Abstract
The inverse scattering transformation for a generalized derivative nonlinear Schrödinger (GDNLS) equation is studied via the Riemann–Hilbert approach. In the direct scattering process, we perform the spectral analysis of the Lax pair associated with a matrix spectral problem for the GDNLS equation. Then, the corresponding Riemann–Hilbert problem is constructed. In the inverse scattering process, we obtain an N-soliton solution formula for the GDNLS equation by solving the Riemann–Hilbert problem with the reflection-less case. In addition, we express the N-soliton solution of the GDNLS equation as determinant expression.
1. Introduction
The nonlinear Schrödinger (NLS) equation,is an important integrable model in physics which was first derived by Zakharov in his study of modulation stability of deep waves [1]. Also, it makes sense to describe soliton propagation in nonlinear fiber optics, water waves, plasma physics, etc. So far, a number of research areas on the NLS equation have achieved remarkable results. For example, soliton solutions of the NLS equation have been obtained via the mathematical and physical methods, such as the inverse scattering transformation (IST) [2], the Bäcklund transformation (BT), and the Darboux transformation (DT). In addition, the dynamic properties of COVID-19 analyzed by mathematical and physical methods have also been widely concerned by scholars [3–7]. With the development of the Riemann–Hilbert (RH) problem, soliton solutions of the NLS equation can be derived from IST via a particular matrix RH problem. Researchers found that it was simpler and more effective than the original IST. On the one hand, scholars have used the RH approach to obtain the solutions to many integrable equations, such as the coupled derivative NLS (DNLS) equation [8], the coupled higher-order NLS equation [9], the generalized Camassa–Holm equation [10], the short pulse (SP) equation [11], and other integrable equations [12–25]. On the other hand, the RH method has been considered to be an effective way to work the initial-boundary value (IBV) problems of the integrable nonlinear evolution PDEs [26–29].
In 1987, Clarkson and Cosgrove [30] proposed a generalized derivative NLS (GDNLS) equation in the form ofwhere is the amplitude of the complex field envelope and denotes the complex conjugation, and are the real parameters and . Equation (2) has several applications in optical fibers [31–33], nonlinear optics [34, 35], weakly nonlinear dispersion water waves [36–38], quantum field theory, plasma physics, etc. However, it is well known that equation (2) has Painlevé property only if holds. At this time, equation (2) is reduced to an integrable GDNLS model as follows:
Given , equation (3) becomes the DNLS-I (Kaup–Newell) equation,and if , equation (3) becomes the DNLS-II (Chen–Lee–Liu) equation,
Recently, the conservation laws [39] and the IBV problems [40] of equation (3) on the half-line have been reported. In this paper, based on the IST idea, we use the RH method to solve the initial value problem of the GDNLS equation, and then, the scattering data from the scattering problem is obtained, and the N-soliton solution of the potential function is constructed.
This paper is organized as follows. In Section 2, based on the Lax pair of the GDNLS equation, we perform spectral analysis, and construct the RH problem in terms of the Jost solutions and scattering data . In Section 3, we apply the Plemelj formula to solve the regular and the nonregular RH problem, respectively. In the inverse scattering process, we reconstruct the potential and derive the determinant expression of the N-soliton solution for the GDNLS equation. Furthermore, we display explicitly one-soliton solution in the section. Finally, we summarise the content of the paper.
2. The Construction of Riemann–Hilbert Problem
In this section, we perform direct scattering transformation for the GDNLS equation. According to Hu et al. [40], the Lax pair of equation (2) iswherewithwhere is a matrix function of the complex isospectral parameter, is the spectral parameter, is named potential function, and denotes complex conjugation.
It will be convenient to express aswhere is introduced as a new matrix spectral function. Inserting equation (9) into equations (6a) and (6b), the spectral problem equations (6a) and (6b) becomeswhere is the commutator.
In the following direct scattering process, we first treat equation (10a) as a spatial scattering problem at a fixed time . Now, we concentrate on Jost solutions for the part of the Lax pair equation (10a) with the following asymptotic property at large distance:
Applying Abel’s formula, and using the boundary conditions equations (11a) and (11b), we see thatfor all .
Based on Volterra integral equations, can be uniquely cast aswhere represents a matrix operator acting on matrix by
It is important to analyze the analytic properties of the Jost solutions , thus we partition into columns, namely, . Now, we assume (in fact, result is the same thing if ), then, it is easy to find that are continuous for and analytic for ; are continuous for and analytic ; here,and .
In order to formulate a Riemann–Hilbert problem for the GDNLS equation, we construct a matrix function which is analytic for :where .
In addition, we see that the large- asymptotic of is
We can check that the inverse matrices satisfy this adjoint equation. On the other hand, we obtain an analytic counterpart of in by the adjoint scattering equation of equation (11a):
If we express as a collection of rows
Then, by applying the same techniques as above, we can show thatwhere are continuous for and analytic for ; meanwhile, and are continuous for and analytic for . Also, the large- asymptotic behavior of is
In addition, defining , we can get that and are both solutions of the linear equations (6a) and (6b) and are linearly related by a scattering matrix .
In view of equation (12) and equation (21), we see that satisfies
Furthermore, from the properties of and , has the following analyticity structures:where the superscripts refer to which the scattering elements are analytic in. Elements without superscripts indicate that such elements do not allow analytical extensions to in general.
Denoting the limit of taken from the left-hand side of as , and the limit of from the right-hand side of as , respectively, we arrive at a Riemann–Hilbert problem for the GDNLS equation.where the jump matrix is as follows:and the canonical normalization condition for this Riemann–Hilbert problem is
To finish the direct scattering transformation, we determine the time evolution of the scattering matrix . By inserting into equation (10b), taking the limit , and recalling the boundary condition equation (11a) for as well as the fact that , as , we obtain
Equation (28) leads to
Equation (29a) shows that and are time independent, and the time evolution for the scattering data and are given by
3. Solution for the Riemann–Hilbert Problem
In this section, we discuss how to solve a regular Riemann–Hilbert problem and a nonregular Riemann–Hilbert problem. The Riemann–Hilbert problem (25) constructed in above section is regular when and in the respective planes of analyticity and is nonregular when and at certain discrete locations of .
3.1. Solution for the Regular Riemann–Hilbert Problem
Recalling the definitions of and , we see that
If in their analytic domain, we can solve this Riemann–Hilbert problem by Plemelj formula. Under the canonical normalization equation (17) and equation (21), the solution to the regular Riemann–Hilbert problem is provided by the following integral equation:where
Moreover, we can prove that the solution to the regular Riemann–Hilbert problem under the boundary condition is unique. This can be easily proved as follows. Suppose (25) has two sets of solutions and , then, , thus
Since is analytic in and is analytic in . On the , they are equal to each other. Thus, they together define a matrix function which is analytic in the whole plane. Due to the boundary condition equation (17) and equation (21), this analytic matrix function is equal to the unit matrix everywhere by Liouville theorem, i.e.,
Therefore, , which means that the solution to the regular Riemann–Hilbert problem (25) is unique.
3.2. Solution for the Nonregular Riemann–Hilbert Problem
In the more general case, the Riemann–Hilbert problem (25) is not regular and we can transform a nonregular Riemann–Hilbert problem into a regular one. In view of equation (6b), we can see that zeros of and are determined by and , respectively.
To specify these zeros, we easily find that there is a symmetry relation for in equation (10a):where the superscript means the Hermitian conjugation. According to equation (18), we get
Then, it follows that
From , we also verify that satisfies the involution property:
Obviously, equation (39) leads to
In addition, there is one more symmetry relation,
Owing to equation (10a), it is easy to see that
Applying equations (41) and (42) to equation (46), we can get
Equation (43) implies that
Furthermore, we deduce a property for and :
The solution to the Riemann–Hilbert problem is associated to the zeros of det and det as well as the kernel structures of and at the zeros. So we assume all zeros are simple, and see that if is a zero of det , then is a zero of det , and is also a zero of det . Firstly, we suppose that det has zeros , which satisfy , thus det possesses zeros which satisfy . In this case, each ker and ker contains only a single column vector and row vector , respectively
Now, we derive the column vectors and the row . Using equations (38) and (46), we deduce that
On the other hand, we get the spatial evolutions of the vectors . By taking the -derivative to the first equations (46) and (10a), we get
Similarly, we have
By solving equations (48) and (49) explicitly, we getwhere and is a nonzero complex constant vector. Therefore, according to the above relationship between each pair of and , the column vector and row vector are determined as follows:and
Based on above results, we reduce the nonregular Riemann–Hilbert problem to the regular Riemann–Hilbert problem by the following theorem.
Theorem 1. The solution to the nonregular Riemann–Hilbert problem (25) with zeros equation (46) under the canonical normalization condition equations (17) and (21) iswherewhere is an matrix with its the element given byand is the unique solution to the following regular Riemann–Hilbert problem.where are analytic in and as .
This theorem was obtained by Zakharov and Shabat in 1979 (see [41]).
In order to obtain N-soliton solutions for GDNLS, we choose the jump matrix , i.e., . In this case, the corresponding scattering problem is called reflection-less. This solution for this particular the Riemann–Hilbert problem is written out explicitly as follows.where is a matrix whose entries are as follows:
4. Inverse Scattering Transform
Now we know that the Riemann–Hilbert problem (25) can be solved from the given scattering data by applying the formulas (58a) and (58b). In this section, we reconstruct the potential by means of the inverse scattering transform of the GDNLS equation. Indeed, due to that is solution of the scattering problem (10a), we expand at large asand substitute it into equation (10a). By comparing terms, we can get
Then, the solution can be obtained aswhere and is the -entry of . Here, the matrix function to the Riemann–Hilbert problem which is reflection-less case, one can be derived in equation (58a):
The corresponding solution is
5. N-Soliton Solutions
In what follows, we substitute equations (52) and (53) into equation (64), then, a N-soliton solution formula for the GDNLS equation is given bywhere with
Recalling that the determinant of a matrix can be expressed as the sum of its elements along a row or column multiplying their corresponding cofactor, we rewrite the solution (66) aswhereand and . When , the single-soliton solution iswith
By choosing different parameters, we can obtain some plots which are displayed to particularly describe the dynamic behaviors for single-soliton solution equation (69). Figure 1 illustrates that the single-soliton solution is a M-type soliton.
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6. Conclusions
In this paper, we started with the spectral analysis of the Lax pair of the GDNLS equation and formulated a Riemann–Hilbert problem on the , which admits simple zero points generated by the roots of . By taking spectral analysis, we found that is an odd function. So, the zero points are paired. According to the symmetry relations of zero points, a transformation was established, which eliminates the zero points and makes the Riemann–Hilbert problem be regular. By solving particular Riemann–Hilbert problem with vanishing scattering coefficients, we successfully constructed as a simple representation for the N-soliton solution for the GDNLS equation. Moreover we reduced the N-soliton solution for the GDNLS equation to a ratio of determinant and determinant.
Data Availability
No data were used to support this study.
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 under the grants 12147115 and 11975145, Natural Science Foundation of Anhui Province under the grant 2108085QA09, Scientific Research Startup Fund Project of Chuzhou University under grant 2022qd038 and University Natural Science Research Project of Anhui Province under the grant KJ2021A1094.