Skip to main content
SpringerLink
Log in

Multi-Component Coupled Fokas-Lenells Equations and Theirs Localized Wave Solutions

  • Published:
Acta Applicandae Mathematicae Aims and scope Submit manuscript

Abstract

As the first negative flow of the integrable generalization of the nonlinear Schrödinger equation, the Fokas-Lenells equation has attracted extensive attention in recent years. In this paper, we derive the general structure of the multi-component coupled Fokas-Lenells equations which have Lax representation in matrix form. Then we construct a basic theory of the general form of Lax pairs and Darboux transformations (classical and generalized) for the previously mentioned equation. As applications, we study two examples in detail, both of the four-component and the three-component coupled Fokas-Lenells equations can be reduced to the ubiquitous Fokas-Lenells equation. Furthermore, we apply the basic theory to obtain kinds of localized wave solutions, that is to say we use the classical Darboux transformation to obtain soliton solutions and use the generalized Darboux transformation to obtain soliton-positon solutions, rogue wave solutions and breather solutions. At last these localized wave solutions are illustrated by three-dimensional structure plots and two-dimensional density plots, as well as their dynamic properties are discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

Data Availability

All data analyzed during this study are included, and the manuscript has no associated data.

References

  1. Fokas, A.S.: On a class of physically important integrable equations. Physica D 87, 145–150 (1995)

    Article  MathSciNet  Google Scholar 

  2. Lenells, J., Fokas, A.S.: An integrable generalization of the nonlinear Schrödinger equation on the half-line and solitons. Inverse Probl. 25, 115006 (2009)

    Article  Google Scholar 

  3. Lenells, J.: Exactly solvable model for nonlinear pulse propagation in optical fibers. Stud. Appl. Math. 123, 215–232 (2009)

    Article  MathSciNet  Google Scholar 

  4. Lenells, J., Fokas, A.S.: On a novel integrable generalization of the nonlinear Schrödinger equation. Nonlinearity 22, 11–27 (2009)

    Article  MathSciNet  Google Scholar 

  5. Fordy, A.P., Kulish, P.P.: Nonlinear Schrödinger equations and simple Lie algebras. Commun. Math. Phys. 89, 427–443 (1983)

    Article  Google Scholar 

  6. Ivanov, R.I., Gerdjikov, V.S.: Multicomponent Fokas-Lenells equations on Hermitian symmetric spaces. Nonlinearity 34, 939963 (2021)

    MathSciNet  MATH  Google Scholar 

  7. Vekslerchik, V.E.: Lattice representation and dark solitons of the Fokas-Lenells equation. Nonlinearity 24, 1165–1175 (2011)

    Article  MathSciNet  Google Scholar 

  8. Matsuno, Y.: A direct method of solution for the Fokas-Lenells derivative nonlinear Schrödinger equation: I. Bright soliton solutions. J. Phys. A, Math. Theor. 45, 235202 (2012)

    Article  Google Scholar 

  9. Matsuno, Y.: A direct method of solution for the Fokas-Lenells derivative nonlinear Schrödinger equation: II. Dark soliton solutions. J. Phys. A, Math. Theor. 45, 475202 (2012)

    Article  Google Scholar 

  10. Lü, X., Peng, M.S.: Nonautonomous motion study on accelerated and decelerated solitons for the variable-coefficient Lenells-Fokas model. Chaos 23, 013122 (2013)

    Article  MathSciNet  Google Scholar 

  11. Chabchoub, A., Hoffmann, N.P., Akhmediev, N.: Rogue wave observation in a water wave tank. Phys. Rev. Lett. 106, 204502 (2011)

    Article  Google Scholar 

  12. He, J.S., Zhang, H.R., Wang, L.H., Porsezian, K., Fokas, A.S.: Generating mechanism for higher-order rogue waves. Phys. Rev. E 87, 052914 (2013)

    Article  Google Scholar 

  13. Xu, S.W., He, J.S., Cheng, Y., Porseizan, K.: The \(n\)-order rogue waves of Fokas-Lenells equation. Math. Methods Appl. Sci. 38, 1106–1126 (2015)

    Article  MathSciNet  Google Scholar 

  14. Zhang, Y., Yang, J.W., Chow, K.W., Wu, C.F.: Solitons, breathers and rogue waves for the coupled Fokas-Lenells system via Darboux transformation. Nonlinear Anal., Real World Appl. 33, 237–252 (2017)

    Article  MathSciNet  Google Scholar 

  15. Ling, L.M., Feng, B.F., Zhu, Z.N.: General soliton solutions to a coupled Fokas-Lenells equation. Nonlinear Anal., Real World Appl. 40, 185–214 (2018)

    Article  MathSciNet  Google Scholar 

  16. Wang, M.M., Chen, Y.: Dynamic behaviors of mixed localized solutions for the three-component coupled Fokas-Lenells system. Nonlinear Dyn. 98, 1781–1794 (2019)

    Article  Google Scholar 

  17. Li, Y.H., Geng, X.G., Xue, B., Li, R.M.: Darboux transformation and exact solutions for a four-component Fokas-Lenells equation. Results Phys. 31, 105027 (2021)

    Article  Google Scholar 

  18. Zhou, Z.X.: Darboux transformations and global solutions for a nonlocal derivative nonlinear Schrödinger equation. Commun. Nonlinear Sci. Numer. Simul. 62, 480–488 (2016)

    Article  Google Scholar 

  19. Wang, Y., Xiong, Z.J., Ling, L.M.: Fokas-Lenells equation: three types of Darboux transformation and multi-soliton solutions. Appl. Math. Lett. 107, 106441 (2020)

    Article  MathSciNet  Google Scholar 

  20. Matveev, V.B., Salle, M.A.: Darboux Transformations and Solitons. Springer Series in Nonlinear Dynamics, pp. 1–131. Springer, Berlin (1991)

    Book  Google Scholar 

  21. Guo, B.L., Ling, L.M., Liu, Q.P.: Nonlinear Schrödinger equation: generalized Darboux transformation and rogue wave solutions. Phys. Rev. E 85, 026607 (2012)

    Article  Google Scholar 

  22. Wen, X.Y., Yang, Y.Q., Yan, Z.Y.: Generalized perturbation (\(n\), \(M\))-fold Darboux transformations and multi-rogue-wave structures for the modified self-steepening nonlinear Schrödinger equation. Phys. Rev. E 92, 012917 (2015)

    Article  MathSciNet  Google Scholar 

Download references

Funding

Partial financial support was received from the National Nature Science Foundation of China (No. 11701334) and the “Jingying” Project of Shandong University of Science and Technology.

Author information

Authors and Affiliations

  1. College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, 266590, Shandong, P.R. China

    Qiulan Zhao, Huijie Song & Xinyue Li

Authors
  1. Qiulan Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Huijie Song
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xinyue Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qiulan Zhao.

Ethics declarations

Competing Interests

The authors declare that they have no conflict of interest.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendices

Appendix A

Proof of Theorem 1.

Proof

We choose the form of the Darboux matrix \(T^{[n]}\) as

$$\begin{aligned} T^{[n]}(\lambda )=D_{n}+D_{n-1}+C\lambda ^{-n}, \end{aligned}$$
(A.1)

where

$$\begin{aligned} \begin{aligned} D_{n}=\left ( \textstyle\begin{array}{c@{\quad}c} D_{11}^{\left [ n \right ]} & 0 \\ 0 & D_{22}^{\left [ n \right ]} \\ \end{array}\displaystyle \right ),\quad \end{aligned} \begin{aligned} D_{n-1}=\left ( \textstyle\begin{array}{c@{\quad}c} 0 & D_{12}^{\left [ n-1 \right ]} \\ D_{21}^{\left [ n-1 \right ]} &0 \\ \end{array}\displaystyle \right ), \end{aligned} \begin{aligned} C=\left ( \textstyle\begin{array}{c@{\quad}c} I_{m} & 0 \\ 0 & I_{n} \\ \end{array}\displaystyle \right ),\quad \end{aligned} \end{aligned}$$

with

$$\begin{aligned} D_{ij}^{[n]}={A}_{ij}^{(n)}\lambda ^{n}+A_{ij}^{(n-2)}\lambda ^{n-2}+ \cdots +A_{ij}^{[-(n-4)]}\lambda ^{-(n-4)}+A_{ij}^{[ -( n-2)]} \lambda ^{-( n-2)}\quad (i=j), \end{aligned}$$
$$\begin{aligned} D_{ij}^{\left [ n-1 \right ]}={B}_{ij}^{\left ( n-1 \right )}{ \lambda }^{n-1}+B_{ij}^{\left ( n-3 \right )}{\lambda }^{n-3}\text{+} \cdots \text{+}B_{ij}^{\left [ -\left ( n-\text{3} \right ) \right ]}{ \lambda }^{-\left ( n-3 \right )}\text{+}B_{ij}^{\left [ -\left ( n-1 \right ) \right ]}{\lambda }^{-\left ( n-1 \right )}\quad (i\neq j). \end{aligned}$$

Here the elements in \(D_{n}\) and \(D_{n-1}\) are undetermined functions of \(x\) and \(t\), \(n\) is any positive integer and

$$\begin{aligned} \begin{aligned} A_{11}=\left ( \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} a_{1,1} &a_{1,2}&\cdots &a_{1,m} \\ a_{2,1} &a_{2,2}&\cdots &a_{2,m} \\ \vdots &\vdots &&\vdots \\ a_{m,1} &a_{m,2}&\cdots &a_{m,m} \\ \end{array}\displaystyle \right ),\quad \end{aligned} \begin{aligned} B_{12}=\left ( \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} b_{1,m+1} &b_{1,m+2}&\cdots &b_{1,m+z} \\ b_{2 ,m+1} &b_{2,m+2}&\cdots &b_{2,m+z} \\ \vdots &\vdots &&\vdots \\ b_{m ,m+1} &b_{m,m+2}&\cdots &b_{m,m+z} \end{array}\displaystyle \right ),\quad \end{aligned} \end{aligned}$$
$$\begin{aligned} \begin{aligned} B_{21}\!=\!\left (\! \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} b_{m+1,1} &b_{m+1,2}&\cdots &b_{m+1,m} \\ b_{m+2,1} &b_{m+2,2}&\cdots &b_{m+2,m} \\ \vdots &\vdots &&\vdots \\ b_{m+z,1} &b_{m+z,2}&\cdots &b_{m+z,m} \end{array}\displaystyle \! \right ),\quad \end{aligned} \begin{aligned} A_{22}\!=\!\left (\! \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} a_{m+1,m+1} &a_{m+1,m+2}&\cdots &a_{m+1,m+z} \\ a_{m+2,m+1} &a_{m+2, m+2}&\cdots &a_{m+2,m+z} \\ \vdots &\vdots &&\vdots \\ a_{m+z,m+1} &a_{m+z,m+2}&\cdots &a_{m+z,m+z} \\ \end{array}\displaystyle \! \right )\!. \end{aligned} \end{aligned}$$

Plugging Eq. (A.1) into Eqs. (2.8) and contrasting the coefficients of \(\lambda ^{j}\) \((j=-(n+1), -(n-1)\cdots n-1, n+1)\) on both sides of the Eqs. (2.8) we can get

$$\begin{aligned} \left \{ \textstyle\begin{array}{l@{\quad}l} \lambda ^{n+1}:&Q_{x}^{[n]}A_{22}^{(n)}=A_{11}^{(n)}Q_{x}+2iB_{12}^{(n-1)}, \quad R_{x}^{[n]}A_{11}^{(n)}=A_{22}^{(n)}R_{x}-2iB_{21}^{(n-1)}, \\ \lambda ^{j}:&Q_{x}^{[n]}B_{21}^{(j-1)}=B_{12}^{(j-1)}R_{x}A_{11x}^{(j)}, \quad R_{x}^{[n]}B_{12}^{(j-1)}=B_{21}^{(j-1)}Q_{x}A_{22x}^{(j)}, \\ &(j=-(n-2),-(n-4),\ldots n-2, n), \\ \lambda ^{j-1}:&Q_{x}^{[n]}A_{22}^{(j-2)}=A_{11}^{(j-2)}Q_{x}+2iB_{12}^{(j-3)}+B_{12x}^{(j-1)}, \\ & R_{x}^{[n]}A_{11}^{(j-2)}=A_{22}^{(j-2)}R_{x}-2iB_{21}^{(j-3)}+B_{21x}^{(j-1)}, \\ &(j=-(n-4),-(n-6),\ldots n-2, n), \\ \lambda ^{-(n-1)}:&Q_{x}^{[n]}=Q_{x}+B_{12x}^{[-(n-1)]},\quad R_{x}^{[n]}=R_{x}+B_{21x}^{[-(n-1)]}, \end{array}\displaystyle \right . \end{aligned}$$

and

$$\begin{aligned} \left \{ \textstyle\begin{array}{l@{\quad}l} \lambda ^{n+1}:&Q_{x}^{[n]}A_{22}^{(n)}=A_{11}^{(n)}Q_{x}+2iB_{12}^{(n-1)}, \quad R_{x}^{[n]}A_{11}^{(n)}=A_{22}^{(n)}R_{x}+2iB_{21}^{(n-1)}, \\ \lambda ^{n}: &-2i\beta Q^{[n]}R^{[n]}A_{11}^{(n)}+\alpha Q_{x}^{[n]}B_{21}^{(n-1)}=-2i \beta A_{11}^{(n)}QR+\alpha B_{12}^{(n-1)}R_{x}+A_{11t}^{(n)}, \\ &2i\beta R^{[n]}Q^{[n]}A_{22}^{(n)}+\alpha R_{x}^{[n]}B_{12}^{(n-1)}=2i \beta A_{22}^{(n)}RQ+\alpha B_{21}^{(n-1)}Q_{x}+A_{22t}^{(n)}, \\ \lambda ^{n-1}:& \alpha Q_{x}^{[n]}A_{22}^{(n-2)}+2i\beta Q^{[n]}A_{22}^{(n)}-2i \beta Q^{[n]}R^{[n]}B_{12}^{(n-1)}=\alpha A_{11}^{(n-2)}Q_{x}+2i \beta A_{11}^{(n)}Q \\ &+2i\alpha B_{12}^{(n-3)}-2i\gamma B_{12}^{(n-1)}+2i\beta B_{12}^{(n-1)}RQ+b_{12t}^{(n-1)}, \\ &\alpha R_{x}^{[n]}A_{11}^{(n-2)}-2i\beta R^{[n]}A_{11}^{(n)}+2i \beta R^{[n]}Q^{[n]}B_{21}^{(n-1)}=\alpha A_{22}^{(n-2)}R_{x}-2i \beta A_{22}^{(n)}R \\ &-2i\alpha B_{21}^{(n-3)}+2i\gamma B_{21}^{(n-1)}-2i\beta B_{21}^{(n-1)}QR+b_{21t}^{(n-1)}, \\ \lambda ^{j-2}: &-2i\beta Q^{[n]}R^{[n]}A_{11}^{(j-2)}+\alpha Q_{x}^{[n]}B_{21}^{(j-3)}+2i \beta Q^{[n]}B_{21}^{(j-1)}=-2i\beta A_{11}^{(j-2)}QR \\ &+\alpha B_{12}^{j-3}R_{x}-2i\beta B_{12}^{(j-1)}R+A_{11t}^{(j-2)}, \\ &-2i\beta R^{[n]}Q^{[n]}A_{22}^{(j-2)}+\alpha R_{x}^{[n]}B_{12}^{(j-3)}-2i \beta R^{[n]}B_{12}^{(j-1)}=2i\beta A_{22}^{(j-2)}RQ \\ &+\alpha B_{21}^{j-3}Q_{x}+2i\beta B_{21}^{(j-1)}Q+A_{22t}^{(j-2)}, \\ &j=(-(n-4), -(n-6),\ldots n-2, n), \\ \lambda ^{j-3}: &\alpha Q_{x}^{[n]}A_{22}^{(j-4)}+2i\beta Q^{[n]}A_{22}^{(j-2)}-2i \beta Q^{[n]}R^{[n]}B_{12}^{(j-3)}=\alpha A_{11}^{(j-4)}Q_{x} \\ & +2i \beta A_{11}^{(j-2)}Q +2i\alpha B_{12}^{(j-5)}-2i\gamma B_{12}^{(j-3)}+2i\beta B_{12}^{(j-3)}RQ+2i \beta B_{12}^{(j-1)} \\ & +B_{12t}^{(j-3)}, \\ &\alpha R_{x}^{[n]}A_{11}^{(j-4)}-2i\beta R^{[n]}A_{11}^{(j-2)}+2i \beta R^{[n]}Q^{[n]}B_{21}^{(j-3)}=\alpha A_{22}^{(j-2)}R_{x}\!-\!2i \beta A_{22}^{(j-2)}R \\ &-2i\alpha B_{21}^{(j-5)}+2i\gamma B_{21}^{(j-3)}-2i\beta B_{21}^{(j-3)}QR-2i \beta B_{21}^{(j-1)}+B_{21t}^{(j-3)}, \\ &(j=-(n-6), -(n-8),\ldots n-2,n), \\ \lambda ^{-(n-1)}:& \alpha Q_{x}^{[n]}+2i\beta Q^{[n]}A_{22}^{[-(n-2)]}-2i \beta Q^{[n]}R^{[n]}B_{12}^{[-(n-1)]}=\alpha Q_{x}+2i\beta A_{11}^{[-(n-2)]}Q \\ &-2i\gamma B_{12}^{[-(n-1)]}+2i\beta B_{12}^{[-(n-1)]}RQ+2i\beta B_{12}^{[-(n-3)]}+B_{12t}^{[-(n-1)]}, \\ &\alpha R_{x}^{[n]}-2i\beta R^{[n]}A_{11}^{[-(n-2)]}+2i\beta R^{[n]}Q^{[n]}B_{21}^{[-(n-1)]}= \alpha R_{x}-2i\beta A_{22}^{[-(n-2)]}R \\ &+2i\gamma B_{21}^{[-(n-1)]}-2i\beta B_{21}^{[-(n-1)]}QR-2i\beta B_{21}^{[-(n-3)]}+B_{21t}^{[-(n-1)]}, \\ \lambda ^{-n}:&Q^{[n]}R^{[n]}-Q^{[n]}B_{21}^{[-(n-1)]}=QR+B_{12}^{[-(n-1)]}R, \\ &R^{[n]}Q^{[n]}-R^{[n]}B_{12}^{[-(n-1)]}=RQ+B_{21}^{[-(n-1)]}Q, \\ \lambda ^{-(n+1)}: &Q^{[n]}=Q+B_{12}^{[-(n-1)]},\quad R^{[n]}=R+B_{21}^{[-(n-1)]}. \end{array}\displaystyle \right . \end{aligned}$$

By this straightforward and complicated calculation we can get Eqs. (2.9). The proof is completed. □

Appendix B

The specific values of \(\Gamma _{1}\) to \(\Gamma _{20}\) in Eq. (2.16) are as follows.

\(\Gamma _{1}=\lambda _{m+z}^{n}{\phi }_{m+z,1}^{(0)}\), \(\Gamma _{2}=\lambda _{m+z}^{-(n-2)}{\phi }_{m+z,1}^{(0)}\), \(\Gamma _{3}=\lambda _{m+z}^{-(n-2)}{\phi }_{m+z,m}^{(0)}\), \(\Gamma _{4}=\lambda _{m+z}^{-(n-1)}{\phi }_{m+z,m+1}^{(0)}\),

\(\Gamma _{5}=\lambda _{m+z}^{-(n-1)}{\phi }_{m+z,m+z}^{(0)}\), \(\Gamma _{6}=\lambda _{1}^{n}\phi _{1,1}^{(1)}+n\lambda _{1}^{n-1} \phi _{1,1}^{(0)}\), \(\Gamma _{7}=\lambda _{1}^{-(n-2)}\phi _{1,1}^{(1)}-(n-2)\lambda _{1}^{-(n-1)} \phi _{1,1}^{(0)}\),

\(\Gamma _{8}=\lambda _{1}^{-(n-2)}\phi _{1,m}^{(1)}-(n-2)\lambda _{1}^{-(n-1)} \phi _{1,m}^{(0)}\), \(\Gamma _{9}=\lambda _{1}^{-(n-1)}\phi _{1,m+1}^{(1)}-(n-1)\lambda _{1}^{-n} \phi _{1,m+1}^{(0)}\),

\(\Gamma _{10}=\lambda _{1}^{-(n-1)}\phi _{1,m+z}^{(1)}-(n-1)\lambda _{1}^{-n} \phi _{1,m+z}^{(0)}\),

\(\Gamma _{11}=\lambda _{1}^{n}\phi _{1,1}^{(n-1)}+n\lambda _{1}^{n-1} \phi _{1,1}^{(n-2)}+\dfrac{n(n-1)}{2!}\lambda _{1}^{n-2}\phi _{1,1}^{(n-3)}+ \cdots +\dfrac{n(n-1)\cdots 2}{(n-1)!}\lambda _{1}\phi _{1,1}^{(0)}\),

\(\begin{array}[t]{lcl}\Gamma _{12}&=&\lambda _{1}^{-(n-2)}\phi _{1,1}^{(n-1)}-(n-2)\lambda _{1}^{-(n-1)} \phi _{1,1}^{(n-2)}+\cdots \\ &&{} +(-1)^{(n-1)} \dfrac{(n-2)(n-1)\cdots (2n-4)}{(n-1)!}\lambda _{1}^{-(2n-3)}\phi _{1,1}^{(0)},\end{array}\)

\(\begin{array}[t]{lcl}\Gamma _{13}&=&\lambda _{1}^{-(n-2)}\phi _{1,m}^{(n-1)}-(n-2)\lambda _{1}^{-(n-1)} \phi _{1,m}^{(n-2)}+\cdots \\ &&{}+(-1)^{(n-1)} \dfrac{(n-2)(n-1)\cdots (2n-4)}{(n-1)!}\lambda _{1}^{-(2n-3)}\phi _{1,m}^{(0)},\end{array}\)

\(\begin{array}[t]{lcl}\Gamma _{14}&=&\lambda _{1}^{-(n-1)}\phi _{1,m+1}^{(n-1)}-(n-1) \lambda _{1}^{-n}\phi _{1,m+1}^{(n-2)}+\cdots \\ &&{}+(-1)^{(n-1)} \dfrac{(n-1)n\cdots (2n-3)}{(n-1)!}\lambda _{1}^{-(2n-2)}\phi _{1,m+1}^{(0)},\end{array}\)

\(\begin{array}[t]{lcl}\Gamma _{15}&=&\lambda _{1}^{-(n-1)}\phi _{1,m+z}^{(n-1)}-(n-1) \lambda _{1}^{-n}\phi _{1,m+z}^{(n-2)}+\cdots \\ &&{} +(-1)^{(n-1)} \dfrac{(n-1)n\cdots (2n-3)}{(n-1)!}\lambda _{1}^{-(2n-2)}\phi _{1,m+z}^{(0)},\end{array}\)

\(\Gamma _{16}=\lambda _{m+z}^{n}\phi _{m+z,1}^{(n-1)}+n\lambda _{m+z}^{n-1} \phi _{m+z,1}^{(n-2)}+\dfrac{n(n-1)}{2!}\lambda _{1}^{n-2}\phi _{1,1}^{(n-3)}+ \cdots +\dfrac{n(n-1)\cdots 2}{(n-1)!}\lambda _{m+z}\phi _{m+z,1}^{(0)}\),

\(\begin{array}[t]{lcl}\Gamma _{17}&=&\lambda _{m+z}^{-(n-2)}\phi _{m+z,1}^{(n-1)}-(n-2) \lambda _{m+z}^{-(n-1)}\phi _{m+z,1}^{(n-2)}+\cdots \\ &&{}+(-1)^{(n-1)} \dfrac{(n-2)(n-1)\cdots (2n-4)}{(n-1)!}\lambda _{m+z}^{-(2n-3)}\phi _{m+z,1}^{(0)},\end{array}\)

\(\begin{array}[t]{lcl}\Gamma _{18}&=&\lambda _{m+z}^{-(n-2)}\phi _{m+z,m}^{(n-1)}-(n-2) \lambda _{m+z}^{-(n-1)}\phi _{m+z,m}^{(n-2)}+\cdots \\ &&{}+(-1)^{(n-1)} \dfrac{(n-2)(n-1)\cdots (2n-4)}{(n-1)!}\lambda _{m+z}^{-(2n-3)}\phi _{m+z,m}^{(0)},\end{array}\)

\(\begin{array}[t]{lcl}\Gamma _{19}&=&\lambda _{m+z}^{-(n-1)}\phi _{m+z,m+1}^{(n-1)}-(n-1) \lambda _{m+z}^{-n}\phi _{m+z,m+1}^{(n-2)}+\cdots \\ &&{}+(-1)^{(n-1)} \dfrac{(n-1)n\cdots (2n-3)}{(n-1)!}\lambda _{m+z}^{-(2n-2)}\phi _{m+z,m+1}^{(0)},\end{array}\)

\(\begin{array}[t]{lcl}\Gamma _{20}&=&\lambda _{m+z}^{-(n-1)}\phi _{m+z,m+z}^{(n-1)}-(n-1) \lambda _{m+z}^{-n}\phi _{m+z,m+z}^{(n-2)}+\cdots \\ &&{}+(-1)^{(n-1)} \dfrac{(n-1)n\cdots (2n-3)}{(n-1)!}\lambda _{m+z}^{-(2n-2)}\phi _{m+z,m+z}^{(0)}.\end{array}\)

Appendix C

(I)

$$\begin{aligned} p^{[1]}=e^{\frac{ix+7it}{2}} \dfrac{\Theta _{1}e^{\Xi _{1}}+\Theta _{2}e^{\Xi _{2}}+\Theta _{3}e^{\Xi _{3}}+\Theta _{4}e^{\Xi}_{4}+\Theta _{5}e^{\Xi _{5}}+\Theta _{6}e^{\Xi _{6}}}{\Omega _{1}e^{\Xi _{1}}+\Omega _{2}e^{\Xi _{2}}+\Omega _{3}e^{\Xi _{3}}+\Omega _{4}e^{\Xi}_{4}+\Omega _{5}e^{\Xi _{5}}+\Omega _{6}e^{\Xi _{6}}}, \end{aligned}$$

where

\(\Theta _{1}=-(90+45i)\sqrt{3+8i}+(306-163i)\sqrt{3-8i}-297+566i+(35+30i) \sqrt{73}\),

\(\Theta _{2}=-(90+45i)\sqrt{3+8i}+(-306+163i)\sqrt{3-8i}-297+566i-(35+30i) \sqrt{73}\),

\(\Theta _{3}=(90+45i)\sqrt{3+8i}+(306-163i)\sqrt{3-8i}-297+566i-(35+30i) \sqrt{73}\),

\(\Theta _{4}=(90+45i)\sqrt{3+8i}-(306-163i)\sqrt{3-8i}-297+566i+(35+30i) \sqrt{73}\),

\(\Theta _{5}=-1280-640i-(40+220i)\sqrt{3+8i}\), \(\Theta _{6}=-1280-640i+(40+220i)\sqrt{3+8i}\),

\(\Omega _{1}=-(180+90i)\sqrt{3 + 8i}-(540 + 70i)\sqrt{3-8i}+1710-20i+(70 + 60i)\sqrt{73}\),

\(\Omega _{2}=-(180+90i)\sqrt{3 + 8i}+(540 + 70i)\sqrt{3-8i}+1710-20i-(70 + 60i)\sqrt{73}\),

\(\Omega _{3}=(180+90i)\sqrt{3 + 8i}-(540 + 70i)\sqrt{3-8i}+1710-20i-(70 + 60i)\sqrt{73}\),

\(\Omega _{4}=(180+90i)\sqrt{3 + 8i}+(540 + 70i)\sqrt{3-8i}+1710-20i+(70 + 60i)\sqrt{73}\),

\(\Omega _{5}=2560+1280i-(80 + 440i)\sqrt{3+8i}\), \(\Omega _{6}=2560+1280i+(80 + 440i)\sqrt{3+8i}\),

\(\Xi _{1}=(32it+25x)\sqrt{3-8i}/100-(32it-25x)\sqrt{3+8i}/200\),

\(\Xi _{2}=-(32it+25x)\sqrt{3+8i}/200+(23-32i)t\sqrt{3-8i}/100\),

\(\Xi _{3}=(32it+25x)\sqrt{3-8i}/100+(23+32i)t\sqrt{3+8i}/200\),

\(\Xi _{4}=(23-32i)t\sqrt{3-8i}/100+(23+32i)t\sqrt{3+8i}/200\),

\(\Xi _{5}=(23t+25x)\sqrt{3-8i}/200-(32it-25x)\sqrt{3+8i}/200\),

\(\Xi _{6}=(23t+25x)\sqrt{3-8i}/200+(32it-25x)\sqrt{3+8i}/200\).

(II)

$$\begin{aligned} p^{[1]}=e^{\frac{ix+3it}{2}} \dfrac{\Theta _{1}e^{\Xi _{1}}+\Theta _{2}e^{\Xi _{2}}+\Theta _{3}e^{\Xi _{3}}+\Theta _{4}e^{\frac{4i}{25}(\sqrt{3-8i}-\sqrt{3+8i})}}{\Theta _{5}e^{\Xi _{1}}+\Theta _{6}e^{\Xi _{2}}+\Theta _{7}e^{\Xi _{3}}+\Theta _{8}e^{\frac{4i}{25}(\sqrt{3-8i}-\sqrt{3+8i})}}, \end{aligned}$$

where

\(\Theta _{1}=-(75+50i)\sqrt{3-8i}+(-133+26i)\sqrt{3+8i}+114+527i+(10-5i) \sqrt{73}\),

\(\Theta _{2}=-(75+50i)\sqrt{3-8i}+(133-26i)\sqrt{3+8i}+114+527i+(-10+5i) \sqrt{73}\),

\(\Theta _{3}=(75+50i)\sqrt{3-8i}+(-133+26i)\sqrt{3+8i}+114+527i+(-10+5i) \sqrt{73}\),

\(\Theta _{4}=(75+50i)\sqrt{3-8i}+(133-26i)\sqrt{3+8i}+114+527i+(10-5i) \sqrt{73}\),

\(\Theta _{5}=-(75+50i)\sqrt{3-8i}+(-5+90i)\sqrt{3+8i}+50-625i+(10-5i) \sqrt{73}\),

\(\Theta _{6}=-(75+50i)\sqrt{3-8i}+(5-90i)\sqrt{3+8i}+50-625i+(-10+5i) \sqrt{73}\),

\(\Theta _{7}=(75+50i)\sqrt{3-8i}+(-5+90i)\sqrt{3+8i}+50-625i+(-10+5i) \sqrt{73}\),

\(\Theta _{8}=(75+50i)\sqrt{3-8i}+(5-90i)\sqrt{3+8i}+50-625i+(10-5i) \sqrt{73}\),

\(\Xi _{1}=[(73-32i)t+25x]\sqrt{3-8i}/200+{[(73+32i)t+25x]\sqrt{3+8i}}/{200}\),

\(\Xi _{2}=[(73-32i)t+25x]\sqrt{3-8i}/200-4it\sqrt{3+8i}/25\),

\(\Xi _{3}=[(73+32i)t+25x]\sqrt{3+8i}/200+4it\sqrt{3-8i}/25\).

(III)

$$\begin{aligned} p^{[1]}=e^{\frac{ix+3it}{2}} \dfrac{\Theta _{1}e^{\Xi _{1}}+\Theta _{2}e^{\Xi _{2}}+\Theta _{3}e^{\Xi _{3}}+\Theta _{4}e^{\frac{-8it}{867}(\sqrt{2665+624i}-\sqrt{-2665+624i})}}{\Theta _{5}e^{\Xi _{1}}+\Theta _{6}e^{\Xi _{2}}+\Theta _{7}e^{\Xi _{3}}+\Theta _{8}e^{\frac{-8it}{867}(\sqrt{2665+624i}-\sqrt{-2665+624i})}}, \end{aligned}$$

where

\(\Theta _{1}=-(3485+7004i)\sqrt{-2665-624i}+(-7971+5772i)\sqrt{-2665+624i}+154596+563847i+(884 -221i)\sqrt{44329}\),

\(\Theta _{2}=-(3485+7004i)\sqrt{-2665-624i}+(7971-5772i)\sqrt{-2665+624i}+154596+563847i+(-884 +221i)\sqrt{44329}\),

\(\Theta _{3}=(3485+7004i)\sqrt{-2665-624i}+(-7971 + 5772i)\sqrt{-2665 + 624i} + 154596 + 563847i + (-884 + 221i)\sqrt{44329}\),

\(\Theta _{4}=(3485+7004i)\sqrt{-2665-624i}+(7971 -5772i)\sqrt{-2665 + 624i} + 154596 + 563847i + (884-221i)\sqrt{44329}\),

\(\Theta _{5}=-(3485+7004i)\sqrt{-2665-624i}+ (221+7820i)\sqrt{-2665 + 624i} + 181220 - 378233i + (884 - 221i)\sqrt{44329}\),

\(\Theta _{6}=-(3485 + 7004i)\sqrt{-2665 - 624i} - (221 + 7820i)\sqrt{-2665 + 624i} + 181220 - 378233i + (-884 + 221i)\sqrt{44329}\),

\(\Theta _{7}=(3485 + 7004i)\sqrt{-2665 - 624i}+ (221 + 7820i)\sqrt{-2665 + 624i} + 181220 - 378233i + (-884 + 221i)\sqrt{44329}\),

\(\Theta _{8}=(3485 + 7004i)\sqrt{-2665 - 624i}- (221 + 7820i)\sqrt{-2665 + 624i} + 181220 - 378233i + (884 - 221i)\sqrt{44329}\),

\(\Xi _{1}=[(4707 - 1024i)t + 867x]\sqrt{-2665 - 624i}/110976+[(4707+1024i)t + 867x] \sqrt{-2665+624i}/110976\),

\(\Xi _{2}=[(4707 - 1024i)t + 867x]\sqrt{-2665 - 624i}/110976-8it\sqrt{-2665 + 624i}/867\),

\(\Xi _{3}=[(4707+1024i)t + 867x]\sqrt{-2665+624i}/110976+8it\sqrt{-2665 - 624i}/867\).

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhao, Q., Song, H. & Li, X. Multi-Component Coupled Fokas-Lenells Equations and Theirs Localized Wave Solutions. Acta Appl Math 181, 17 (2022). https://doi.org/10.1007/s10440-022-00535-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10440-022-00535-5

Keywords

Mathematics Subject Classification

Search

Navigation