The solutions of classical and nonlocal nonlinear Schr¨odinger equations with nonzero backgrounds: Bilinearisation and reduction approach

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Introduction
It is common knowledge that the nonlinear Schrödinger-type equations admit carrier waves and solitons, and that breathers and other solutions (e.g.rogue waves) are the modulations of carrier waves.Meanwhile, many (1+1)-dimensional soliton equations admit solitons with either zero or nonzero asymptotic behaviours as |x| → ∞.As for the one of the most popular nonlinear integrable models, the focusing nonlinear Schrödinger (NLS) equation, where i is the imaginary unit, |q| 2 = qq * and q * stands for the complex conjugate of q, early investigations of the solutions of this equation with nonzero boundary conditions were due ]ocnmp[ Da-jun Zhang, Shi-min Liu and Xiao Deng to Kuznetsov [40], Kawata and Inoue [38,39] and Ma [47].They solved the NLS equation ( 1) with nonzero boundary conditions, i.e., |q(x, t)| → const.as x → ±∞, by means of the inverse scattering transform.Faddeev and Takhtajan have also done important work in this area (see for instance the monograph Ref. [25] and references therein).Besides, the NLS equation with different asymmetric nonzero boundary conditions has been studied in [20,37,14,21,59].The defocusing NLS equation, has dark solitons with nonzero boundary condition (|q| goes to a positive constant as |x| → ∞).Zakharov and Shabat are pioneers who studied the two NLS equations using tools of integrability [67,68].Hirota derived bright soliton solution for the equation (1) and dark soliton solution for (2) by using bilinear method, respectively in [34] and [36].
For more references about the integrability of NLS equations one can refer to [8] and the references therein.
From the point of view of the Darboux transformation, for any seed solution q 0 of the NLS equation (1) (i.e.q 0 satisfies (1)), the envelope of the solution q generated from the Darboux transformation has a form (see equation (4.3.10) in [48]) In this context, when q 0 ̸ = 0, we say the resulting solution q is a solution with a nonzero background q 0 .Various methods for the systematic construction of solutions of equation ( 1) with a plane wave background q 0 = √ αe −2iαt have been established, where α is a positive constant.One can replace q with qe −2iαt in equation (1), which leads it to the form iq t = q xx + 2(|q| 2 − α)q. (4) Mathematically, this implies, compared with (1), that the envelope |q| gains a positive lift √ α such that |q| → √ α as |x| → ∞.However, the plane wave background does bring interesting behavior of |q| more than that.The simplest solution (corresponding to one-soliton) of the equation ( 4) is a breather [39,40,47], not the usual soliton.In a special limit the breather yields a localized rational solution [52], which is nowadays used to describe a rogue wave.The rational solution was also derived by Matveev and Salle via the Darboux transformation (see §4.3 in [48], where the rogue wave is called "exulton" solution).The second order rational solution of equation ( 4) was derived in 1985 in [10], using a similar way as in [52].The rational solution of arbitrary order of the NLS equation was first constructed in 1986 in [24], where explicit formula of the solution was presented in an elegant way and nonsingular property of the solution was proved as well (see also [23] for an alternative proof)."Rogue waves" is the name given by oceanographers to isolated large amplitude waves, which occur more frequently than expected for normal, Gaussian distributed, statistical events (cf.[51]).After rogue waves was observed in optic experiment in 2007 [57], it started to draw new attention and the research on rogue waves has become a hot topic.One can refer to the review [51] for more references.Mathematically, higher order rational solutions of the NLS equation ( 4) can be obtained using the Darboux transformation via a special limit procedure [29,32], from a bilinear approach using reduction of the Kadomtsev-Petviashvili τ functions [50], and from inverse scattering transform [13].There are also some research of the NLS equation on the elliptic function background, e.g.[17,26].
In this paper, we will derive solutions with a plane wave background for the NLS equation (1) and (2) and their nonlocal versions by using bilinear method but in a completely different way from [50].
Our idea is to solve the second order Ablowitz-Kaup-Newell-Segur (AKNS) coupled equations ir t = − r xx + 2r 2 q (5b) as an unreduced system, which, for instance, yields the NLS equation ( 1) via reduction r = −q * .We can bilinearize this unreduced system and present solutions of the bilinear equations in terms of double Wronskians.Then, we impose constraints on the column vectors of the double Wronskians so that the desired reduction holds and thus we get solutions to the reduced equation.Such an idea was first proposed in [18,19] for obtaining solutions for the nonlocal integrable equations.Nonlocal integrable systems were first systematically proposed by Ablowitz and Musslimani in 2013 [5] and have drawn intensive attention (e.g.[72,1,63,65,7,45,15,4,53,30,46,54,55]).The bilinearisation-reduction approach has proved effective in deriving solutions not only to the nonlocal systems but also to the classical equations (e.g.[22,42,43,44,56,60,61]).In this paper, we introduce transformation for the unreduced system (5).Here (q 0 , r 0 ) are an arbitrary set of solution of (5).It will be seen that for the NLS equation (1), |q 0 | does act as a background of the envelope |q|, see equation (4.3.10) in [48] and equation (73) in this paper).In this context we also call (q 0 , r 0 ) a set of background solution of the system (5).We will employ (6) to bilinearize the unreduced system (5) and present (quasi) double Wronskian solutions to the bilinear equations.Then we will implement reduction technique to obtain solutions to the reduced equations listed in ( 9)- (12).The paper is organized as follows.In Sec.2 we recall the classical and nonlocal reductions of the unreduced AKNS system (5).In Sec.3 we derive the bilinear form of (5) with a set of background solution (q 0 , r 0 ) and derive (quasi) double Wronskian solutions to the bilinear equations.In Sec.4 the reduction technique is implemented and explicit form of solutions with plane wave background solutions are obtained for the reduced equations.Then in Sec.5 we investigate dynamics of some obtained solutions for the classical NLS equation and the nonlocal NLS equations with nonzero backgrounds.Finally, Sec.6 serves for presenting conclusions.

The second order AKNS system and its reductions
The second order coupled AKNS equations (5), where q = q(x, t) and r = r(x, t) are functions of (x, t) ∈ R 2 , has been studied as a classical coupled system in past decades.Recently it was found that this system is related to the cubic nonlinear Klein-Gordon ]ocnmp[ Da-jun Zhang, Shi-min Liu and Xiao Deng equation, see [7].Its Lax pairs consist of the well known AKNS (or Zakharov-Shabat (ZS)-AKNS) spectral problem [67,2,3], and the corresponding time evolution part in which λ is spectral parameter, λ t = 0, Φ and Ψ are wave functions.
3 Bilinearisation and solutions of the AKNS system (5) In this section, we develop the double Wronskian technique to construct exact solutions of the second order AKNS system (5) with nonzero background solution (q 0 , r 0 ).

Bilinearisation
Suppose that (q 0 , r 0 ) are a set of solution to the second order AKNS system (5).To introduce nonzero backgrounds, we consider the dependent variable transformation (i.e. ( 6)) with which the system (5) can be decoupled into the following bilinear form of f, g and h, where D x and D t are the well known Hirota bilinear operators defined as [35] Note that when q 0 = r 0 = 0 the above bilinear form (14) degenerates to the case of zero background (cf.equations (1.5.1)-(1.5.3) in [16]).
To have solutions of ( 14), we expand f, g and h as the series where ε is an arbitrary number, {f (j) , g (k) , h (l) } are functions to be determined.Consider a special case, where A 0 is an arbitrary constant.By calculation we can find out 1-and 2-soliton solutions, which agree with the following expressions with (µ 1 , µ 2 ) = (1, 0) and (µ 1 , µ 2 ) = (1, 1): where (for j = 1, 2) We should mention that the reduction A 0 = 0 of the above expression does not gives rise to a significant solution for the unreduced system (5) with a zero background.Note that in [41] the system (5) was also bilinearized via the following transformation and the bilinear form is , where λ ∈ R. One-soliton and two-soliton solutions they derived (see equation ( 58) and (82) in [41]) are shown to be associated with a more general plane wave background (see (48), which degenerates to (16) when a = 0 and A 0 = B 0 ).

]ocnmp[
Da-jun Zhang, Shi-min Liu and Xiao Deng 3.2 Quasi double Wronskian solutions of the AKNS system (5) We now derive double Wronskian solutions of the second order AKNS system (5).
Theorem 1.The bilinear system (14) has solutions (21), where ϕ and ψ in (20) satisfy ( 18) and (19), and (q 0 , r 0 ) are given solutions of the system (5).Furthermore, matrix A and any matrix that is similar to it lead to the same solution of the AKNS system (5) through the transformation (13).
The proof will be sketched in Appendix A. Later we only need to consider the canonical forms of A, i.e.A being diagonal or a Jordan block.
Strictly speaking, the above f, g, h in (21) are not double Wronskians that are defined by arranging columns by increasing the order of derivatives of ϕ and ψ.We may call them quasi double Wronskians.Note that when A is diagonal the results in Theorem 1 are the same as those derived from the Darboux transformation (cf.§4.2 in [48]).When A is a Jordan block, the corresponding solutions can be obtained using a limit procedure from those solutions which are derived from a diagonal matrix A (e.g.§4.3 in [48]).We also note that when the background solution (q 0 , r 0 ) is independent of x, we may covert f, g, h given in (21) to double Wronskians.
where | n; m| denotes a (m + n + 2) double Wronskian defined as (see [49]) ϕ and ψ are (2m + 2)-th order column vectors.When ϕ and ψ meet the condition where A is a (2m + 2) × (2m + 2) complex matrix, (q 0 , r 0 ) satisfy (5) but are independent of x, then f, g, h defined in (22) are solutions to the bilinear equations (14).Furthermore, matrix A and any matrix that is similar to it lead to the same solution to the AKNS system (5) through the transformation (13).
The proof will be given in Appendix B.

Reduction and solutions
For convenience we call (5) the unreduced system and ( 9)-( 12) the reduced equations.In the previous section we have already obtained solutions in terms of quasi double Wronskians (21) (see Theorem 1 for the unreduced system ( 5)).In this section we implement reductions by imposing constraints on A and ψ so that ( 21) can provides solutions to the reduced equations ( 9)- (12).Such a reduction technique was first introduced in [18,19].

Reduction of the Wronskian solution
Let us directly present results and then prove them.
(1) The classical NLS equation (9) has solution where q 0 is a solution of equation ( 9) such that r * 0 = δq 0 , vector ϕ is a solution of matrix equations and A and T obey the relation (2) For the reverse-space nonlocal NLS equation (10), its solution is given by ]ocnmp[ Da-jun Zhang, Shi-min Liu and Xiao Deng where q 0 is a solution of equation (10) such that r * 0 (x) = δq 0 (−x), vector ϕ is a solution of matrix equations and A and T obey the relation (3) For the reverse-time nonlocal NLS equation (11), its solution is given by where q 0 is a solution of equation ( 11) such that r 0 (t) = δq 0 (−t), ϕ is a solution of matrix equations and A and T obey the relation (4) For the reverse-space-time nonlocal NLS equation (12), its solution is given by where q 0 is a solution of equation ( 12) such that r 0 (x, t) = δq 0 (−x, −t), vector ϕ is a solution of matrix equations and A and T obey the relation Proof.We employ the classical NLS equation (9) as an illustrating example.Introduce constraint on ψ, where T is a certain matrix in C (2m+2)×(2m+2) .First, it can be verified that when r 0 = δq * 0 and A and T satisfy (26), the constraint (36) reduces ( 18) and ( 19) to (25).In fact, taking (18) as an example, under (36) and r 0 = δq * 0 , we rewrite (18) as where (37a) is nothing but (25a).Making use of ( 26), equation (37b) multiplied by δT * from the left gives rise to the complex conjugate of (37a).This indicates (18) reduces to (25a).In a similar way one can find (19) reduces to (25b) in this case.
Next, with the constraint (36), we can rewrite the quasi double Wronskians (21) as Making use of ( 26) we find that Then, switching the first (m + 1) columns and the last (m + 1) columns and picking the parameter δ out yield In a similar way we can prove Thus we have which is the reduction by which we get (9) from (5).
The proof of nonlocal cases is similar to the classical one.For the reverse-space nonlocal NLS equation ( 10), the reduction is implemented by taking together with (29).Here and below we note that we do not write out independent variables unless the inverse of them are involved.Relations between Wronskians are i.e. r = δq * (−x), which reduces the unreduced system (5) to equation (10).

Matrices A and T
We look for explicit forms of A and T in Theorem 3. Equations ( 26) and ( 29) can be unified to be and equations ( 32) and ( 35) can be unified to be Consider special solutions to these matrix equations, i.e.A and T are block matrices where T i and K i are (m + 1) × (m + 1) matrices.Then solutions to equations ( 42) and ( 43) can be listed out as in the Table 1 and 2, cf.[18].In addition, equation ( 29) admits real solution in the form (44) for the case δ = −1, where or Table 1.T and A for equation (42) (σ, δ) T A Table 2. T and A for equation (43) (σ, δ) T A (1, 1) Besides, equation ( 26) with δ = 1 can have pure imaginary solution ( 44) and (45) where in (45a Due to the fact that A and any matrix that is similar to it generate same solutions to the system (5) (see Theorem 1), we only need to consider the canonical forms 3 of A. That is, K m+1 can either be or

Plane wave background solution q 0
Considering the expression (13) (and also (73)) we can call q 0 and r 0 to be background solutions of q and r, respectively.The unreduced system (5) admits a set of plane wave solutions 3 A general case for Km+1 is the block diagonal form where each J h j (kj) is an hj × hj Jordon block matrix defined as (47), Diag(ks+1, • • • , ks+n) is an n × n diagonal matrix and n + s j=1 hj = m + 1.In this case, ϕ is just composed accordingly since ( 18) and (19) are linear system of ϕ and ψ.Thus, we will only consider two limiting cases, (46) and (47).Note that matrix A corresponds to the eigenvalues of the AKNS spectral problem (7), which means ( 46) is for the case of simple distinct eigenvalues and ( 47) is for the case of one eigenvalue of geometric multiplicity one and algebraic multiplicity m + 1.
]ocnmp[ Da-jun Zhang, Shi-min Liu and Xiao Deng where A 0 , B 0 and a are arbitrary complex constants.It is easy to find that the reduced classical and nonlocal NLS equations ( 9)-( 12) admit the following solutions, respectively, Our purpose is to write out explicit Wronskian vectors ϕ that respectively satisfy the conditions ( 25), ( 28), ( 31) and ( 34) for given background solutions q 0 .We are going to consider the simple case where q 0 are given in (49).If making use of some symmetries, we may start from a simpler background solution instead of ( 49).
• Scaling symmetry: if q(x, t) solves the NLS equation ( 9) with a background solution A 0 q 0 = e 2iδY is a solution.
• Scaling symmetry: if q(x, t) solves the reverse-space NLS equation (10) with a background solution q 0 (x, t)  • Scaling symmetry: if q(x, t) solves the reverse-time NLS equation (11) with the background solution q 0 (x, t) given in (49c), then Q(X, Y ) = 1 A 0 q(x, t), where X = A 0 x and Y = A 2 0 t, also solves the reverse-time NLS equation (11) with Q(X, Y ), i.e.
• Scaling symmetry: if q(x, t) solves the reverse-space-time NLS equation ( 12) with a background solution q 0 (x, t) = A 0 e 2iδA 2 0 t , then Q(X, Y ) = 1 A 0 q(x, t), where X = A 0 x and Y = A 2 0 t, also solves the reverse-space-time NLS equation ( 54), of which A 0 q 0 = e 2iδY is a solution.Based on these symmetries of equations ( 9)-( 12), we only need to consider the unified background solution q 0 given in (50).

Wronskian column vectors ϕ and ψ
4.4.1 Vectors ϕ and ψ for the unreduced system (5) We start with a pair of background solutions of the unreduced system (5).Note that the background solution (q 0 , r 0 ) agrees with the reductions used in the equations ( 9)- (12).Substituting (q, r) = (q 0 , r 0 ) into the matrix equations ( 7) and ( 8), we find the following solutions of wave functions, in which c and d are constants (or functions of λ).Define ]ocnmp[ Da-jun Zhang, Shi-min Liu and Xiao Deng Then, the quasi double Wronskians (21) composed by the above ϕ and ψ provide solutions to the unreduced system (5) via the transformation (13) where the background solutions take (55).With regards to (57), the matrix 18) and (19).One can also take to get multiple pole solutions corresponding to A = J 2m+2 (k 1 ), defined as in (47).
To describe the relations between c j and d j in a more succinct and symmetric way, we rewrite (56a) and (56b) as where we have introduced ĉ, d ∈ C and taken in ( 56) that Then, consider diagonal case where For the reverse-space-time NLS equation ( 12), we have where Φ is defined in (66a), k j , h j ∈ C, ĉ+ j and ĉ− j can be arbitrary complex functions of k j and h j , respectively.For the classical NLS equation ( 9) with δ = 1 and the reverse-space NLS equation (10) with δ = −1, we have where k j , h j ∈ iR for equation ( 9) with δ = 1 and k j , h j ∈ R for equation (10) with δ = −1. When we have where ϕ ± (1) are defined as in ( 68), ( 69), ( 70) or (71), varying with the equation considered.

Examples of dynamics of solutions
Nonzero background can bring new features for the classical and nonlocal NLS equations.
In this section we analyze some solutions and illustrate their dynamics.The classical NLS equation ( 1) and reverse-space nonlocal NLS equation ( 10) with δ = −1 will serve as main models.

The classical focusing NLS equation
It follows from the transformation (13) and the bilinear form ( 14) that the envelope |q| of the solution to the focusing NLS equation ( 1) with a background solution q 0 can be expressed as (also see [12,58]) where f = | ϕ m ; T ϕ * m | is the quasi double Wronskian.For the focusing NLS equation (1), A = Diag(K m+1 , −K * m+1 ), T is given by ( 61) with γ = −1, ϕ is given by ( 62) with (ϵ, α, β) = (1, 1, 1).In principle, solutions to equation ( 1) can be determined by the eigenvalue structure of K m+1 .One can investigate these solutions according to the canonical form of K m+1 .

Breathers
Case 1: K m+1 being a complex diagonal matrix When K m+1 is a diagonal matrix (46), following (62) we have ϕ = (ϕ + , ϕ − ) T where the entries in ϕ ± are Φ and Ψ are given as (56).When the background solution takes q 0 = e −2it , we have where we have taken c j = e −ξ (0) j , d j = e ξ (0) j with ξ (0) j being an arbitrary functions of k j .When m = 0 we have from (24b) that where Note that in this case we have which is positive definite when ϕ + (1) and ψ + (1) are defined as in (74).By calculation we find where and 1R , ξ 1I ∈ R. Since k 2 1 − 1 is a double-valued function of k, here we consider the branch without loss of generality.Further we introduce such that (77) is rewritten as Noticing that A 1 > |A 2 | > 0 for all k 1 ̸ = 0, from the above expression and (73), |q| 2 behaves like a wave traveling along the line X 1 = 0 and oscillating periodically with a period determined by 2X 2 + θ = 2jπ, j ∈ Z.Note that the case a 1 = 0 or a 1 = ±1, b 1 = 0 yields |q| 2 = 1, which is trivial and we do not consider.
To see more details, we rewrite (78) in terms of the following coordinates, which gives rise to ]ocnmp[ Da-jun Zhang, Shi-min Liu and Xiao Deng In terms of (79) we can see that (73) with (80) provides a breather traveling along the straight line z = constant and oscillating with a period with respect to x, An illustration is given in Fig. 1(a), which describes a moving breather.Such a breather is also known as the Tajiri-Watanabe breather (see Fig. 4 in [58]).In 1998 Tajiri and Watanabe derived and studied breathers of the focusing NLS equation using Hirota's bilinear method [58].Back to the expression (78).Stationary breathers appear when b 1 = 0.More precisely, when |a 1 | > 1 and b 1 = 0, which leads to u 11 = a 2  1 − 1 and u 12 = 0, we have B 1 = 0 and then X 1 (x, t) = u 11 x + ξ (0) 1R .In this case we can have a breather stationary with respect to x, where It follows that a stationary breather oscillating in time with period , which is known as the Kuznetsov-Ma breather [40,47].It is described in Fig. 1(b).In another case where |a 1 | < 1 and b 1 = 0, which leads to u 11 = 0 and u 12 = 1 − a 2 1 , from (78) we have 1 .This will gives rise to a breather traveling along the line t = − and being periodic with respect to x with the period . Such a breather is known as the Akhmediev breather [11], which was first studied by Akhmediev in [11] and then bear his name.Stability of the Akhmediev and Kuznetsov-Ma breathers was studied recently [28,31].The Akhmediev breather is perpendicular to the Kuznetsov-Ma breather, as depicted in Fig. 1 (b) and (c).
The envelope of two-breather solution can be obtained via (73) by taking m = 1 in quasi double Wronskians (24b), i.e.
in which ϕ + (j) and ψ + (j) are defined as in (74).There are various types of two-breather interactions.As examples Fig. 2 illustrates interactions between two Tajiri-Watanabe breathers, interaction of the Akhmediev breather and Kuznetsov-Ma breather and interaction of two Akhmediev breathers, in Fig. 2   Case 2: K m+1 being a Jordan matrix Let ϕ + (1) and ψ + (1) be defined as in (74), and we define The corresponding f composed by the above elements yields breathers when K m+1 is the Jordan matrix J m+1 (k 1 ) as given in (47).For the simplest Jordan block solution of the focusing NLS equation (1) with the background solution q 0 = e −2it , we have m = 1 and f composed by Such a breather is described in Fig. 3.

Rational solutions and rogue waves
Rational solutions can be obtained as a limit case of breathers when taking k j → 1.This can be seen from the expression (74).Since the Akhmediev breathers and Kuznetsov-Ma breathers are generated when b j = 0, rational solutions can also be understood as a limit of these two types of breathers.In principle, for getting rational solutions, in A we should take K m+1 = J m+1 (1), but the limit procedure needs to be elaborated.Let us consider (56) and rewrite them in the form ]ocnmp[ Da-jun Zhang, Shi-min Liu and Xiao Deng where s j are arbitrary complex parameters.We denote the above Φ(κ, c, d) and Ψ(κ, c, d) with (87) by Φ odd and Ψ odd respectively.Expand them in terms of κ as in which Define where T takes the form ( 61) with γ = −1.It can be verified that ϕ odd satisfies equation (25) where q 0 = e −2it , δ = −1, T is given by ( 61) with γ = −1, and A = Diag(K m+1 , −K * m+1 ) with Note also that A and T satisfy ( 26) with δ = −1.Thus, the quasi double Wronskians provide rational solutions to the focusing NLS equation ( 1) via ( 13) and the envelope via (73).
The first order rational solution (for m = 0) is where x = x + s 0 −2s 1 2s 0 with s 0 , s 1 being coefficients of c(k).Here we take s 0 , s 1 ∈ R for simplicity.We refer to it as the Peregrine soliton since it was first derived by Peregrine in [52].Its envelope |q| is localized in both space and time.It is also known as a rogue wave of the focusing NLS equation.The maximum value of |q| is 3, occurring at (x, t) = (− s 0 −2s 1 2s 0 , 0), three times hight of the background |q 0 | = 1.The envelope is depicted in Fig. 4(a).
The general second order rational solution can be obtained from where We skip explicit expression of q.The envelope of a typical second order rational solution is shown in Fig. 4 the maximum amplitude of a nth-order rogue wave with one central main peak is 2n + 1 times of the height of the amplitude of the background plane wave [9,62], (also see [62] where rogue wave with such pattern is called a "fundamental rogue wave").The envelope of another typical second order rational solution has three peaks, as shown in Fig. 4(c).The third order rational solution is obtained by taking m = 2 in (92).Without presenting formulae, we depict some different patterns of the envelope of these solutions in Fig. 5. Fig. 5 (a) shows the pattern where there is only one central main peak, Fig. 5 (d) and Fig. 5 (e) show the pattern consisting basically of 6 well-separated fundamental part on a unit background, which are located on a triangle and a pentagon, respectively.Another two interesting patterns are shown in Fig. 5 (b) and Fig. 5 (c).Thus, it indicates that higher-order rogue waves contain richer structures.Note that recently it was found the pattern of rogue waves is related to the roots of Yablonskii-Vorob'ev polynomials [64,66].
Apart from (90), one can also introduce Wronskian entries by imposing c(κ such that Φ(κ, c, d) and Ψ(κ, c, d) given in (86) (denoted by Φ evev and Ψ even , respectively) can be expanded as where Then the vectors for the quasi double Wronskian are taken as where T is given by ( 61) with γ = −1.In this case m = 0 does not lead to a nontrivial solution but the solutions obtained by taking m = 1 and m = 2 correspond to (93) and (94a), which are the first order and second order rational solutions derived using ϕ odd and ψ odd .One may conjecture that the m-th order rational solution derived using ϕ odd and ψ odd corresponds to the (m+1)-th order rational solution derived using ϕ even and ψ even .Similar connection is proved in the rational solutions of the discrete KdV-type equations, see [69].We also note that the parameters {s j } (or c(κ)) play the same roles as the lower triangular Toeplitz matrices, cf.[70,71].An (m + 1)-th order lower triangular Toeplitz matrix P m+1 is defined as which commutes with K m+1 defied in (91).For the block diagonal matrix Q = Diag(P m+1 , P * m+1 ), ]ocnmp[ Da-jun Zhang, Shi-min Liu and Xiao Deng when T is given by ( 61) with γ = −1, and A = Diag(K m+1 , −K * m+1 ) with (91), we have This indicates that for any ϕ that satisfies (25) with the above A and T , ϕ = Qϕ is also a solution of (25).Moreover, if ϕ odd in (90a) is derived with c(κ) = 1, then in ϕ = Qϕ, the parameters {t j } play the exactly same roles as {s j }.In [70], for the KdV equation, the relation between P m+1 and c(κ) is described, see Sec.2 of [70].

The defocusing reverse-space nonlocal NLS equation
In this section we investigate solutions of the defocusing reverse-space nonlocal NLS equation with the background solution q 0 = e 2it .This is the equation ( 10) with δ = 1.Note that the reverse-space nonlocal NLS equation ( 10) is considered as a model with parity-time symmetry (see [5]).Efforts of finding physical applications of NLS type nonlocal integrable systems can also be found in [7,46,65], etc.

Solitons and doubly periodic solutions
Solution to equation (100) with a background solution q 0 is written as where we take q 0 = e 2it .Consider the simplest case, m = 0. From the results in Table 1 and in Sec.4.4.2, we have where and in Φ(k, c, d) and Ψ(k, c, d) defined in (56) we have taken δ = 1, with α 1 and β 1 as arbitrary functions of k.
The envelope |q| of some solutions resulting from (102) is depicted in Fig. 6, which exhibits features of two-soliton interactions, although the solution is from the simplest case, m = 0.In the following we implement asymptotic analysis so as to understand ]ocnmp[ Da-jun Zhang, Shi-min Liu and Xiao Deng with characteristics trajectory : ; when bt → ±∞, the |q| 2 of X 2 -soliton asymptotically travels on a background |q| 2 = 1 and with characteristics top trace : .
Each soliton gets a phase shift −2 ln |b| due to interaction.
The value of amplitude of each soliton can be either larger or less than the background |q 0 | = 1.This indicates various types of interactions.Fig. 6 exhibits three types of interactions.It is also notable that the value of amplitude of each soliton can be even equal to the background |q 0 | = 1, which means the soliton can vanish on the background.This happens when β I = 0 for the X 1 -soliton and when (2b 2 − 1)β I − 2b √ 1 − b 2 β R = 0 for the X 2 -soliton.Illustrations are given in Fig. 7.Note that such a behavior usually appear in some coupled system and known as "ghost soliton", cf.[33].In this case the one-soliton solution of equation (100) resulting from (102) can be written as where ]ocnmp[ Da-jun Zhang, Shi-min Liu and Xiao Deng Solution (103) is doubly periodic with respect to both x and t and the periods are The solution is plotted in Fig. 8.Although there are some results on doubly periodic solutions, which are constructed by Jacobi elliptic functions in general, to our knowledge, the doubly periodic solution (103) to the defocusing reverse-space NLS equation (100) is not reported before.

Rational solutions
Similar to the classical case, rational solutions can be seen as a limit case of breathers when taking k j → i, but the limit procedure should also be elaborated.
The first order rational solution (for m = 0) is provided by with where we have taken c = s 0 + s 1 κ.Explicit form of the first order rational solution is given by where To understand the dynamics we investigate asymptotic behaviors of the above rational solution.We introduce a new coordinate (X 1 = x + 2t, t) , then rewrite (110) in this coordinate, keep X 1 to be constant and let t → ±∞.It follows that for convenience, we call it in terms of which we rewrite (110).Then keeping X 2 to be constant and letting t → ±∞, we find This implies that, when t → ±∞, there is a wave (X 2 -soliton for short) traveling on the background plane 2 and without phase shift due to interaction.The wave can be either above or below or vanishes in the background plane, depending on the sign of Im(s * 0 s 1 ).We summarize these asymptotic behaviors in the follow theorem.
Theorem 5. When t → ±∞, the envelope |q| 2 of X 1 -soliton asymptotically travels on a background |q 0 | 2 = 1 with characteristics trajectory : and the envelope |q| 2 of X 2 -soliton asymptotically travels on a background |q 0 | 2 = 1 with characteristics top trace : Asymptotically, no phase shift occurs for each soliton.
Various types of interactions are illustrated in Fig. 9, which coincide with the above results of asymptotic analysis.Note that, considering the signs of |s 0 | 2 + Im(s * 0 s 1 ) and −8Im(s * 0 s 1 ), it is impossible to have both waves below the background plane, neither one wave below the background plane and another vanishing.

The defocusing reverse-time nonlocal NLS equation
For the defocusing reverse-time NLS equation with nonzero background q 0 = e 2it , we can analyze solutions resulting from k 1 = ib, b ∈ R, as we have done in Sec.5.2 for the reverse-space nonlocal NLS equation (100).However, it turns out that the analysis procedure of these solutions and their dynamics are all similar to those in Sec.5.2 for equation (100).Let us explain the statement below.Rewrite (102b) and (102c) as where and for the reverse-time NLS equation (113), we have where T = 0 1 γ 0 with γ = ±1, the subscripts [x] and [t] stand for the reverse-space and reverse-time, respectively.It can be verified that, when k 1 = ib, b ∈ R (as we have taken in Sec.5.2), we have It then follows that and where B = 1 0 0 −1 .These relations indicate that |q| 2 resulting from the above ϕ and ψ for the reverse-time NLS equation (113) are similar to those of the reverse-space NLS equation (100).We skip presenting illustrations.

The defocusing reverse-space-time nonlocal NLS equation
For the solutions of the nonlocal defocusing reverse-space-time NLS equation (i.e.(12) with δ = 1), with the plane wave background q 0 = e 2it , the matrices A and T take the form (see Table 2) Solution to equation (120) with the background solution q 0 = e 2it is written as ] we take ĉ+ 1 = ĉ− 1 = 1 and denote The envelope of (125) behaves like breather which travels along the line x = − 2(a 1 v 12 +b 1 v 11 )t v 11 .An illustration is given in Fig. 11(a).Note that a 1 = 0 yields trivial solution and b 1 = 0 leads to the solution (124).We can also calculate two-soliton solution from (123) of this case, where we take h j = −k * j , j = 1, 2. Its envelope describes a head-on collision of two breathers, as shown in Fig. 11  Finally, H m+1 = K * m+1 with k i ∈ C, we claim that dynamics of solutions are similar to the reverse-space case.The explanation is similar to what we have done in Sec.5.
In the special case c 1 = 1, d 1 = i, ĉ1 = 1, the following relations hold: where C = 1 0 0 −i .Besides, the construction of f and g, the same A = k 1 0 0 k * 1 is used.This indicates that in the case H m+1 = K * m+1 , the analysis of dynamics of solutions for the revere-space-time NLS equation will be similar to those of the revere-space NLS equation that has been investigated in Sec.5.2.We skip it.

Conclusions
In this paper, by means of the bilinearisation-reduction approach, solutions for the classical and nonlocal NLS equations with nonzero backgrounds were constructed in a systematical way.Solutions are presented in terms of quasi double Wronskians.Asymptotic analysis and illustrations were provided to understand dynamics of solutions, in particular, breathers and rogue waves of the classical focusing NLS equation ( 9) and solitons and rational solutions of the reverse-space nonlocal NLS equation (10).One can see that the nonzero backgrounds bring more interesting behaviors in the dynamics of solutions.In addition, although the solutions are given in terms of quasi double Wronskians (not standard double Wronskians), the reduction technique is still effective.In light of Theorem 2 one can also use the double Wronskians given in Theorem 2 if q 0 is independent of x.This bilinearisation-reduction technique can also be extended to the other integrable equations with nonzero backgrounds, which will be investigated elsewhere.
is a solution.