Abstract
The Landau-Ginzburg-Higgs (LGH) equation explains the ocean engineering models, superconductivity and drift cyclotron waves in radially inhomogeneous plasma for coherent ion-cyclotron waves. In this paper, with a simple modification of the Ablowitz-Kaup-Newell-Segur (AKNS) formalism, the integrability of LGH equation is proved by deriving the Lax pair. Hence for that, the inverse scattering transformation (IST) is applied, and the travelling wave solutions are obtained and graphically represented in 2d and 3d profiles.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Avoid common mistakes on your manuscript.
1 Introduction
Nonlinear evolution equations (NLEE’s) are one of the most important fields of ocean engineering models, modern physics, through which physical phenomena and other fields of modern physics are modeled. Accordingly, many effective methods have been established to obtain exact and explicit solutions of this type of equation, such as: the improved Bernoulli sub-equation function method [1, 2], the modified simple equation method [3], solitary wave ansatz [4], the extended tanh-function method [5, 6], the sine–Gordon expansion method [7], the extended mapping method [8], the first integral method [9], the improved Kudryashov method [10, 11], the hyperbola function method [12], the improved tanh-method [13], Hirota's bilinear method [14], Lie symmetry analysis [15, 16], Bäcklund transformation method [17], Darboux transformation method [18], inverse scattering transformation (IST) method [19,20,21,22], the \(\left( {G^{\prime } /G} \right)\)-expansion method [23,24,25], the \(\left( {G^{\prime } /G,\;1/G} \right)\)-expansion method [26, 27], and reference therein.
One of the more general classes of NLEE’s with a nonlinear term of any order is [28]:
where \({a}_{1},{a}_{2},{a}_{3},{a}_{4},\) and \(p\ne 1\) are arbitrary constants. When \(p=3\) and \({a}_{4}=0\), the following special case is stated
Depending on the arbitrary constants \(a,b\) and \(c\), typical forms of Eq. (2) are specified, one of them is the Landau-Ginzburg-Higgs (LGH) equation. When \(a=-1, b={-m}^{2}\) and \(c={n}^{2}\), the LGH equation is stated as
where \(u(X,T)\) symbolizes the electrostatic potential of the ion-cyclotron wave, \(X\) and \(T\) stand for the nonlinearized spatial and temporal coordinates and \(m\) and \(n\) are real parameters. The LGH Eq. (3) was formulated by Lev Devidovich Landau and Vitaly Lazarevich Ginzburg with broad applications for the internal processes of complex physical phenomena which occur to explain superconductivity and drift cyclotron waves in radially inhomogeneous plasma for coherent ion-cyclotron waves [11]. Another typical form of Eq. (2) can be stated, such as \({\phi }^{4}\) equation(\(a=-1, b=1, c=-1\)), Klein–Gordon equation (\(a=-1,b={m}^{2},c=n\)), Duffing equation (\(a=0\) with \(b\) and \(c\) arbitrary), Sine–Gordon equation (\(a=-1,b=1,c=-1/6\)) [8].
It is worth mentioning that there are many attempts in the literature to obtain the exact solutions of Eq. (2), as well as the special case of it, LGH Eq. (3), using different analytical methods. Considering the more general model in Eq. (2), there are different schemes to obtain the exact and explicit solutions, such as the extended mapping method, the hyperbola function method, the Improved tanh-method, the modified extended tanh-function method, a direct and unified algebraic method [8, 12, 13, 16, 29], while for the particular case, LGH Eq. (3), the travelling wave solutions have been investigated in different contexts using different approaches such as solitary wave ansatz method in [4], the first integral method in [9], the \(\left( {G^{\prime } /G,\;1/G} \right)\)-expansion method in [26], the improved Bernoulli sub-equation function (IBSEFM) method in [1], the sine–Gordon expansion (SGE) method in [7], the extended tanh scheme in [6], the generalized Kudryashov technique in [11].
In this paper, under some conditions, the integrability of LGH Eq. (3) is proved by deriving the Lax pair using AKNS scheme. It is worth noting that, there are many different methods for deducing the Lax pair for integrable NLEE's, such as the prolongation method [30], the extended homogeneous balance method [31], the singular manifold method [32,33,34], and the AKNS approach [21, 22, 35]. Accordingly, and using the inverse scattering transformation (IST) method, we obtain a closed form solution to Eq. (3) of type Kink soliton solution.
The residue of the paper is organized as follows: in Sect. 2, we investigate the Lax pair for Eq. (3) using AKNS approach. The inverse scattering transformation is applied to Eq. (3) in Sect. 3. In Sect. 4, the kink type soliton solution is obtained and graphically represented in 2d and 3d plots and a comparison between our solution and different solutions in the literature is represented in tabularized form.
2 The derivation of Lax pair
In this section, we derive the Lax pair in matrix form for Eq. (3) by applying a simple modification of the standard AKNS formalism.
Make the following transformation to Eq. (3):
Then by chain rule, we have
From which we have
According to the above, the differential terms can be written in the form
Then Eq. (3) become
Consider the following linear spectral problems
where \(\psi \left(x,t\right)={\left({\psi }_{1}\left(x,t\right),{\psi }_{2}\left(x,t\right)\right)}^{T}\) and \(\lambda \) is the spectral parameter with \({\lambda }_{t}=0\).
From Eq. (9) we have
and from Eq. (10) we have
From Eqs. (11) and (13) we have
From Eqs. (12) and (14) we have
The compatibility condition \({\psi }_{1xt}={\psi }_{1tx}\) yields
While the compatibility condition \({\psi }_{2xt}={\psi }_{2tx}\) yields
Equating the coefficients of \({\psi }_{1}\) and \({\psi }_{2}\) to zero, we obtain the following system of equations
Now, expand \(A, B\) and \(C\) as follows
Then
Substituting from Eq. (23) into Eq. (21), we obtain the following system of equations
Equating the coefficients of \({\lambda }^{0}\) to zero gives
While equating the coefficients of \({\lambda }^{-1}\) to zero gives
Multiply Eq. (30) by \(2i\), and use Eqs. (26, 27), we have
Suppose the following quantities for \(\alpha , a\left(x,t\right), b\left(x,t\right), c\left(x,t\right),q(x,t)\) and \(r(x,t)\)
Under these considerations, Eq. (30) become
i.e.,
which is the LGH equation given in Eq. (3). We also noted that under the considerations given in Eqs. (31–34), Eqs. (29) and (30) are satisfied. Therefore, The Lax pair for LGH Eq. (3) can be written as
Remark (1)
Under the assumptions given in Eqs. (31–34), at \(\left|x\right|\to \infty \) with the initial condition \(u\to 0\), the limits of \(A, B\) and \(C\) defined in Eq. (22) are evaluated as.
3 The inverse scattering transformation for LGH Eq. (3)
In this section, the inverse scattering transform (IST) procedures will be followed for Eq. (3). Starting from Eq. (9), which may be written in the form
Assume that \(q\left(x,t\right), r(x,t)\) and it's derivative with respect to \(x\) are decay sufficiently rapidly as.
\(\left|x\right|\to \infty \), then we can introduce the following four solutions to Eq. (9), which are defined by their asymptotic behaviors at infinity as
These solutions may be written in matrix form as
Since solutions given in Eq. (42) and Eq. (43) are linearly dependent, where there Wronskian denoted \(W\left({\psi }_{+},{\psi }_{-}\right)\) and \(W\left({\overline{\psi }}_{+},{\overline{\psi }}_{-}\right)\) is equal to zero
Then solutions \({\Psi }_{+}\) and \({\Psi }_{-}\) may be connected via the scattering matrix denoted \(S(\lambda )\) as follows
i.e.,
where
The solution \({{\varvec{\Psi}}}_{+}\left(x;\lambda \right)\) may always be represented by an integral over an appropriate Kernel, while \({{\varvec{\Psi}}}_{-}\) can be obtained using the relation (47)
i.e.,
where at \(\left|x\right|\to \infty \) and \(\underset{x}{\overset{\infty }{\int }}K(x,\lambda ){\Psi }_{0}\left(x;\lambda \right)dy=0\) we have \({{\varvec{\Psi}}}_{+}\left(x;\lambda \right)={{\varvec{\Psi}}}_{0}\left(x;\lambda \right)\) which describe the behavior given in Eqs. (42,43). To find the equations satisfied by the elements of \(K(x;\lambda )\), it is necessary to ensure that \({{\varvec{\Psi}}}_{+}\left(x;\lambda \right)\) is indeed a solution of (41).
and
using the Leibniz integral rule for differentiation under the integral sign which is defined as
where
Using integration by parts for \(\underset{x}{\overset{\infty }{\int }}K{{\varvec{\Psi}}}_{0}dy\) we have
Then Eq. (55) become
Equation (58) is satisfied if \(K(x,y)\) is a solution of the following system of equations
From Eq. (59) we obtain the equations for \(q(x)\) and \(r(x)\) as
From Eq. (47) we have
Then
Assume that
Then Eq. (47) may be rewritten as
Substituting from Eq. (50) into Eq. (66) gives
To get an integral equation for \(K\), we multiply Eq. (67) by \(\frac{1}{2\pi }{{\varvec{\Psi}}}_{0}\left(z;\lambda \right)\) for \(z>x\), then we have
Integrate with respect to \(\lambda \) a long appropriate contour in the complex \(\lambda \)-plane from \(-\infty \) to \(+\infty \). This contour is indented into the upper half-plane for terms involving \({e}^{i\alpha \lambda z}\) and into the lower half-plane for \({e}^{-i\alpha \lambda z}\), we call these contours \({C}_{+}\) and \({C}_{-}\), respectively. One can arrive to the matrix Marchenko equation
where
where \(\rho \left(\lambda \right)\) and \(\overline{\rho }\left(\lambda \right)\) are defined as reflection coefficients, while \({c}_{n}\left(t\right)\) and \({\overline{c} }_{m}\left(t\right)\) are defined as the normalizing coefficients given by
The solution of the matrix Marchenko Eq. (69) gives \(K(x,z)\) from which and by using Eqs. (61, 62) we can recover the potentials \(q(x)\) and \(r(x)\).when \(A\sim \delta (\lambda )\), \(B\sim 0\) and \(C\sim 0\) as \(\left|x\right|\to \infty \) and \(u\to 0\), the time evolution of scattering data can be evaluated as follow (see e.g. [20, 21])
Then from Remark (1), one can obtain the time evolution of the scattering data for Eq. (3) as follows
4 Travelling wave solutions for LGH Eq. (3)
In this section, we consider the reflectionless potential \(\rho \left(\lambda \right)=\overline{\rho }\left(\lambda \right)=0\) and \(N=\overline{N }=1\). Substitute by these considerations in Eqs. (70–78) we have
Substituting Eqs. (79, 80) with the help of Eq. (70) into Eq. (69), we obtain the following system of equations
Starting from Eqs. (81) and (82), assume that
Then Eq. (81) become
Inserting Eq. (86) in Eq. (82), we have
Assume that \({\lambda }_{1}-{\overline{\lambda }}_{1}\) is pure imaginary, then Eq. (87) become
Inserting Eq. (88) into Eq. (85), we obtain
Inserting Eq. (89) into Eq. (88), we obtain
Then from Eq. (61)
In similar way, by solving Eqs. (83, 84) and using Eq. (62) we obtain the potential \(r(x,t)\) as
Case (1)
Let \({\lambda }_{1}=i\beta , \overline{{\lambda }_{1}}=-i\beta \), then according to the symmetry \(q=-r\) we have from Eq. (91) and Eq. (92) that \(\overline{{c }_{1}}\left(0\right)=-{c}_{1}\left(0\right)\), let \({c}_{1}\left(0\right)=2i\beta \), then at these symmetries, and since \(\alpha ={m}^{2}\), the potentials \(q(x,t)\) in Eq. (91) and \(r(x,t)\) in Eq. (92) can be written as
And
Since \(q(x,t)=\frac{-\sqrt{6}n}{2m}{u}_{x}\) then
Use the transformation (4) we obtain the solution \({u}_{1}\left(X,T\right)\) for the LGH Eq. (3) as
where \(m\) and \(n\) are real parameters.
Case (2)
Let \({\lambda }_{1}\) and \({\overline{\lambda }}_{1}\) as defined in case (1) and assume that \({c}_{1}\left(0\right)=-{\overline{c} }_{1}\left(0\right)=2\beta \), then the potentials \(q(x,t)\) in Eq. (91) and \(r(x,t)\) in Eq. (92) can be written as
and
Since \(q(x,t)=\frac{-\sqrt{6}n}{2m}{u}_{x}\) then after integration with respect to \(x\)
Use the transformation (4) we obtain another solution, \({u}_{2}\left(X,T\right)\), for the LGH Eq. (3) as
In Figs. 1 and 2 different solutions extracted from Eq. (96) are graphically represented with different values of \(\beta ,m\) and \(n\)
In Figs. 3 and 4 different solutions extracted from Eq. (100) are graphically represented with different values of \(\beta ,m\) and \(n\)
As we mentioned earlier that there exist some attempts for obtaining travelling wave solutions for LGH Eq. (3) in the literature.
5 Conclusion
In this article, we have investigated the Landau-Ginzburg-Higgs (LGH) equation which explain the superconductivity and drift cyclotron waves in radially inhomogeneous plasma for coherent ion-cyclotron waves by the aid of the inverse scattering transformation (IST) method. The integrability has been proved by deriving the Lax pair under a simple modification of Ablowitz-Kaup-Newell-Segur (AKNS) formalism. Different types of travelling wave solutions have been established and graphically represented in 2d and 3d profiles.
Data availability
The data supporting the findings of this work can be openly accessed at www.github.com/haller-group/ SSMLearn/tree/main/fastSSM.
References
Ismael, H.F., Akkilic, A.N., Murad, M.A.S., Bulut, H., Mahmoud, W., Osman, M.S.: Boiti–Leon–Manna–Pempinelli equation including time-dependent coefficient (vcBLMPE): a variety of nonautonomous geometrical structures of wave solutions. Nonlinear Dyn. 110, 1–14 (2022)
Abdel-Gawad, H. I.: Towering and internal rogue waves induced by two-layer interaction in non-uniform fluid. A 2D non-autonomous gCDGKSE. Nonlinear Dyn. 34, 1–18 (2022)
Arnous, A.H., Ullah, M.Z., Asma, M., Moshokoa, S.P., Mirzazadeh, M., Biswas, A., Belic, M.: Nematicons in liquid crystals by modified simple equation method. Nonlinear Dyn. 88(4), 2863–2872 (2017)
Zhu, H., Chen, L.: Vector dark-bright second-order rogue wave and triplets for a (3+ 1)-dimensional CNLSE with the partially nonlocal nonlinearity. Nonlinear Dyn. 51, 1–10 (2022)
Muniyappan, A., Suruthi, A., Monisha, B., Sharon Leela, N., Vijaycharles, J.: Dromion− like structures in a cubic− quintic nonlinear Schrödinger equation using analytical methods. Nonlinear Dyn. 104(2), 1533–1544 (2021)
Yang, J., Jiang, J.Z., Neild, S.A.: Dynamic analysis and performance evaluation of nonlinear inerter-based vibration isolators. Nonlinear Dyn. 99(3), 1823–1839 (2020)
Kuetche, S.G., Nana, L.: Higher-order spectral filtering effects on the dynamics of stationary soliton in dissipative systems in the presence of linear and nonlinear gain/loss. Nonlinear Dyn. 105(3), 2559–2573 (2021)
Wen, Z.: Bifurcations and exact traveling wave solutions of a new two-component system. Nonlinear Dyn. 87(3), 1917–1922 (2017)
Tariq, H., Akram, G.: New traveling wave exact and approximate solutions for the nonlinear Cahn-Allen equation: evolution of a nonconserved quantity. Nonlinear Dyn. 88(1), 581–594 (2017)
Gaber, A.A., Aljohani, A.F., Ebaid, A., Machado, J.T.: The generalized Kudryashov method for nonlinear space–time fractional partial differential equations of Burgers type. Nonlinear Dyn. 95(1), 361–368 (2019)
Ma, H., Gao, Y., & Deng, A.: Nonlinear superposition of the (2+1)-dimensional generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt equation. Nonlinear Dyn. 46, 1–14 (2022)
Lin, H., He, J., Wang, L., Mihalache, D.: Several categories of exact solutions of the third-order flow equation of the Kaup-Newell system. Nonlinear Dyn. 100(3), 2839–2858 (2020)
Zhang, H.Q., Chen, F., Pei, Z.J.: Rogue waves of the fifth-order Ito equation on the general periodic travelling wave solutions background. Nonlinear Dyn. 103(1), 1023–1033 (2021)
Ali, M.R., Sadat, R.: Construction of Lump and optical solitons solutions for (3+ 1) model for the propagation of nonlinear dispersive waves in inhomogeneous media. Opt. Quantum Electron. 53(6), 1–13 (2021)
Ali, M.R., Ma, W.X., Sadat, R.: Lie symmetry analysis and wave propagation in variable-coefficient nonlinear physical phenomena. East Asian J. Appl. Math. 12(1), 201–212 (2022)
Wazwaz, A.M., Kaur, L.: Complex simplified Hirota’s forms and Lie symmetry analysis for multiple real and complex soliton solutions of the modified KdV–Sine-Gordon equation. Nonlinear Dyn. 95(3), 2209–2215 (2019)
Suyalatu, D., et al.: Bäcklund transformation and multi-soliton solutions for the discrete Korteweg–de Vries equation. Appl. Math. Lett. 125, 107747 (2022)
Ali, Mohamed R.: Solution of KdV and boussinesq using Darboux transformation. Commun. Math. Model Appl. 3, 16–27 (2018)
Chen, Y, and Xue-Wei Y.: Inverse scattering and soliton solutions of high-order matrix nonlinear Schrödinger equation. Nonlinear Dyn. 1–11 (2022)
Song, C.Q., Zhao, H.Q.: Dynamics of various waves in nonlinear Schrödinger equation with stimulated Raman scattering and quintic nonlinearity. Nonlinear Dyn. 99(4), 2971–2985 (2020)
Depollier, C., Fellah, Z.E.A., Fellah, M.: Propagation of transient acoustic waves in layered porous media: fractional equations for the scattering operators. Nonlinear Dyn. 38(1), 181–190 (2004)
Ali, M.R., et al.: Mathematical examination for the energy flow in an inhomogeneous Heisenberg ferromagnetic chain. Optik 271, 170138 (2022)
Ali Akbar, M., et al.: Abundant exact traveling wave solutions of generalized bretherton equation via improved (G′/G)-expansion method. Commun. Theor. Phys. 57, 173 (2012)
Du, X., Zhang, X.: Influence of ocean currents on the stability of underwater glider self-mooring motion with a cable. Nonlinear Dyn. 99(3), 2291–2317 (2020)
Ding, Y., et al.: Abundant complex wave solutions for the nonautonomous Fokas-Lenells equation in presence of perturbation terms. Optik 181, 503–513 (2019)
Wang, J.: The extended Rayleigh-Ritz method for an analysis of nonlinear vibrations. Mech. Adv. Mater. Struct. 29(22), 3281–3284 (2022)
Wu, J.: A new approach to investigate the nonlinear dynamics in a (3+ 1)-dimensional nonlinear evolution equation via Wronskian condition with a free function. Nonlinear Dyn. 103(2), 1795–1804 (2021)
Tang, X.Y., Cui, C.J., Liang, Z.F., Ding, W.: Novel soliton molecules and wave interactions for a (3+ 1)-dimensional nonlinear evolution equation. Nonlinear Dyn. 105(3), 2549–2557 (2021)
Wazwaz, A.M., Xu, G.Q.: Kadomtsev-Petviashvili hierarchy: two integrable equations with time-dependent coefficients. Nonlinear Dyn. 100(4), 3711–3716 (2020)
Najafi, R., Bahrami, F., Hashemi, M.S.: Classical and nonclassical Lie symmetry analysis to a class of nonlinear time-fractional differential equations. Nonlinear Dyn. 87(3), 1785–1796 (2017)
Zhou, T.Y., Tian, B., Chen, Y.Q., Shen, Y.: Painlevé analysis, auto-Bäcklund transformation and analytic solutions of a (2$$+ $$+ 1)-dimensional generalized Burgers system with the variable coefficients in a fluid. Nonlinear Dyn. 108(3), 2417–2428 (2022)
Zhao, X., Tian, B., Tian, H.Y., Yang, D.Y.: Bilinear Bäcklund transformation, Lax pair and interactions of nonlinear waves for a generalized (2+ 1)-dimensional nonlinear wave equation in nonlinear optics/fluid mechanics/plasma physics. Nonlinear Dyn. 103(2), 1785–1794 (2021)
Wang, J., Wu, R.: The extended Galerkin method for approximate solutions of nonlinear vibration equations. Appl. Sci. 12(6), 2979 (2022)
Hua, Z., Li, J., Li, Y., Chen, Y.: Image encryption using value-differencing transformation and modified ZigZag transformation. Nonlinear Dyn. 106(4), 3583–3599 (2021)
Shi, B., Yang, J., Wang, J.: Forced vibration analysis of multi-degree-of-freedom nonlinear systems with the extended Galerkin method. Mech. Adv. Mater. Struct. (2021). https://doi.org/10.1080/15376494.2021.20239221-9
Acknowledgements
We are grateful to Kerstin Avila and Bastian Bäuerlein (U. Bremen) for making their experimental surface profile data from Ref. [8] available to us. We also wish to thank Melih Eriten (U. Wisconsin) for supplying the resonant beam experimental data from Ref. [11].
Funding
Open access funding provided by The Science, Technology & Innovation Funding Authority (STDF) in cooperation with The Egyptian Knowledge Bank (EKB). Open access funding provided by Swiss Federal Institute of Technology Zurich. The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.
Author information
Authors and Affiliations
Contributions
MRA: designed the research, carried out the research. MAK, developed the software and analyzed the examples. SMM wrote the paper, lead the research team.
Corresponding author
Ethics declarations
Conflict of interest
The authors have no relevant financial or non-financial interests to disclose.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Ali, M.R., Khattab, M.A. & Mabrouk, S.M. Travelling wave solution for the Landau-Ginburg-Higgs model via the inverse scattering transformation method. Nonlinear Dyn 111, 7687–7697 (2023). https://doi.org/10.1007/s11071-022-08224-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-022-08224-6