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Multiple-Pole Solutions to a Semidiscrete Modified Korteweg-de Vries Equation.

1. Introduction

In the theory of inverse scatter transform, soliton solutions are closely related to the poles of the transmission coefficient (we called here the zero of the Riemann-Hilbert problem). In contrast with the study of the multisoliton solutions from simple poles, relatively few multiple-pole solitons are discussed, and mainly on continuous integrable equations [1-8]. It is known that multiple-pole solutions can be obtained directly by other methods, for example, bilinear method and Cauchy matrix approach; these solutions can also be done by coalescing several simple poles with a procedure via taking limits of the poles [9-12].

In this paper, we consider the multiple-pole soliton solutions of a semidiscrete modified Korteweg-de Vries (sdmKdV) equation [13-15]

i[u.sub.t] (n, t) = (l + [u.sup.2] (n, t)) [u (n + 1, t) - u(n - 1, t)], (1)

in virtue of the Riemann-Hilbert approach under a different symmetry condition.

As we know, for a complex function f(z), if it is analytic inside the circle [absolute value of z] < 1, then by using the Cauchy-Riemann equations [f.sup.*]([bar.z]) is analytic outside the circle [absolute value of z] > 1, where the star denotes the complex conjugate and [bar.z] = l/[z.sup.*]. It is worth mentioning that f([z.sup.-1]) is also analytic outside the circle [absolute value of z] > 1, if f(z) is analytic inside the circle [absolute value of z] < 1. With application of the latter fact, for the sdmKdV equation (1) with a modified linear spectral problem, we introduce a new symmetry condition as [f.sup.T]([z.sup.-1]) = [f.sup.-1](z) about the associated eigenfunction. The symmetry condition determines the distribution of the relevant eigenvalues of the discrete spectrum, which are deeply related to the soliton solutions of the integrable equations. In the paper, we derive some multiple-pole soliton solutions according to the new symmetry condition.

The Riemann-Hilbert (RH) problem plays an important role in the scattering theory. For integrable nonlinear equations, the nonregular RH problem will be concerned. After a procedure of regularization of the RH problem with zeros relating to the eigenvalues of discrete spectrum, the so-called soliton matrix will involve a set of relevant zero eigenvectors and is related to the potential functions. The soliton matrix for the case of canonical normalization condition of the RH problem, in general, satisfies the relevant linear spectral problem. Thus the zero eigenvectors can be obtained, and then the soliton matrix and the potential will be given. However, for the sdmKdV equation (1), the normalization condition of the nonregular RH problem is noncanonical, so the soliton matrix will not admit the associated linear spectral problem, which is different from that of the continuous case [7, 8]. In this paper, we choose a special solution of regular RH problem and use the solution of the RH problem (or the sectionally analytic function) to determine the zero eigenvectors.

The paper is organized as follows. In Section 2, we discuss the spectral analysis of the spectral problem of the sdmKdV equation under a new symmetry condition, and in Section 3, we use them to build a nonregular RH problem and then discuss its regularization. Finally, in Section 4, we establish the relationship between the potential and the soliton matrix and then derive the multiple-pole solitons of the sdmKdV equation. The figures of the simplest multiple-pole soliton solution show some novel properties.

2. Spectral Analysis of the Focusing Ablowitz-Ladik Equation

Equation (1) can be represented as the compatibility condition of the following linear system [16]:

J(n + 1, t) = [[gamma].sub.n] (I + [Q.sub.n])ZJ(n, t)[Z.sup.-1], (2)

[J.sub.t] (n, t) = -i[lambda] [[[sigma].sub.3], J (n, t)] -i[[??].sub.n]J (n, t), (3)

where z is a spectral parameter and

[mathematical expression not reproducible]. (4)

To investigate the analyticity of the associated eigenfunction, we introduce the Jost functions [J.sub.[+ or -]](n, z) with the following boundary conditions:

[J.sub.[+ or -]] (n, z) = I, n [right arrow] [infinity]. (5)

Since [J.sub.+](n, z) and [J.sub.-](n, z) are both solutions of the linear equation (2), they are not independent and are linearly related by the scattering matrix S(z) as

[mathematical expression not reproducible]. (6)

By the famous Cauchy- Riemann equations, we know that, for a complex function f(z), f([z.sup.-1]) is analytic in [absolute value of z] > 1, if f(z) is analytic in [absolute value of z] < 1. By the latter fact, we consider a different symmetry condition. In this case, the symmetry conditions can be obtained according to the discrete spectral problem (2) and the asymptotic condition (5) as

[mathematical expression not reproducible]. (7)

Note, in (2), det ([[gamma].sub.n](I + [Q.sub.n])) = 1, then det[J.sup.[+ or -]](x, z) is independent of n. As a result,

det [J.sub.[+ or -]] (n, z) = 1, det S(z) = 1, (8)

in terms of (5) and (6).

It is easy to see that the linear spectral problem (2) is a recurrence equation. Using the above properties of the Jost functions, it follows that the first column of [J.sub.-] (n, z), denoted by [J.sup.[1].sub.-](n, z), and the second column [J.sup.[1].sub.+] (n, z) of [J.sub.+](n, z) are analytic outside the unit circle (i.e., [absolute value of z] > 1) [14]. With these vectors in hand, we define a new matrix function [[PHI].sub.+] (n, z) by

[[PHI].sub.+] (n, z) = ([J.sup.[1].sub.-], [J.sup.[2].sub.+]), (9)

which is a solution of the spectral problem (2) and is analytic as a whole in the region [absolute value of z] > 1, while another sectionally analytic matrix function [[PHI].sup.-1.sub.-] (n, z) can be defined by virtue of the symmetry condition (7) as

[mathematical expression not reproducible], (10)

where [([J.sup.-1.sub.-]).sub.[1]] denotes the first row of the matrix function [J.sup.-1.sub.-] and [([J.sup.-1.sub.+]).sub.[2]] is the second row of [J.sup.-1.sub.+]. We note that the function [[PHI].sup.-1.sub.-](n, z) is a solution of the adjoint spectral problem of (2) and is analytic in the region [absolute value of z] < 1. It follows from (6) that

[[PHI].sub.+] (n, z) = [J.sub.[+ or -]] (n, z) [Z.sup.n][S.sub.[+ or -]] (z) [Z.sup.-n], (11)

where

[mathematical expression not reproducible]. (12)

Similarly, we have

[mathematical expression not reproducible]. (13)

Hence, (11)-(13) imply that

[mathematical expression not reproducible], (14)

in view of (8). It is worth noting that the symmetry condition about the Jost functions in (7) implies the symmetry condition about the sectionally analytic function:

[[PHI].sup.T.sub.+] (n, [z.sup.-]) = [[PHI].sup.-1.sub.-] (n, z). (15)

Next, we shall discuss the asymptotic behavior of the functions [[PHI].sub.+](n, z) and [[PHI].sup.-1.sub.-](n, z) in the z plane. Let the solution of (2) have the following asymptotic behaviors:

[mathematical expression not reproducible], (16)

and

[mathematical expression not reproducible]. (17)

Then, on use of the spectral problem (2) and its adjoint problem, the entries of the matrix [J.sup.(j)](n) and [J.sub.(j)] (n), j = 0, 1, satisfy the following equations:

[mathematical expression not reproducible], (18)

and [J.sup.(0).sub.12](n) = 0, [J.sub.(0)21](n) = 0,

[mathematical expression not reproducible]. (19)

We note that the Jost functions [J.sub.[+ or -]](n, z) have the same asymptotic behaviors as above.

Hence, asymptotic formulae for the sectionally analytic functions [[PHI].sub.+](n, z) and [[PHI].sup.-1.sub.-] (n, z) can be derived from (16) to (19) as

[mathematical expression not reproducible], (20)

and

[mathematical expression not reproducible], (21)

in view of the symmetry condition (7). Here the functions [v.sub.[+ or -]](n) are defined as

[mathematical expression not reproducible], (22)

and [[gamma].sub.l] is defined in (4).

It is noted that the potential u(n) can be constructed from (18) as

[mathematical expression not reproducible], (23)

where the expansion

[mathematical expression not reproducible] (24)

has been used.

3. Regularization for the Riemann-Hilbert Problem with Higher-Order Zeros

From the representations (11) and (13) of the sectionally analytic functions, we find that they satisfy the jump condition on the unit circle, which gives

[mathematical expression not reproducible]. (25)

To obtain a unique solution of the Riemann-Hilbert (RH) problem (25), a normalization condition should be given. Here we take the condition (20) as the normalization condition, where [mathematical expression not reproducible] is a constant. Note that this condition is noncanonical.

In order to derive the multiple-pole soliton solutions of the sdmKdV equation (1), we take G(z) = I, and suppose that det [[PHI].sub.+] (n, z) = [a.sub.+] (z) has one zero [z.sub.1] of order N, ([absolute value of [z.sub.1]] > 1). Then det [[PHI].sup.-1.sub.-] (n,z) = [a.sub.-] (z) has one zero [z.sup.-.sub.1] of order N, in terms of (7). It is noted that the zero [z.sub.1] of the RH problem (25) is an elementary higher-order zero for the 2x2 matrix [[PHI].sub.+] (n, z), and the RH problem can be solved by means of its regularization. Thus, we have

[[PHI].sub.[+ or -]] (n, z) = [[phi].sub.[+ or -]] (n, z) [GAMMA] (n, z), (26)

where the soliton matrix and its inverse are given as [7]

[mathematical expression not reproducible], (27)

[mathematical expression not reproducible], (28)

where

[mathematical expression not reproducible], (29)

and

[mathematical expression not reproducible]. (30)

Here vectors |[p.sub.j]>, <[[bar.p].sub.j], <[q.sub.j]|, <[[bar.q].sub.j]>, (j = 1 ..., N) are independent of z. We note that, in these vectors, only half of them (i.e., |[p.sub.j]> and <[[bar.p].sub.j]|, (j = 1 ..., N)) are independent. It is readily verified that these vectors satisfy the following equations:

[mathematical expression not reproducible], (31)

where the block Toeplitz matrix function T([GAMMA](z)) is defined as

[mathematical expression not reproducible]. (32)

For the sake of convenience, we introduce the following transformations:

[mathematical expression not reproducible], (33)

Then (27) and (28) can be rewritten as

[mathematical expression not reproducible], (34)

[mathematical expression not reproducible], (35)

where

[mathematical expression not reproducible], (36)

[mathematical expression not reproducible]. (37)

From (28) and (34), we find

[mathematical expression not reproducible], (38)

where [[GAMMA].sup.(1)] (n) is defined by the expansion

[GAMMA](n, z) = I + [1/z] [[GAMMA].sup.(1)] (n) + ..., z [right arrow] [infinity]. (39)

4. Multiple-Pole Solitons

In this section, we shall construct the soliton solution of the sdmKdV equation by means of the soliton matrix in (34). To this end, we find, from (24), (39), and (26), that

[mathematical expression not reproducible], (40)

The potential u(n) can be rewritten as

u(n) = -[[v.sub.-] (n)/[v.sub.+] (n)] [[GAMMA].sup.(1).sub.12], (41)

in view of (20). From (22), we find that [v.sub.+](n) and [v.sub.-](n) are still representation of the potential u(n). Hence, we need to establish the relationship between [v.sub.[+ or -]](n) and the soliton matrix [GAMMA]. Note that G(z) = I implies [[PHI].sub.+] = [[PHI].sub.-], from which we can consider the asymptotic behavior of [[PHI].sub.+] near z = 0. Indeed, the asymptotic formulae (20) and (21) imply that

[mathematical expression not reproducible]. (42)

Thus from (26) we obtain

[mathematical expression not reproducible], (43)

which implies that [v.sub.+](n)/[v.sub.-](n) = v[[GAMMA].sub.11] (n, z = 0). Here v = [v.sub.+](n)[v.sub.-](n). As a result, the potential u(n) takes the form

u(n) = -[1/v][[[GAMMA].sup.(1).sub.12] (n)/[[GAMMA].sup.11] (n, z = 0)]. (44)

On the other hand, G(z) = I implies [a.sub.+] (z)[a.sub.+]([z.sup.-]) = 1, and from the zeros of [a.sub.+] (z), we know that

[a.sub.+](z) = [[(z - [z.sub.1]).sup.N]/[(z[z.sub.1] - 1).sup.N]]. (45)

Because [a.sub.+](z) [right arrow] v, z [right arrow] [infinity], then we have

[1/v] = [z.sup.N.sub.1]. (46)

Now we need to determine the vectors |[p.sub.j]> and <[[bar.p].sub.j]| in (34). It is noted that the soliton matrix [GAMMA](z) does not satisfy the linear system (2) and (3), if the normalization condition of the RH problem is noncanonical. Because [[PHI].sub.+] (n, z) = [[PHI].sup.(0).sub.+](n)[GAMMA](n, z) and [[PHI].sup.(0).sub.+](n) is independent of the spectral variable z, we obtain

[mathematical expression not reproducible], (47)

in terms of (32). It is easy to verify that [PHI](z) admits

[mathematical expression not reproducible], (48)

where [U.sub.n] = diag[([[U.sub.n], ..., [U.sub.n]).sup.(2N)x(2N)], ([U.sub.n] = [[gamma].sub.n] (I + [Q.sub.n])), is a diagonal block matrix, and

[mathematical expression not reproducible]. (49)

Equations (47) and (48) imply

[mathematical expression not reproducible]. (50)

Thus, we take, for simplicity,

[mathematical expression not reproducible], (51)

or equivalently

[mathematical expression not reproducible]. (52)

Here |[p.sub.j][>.sub.0], j = 1, ..., N are some constant vectors, and

E([z.sub.1]) = [Z.sup.n] ([z.sub.1]) exp {-i[lambda] ([z.sub.1]) t[[sigma].sub.3]}. (53)

In addition, from the symmetry condition (15), we obtain

(<[[bar.p].sub.1]|, ..., <[[bar.p].sub.N]|) = ([|[p.sub.1]>.sup.T], ..., [|[P.sub.N]>.sup.T]). (54)

With the vectors |[p.sub.j]> and <[[bar.p].sub.j]| in hand, we shall construct the multiple-pole solitons of sdmKdV (1) by virtue of equations (44), (38), and (34). As an application, we consider the simplest multiple-pole soliton, that is, N = 2. In this case, (52) reduces to

[mathematical expression not reproducible]. (55)

Let

[mathematical expression not reproducible], (56)

and (55) can be rewritten as

[mathematical expression not reproducible], (57)

where

[mathematical expression not reproducible]. (58)

Thus,

[mathematical expression not reproducible], (59)

where the abbreviations f = f(n, t) and [f.sub.0] = f(n, 0) have been used. It is easy to find that

[mathematical expression not reproducible]. (60)

Hence, we have

[mathematical expression not reproducible]. (61)

Furthermore, we have

[mathematical expression not reproducible], (62)

[mathematical expression not reproducible], (63)

where

[mathematical expression not reproducible]. (64)

We now take [z.sub.1] = [e.sup.k], ([absolute value of k] > 0) and f = [e.sup.[theta]/2]; the multiple-pole soliton for N = 2 has the form

[mathematical expression not reproducible], (65)

in terms of v = [z.sup.-2.sub.1]. Here 0 and [[beta].sub.12] are defined by

[mathematical expression not reproducible], (66)

and [[theta].sub.0] and [[beta].sub.0] are taken to be real constants. We note that the parameter k is a complex number. In particular, if we take [mathematical expression not reproducible], then

[mathematical expression not reproducible], (67)

and

[mathematical expression not reproducible]. (68)

In this case, the multiple-pole soliton (65) has a different form

[mathematical expression not reproducible], (69)

where

[mathematical expression not reproducible]. (70)

From (69), one finds that if [mathematical expression not reproducible], solution (65) is a real function for odd n or an imaginary function for even n (see Figure 2) and reaches its peak near n = t = 0 for the parameters chosen as [k.sub.1] = 0.1, [[theta].sub.0] = [[beta].sub.0] = 0 (see Figure 1).

We note that the multiple-pole soliton (65) possesses the character of two-soliton type solution (see the densityplot of (65) in Figure 2). The collision of the "two-soliton" is illustrated in Figures 1 and 2. However, each "single-soliton" in Figure 2 shows the character of a breather. In addition, one "single-soliton" is a transverse wave, and another is a longitudinal wave. Figure 3 shows the local pictures of the collision.

5. Conclusion

In this paper, we have considered a focusing Ablowitz-Ladik equation associated with a modified linear spectral problem and its adjoint problem. By means of the dressing method, we have constructed the nonregular Riemann-Hilbert problem with one zero of order N, which is closely related to the spectrum of discrete eigenvalues. The multiple-pole soliton solutions have been obtained in virtue of the regularization of the Riemann-Hilbert problem. For application, we have presented the simplest multiple-pole soliton solution and studied the relevant dynamics. In view of the analysis, we clearly see that the adaptive method is efficient for solving the nonregular Riemann-Hilbert problem with more higher-order zeros.

https://doi.org/10.1155/2019/5468142

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science

Foundation of China (Projects 11471295).

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Zhixing Xiao, (1) Kang Li, (2) and Junyi Zhu [ID] (3)

(1) The High School, Huanghe S&T College, Zhengzhou, Henan 450006, China

(2) The 79th Middle School, Zhengzhou, Henan 450000, China

(3) School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, Henan 450001, China

Correspondence should be addressed to Junyi Zhu; jyzhu@zzu.edu.cn

Received 24 December 2018; Revised 14 February 2019; Accepted 5 March 2019; Published 2 May 2019

Academic Editor: Jorge E. Macias-Diaz

Caption: Figure 1: Solution (65) with [z.sub.1] = i[e.sup.0.1], [[theta].sub.0] = [[beta].sub.0] = 0.

Caption: Figure 2: Densityplot of solution (65) with [z.sub.1] = i[e.sup.0.1], [[theta].sub.0] = [[beta].sub.0] = 0.

Caption: Figure 3: Collision of solution (65) with [z.sub.1] = i[e.sup.0.1], [[theta].sub.0] = [[beta].sub.0] = 0.
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Title Annotation:Research Article
Author:Xiao, Zhixing; Li, Kang; Zhu, Junyi
Publication:Advances in Mathematical Physics
Date:May 31, 2019
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