Abstract
Multiple-pole soliton solutions to a semidiscrete modified Korteweg-de Vries equation are derived by virtue of the Riemann-Hilbert problem with higher-order zeros. A different symmetry condition is introduced to build the nonregular Riemann-Hilbert problem. The simplest multiple-pole soliton solution is presented. The dynamics of the solitons are studied.
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]in virtue of the Riemann-Hilbert approach under a different symmetry condition.
As we know, for a complex function , if it is analytic inside the circle , then by using the Cauchy-Riemann equations is analytic outside the circle , where the star denotes the complex conjugate and . It is worth mentioning that is also analytic outside the circle , if is analytic inside the circle . With application of the latter fact, for the sdmKdV equation (1) with a modified linear spectral problem, we introduce a new symmetry condition as 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]: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
By the famous Cauchy-Riemann equations, we know that, for a complex function , is analytic in , if is analytic in . 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) asNote, in (2), , then is independent of . As a result,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 , denoted by , and the second column of are analytic outside the unit circle (i.e., ) [14]. With these vectors in hand, we define a new matrix function bywhich is a solution of the spectral problem (2) and is analytic as a whole in the region , while another sectionally analytic matrix function can be defined by virtue of the symmetry condition (7) aswhere denotes the first row of the matrix function and is the second row of . We note that the function is a solution of the adjoint spectral problem of (2) and is analytic in the region . It follows from (6) thatwhereSimilarly, we haveHence, (11)-(13) imply thatin 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 and in the plane. Let the solution of (2) have the following asymptotic behaviors:andThen, on use of the spectral problem (2) and its adjoint problem, the entries of the matrix and , satisfy the following equations:and ,We note that the Jost functions have the same asymptotic behaviors as above.
Hence, asymptotic formulae for the sectionally analytic functions and can be derived from (16) to (19) asandin view of the symmetry condition (7). Here the functions are defined asand is defined in (4).
It is noted that the potential can be constructed from (18) aswhere the expansionhas 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 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 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 , and suppose that has one zero of order , . Then has one zero of order , in terms of (7). It is noted that the zero of the RH problem (25) is an elementary higher-order zero for the matrix , and the RH problem can be solved by means of its regularization. Thus, we havewhere the soliton matrix and its inverse are given as [7]whereandHere vectors are independent of . We note that, in these vectors, only half of them (i.e., and ) are independent. It is readily verified that these vectors satisfy the following equations:where the block Toeplitz matrix function is defined asFor the sake of convenience, we introduce the following transformations:Then (27) and (28) can be rewritten aswhere
where is defined by the expansion
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), thatThe potential can be rewritten asin view of (20). From (22), we find that and are still representation of the potential . Hence, we need to establish the relationship between and the soliton matrix . Note that implies , from which we can consider the asymptotic behavior of near . Indeed, the asymptotic formulae (20) and (21) imply that Thus from (26) we obtainwhich implies that . Here . As a result, the potential takes the formOn the other hand, implies , and from the zeros of , we know thatBecause , then we have
Now we need to determine the vectors and in (34). It is noted that the soliton matrix does not satisfy the linear system (2) and (3), if the normalization condition of the RH problem is noncanonical. Because and is independent of the spectral variable , we obtainin terms of (32). It is easy to verify that admitswhere , is a diagonal block matrix, and
Equations (47) and (48) imply Thus, we take, for simplicity,or equivalently
Here are some constant vectors, andIn addition, from the symmetry condition (15), we obtain
With the vectors and 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, . In this case, (52) reduces toLet and (55) can be rewritten aswhere Thus,where the abbreviations and have been used. It is easy to find that Hence, we haveFurthermore, we havewhere
We now take and ; the multiple-pole soliton for has the formin terms of . Here and are defined by and and are taken to be real constants. We note that the parameter is a complex number. In particular, if we take (i.e., ), then andIn this case, the multiple-pole soliton (65) has a different formwhere
From (69), one finds that if , solution (65) is a real function for odd or an imaginary function for even (see Figure 2) and reaches its peak near for the parameters chosen as (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 , 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.
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).