Bi-Dbar-Approach for a Coupled Shifted Nonlocal Dispersionless System

We propose a Bi-Dbar approach and apply it to the extended coupled shifted nonlocal dispersionless system. We introduce the nonlocal reduction to solve the coupled shifted nonlocal dispersionless system. Since no enough constraint conditions can be found to curb the norming contants in the Dbar data, the “solutions” obtained by the Dbar dressing method, in general, do not admit the coupled shifted nonlocal dispersionless system. In the Bi-Dbar approach to the extended coupled shifted nonlocal dispersionless system, the norming constants are free. The constraint conditions on the norming constants are determined by the general nonlocal reduction, and the solutions of the coupled shifted nonlocal dispersionless system are derived.


Introduction
The coupled integrable dispersionless system can be used to describe the marginally unstable baroclinic wave packets in geophysical fluids and ultra-short pulses in nonlinear optics. The nonlinear coupled dispersionless system is related to the short pulse equation, the Pohlmeyer-Lund-Regge equation and to the Sine-Gordon equation if the potential function q is real [1][2][3]. For the complex case, the coupled dispersionless system [1,4] is closely connected to the coupled AB system [5][6][7][8][9][10][11]. The coupled dispersionless systems have been studied by many methods [4,10,12] with interesting explicit solutions, including amplitude growing solitons, decaying solitons, stationary solitons, and loop soliton solutions [3,13].
(Dbar)-problem is a very effective tool to study nonlinear evolution equations and to give their explicit solutions [14][15][16][17][18][19][20][21][22][23][24][25][26]. The Dbar approach to the NLS equation with nonzero boundary condition was discussed in [21,27]. The Dbar-problem to investigate the coupled nonlocal NLS equation is first considered in our recent work [28]. In this paper, we apply the Dbar-approach to study the following extended complex coupled shifted nonlocal dispersionless (ECCSND) system where w(x, t) = 1 − −1 x q(x, t)q(x 0 − x, t 0 − t) t . So, the ECCSND system can be regarded as a sub-critical case in [5][6][7][8]. It is noted that, for the ECCSND system (1), if {q(x, t),q(x, t)} is a set of solution, so is {q( Equation (1) reduces to the following complex coupled shifted nonlocal dispersionless (CCSND) system if q(x, t) =q(x, t) and w(x 0 − x, t 0 − t) = w(x, t) . If x 0 = t 0 = 0 , Eq. (2) can be further reduced to the complex coupled nonlocal dispersionless (CCND) system [29][30][31]. In fact, the CCND system (or reverse space-time nonlocal version) and the CCSND one are equivalent on certain translation. However, the shifted items play some roles on the norming constants which may let to some special properties which were shown in this paper. We note that the latter is able to be cast to the real reverse space-time nonlocal sine-Gordon ( = −1) and sinh-Gordon ( = 1) [29], if both q and w are real functions.
In review of the Riemann-Hilbert problem to nonlocal integrable equations, discrete spectrum and norming constants play an important role to construct their solutions. The left and right scattering problems are considered to obtain the potential reconstruction, symmetry conditions are introduced to determine the constraint conditions about the discrete spectrum, and the trace formula are used to find the constraints on the norming constants. Then solutions of the nonlocal integrable equations are obtained [32].
The Dbar approach is mainly discussed in the spectral space, so only the constraint conditions of the discrete spectrum can be found. It is difficult to find the relation between the Dbar problem and the trace formula, and no enough constraint conditions on the norming constrants can be found. As a result, the obtained the formal solutions do not admit the nonlocal integable equations. (1)

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Certain remedies are needed to find the constraint conditions on the norming constants from the obtained formal solutions. We propose a Bi-Dbar problem to consider the extended complex coupled shifted nonlocal dispersionless (ECCSND) system (1). The advantage to solve the extended system is that it only needs the constraint conditions about the discrete spectrum. Hence, solutions of the ECCSND system can be obtained as same as the usual dressing method. The general discrete spectrum and the norming constants imply the solutions, and the special Dbar data give special solution. With the explicit solutions of the ECCSND system (1) in hand, we introduce the general nonlocal reduction to determine the constraint conditions on the norming constants and to construct the solutions of the CCSND system (2).
The (shifted) nonlocal integrable equations reveal the models that the associated field at certain physical point is also determined by other relevant points. For the (shifted) nonlocal equations, it is important to reveal the inner link between these points, which are difficult to find from the results of the classical integrable equations. We know that the field or the solution to a nonlinear equation is recovered from the spectral data which need carefully investigated by the inverse spectral transform. These information can not be caught from the simple transformation between the ECCSND equation and the classical coupled dispersionless system. That is why the NLS equation has been well studied, while carefully discussions for nonlocal NLS was considered. That is also the motivation to carry out our work.
Organization of this paper is as follows. In Sect. 2, we introduce two local Dbar problems (called Bi-Dbar problems), and derive two Lax pairs of the ECCSND equation with different potential matrices. In Sect. 3, we derive the focusing( = −1 )/defocusing( = 1 ) ECCSND system and its conservation laws. In Sect. 4, we present the explicit solutions for the focusing ECCSND system. In Sect. 5, we discuss the nonlocal reductions to the CCSND system in detail. In last section, some conclusions and discussions are given.

Bi-Dbar-Problem and Dressing Method
Consider the first local Dbar-problem with the normalization condition where R(k) is the spectral transform matrix. The Dbar-problem (3) and (4) equivalent to the following integral equation where the Cauchy-Green operation in complex plane is defined as The aim of dressing method is construct the relation between the ECCSND potential and the solution of the Bi-Dbar-problem. To this end, a good way is to construct two different Lax pairs of the ECCSND system from each Dbar-problem. It is noted that the Dbar-problem is defined in the spectral space, while the ECCSND system is in the physical space. Thus we need to introduce the physical variables x, t into the function (k) , which can be done by extending the spectral transform matrix to be the form R(k; x, t), and letting where (k) = −i (k) . We note that the solution of the system (7) and (8) is not unique.
Under the dressing procedure [20,23,24], we find that and where It is noted that [24] which reduces to . From the definition of Q in (9) and the properties of the Dbar problem, we find [24]

Thus, we obtain
Since U 2 = I , then V 2 = 1 , which implies a normalization condition. We note that Eqs. (14) and (15) imply the classical coupled dispersionless system [33]. We know that, for the Dbar approach to nonlinear integrable equations, it is important to consider the associated symmetry conditions which determine the properties of the scattering data. For the ECCSND system, we need to consider the second local Dbar problem where the new spectral transform matrix R (k;x, t) is another different solution of the evolution system (7) and (8). It is noted that the local Dbar problem is determined by the spectral transform matrix, and different spectral transform matrix defines different Dbar poblem. Thus two local Dbar problems (3) and (16) Journal of Nonlinear Mathematical Physics (2023) 30:578-599 In addition, to get the ECCSND system, one also needs to introduce a symmetry condition about the two potentials then the symmetry condition of the eigenfunction takes the following form Thus, we have Under these symmetry conditions, Eqs. (13), (14) and (21) give the ECCSND system (1).

The Conservation Laws
From (4) and (9), we know that (k;x, t) has the following asymptotic behaviors where a 0 (x, t) = I , and with a [d] and a [o] are the diagonal and off-diagonal part of the matrix a n , respectively. Here In particular, (28) implies that Let the trace of the spectral transform matrix R(k; x, t) is zero, then the Dbar problem (3) implies ̄d et (k;x, t) = 0, and further det (k;x, t) = 1 in view of the asymptotic behaviors (27). Since −1 = 2 T 2 , the temporal linear spectral problem (10) can be rewritten as Substituting the expansions (27) into (30), and taking O(k −n ), (n = 0, 1, 2, ⋯) terms, we obtain and The diagonal part of (31) implies which gives the third equation in (1). This equation can be regarded as the first conservation law.
The O(k −1 ) terms of the expansion of (30), or n = 1 in (32), takes the form which can be rewritten as the off-diagonal part and the diagonal part Substituting (29) into (35), we obtain which gives the second conservation law. Here and after, we take the integral constants of a [d] n to be zero. Similarly, the diagonal part (36) reduces to which gives the first two equations in (1) in terms of (33). We note that the diagonal parts of other equations in (36) will also give the ECCSND system in (1), and the off-diagonal parts will reduce to more conservation laws. For example, the off-diagonal parts of (36) for n = 2 , that is which implies the third conservation law where = −1 x Q 2 and a 2 is given in (29).

The Solutions of the ECCND System
In this section, we give the explicit solutions of ECCSND system in the focussing case = −1.
According to the above symmetry conditions, we let the first spectral transform matrix R(k; x, t) has the following form and take the second spectral transform matrix be of the form where and for any function f ((⋅); From Eqs. (41) and (9), as well as (42) and (18), we get the reconstruction of the potentials about the Dbar data and the eigenfunctions. Here, the norming constants c j and d j are free. In addition, we can show that r(x, t) = −q(x 0 − x, t 0 − t) and r(x, t) = −q(x 0 − x, t 0 − t) , in terms of the symmetry condition (26)

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The closed form expressions for the explicit solution for the ECCSND system (1)  are where Here the symmetry matrices Ω and Ω are given by and with Λ is the Ñ × N Cauchy type matrix with Λ lj = 1∕( l +k j ) . We note that g j (x, t) = g j (−x, −t),h l (x, t) = h l (−x, −t) . It is noted that, for some scattering data S = {{c j , k j } N j=1 ;{d l , l }Ñ l=1 } , the symmetry matrices Ω and Ω may degenerate in x-t plane 2 at certain point sets denoted by P S and P S . In general, q(x, t) and q(x, t) have some singularities at the points in P S and P S , respectively. It is important to express the solution of the ECCSND system in the form (44), (45). Because it makes us possible and easier to give the general nonlocal reduction of the ECCSND system, which will be discussed in the next section.
It is remarked that w(x, t) is given by (33), which involves the differential with respect to t and integral about x. So, we may find another way to give the expression of w(x, t). In fact, from (10) and (15), we get w = ( 12 21 + 11 22 )| k=0 . By virtue of det = 1 obtained from the Dbar problem with zero trace R, we find the expression of w(x, t) as For N =Ñ = 1 , we find where g j , h j and ĝ j ,ĥ j are defined in (48). In particular, if k 1 = ib 1 , 1 = i 1 and d 1 = |d 1 |e 2i 1 , c 1 = |c 1 |e 2i 1 , then (51) reduces to the following forms and where If 1 − 1 ≠ n , n ∈ ℤ , then −1 ≤ cos 2( 1 − 1 )) < 1 . Solutions q and q in (52) with b 1 1 < 0 show amplitude changing solitons, while w in (53) reduces to a one-soliton, (bright/dark one-soliton depends on the value of 1 and 1 , if other parameters b 1 , 1 , |c 1 |, |d 1 | are fixed). Since (50) then q and q are growing solitons for | 1 | > |b 1 | and decaying soliton for | 1 | < |b 1 | . While if 1 = −b 1 , then q and q are one-solitons, and on different lines. If 1 − 1 = n , the solutions q, w and q in (52) with b 1 1 < 0 have singularities on the parallel lines l j , j = 1, 2 , respectively, where with We note that, if b 1 1 > 0 , q and q will be unstable, but w is still stable and will give a one-soliton or w-shaped wave, which depends on the choice of 1 and 1 .
Similarly, for N =Ñ = 2 , we obtain the solutions q and q in (44) with and In addition, for the solution w in (49), we get and Figure 1 shows the interaction of decaying two-soliton q(x, t) (left),q(x, t) (center) and w(x, t) (right) in (55)-(58) and (44).
As an example, we discuss the asymptotic behavior of w. For convenience, we choose j = i j and k j = ib j and b 2 < b 1 < 0 < 1 < 2 associated with the parameters in Fig. 1. In this case, M = detΩ detΩ and M b = detΩ b detΩ b take the following forms (57) (59) Fig. 1 The interaction of decaying two-soliton q(x, t),q(x, t) and two-soliton of w(x, t) in (55) and (44) where X l = j x − l t + l0 , (l = 1, 2, 3) and  We note that the wave train along the direction x = 1 − 3 1 − 3 t may has singularities at X 1 − X 3 + ± 2 = 0 , and the amplitude is changed after interaction, see Fig. 1(right). From the above figures, we find that the two wave trains of q(x, t) and q(x, t) locate on different lines which are determined by the Dbar data { l , d l } and {k j , c j } . So we can carefully choose certain Dbar data to ensure q(x, t) =q(x, t).

Nonlocal Reduction to the CCSND System
In general, q(x, t) and q(x, t) given in (44) are not equal to each other. From (48), we find that q(x, t) =q(x, t) implies the following conditions k j = −k j , l = −̄l and N =Ñ , as well as |c j |, |d l | dependenting on {k j , l } as well as x 0 , t 0 . To find the constraint conditions on |c j | and |d l | for q(x, t) =q(x, t) , we take k j = ib j , l = i l , j, l = 1, 2, … , N . It is remarked that Since the matrices Ω and Ω in (45) are symmetry matrix and the matrix ) is a Cauchy matrix. Using the Cauchy-Binet formula, we give the determinants in (44) as the following form [23,24] and − 2 = ln l, l � ∈ {j 1 , j 2 , ⋯ , j }, s, s � ∈ {r 2 , ⋯ , r }, m, m � ∈ {r 1 , r 2 , ⋯ , r }, n ∈ {1, 2, ⋯ , N}�{j 1 , ⋯ , j }, where ∑ 12 denotes the summation for indices If the determinants in (44) admit which means that detΩ a det Ω + det Ω a detΩ = 0 , then q(x, t) =q(x, t) , and the ECCND system (1) reduces to the CCSND system (2). In addtion, from (64), (65) and (66), the constraint conditions about |c j |, |d j | on j , b j and x 0 , t 0 can be derived. For N =Ñ = 1 , we have, from (64), (65), detΩ a = det Ω a = −1 and which can also be obtained from (51). Then detΩ = − det Ω implies that in terms of (54). In this case, we take c 1 = |c 1 |e 2i 1 , d 1 = |d 1 |e 2i 1 and , then the solution of CCSND equation (2) takes the form of where In addition, which admits w(x, t) = w(x 0 − x, t 0 − t) in terms of It will give a amplitude-changing single-soliton in the case of 1 b 1 < 0 , see Fig. 2 (left). While if 1 b 1 > 0 , q will be unstable and not the amplitude-changing single-soliton, but w can still give a soltion, see Fig. 2 (center and right). Furthermore, if b 1 = − 1 , (68) reduces to a single-soliton with wave train on the line It can be found that the arguments of c 1 and d 1 play an important role on the wave height, for example, the wave height of |W| is When x 0 = t 0 = 0 , the present solutions can reduce to the solutions given in [31]. In particular, if 1 − 1 = n , n ∈ ℕ , then we have which have singularities on the straight line For N =Ñ = 2 , using (64), (65) or equivalent (55) and (56), we find, from detΩ a = − det Ω a , that and detΩ = det Ω implies that (72) |c 1 c 2 |ẽ1 (x 0 ,t 0 )+̃2(x 0 ,t 0 ) = |d 1 d 2 |e 1 (x 0 ,t 0 )+ 2 (x 0 ,t 0 ) , For convenience, we let c j = |c j |e 2i j , d j = |d j |e 2i j . Then we find that solution of CCSND equation (2) is where and Here and the constants p j satisfy the following relations (73) The interaction of solution (74) with 2 − 1 = b 2 − b 1 is shown in Fig. 3, and 2 − 1 = b 1 − b 2 is shown in Fig. 4. In the two Figures, q(x, t) is decaying two-soliton, while Fig. 3 (right) shows the interaction of bright two-soliton of w(x, t) in the case of (79) In particular, if we take b 1 = − 2 , b 2 = − 1 and 1 = 2 + ∕2, 2 = 1 + ∕2 , then (74) reduces to two-soliton, see Fig. 5 (left). It is interesting that the interaction of the two-soliton q is inelastic.
We have discussed the reduction conditions for the ECCSND equation (1) to the CCSND equation (2) in the case of N =Ñ = 1, 2 , and have given the explicit solutions of the CCSND equation (2). It is remarked that the general reduction conditions for the ECCSND equation (1) to the CCSND equation (2) can also be obtained [28] and

Conclusions and Discussions
A special linear differential system about the spectral transform matrix was introduced to define the Bi-Dbar-problem. The potentials of the ECCSND equation were recovered from the Dbar data, that is, finding two Lax pairs of the ECCSND equation with different potential matrices, by the dressing method. The conservation laws and the ECCSND equation were given. The coupled dispersionless system can be reduced from the AB system The other explicit solutions can be obtained similarly from those of the CCSND system.