Abstract
In this paper, the non-local reverse space−time fifth-order non-linear Schrödinger(NLS) equation has been investigated, which is proposed by the non-local reduction of Ablowitz–Kaup–Newell–Segur (AKNS) scattering problems. The determinant representation of the Nth Darboux transformation for the non-local reverse space−time fifth-order NLS equation is obtained. Some interesting non-linear wave solutions, including soliton, complexiton, and rogue wave solutions, are derived by the Darboux transformation. Moreover, the dynamics of non-linear wave solutions are illustrated with the corresponding evolution plots, and the results show that the non-local fifth-order NLS equation has new different properties from the local case.
1 Introduction
Integrable systems play an important role in non-linear science fields such as non-linear optics [1, 2], ocean physics [3], Bose–Einstein condensates [4], and even financial markets [5]. The investigation of various physically meaningful non-linear wave solutions is still one of the active areas of research in the field of integrable systems. In the past decades, many powerful methods and techniques have been proposed to construct various non-linear wave solutions and to study their underlying dynamics, such as Darboux transformation [6, 7], inverse scattering [8, 9], bilinear transformation [10], and Riemann–Hilbert approaches [11, 12]. Recently, Ablowitz and Musslimani proposed a new integrable non-local non-linear Schrödinger (NLS) equation under a reduction of the Ablowitz–Kaup–Newell–Segur (AKNS) system, and some non-linear wave solutions are constructed by the inverse scattering method [13]. Subsequently, much more non-local integrable systems including reverse space–time and reverse time cases are further investigated [14]. At the same time, the physical background of non-local integrable equations is also investigated from various related fields, such as multi-place systems [15], magnetic structures [16], nanomagnetic artificial materials [17], and loop quantum cosmology [18] [19, 20].
The NLS equation [21] is a fundamental prototype and plays a pivotal role in many fields of physics, such as fluid mechanics [22], plasmas [23], Bose–Einstein condensates [24], and deep water waves [25]. However, the NLS equation only contains the lowest-order dispersion term and the lowest-order non-linear effect. Under the necessary physical conditions, various higher-order dispersions and non-linear effects must be taken into account, such as ultrashort pulses in optical fibers [26], where the effects of higher-order dispersions should be considered. Therefore, some higher-order NLS equations, including Hirota [27], Lakshmanan–Porsezian–Daniel (LPD) [21, 28], and quintic NLS equations [29], have been constructed, and their corresponding integrable properties and dynamics have been studied.
In this paper, we consider the scattering problem as follows:where , λ is the spectral parameter, and U, V0, L, M, and N are given bywhereUnder the symmetry reduction q (x, t) = r*(x, t), the generalized integrable fifth-order NLS equation [30],wherecan be obtained from the compatibility condition of the linear spectral problem (1), i.e., the zero-curvature equation, . However, a new integrable reverse space–time non-local fifth-order NLS equation,can be obtained under the symmetry reduction,
Considering the importance of such non-local equations in multi-place physical systems [15], it is significant and has far-reaching importance in constructing exact solutions to the equations and aids in studying the dynamical properties of the solutions. To the best of our knowledge, such reverse space–time non-local equations have not been investigated. This paper is organized as follows: in Section 2, the one-fold and N-fold Darboux transformation of Eq. 5 are presented; in Section 3, soliton, complexiton, and rogue wave solutions are derived through the Darboux transformation and their corresponding dynamical properties and evolutions are discussed; and in Section 4, some conclusions and discussions are drawn.
2 Darboux transformation for the reverse space–time non-local fifth-order NLS equation
The Darboux transformation method is a very effective tool for constructing exact solutions of integrable non-linear equations in the soliton theory. In order to derive the Darboux transformation of the reverse space–time non-local fifth-order NLS in Eq. 5, we first introduce a gauge transformation of the linear spectral problem (1),under which the linear spectral problem (1) can be deformed as follows:The next pivotal step is to construct the Darboux matrix T[1] in such a form that in Equation 8 have the same form as U and V in (1) and the old potentials r and q are replaced by the new potentials . Supposewhere are functions of x and t. Substituting Eq. 9 into Eq. 8, it is evident that the relationships between two potentials in the two linear spectral problems (1, 8) can be given asIn addition, combined with symmetry reduction (6), there isWe see that and are two eigenfunctions corresponding to the eigenvalue λ = λj (j = 1, 2). From the gauge transformation, there exist constants such that
where
Then, the gauge transformation can be given as follows:where can satisfy
along with
By tedious calculations and using the identities (15), it can be verified that have the same forms as U and V under the symmetry reduction (6). To construct the N-fold Darboux transformation of Eq. 5, a more generalized higher-order gauge transformation can be given as follows:where
from which the following relationships can be obtained:
Combined with symmetry reduction (6), there is
Similar to the case of one-fold Darboux transformation, we construct the following equations:
withFrom algebraic Eq. 21, the determinant representation of the N-fold Darboux matrix TN can be derived by Cramer’s rule, from which the determinant representations of and can be given as follows:Here,This is the N-fold Darboux transformation of Eq. 5. Moreover, the existence of the symmetry reduction condition Eq. 6 implies that the Darboux transformation of the non-local reverse space−time fifth-order NLS Eq. 5 is very different from the Darboux transformation of the classical fifth-order NLS Eq. 4, although both of them have the same form.
3 Non-linear wave solutions of the reverse space–time non-local fifth-order NLS equation
3.1 One-soliton solutions from zero seed solution
To construct a soliton solution of the non-local Eq. 5, we take a zero seed solution, under which the corresponding eigenfunctions of the linear spectral problem (1) can be given as follows:
Then, the following relationships can be obtained:
under which the conditions for symmetry reduction (6) can be obtained as follows:Without the loss of generality, we take γ1 = −1 and γ2 = 1 and , and the solution can be simplified as follows:
where
The soliton solution can be obtained as follows:
This is under the condition that k1 = k2 and w1 = −w2. Evidently, the propagation direction of a soliton (34) is determined by the value of . In Figure 1A, the evolution of a soliton solution (34) is illustrated, and the corresponding evolution profiles of the real and imaginary parts are shown in Figures 1B, C, which exhibit the characteristics of a breather. On the other hand, by taking k2 = −2k1 and w2 = 0, the complexiton solution can be given as follows:where
FIGURE 1
It can be seen from Figure 2A that the solution (35) propagates to the left along the x-axis under the condition that . Figures 2B, C show the evolution characteristics of the real and imaginary parts of the solution (35). The propagation states of the solution (35) at three different times are shown in Figure 2D.
FIGURE 2
3.2 Two-soliton solutions from zero seed solution
Two-fold exact solutions of Eq. 5 can be derived from the Darboux transformation (23) by taking N = 2. In order to satisfy the constraint condition (20), we take γ1 = −1, γ2 = 1, γ3 = −1, and γ4 = 1 and consider the case that the eigenvalues are two pairs of conjugate complexes, i.e., . Then, the solution can be obtained aswhereand ξjR and ξjI are defined by (33) previously. In Figure 3A, the two-soliton solution behaves as an interaction of two bright solitons; after that, they stably propagate with original shapes and velocities. The corresponding evolutions of real and imaginary parts of the solution are shown in Figures 3B, C, which are all two-order breather solutions.
FIGURE 3
3.3 One-soliton solutions from non-zero seed solution
In order to construct the rogue wave solution of Eq. 5 by the Darboux transformation (10), the seed solution is taken as the plane wave solution,where a and c are an arbitrary constant and , respectively. Substituting the seed solution (39) into the linear spectral problem (1) with or , the eigenfunctions can be obtained as follows:orwhereand C1, C2, C3, and C4 are arbitrary constants. For simplicity, taking and and considering their relationship (13), we have the following:To satisfy the constraint condition (11), we take γ1 = 1 and γ2 = −1. Then, the rogue wave solution can be given as follows:whereThe density evolution and 2D contour plots for the rogue wave solution (45) under appropriate parameters are shown in Figures 4A, B, and the typical amplitude |r[1]| profile with t = 0 is illustrated in Figure 4C.
FIGURE 4
4 Conclusion
In this paper, the reverse space–time non-local fifth-order NLS Eq. 5 is studied by Darboux transformations. Based on the scattering problem, the N-fold Darboux transformation of the equation is constructed. By selecting different seed solutions, we have presented soliton, complexiton, and rogue wave solutions of Eq. 5, whose non-linear dynamics and evolutions are discussed. However, the computational effort increases rapidly due to the increase of the order of the Darboux transformation and the presence of the symmetric reduction condition; so, more interesting and physically meaningful non-linear wave solutions are difficult to be derived, such as breather and higher-order rogue wave solutions. At the same time, whether the equation has other integrable properties, such as Bäcklund transformations, Hamilton structures, and infinite conservation laws, will be studied in the near future.
Statements
Data availability statement
The raw data supporting the conclusion of this article will be made available by the authors, without undue reservation.
Author contributions
XS: conceptualization, formal analysis, investigation, methodology, and writing—original draft. YY: conceptualization, methodology, and writing—review and editing. All authors contributed to the article and approved the submitted version.
Funding
This work is supported by the Zhoushan Science and Technology Program (No. 2021C21023).
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
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Summary
Keywords
non-local fifth-order non-linear Schrödinger equation, Darboux transformation, soliton, rogue wave, integrable system
Citation
Shi X and Yang Y (2023) Exact solutions and Darboux transformation for the reverse space–time non-local fifth-order non-linear Schrödinger equation. Front. Phys. 11:1179961. doi: 10.3389/fphy.2023.1179961
Received
05 March 2023
Accepted
16 March 2023
Published
31 March 2023
Volume
11 - 2023
Edited by
Xiangpeng Xin, Liaocheng University, China
Reviewed by
Junchao Chen, Lishui University, China
Xiaoen Zhang, South China University of Technology, China
Updates
Copyright
© 2023 Shi and Yang.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Yunqing Yang, yqyang@amss.ac.cn
This article was submitted to Mathematical Physics, a section of the journal Frontiers in Physics
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.