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- Open Access
Solutions induced from bright solitons for nucleon–meson model
- Qihong Shi^{1}Email authorView ORCID ID profile,
- Suqin Chang^{1},
- Xiaobing Zhang^{1} and
- Jianxiong Cao^{1}
https://doi.org/10.1186/s13662-018-1831-4
© The Author(s) 2018
- Received: 10 May 2018
- Accepted: 2 October 2018
- Published: 10 October 2018
Abstract
In this work, the classical \((1+1)\)-dimensional Klein–Gordon–Schrödinger (KGS) system is studied. The ansatz and the homogeneous balance principle are employed in searching for particular soliton solutions, such as bright and dark solitons. Regrettably, dark solitons cannot be captured. However, this procedure leads to a series of new singular solitons and explicit periodic wave solutions.
Keywords
- KGS system
- Regular solitons
- Singular solitons
- Periodic solutions
1 Introduction
For KGS system, Li–Yang–Wang [14] and Wang–Zhou [12] investigated the periodic wave solutions by using F-expansion method, and Wang–Xia [18] applied Exp-function method to get some generalized solitary wave solutions. Based on a series hypothesis, Darwish [13] has recently obtained multiple exact explicit solutions of the KGS system (1.1)–(1.2). However, to the best of our knowledge, there are few report on the existence of some particular physical waves, such as bright and dark solitons [19]. The main reason is that these methods always omit the original properties of this system.
Our original purpose in this paper is to find a particular type solitary wave solution. For this, we assume that the solution \((u,n)\) for the system (1.1)–(1.2) has the form \(u=f(\tau )^{p_{1}}\exp\{i\phi\}\) and \(n=g(\tau)^{p_{2}}\), respectively, where both \(f(\tau)\) and \(g(\tau)\) are the basis wave functions, \(p_{1}\) and \(p_{2}\) will be determined later by balancing the coefficients. In this paper we explicitly obtain the bright 1-solitons and the bell-shaped regular solitons with the appropriate choice of f and g. Moreover, taking into account the superposition of wave components, we also get the interacting wave \((u,n)\) having the form \(u=[f_{1}(\tau)^{p_{1}}+f_{2}(\tau )^{p_{1}}]\exp\{i\phi\}\) and \(n=g_{1}(\tau)^{p_{2}}+g_{2}(\tau)^{p_{2}}\), respectively. It is regretful that the usual dark soliton cannot be derived, because it strongly depends on the coefficients for Yukawa interaction terms. But fortunately, it may induce a large class of periodic wave solutions that would not be provided by the exposed standard methods.
2 Mathematical analysis
3 Solitary wave
In this section we consider the regular solitary wave for the KGS system (1.1)–(1.2) and present some particular solutions which play important roles in the physical traveling wave theory.
Case 1
Remark 3.1
Case 2
Remark 3.2
This solution has a strong space location requirement. Indeed, any tiny upper and lower translation is ineffective, that is, if one sets \(u(x,t)=(A_{1}\operatorname{sech}^{p_{1}}\tau +C_{1})\exp\{i\phi\}\) and \(n(x,t)=A_{2}\operatorname{sech}^{p_{2}}\tau+C_{2}\) with real constants \(C_{1}\) and \(C_{2}\), then one can deduce that \(C_{1}=C_{2}=0\) with \(p_{1}=1\) or 2 and \(p_{2}=2\).
Remark 3.3
We will also attempt to find a dark soliton of the form \(u=A_{1}\tanh(\tau)\exp\{i\phi\}\) and \(n=A_{2}\tanh ^{2}(\tau)\). It is regretful that the homogeneous balancing technique cannot produce a nontrivial solution because it strongly depends on the sign of the coefficients for Yukawa interaction terms. However, we can deduce a solution with \(\tanh(\tilde{B}(x-vt))+\tilde{C}\) where B̃ is a pure imaginary number. Noting that \(\tanh(ix)=i\tan x\), we will then have the solution in Sect. 4.
4 Periodic wave
The search for the dark solitons shows the existence of a solution with the form \(\tanh(i\tilde{B}(x-vt))+\tilde{C}\) (see Remark 3.3), which inspires us to exert fresh efforts.
5 Conclusion
In this paper, the ansatz method is applied to the \((1+1)\)-dimensional KGS system for some particular solitary waves, and a series of explicit exact solutions are constructed. Most of them cannot be derived by the existing standard procedures, particularly, the bright solitons and singular wave that play essential role in plasma and fluid physics. More importantly, their amplitudes are not only strongly dependent on the wave speed, but also related to the mass of the meson. As for the existence of dark solitons of another form, we will make further discussion in the forthcoming work.
Declarations
Acknowledgements
The authors wish to thank the referees for their valuable comments.
Funding
This work is supported by the Development Program of Hongliu First-Class Disciplines in Lanzhou University of Technology.
Authors’ contributions
QS conceived of the study and drafted the manuscript. SC carried out the theoretical computation. XZ participated in the design and coordination. JC carried out the modifications and participated in its final design. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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