arbitrary real parameters.Since θ 1 is defined in terms of (K 1 , a, η), the growth rate σ(K 1 ) and the parameter G 1 can be expressed in terms of (K 1 , a, η) or in terms of (θ 1 , a, η) in the following way: , If η = 1, ( 24) is the Narita solution [47] of AL + , discrete analogue of the AB solution (10) of focusing NLS, reducing to it through the scaling where h is the lattice spacing.If η = −1, (24) is, to the best of our knowledge, the novel solution describing the MI present also in the AL − model.The solution (24) oscillates in n and is exponentially localized over the background in t in the following way To study its behavior, we first replace n ∈ Z by x ∈ R in (3); it is legitimate, observing that function N 1 (x, t; K 1 , X 1 , T 1 , ρ, η) solves AL ± with n replaced by x: If η = 1, equation (26) implies that sin θ 1 < cos(K 1 /2) and equation ( 25) that G 1 < 1.It follows that the denominator of N 1 is always positive.Therefore the solution N 1 is always regular in the (x, t) plane for all values of its arbitrary parameters, like in the NLS case.But there is an important difference, since now the maximum of the absolute value of the solution (24), reached at the point (x, t) = (X 1 , T 1 ), reads [6]: implying that the relative maximum (the ratio of the maximum of the amplitude of (24) to the background amplitude |a|) grows as O(|a| 2 ) for |a| 1: unlike the NLS case, for which M/|a| = 1 + 2 sin φ does not depend on a.
We remark that, if X 1 / ∈ Z, the maximum of |N 1 | is not reached in a lattice point (see Figure 3).
this curve is centered at (X 1 , T 1 ) and x-periodic with period 2π/K 1 (see Figures 4).
the solution N 1 blows up at the two points (x, t ± (x)): where arccos is here the smallest positive branch of the inverse of cos, and cosh −1 is the positive branch of the inverse of cosh (see Figures 4).
We remark that the extension l of the singular curve (33) in the x direction is less than 1, since: Consequently, if n X is the integer closest to X 1 and then the appearance of the AW is not singular on the lattice, since the singular curve is located in the region between two subsequent sites.But this situation is not generic.We remark that the solution (24) for η = −1 does not have a continuous limit to defocusing NLS, since the prescription a ∼ hã, h 1 in ( 28) is not consistent with the instability condition |a| > 1.

The case of two unstable modes
The instability of two nonlinearly interacting modes K 1 and K 2 is described by the novel (to the best of our knowledge) two-mode solution of AL ± : where and where cos Also this solution oscillates in n and is exponentially localized in time over the background: ] , as t → ±∞.
In the rest of the paper we shall limit our considerations to the case K 2 = 2K 1 ; then the solution is periodic with period 2π/K 1 .
As in the case of a single mode, in the natural continuous limit (see ( 28)) the solution for η = 1 reduces to the two breather solution of Akhmediev type [22], while it does not have a continuous limit in the case η = −1.
As in the case of a single mode, it would be possible to show the following.i) If η = 1, the solution ( 38) is always regular.If |T 1 − T 2 | > O(1) the two nonlinear modes are separated into two weakly interacting Narita solutions with wave numbers K 1 and 1 the two nonlinear modes appear almost at the same time interacting nonlinearly.If T 1 = T 2 , K 2 = 2K 1 , and X 2 = X 1 + 2π 4 K 1 the two modes are amplitude-locked and phaselocked in a characteristic configuration similar to the one of NLS (see Figures 5)).
The maximum height of |N 2 | can be calculated in terms of elementary functions in two cases: when 1 and the solution describes two separated Narita solutions (24), and when they are amplitude-and phaselocked: In the second case, the maximum height reached by the solution is given by:  Top left: T 1 = −1, T 2 = 1.5, X 1 = 0 and X 2 = 1.|T 2 − T 1 | ≥ 1, and the solution appears as two separate Narita solutions of wave numbers K 1 and K 2 .Top right: T 1 = T 2 = 0, X 1 = 0 and X 2 = 1.Since T 1 = T 2 , the two modes appear together and strongly interact.Bottom: the phase locking choice of the parameters: ii) If η = −1, N 2 develops always singularities at finite time on three closed curves of the (x, t) plane; one curve for the mode K 1 and two curves for K 2 (see Figures 6).

AW recurrence and blow up at finite time of periodic AWs of AL ±
In this section we study the periodic Cauchy problem with period in which the initial condition is a generic periodic perturbation of the background ( 8) (what we call the "periodic AW Cauchy problem"): where and As we shall see in the following, it is convenient to define the parameters σ j = 2a 2 sin(2φ j ), where cos and α j := c j e −iηφ j − c −j e iηφ j , β j = c −j e iηφ j − c j e −iηφ j .
To construct the solution, to leading order and in terms of elementary functions, we use the matched asymptotic expansion technique introduced in [22] for the focusing NLS model.

The AL + case
We first concentrate on the case η = 1, giving rise to a recurrence of regular periodic AWs, in the case of one and two unstable modes.The instability condition |κ| < κ a implies that the first N ≤ p modes ±κ j , 1 ≤ j ≤ N are unstable, where and x is the largest integer less or equal to x.

One unstable mode
In the simplest case of one unstable mode (N = 1) only, corresponding to the case in which the period M satisfies the inequalities only the mode κ 1 is unstable, and the corresponding nonlinear stage of MI is described by the solution (3) for a suitable choice of its arbitrary parameters, obtained using matched asymptotic expansions [22].
Matching the linearized solution (53) for N = 1 and the exact solution (24) in the intermediate time interval 1 , and It follows that the Narita solution N 1 (x, t, κ 1 , X + 1 , t (1) , 2φ 1 , 1) describes the first appearance of the AW.To describe the recurrence of AWs, it is convenient to obtain the first appearance for negative times [22], matching the linearized solution (53) for N = 1 and the Narita solution (24) in the time interval 1 1 , and It follows that the solution N 1 (x, t, κ 1 , X − 1 , t (0) , −2φ 1 , 1) describes the first appearance of the AW also at negative times.Comparing the two consecutive appearances at times t (0) and t (1) , and using the time translation symmetry of AL + , we infer the following periodicity law for the general recurrence of the AL + AWs (see [22] for more details) Summarizing, the n th AW appearance in the FPUT recurrence generated by the Cauchy problem (46), in the case of the single unstable mode κ 1 , is described, in the time interval |t − t (n) | = O(1), by the Narita solution , where and ∆x and ∆t are defined in (60).This is the analytic and quantitative description of the FPUT recurrence of AWs of the AL + equation in term of the initial data through elementary functions.x (1) and t (1) are respectively the first appearance time and the position of the maximum of the absolute value of the AW; ∆x is the x-shift of the position of the maximum between two consecutive appearances, and ∆t is the time interval between two consecutive appearances (see Figure 7).The numerical output is in perfect quantitative agreement with the theory described by ( 60), (61) The quantitative agreement between the above theory and numerical experiments is perfect, as one can see from the following table, in which one compares the values (x (j) , t (j) ) of the position and time of the j th appearance of the AW, for j = 1, . . ., 6, as predicted by ( 60)-( 61) with the values coming from the numerical experiment of Figure 7. E.g., for = 10 −4 , the first disagreement in the 6 th appearance is in the 7 th decimal digit, corresponding to the 9 th significant digit!Numeric Theor (x (1) , t (1) ) (1.87977083, 4.17892477) (1.87977074, 4.17892429) (x (2) , t (2) ) (3.08443357, 12.8077003) (3.08443336, 12.8076998) (x (3) , t (3) ) (4.28909633, 21.43647586) (4.28909597, 21.43647549 ) (x (4) , t (4) ) (5.49375910, 30.06525140) (5.49375859, 30.06525108) (x (5) , t (5) ) (6.69842186 , 38.69402694) (6.69842121, 38.69402668) (x (6) , t (6) ) (0.90308462, 47.32280249) (0.90308383, 47.32280228 ) In addition, using the fact that, at each appearance, the AW is exponentially localized in time, and that, from (29), after each appearance, the background exhibits a 4φ 1 phase shift, the FPUT recurrence can be described by the following expression, uniform in space-time, with t ≤ t We remark, from (61), that the maximum of the AW at the first appearance is located on a lattice point x (1) ∈ Z, if the initial data are such that arg If, in addition, ∆x ∈ Z, i.e., from ( 61), (60) arg then the maxima of the FPUT recurrence are all located on the lattice points.
The distinguished case α 1 β 1 ∈ R. As in the NLS case, a very distinguished situation occurs when the initial data are such that Indeed, from (51),(60): • If α 1 β 1 > 0, then ∆x = 0 and the FPUT recurrence is periodic with period ∆t.
It is easy to verify that with Therefore, in terms of the initial data : and a periodic FPUT recurrence with period 2∆t.We remark that the condition (65) is not generic with respect to the AL dynamics, since it arises imposing the real constraint |c 1 | = |c −1 | on the initial data.But, as we shall see in the forthcoming paper [14], it becomes the generic asymptotic state when the AL + dynamics is perturbed by a small loss or gain.

Two unstable modes
In the case of two unstable modes (N = 2), corresponding to the case in which the period M satisfies the inequalities only the modes κ 1 and κ 2 are unstable, and the corresponding nonlinear stage of MI is described by the solution N 2 for a suitable choice of its arbitrary parameters.
Proceeding as in the case of a single unstable mode, we describe the first appearance for negative times, matching the linearized solution (22) for N = 2 and the solution (38) sin(2φ j ) e −σ j t−iφ j cos κ j (x − X − j ) , N 2 ∼ a e 2i|a| 2 t e i(ρ+2(θ 1 +θ 2 )) 1 + inferring that ρ = −2(θ 1 + θ 2 ), K j = κ j (consequently θ j = φ j and Σ j = σ j ), X j = X − j , and It follows that the solution ) describes the first appearance of the AW at negative times.As in the one mode case, comparing the two consecutive appearances we construct the solution of the Cauchy problem to leading order.Introduce: .
(73) 1, j = 0, 1; these two conditions imply that the two unstable modes appear approximately at the same time for many recurrences.
This solution grows exponentially and generically develops singularities of the type discussed in §3 in the first nonlinear stage, when t = O(log(1/ )).Therefore it does no make sense to study its recurrence properties, but it does make sense to see how a smooth initial condition (46) evolves into a singularity in finite time.
If p = 1, then the period is M = 3, and κ 1 is the only unstable mode.The matching procedure of the previous section leads to the comparison between the linearized solution (76) and the one mode solution (3) Therefore the smooth initial condition u n (0) = a [1 + (c 1 e iκ 1 n + c −1 e −iκ 1 n )] evolves into the generically singular solution whose singularity properties have been studied in §3.Analogously, in the case p = 2, corresponding to the two unstable modes κ j , j = 1, 2 and to the period M = 5, the smooth initial condition c j e iκ j n + c −j e −iκ j n (80) evolves into the generically singular two-breather solution 1 , t (see Figure 6).

Appendix. Darboux transformations and periodic AW solutions
We look for a gauge transformation matrix D n (t, λ) (the so-called Darboux matrix) that preserves the structure and the symmetries of the AL Lax pair.More precisely, let (u n ) and (u n , ψ n ) be solutions of the Lax pair (4).Then we look for the transformation implying the following two equations for the Darboux matrix where (L n , together with the symmetry coming from ( 5), (6).Following [19], we look for the Darboux matrix in the form: for N ∈ N + , corresponding to the Darboux transformation then the symmetry (84) implies the following relations between the matrix elements of D It is well-known that the Darboux matrix describes a one-parameter (the complex parameter λ) family of transformations becoming singular at one or more values of λ.If λ i is a singular point such that det D n (λ i , t) = 0, the symmetries of the Lax pair imply that also −λ i and ±1/λ i are singular points.At the singular points (±λ i , ±1/λ i ) the matrix D n (λ, t) has range 1 and, if ξ n (t, λ), χ n (t, λ) are the columns of the fundamental matrix solution Ψ  n (t, λ) = ξ n (t, λ), χ n (t, λ) , their images must be proportional in the points (±λ i , ±1/λ i ) where the matrix is singular: where γ i is the proportionality factor.Equation (88) implies, for each λ i , the system: where Note that a change of the basis of the eigenvectors ξ n (t, λ), χ n (t, λ) is equivalent to a rescaling of the constant γ i .If the number of singular points λ i is equal to the order N of the Darboux transformation, the relations (89) for i = 1, . . ., N define a determined system of 2N equations for the 2N unknowns a n , l = 1, . . ., N , that can be uniquely solved, leading to the wanted Darboux matrix.
At last, from the the first of equations (83), the dressed solution u [N ] n (t) can be calculated from the solution u where At last: u [1]  n (t) = u  ∆ (2) , where , and ∆ a and ∆ (2) b are obtained substituting in ∆ (2) the second and the fourth columns, respectively, by the vector (−λ 2 1 , −λ 2 2 , −1/λ 1 2 , −1/λ 2 2 ) T .As for the previous case, specializing (91) for N = 2, we obtain the solution: u [2]  n (t) = u  (97) Therefore the building blocks of the solutions (93), (95), are the functions r i (n, t), calculated from (96) and (90) in the following form: where n i = η arg γ i κ i and t i = − log(|γ i |) σ i , and the singularities λ i are expressed in terms of the modes κ i and the angles φ i via (97): Substituting ( 98) and (99) into equations ( 92) and (94) we construct the relevant coefficients of the Darboux matrices under construction; then formulas (93) and (95) give the wanted solutions, equivalent to the one and two breather solutions (24) and (38), through the following relations among the parameters.

Figure 7 :
Figure 7: Density and 3D plots of |u n (t)| coming from the numerical integration of the Cauchy problem of the AWs for the AL + equation (η = 1), in the case of a single unstable mode.We used the 6th order Runge-Kutta method [57] with the initial condition u n (0) = a(1 + (c + e ikn + c − e −ikn )), where M = 7, a = 1.1, = 10 −4 , c + = 0.53 − i 0.86 and c − = −0.26+ i 0.22.The numerical output is in perfect quantitative agreement with the theory described by (60),(61)

n
(t, λ) of the Lax pair (4) for u n = u

(
the previous formulas for the two simplest cases N = 1, 2.N=1If N = 1, the linear system (89) of two equations yields the solution a

If N = 2 ,(
we can write the relevant Darboux matrix elements a The Darboux dressing of the background solution Now we specialize this construction choosing u