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

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, andmainly 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] iut (n, t) = (1 + u2 (n, t)) [u (n + 1, t) − u (n − 1, t)] , (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][2][3][4][5][6][7][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][10][11][12].
As we know, for a complex function ( ), if it is analytic inside the circle | | < 1, then by using the Cauchy-Riemann equations * ( ) is analytic outside the circle | | > 1, where the star denotes the complex conjugate and = 1/ * . It is worth mentioning that ( −1 ) is also analytic outside the circle | | > 1, if ( ) is analytic inside the circle | | < 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 ( −1 ) = −1 ( ) 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 2 Advances in Mathematical Physics (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.

Spectral Analysis of the Focusing Ablowitz-Ladik Equation
Equation (1) can be represented as the compatibility condition of the following linear system [16]: where is a spectral parameter and To investigate the analyticity of the associated eigenfunction, we introduce the Jost functions ± ( , ) with the following boundary conditions: Since + ( , ) and − ( , ) are both solutions of the linear equation (2), they are not independent and are linearly related by the scattering matrix ( ) as ) .
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 − ( , ), denoted by [1] − ( , ), and the second column [2] + ( , ) of + ( , ) are analytic outside the unit circle (i.e., | | > 1) [14]. With these vectors in hand, we define a new matrix function Φ + ( , ) by which is a solution of the spectral problem (2) and is analytic as a whole in the region | | > 1, while another sectionally analytic matrix function Φ −1 − ( , ) can be defined by virtue of the symmetry condition (7) as where ( −1 − ) [1] denotes the first row of the matrix function −1 − and ( −1 + ) [2] is the second row of −1 + . We note that the function Φ −1 − ( , ) is a solution of the adjoint spectral problem of (2) and is analytic in the region | | < 1. It follows from (6) that where + ( ) = ( Similarly, we have Hence, (11)- (13) imply that Advances in Mathematical Physics 3 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: Next, we shall discuss the asymptotic behavior of the functions Φ + ( , ) and Φ −1 − ( , ) in the plane. Let the solution of (2) have the following asymptotic behaviors: and Then, on use of the spectral problem (2) and its adjoint problem, the entries of the matrix ( ) ( ) and ( ) ( ), = 0, 1, satisfy the following equations: We note that the Jost functions ± ( , ) have the same asymptotic behaviors as above.
Hence, asymptotic formulae for the sectionally analytic functions Φ + ( , ) and Φ −1 − ( , ) can be derived from (16) to (19) as and in view of the symmetry condition (7). Here the functions ] ± ( ) are defined as and is defined in (4).
It is noted that the potential ( ) can be constructed from (18) as where the expansion has been used.

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 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 det Φ (0) Note that this condition is noncanonical.
In order to derive the multiple-pole soliton solutions of the sdmKdV equation (1), we take ( ) = , and suppose that det Φ + ( , ) = + ( ) has one zero 1 of order , (| 1 | > 1). Then det Φ −1 − ( , ) = − ( ) has one zero − 1 of order , in terms of (7). It is noted that the zero 1 of the RH problem (25) is an elementary higher-order zero for the 2 × 2 matrix Φ + ( , ), and the RH problem can be solved by means of its regularization. Thus, we have where the soliton matrix and its inverse are given as [7] and ) .
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.

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 , 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 higherorder zeros.

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.