Abstract
The (2+1)-dimensional nonlocal breaking solitons AKNS hierarchy and the nonlocal negative order AKNS hierarchy are presented. Solutions in double Wronskian form of these two hierarchies are derived by means of a reduction technique from those of the unreduced hierarchies. The advantage of our method is that we start from the known solutions of the unreduced bilinear equations, and obtain solitons and multiple-pole solutions for the variety of classical and nonlocal reductions. Dynamical behaviors of some obtained solutions are illustrated. It is remarkable that for some real nonlocal equations, amplitudes of solutions are related to the independent variables that are reversed in the real nonlocal reductions.
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1. Introduction
Recently, -symmetric integrable nonlocal nonlinear Schrödinger equation
was proposed by Ablowitz and Musslimani [1], which is reduced from the second-order Ablowitz–Kaup–Newell–Segur (AKNS) system with reduction . It is -symmetric because of the potential (see [2, 3]). Since then, more and more integrable nonlocal systems, such as nonlocal Davey–Stewartson model, nonlocal Korteweg–de Vries model and so forth, were proposed (e.g. [4–9]). Classical solving methods, such as the inverse scattering transform and Darboux transformation (e.g. [1, 5, 10–14]), have been successfully used to find solutions of nonlocal integrable systems. However, the bilinear method cannot be directly applied to the nonlocal case as it is difficult to define a 'nonlocal' Hirota's bilinear operator. Recently, Chen et al [15–18] developed a reduction technique of double Wronskians on unreduced bilinear equations to obtain solutions to those reduced equations, including nonlocal ones. One advantage of this method is that one only needs to make use of the solutions (most of them are known) of the unreduced bilinear equations. The other advantage is one can obtain N-soliton solutions for the whole hierarchy, rather than implementing reductions solution by solution and equation by equation (see [19]).
The (2+1)-dimensional breaking solitons equations ware first systematically investigated by Bogoyavlenskii [20–26]. Such models can be used to describe two-dimensional interaction of a Riemann wave with transverse long waves (see [20, 21]). Mathematically, these equations can be constructed based on (1+1)-D Lax pairs by imposing an evolution in y-direction in both spectral parameters and eigenfunctions (see [21, 24]). The (2+1)-D breaking AKNS(BAKNS in brief) equations were first proposed in [24], and has been investigated from many aspects (e.g. [27–31]). And recently, bilinear form and double Wronskian solutions of (2+1)-D BAKNS hierarchy were worked out [32]. In this paper, we will take into account of bilinearization-reduction technique and derive double Wronskian solutions for the classical and nonlocal (2+1)-D BAKNS hierarchy and the negative order AKNS hierarchy. Note that the negative order AKNS equations also exhibits interesting dynamical behaviors (see [33]).
This paper is organized as follows. In section 2 we recall the unreduced (2+1)-D BAKNS hierarchy, its bilinear form and double Wronskian solutions. In section 3 we present classical and nonlocal reductions of the (2+1)-D BAKNS hierarchy. In section 4, solutions in double Wronskian form of the reduced (2+1)-D BAKNS hierarchies are derived by means of the reduction technique from those of the unreduced hierarchy. Section 5 is contributed to the negative order AKNS hierarchy. Finally, section 6 consists of concluding remarks.
2. Unreduced (2+1)-D BAKNS hierarchy and solutions
Let us recall the unreduced (2+1)-D BAKNS hierarchy, its bilinear form and double Wronskian solutions.
The unreduced (2+1)-D BAKNS hierarchy reads [27]
where , L is a recursion operator
in which and . This hierarchy is related to the AKNS spectral problem [24]
and
where and
The hierarchy (2) can be rewritten as
with n = 1, 2, 3, ⋯ and setting t0 = y. Introducing transformation
the hierarchy is written as the following bilinear form [32]
for n ≥ 1, where Hirota's bilinear operator D is defined as [34]
Theorem 1. System (7) admits double Wronskian solutions [32]
Here, and are respectively th order column vectors
defined by
where is a constant matrix in and
and respectively denote and Wronski matrices
It is easy to find that for the odd members in the unreduced (2+1)-D BAKNS hierarchy (2)
their solutions are given through (6) and (8) where
and for the even members
their solutions are given through (6) and (8) where
3. Reductions of the (2+1)-D BAKNS hierarchy
3.1. Classical reductions
The odd hierarchy (13) allows a classical complex reduction
where i is the imaginary unit and ∗ stands for complex conjugate. And the representative one is
The even hierarchy (15) allows classical complex reduction
when l = 1, the corresponding equation is
3.2. Nonlocal reductions
For the odd hierarchy (13), first, it allows a real nonlocal reduction
where the simplest one-component equation reads
Equation (13) also allows a complex nonlocal reduction
and the resulted simplest one-component equation is
The even hierarchy (15) also allows two types of reductions. One is real
and the resulted simplest one-component equation is
where
The other reduction is complex
and the resulted simplest one-component equation is
where is given as (27).
It is remarkable that the integration operator should specially take the form
which has played an important role in nonlocal reduction involved with x (see [16, 18]). For example, for the integration term in (5) with n = 1, one can handle it as the following. First, introduce
Then we have
which provides a clear expression for the integration term with nonlocal reduction.
4. Reduction of solutions
4.1. Nonlocal cases
In the following we implement the reduction procedure on the double Wronskian solutions, by which solutions for the reduced nonlocal hierarchies can be obtained from those of the unreduced (2+1)-D BAKNS hierarchy (2) presented in theorem 1.
4.1.1. Reduction (21)
Theorem 2. The nonlocal hierarchy
allows the following solution
where , (i.e. m = n in (9)), defined by (14) and satisfy
in which is a constant matrix determined through
Proof. It follows from (14), (33b) and (34a) that
which gives rise to the the constraint (33a). Thus, f, g, h in (8) (with m = n) are expressed as
where we employed a notation (see [15])
Next, with assumption (34b), we have
and in a similar way
which then give rise to
This is nothing but the reduction (21). Thus we complete the proof.□
Solutions of (34) can be written out by assuming T and A to be block matrices
where Ti and Ki are (n + 1) × (n + 1) matrices and Ki is a complex matrix. We list them in table 1.
One can prove that A and any of its similar forms lead to same solution (32). We only need to consider canonical form of A. When
we get
When is a Jordan matrix
we get
where
As examples, for equation (22) with different (σ, δ), its one-soliton solution (1SS) are
where we have taken .
Next, let us quickly investigate dynamics of which is governed by equation
Its 1SS (42a) is rewritten as
which is a moving wave with an initial phase and a y-dependent amplitude . The top trajectory is , and travel velocity is 2k2. Fixing y, the trajectory of one-solton in coordinate frame {x, t} is depicted as figure 1(a).
The 2SS of equation (43) can be written as
where
and we note that 2SS is non-singular if . As shown in figures 1(b)–(d), there are three types of two-soliton interactions, respectively are soliton-soliton interaction, soliton-anti-soliton interaction and breather soliton. To investigate asymptotic behavior of the first two types solution interactions, we make asymptotic analysis for t going to infinity in the case of . To do that, we first rewrite the 2SS (45) in the following coordinate
fix X1, let , and we get
It shows that along the line X1 = constant, there is a soliton; when , the soliton asymptotically follows
and phase shift after interaction is .
We can also rewrite the 2SS (45) in the coordinate . By fixing X2, we can obtain
It indicates that along the line X2= constant, there is a soliton; when , the soliton asymptotically follows
and the phase shift after interaction is .
Supposing , from the asymptotic analysis, we can obtain that if , it will appear soliton-soliton interaction, such as figure 1(b); if , it will appear soliton-anti-soliton interaction, such as figure 1(c).
We list out solutions for other cases of nonlocal reductions without giving proof.
4.1.2. Reduction (23)
Theorem 3. For the reduction (23), the reduced nonlocal hierarchy
allow a solution
where and are th order column vectors (i.e. in (9)), defined by (14) and satisfy
in which is a constant matrix determined through
When T and A are block matrices (36), solutions to equations (54) are given in table 2.
Table 2. T and A for (54).
T | A | |
---|---|---|
(1, −1) | ||
(1, 1) | ||
(−1, −1) | ||
(−1, 1) |
When is a diagonal matrix (37), the vector φ in (52) can be given as
where
When (39), we have
For the equation (24) with different (σ, δ), its 1SS are
where .
4.1.3. Reduction (25)
Theorem 4. For the reduction (25), the reduced nonlocal hierarchy
allow a solution
where , , defined by (16) and satisfy
in which is a constant matrix satisfying (34).
If T and A are block matrices (36), solutions to (34) haven been given by table 1. In the case that is a diagonal matrix (37), the vector φ in (60) can be given as
where
When (39), we have
Table 1. T and A for (34).
(σ, δ) | T | A |
---|---|---|
(1, −1) | ||
(1, 1) | ||
(−1, −1) | ||
(−1, 1) |
For the equation (26) with different (σ, δ), its 1SS are
where .
As an example we consider the dynamics of , which is governed by equation
where . Its 1SS (65a) can be rewritten as
Depicted as figure 2(a), this is a stationary wave with an initial phase and an amplitude that exponentially increases with time t and y, and the top trajectory is . It is interesting that the amplitude is y and t-dependent, which looks like in nonisospectral case (see [35]) where amplitudes are changed due to time-dependent eigenvalues (spectral parameters). However, here the amplitude changes might be caused by the reversed y and t in the nonlocal reduction, rather than nonspectral parameters as the eigenvalues (see A in table 1) are still constant.
Download figure:
Standard image High-resolution imageThe 2SS of equation (66) can be written as
where
4.1.4. Reduction (28)
Theorem 5. For the reduction (28), the reduced nonlocal hierarchy
allow a solution
where and are defined by (9) with and (16), satisfying
in which is a constant matrix satisfied (54).
When T and A are block matrices (36), solutions to (54) are already given by table 2. In the case that is a diagonal matrix (37), the vector φ in (5) can be written as
where
When (39), we get
For equation (29) with different (σ, δ), its 1SS are
where .
4.2. Classical cases
Theorem 6. For the classical reduction (17), the reduced local hierarchy
allow a solution
where and are defined by (9) with and (14), satisfying
in which is a constant matrix determined through
When T and A are block matrices (36), solutions to equations (79) are given in table 2 with σ = 1. When is a diagonal matrix (37), the vector φ in (77) can be given as
where
When (39), we have
In addition, we present the 1SS of the equation (18)
where .
Theorem 7. For the reduction (19), the reduced local hierarchy
allow a solution
where , expressed as (16) and satisfy the condition (78), in which is a constant matrix satisfied (79).
When T and A are block matrices (36), solutions to (79) are already given by table 2. In the case that is a diagonal matrix (37), the vector φ in (85) can be written as
where
When (39), we get
For equation (20), its 1SS with different δ are given as
where .
5. Negative order AKNS hierarchy
In this section, we list double Wronskian solutions for the negative order AKNS hierarchy and consider several cases of reductions.
5.1. Solutions
For the negative order AKNS hierarchy [33]
where L is defined by (3). The hierarchy can be rewritten as [36]
By the transformation
its bilinear form turns out to be [36]
which allow double Wronskian solution [32]
where φ and ψ are defined as
and are th-order constant column vector.
5.2. Reductions and solutions
The odd members in the hierarchy (91) admit a real reduction
under which we have
and the representative one
In addition, the odd members in the hierarchy (91) also admit a complex reduction
under which we have
and
as the first member.
For those even members in the hierarchy (91), first, they allow a real reduction
which generate
the simplest one is
where . And second, a complex reduction
leads to
in which the representative one is
where .
In the following we list out double Wronskian solutions and we skip proof.
Theorem 8. The system (97) admits the following solutions
where , , are defined by
and satisfy
in which is a constant matrix determined by (34).
For equation (98) with different (σ, δ), its 1SS are
where .
Theorem 9. The system (100) admits solutions
where , , are defined by (109) and satisfy
in which is a constant matrix determined by (54).
1SS of the equation (101) with different (σ, δ) are
where .
Theorem 10. The hierarchy (103) admit
where , are defined by
and satisfy
in which is a constant matrix determined by (34).
For equation (104), its 1SS with different (σ, δ) are
where .
As an example, let us illustrate dynamics of which is governed by the equation
where , (118a) describes a single soliton
with an initial phase , top trajectory , and an amplitude that decreases with time, see figure 3(a).
Download figure:
Standard image High-resolution imageThe 2SS of equation (119) is
where
which we depict in figure 3(b).
Theorem 11. The hierarchy (106) admit solutions
where φ and ψ are th order column vectors (i.e. in (9)), are defined by (116) and satisfy
in which is a constant matrix determined by (54).
The equation (107) with different (σ, δ) has the following 1SS:
where .
6. Conclusions
We have discussed possible nonlocal reductions of the (2+1)-D BAKNS hierarchy and the negative order AKNS hierarchy. By means of the reduction technique (see [15–18]), we derived N-soliton solutions and multiple-pole solutions in double Wronskian form of the reduced hierarchies from those of the unreduced hierarchies. Note that these hierarchies contain integration w.r.t. x, the integration operator with form of (30) is necessary and helpful in considering nonlocal reductions. In addition, this reduction method is based on bilinear forms and double Wronskians, As we can see, it allows us to obtain N-soliton solutions and multiple-pole solutions for the whole reduced hierarchy. Compared with [14, 16, 19] where only some single equations in the (2+1)-D BAKNS hierarchy and the negative order AKNS hierarchy were considered, here we solved the two hierarchies and obtained more solutions. Finally, it is remarkable that for some real nonlocal equations, amplitudes of solutions are related to the independent variables that are reversed in the real nonlocal reductions, e.g. (44), (67) and (120).
Acknowledgments
This work was supported by the NSF of China [grant numbers 11 875 040, 11 631 007, 11 571 225].