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doi: 10.3934/dcdsb.2021259
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Inverse scattering transform for the integrable nonlocal Lakshmanan-Porsezian-Daniel equation

School of Mathematics and Institute of Mathematical Physics, China University of Mining and Technology, Xuzhou 221116, China

Corresponding author: Shou-Fu Tian

*This author is contributed equally as the first author

Received  May 2021 Revised  September 2021 Early access November 2021

Fund Project: The authors are supported by the National Natural Science Foundation of China under Grant No. 11975306, the Natural Science Foundation of Jiangsu Province under Grant No. BK20181351, the Six Talent Peaks Project in Jiangsu Province under Grant No. JY-059, and the Fundamental Research Fund for the Central Universities under the Grant Nos. 2019ZDPY07 and 2019QNA35

In this work, a generalized nonlocal Lakshmanan-Porsezian-Daniel (LPD) equation is introduced, and its integrability as an infinite dimensional Hamilton dynamic system is established. We successfully derive the inverse scattering transform (IST) of the nonlocal LPD equation. The direct scattering problem of the equation is first constructed, and some important symmetries of the eigenfunctions and the scattering data are discussed. By using a novel Left-Right Riemann-Hilbert (RH) problem, the inverse scattering problem is analyzed, and the potential function is recovered. By introducing the special conditions of reflectionless case, the time-periodic soliton solutions formula of the equation is derived successfully. Take $ J = \overline{J} = 1,2,3 $ and $ 4 $ for example, we obtain some interesting phenomenon such as breather-type solitons, arc solitons, three soliton and four soliton. Furthermore, the influence of parameter $ \delta $ on these solutions is further considered via the graphical analysis. Finally, the eigenvalues and conserved quantities are investigated under a few special initial conditions.

Citation: Wei-Kang Xun, Shou-Fu Tian, Tian-Tian Zhang. Inverse scattering transform for the integrable nonlocal Lakshmanan-Porsezian-Daniel equation. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021259
References:
[1]

M. J. Ablowitz and D. E. Baldwin, Nonlinear shallow ocean-wave soliton interactions on flat beaches, Phys. Rev. E., 86 (2012), 036305.  doi: 10.1103/PhysRevE.86.036305.  Google Scholar

[2] M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, 1991.  doi: 10.1017/CBO9780511623998.  Google Scholar
[3]

M. J. Ablowitz and Z. H. Musslimani, Integrable nonlocal nonlinear Schrödinger equation, Phys. Rev. Lett., 110 (2013), 064105.  doi: 10.1103/PhysRevLett.110.064105.  Google Scholar

[4]

M. J. Ablowitz and Z. H. Musslimani, Inverse scattering transform for the integrable nonlocal nonlinear Schrödinger equation, Nonlinearity, 29 (2016), 915-946.  doi: 10.1088/0951-7715/29/3/915.  Google Scholar

[5]

M. J. Ablowitz and Z. H. Musslimani, Integrable nonlocal nonlinear equations, Stud. Appl. Math., 139 (2017), 7-59.  doi: 10.1111/sapm.12153.  Google Scholar

[6]

M. J. AblowitzB.-F. FengX-D. Luo and Z. H. Musslimani, Reverse space-time nonlocal Sine-Gordon/Sinh-Gordon equations with nonzero boundary conditions, Stud. Appl. Math., 141 (2018), 267-307.  doi: 10.1111/sapm.12222.  Google Scholar

[7]

M. J. Ablowitz and Z. H. Musslimani, Integrable space-time shifted nonlocal nonlinear equations, Phys. Lett. A, 409 (2021), 127516.  doi: 10.1016/j.physleta.2021.127516.  Google Scholar

[8] M. J. AblowitzB. Prinari and A. D. Trubatch, Discrete and Continuous Nonlinear Schrödinger Systems, Cambridge University Press, 2004.   Google Scholar
[9]

M. J. Ablowitz and H. Segur, Olitons and the Inverse Scattering Transform, SIAM, 1981.  Google Scholar

[10]

E. M. DianovP. Mamyshev and A. M. Prokhorov, Nonlinear fiber optics, Soviet J. Quantum Elect, 18 (1988), 1.   Google Scholar

[11]

A. S. Fokas, A unified transform method for solving linear and certain nonlinear PDEs, Proc. R. Soc. Lond. Ser. A., 453 (1962), 1411-1443.  doi: 10.1098/rspa.1997.0077.  Google Scholar

[12]

A. S. Fokas, Integrable multidimensional versions of the nonlocal nonlinear Schrödinger equation, Nonlinearity, 29 (2016), 319-324.  doi: 10.1088/0951-7715/29/2/319.  Google Scholar

[13]

X. Geng and H. Liu, The nonlinear steepest descent method to long-time asymptotics of the coupled nonlinear Schrödinger equation, J. Nonlinear. Sci., 28 (2018), 739-763.  doi: 10.1007/s00332-017-9426-x.  Google Scholar

[14]

X. GengH. Liu and J. Zhu, Initial-boundary value problems for the coupled nonlinear Schrödinger equation on the half-line, Stud. Appl. Math., 135 (2015), 310-346.  doi: 10.1111/sapm.12088.  Google Scholar

[15]

F.-J. HeE.-G. Fan and J. Xu, Long-time asymptotics for the nonlocal MKdV equation, Commun. Theor. Phys., 71 (2019), 475-488.  doi: 10.1088/0253-6102/71/5/475.  Google Scholar

[16] R. Hirota, The Direct Methods in Soliton Theory, Cambridge University Press, Cambridge, 2004.  doi: 10.1017/CBO9780511543043.  Google Scholar
[17]

J.-L. Ji and Z.-N. Zhu, Soliton solutions of an integrable nonlocal modified Korteweg-de Vries equation through inverse scattering transform,, J. Math. Anal. Appl., 453 (2017), 973-984.  doi: 10.1016/j.jmaa.2017.04.042.  Google Scholar

[18]

B. B. Kadomtsev and V. I. Petviashvili, On the stability of solitary waves in weakly dispersing media, Sov. Phys. Dokl., 15 (1970), 539.   Google Scholar

[19] Y. S. Kivshar and G. Agrawal, Optical Solitons: From Fibers to Photonic Crystals, Academic press, 2003.   Google Scholar
[20]

M. LakshmananK. Porsezian and M. Daniel, Effect of discreteness on the continuum limit of the Heisenberg spin chain, Phys. Lett. A, 133 (1988), 483-488.  doi: 10.1016/0375-9601(88)90520-8.  Google Scholar

[21]

W. LiuD.-Q. QiuZ.-W. Wu and J.-S. He, Dynamical behavior of solution in integrable nonlocal Lakshmanan-Porsezian-Daniel equation, Commun., Theor. Phys., 65 (2016), 671-676.  doi: 10.1088/0253-6102/65/6/671.  Google Scholar

[22]

S. Y. Lou, Alice-Bob systems, $P_s$-$T_d$-$C$ principles and multi-soliton solutions, arXiv: nlin/1603.03975. Google Scholar

[23]

W.-X. Ma, Riemann-Hilbert problems and $N$-soliton solutions for a coupled mKdV system., J. Geom. Phys., 132 (2018), 45-54.  doi: 10.1016/j.geomphys.2018.05.024.  Google Scholar

[24]

W.-X. Ma, Inverse scattering for nonlocal reverse-time nonlinear Schrödinger equations, Appl. Math. Lett., 102 (2020), 106161.  doi: 10.1016/j.aml.2019.106161.  Google Scholar

[25]

W.-X. Ma and X. Geng, Bäcklund transformations of soliton systems from symmetry constraints, CRM Proceedings and Lecture Notes, 29 (2001), 313-323.  doi: 10.1090/crmp/029/28.  Google Scholar

[26]

V. B. Matveev, Darboux transformation and explicit solutions of the Kadomtcev-Petviaschvily equation, depending on functional parameters, Lett. Math. Phys., 3 (1979), 213-216.  doi: 10.1007/BF00405295.  Google Scholar

[27]

S. Novikov, S. V. Manakov, L. P. Pitaevskii and V. E. Zakharov, Theory of Solitons: The Inverse Scattering Method, Springer Science & Business Media, 1984.  Google Scholar

[28]

W.-Q. PengS.-F. TianT.-T. Zhang and Y. Fang, Rational and semi-rational solutions of a nonlocal (2+1)-dimensional nonlinear Schrödinger equation, Math. Meth. Appl. Sci., 42 (2019), 6865-6877.  doi: 10.1002/mma.5792.  Google Scholar

[29]

K. PorsezianM. Daniel and M. Lakshmanan, On the integrability aspects of the one-dimensional classical continuum isotropic biquadratic Heisenberg spin chain, J. Math. Phys., 33 (1992), 1807-1816.  doi: 10.1063/1.529658.  Google Scholar

[30]

J. RaoY. ZhangA. S. Fokas and J. He, Rogue waves of the nonlocal Davey-Stewartson I equation, Nonlinearity, 31 (2018), 4090-4107.  doi: 10.1088/1361-6544/aac761.  Google Scholar

[31]

J. WangH. Wu and D.-J. Zhang, Solutions of the nonlocal (2+1)-D breaking solitons hierarchy and the negative order AKNS hierarchy, Commun. Theor. Phys., 72 (2020), 045002-45014.   Google Scholar

[32]

D.-S. Wang, D.-J. Zhang and J. Yang, Integrable properties of the general coupled nonlinear Schrdinger equations, J. Math. Phys., 51 (2010), 023510, 17 pp. doi: 10.1063/1.3290736.  Google Scholar

[33]

L. H. WangK. Porsezian and J. S. He, Breather and rogue wave solutions of a generalized nonlinear Schrödinger equation, Phys. Rev. E, 87 (2013), 053202.  doi: 10.1103/PhysRevE.87.053202.  Google Scholar

[34]

W. Weng and Z. Yan, Inverse scattering and N-triple-pole soliton and breather solutions of the focusing nonlinear Schrödinger hierarchy with nonzero boundary conditions, Phys. Lett. A, 407 (2021), 127472.  doi: 10.1016/j.physleta.2021.127472.  Google Scholar

[35]

Z. Yan, An initial-boundary value problem for the integrable spin-1 Gross-Pitaevskii equations with a 4$\times$ 4 Lax pair on the half-line, Chaos., 27 (2017), 053117.  doi: 10.1063/1.4984025.  Google Scholar

[36]

Z. Yan, Integrable PT-symmetric local and nonlocal vector nonlinear Schrödinger equations: A unified two-parameter model, Appl. Math. Lett., 47 (2015), 61-68.  doi: 10.1016/j.aml.2015.02.025.  Google Scholar

[37]

B. Yang and J. Yang, Transformations between nonlocal and local integrable equations, Stud. Appl. Math., 140 (2018), 178-201.  doi: 10.1111/sapm.12195.  Google Scholar

[38]

Y. Yang and E. Fan, Riemann-Hilbert approach to the modified nonlinear Schrödinger equation with non-vanishing asymptotic boundary conditions, Phys. D, 417 (2021), Paper No. 132811, 20 pp, arXiv e-prints, 2019. doi: 10.1016/j.physd.2020.132811.  Google Scholar

[39]

G. Zhang and Z. Yan, The derivative nonlinear Schrödinger equation with zero/nonzero boundary conditions: Inverse scattering transforms and N-double-pole solutions, J. Nonlinear Sci., 30 (2020), 3089-3127.  doi: 10.1007/s00332-020-09645-6.  Google Scholar

[40]

G. Zhang and Z. Yan, Inverse scattering transforms and soliton solutions of focusing and defocusing nonlocal mKdV equations with non-zero boundary conditions, Phys. D, 402 (2020), 132170.  doi: 10.1016/j.physd.2019.132170.  Google Scholar

[41]

G. Zhang and Z. Yan, Focusing and defocusing mKdV equations with nonzero boundary conditions: Inverse scattering transforms and soliton interactions, Phys. D, 410 (2020), 132521.  doi: 10.1016/j.physd.2020.132521.  Google Scholar

[42]

Z.-X. Zhou, Darboux transformations and global explicit solutions for nonlocal Davey-Stewartson I equation, Stud. Appl. Math., 141 (2018), 186-204.  doi: 10.1111/sapm.12219.  Google Scholar

show all references

References:
[1]

M. J. Ablowitz and D. E. Baldwin, Nonlinear shallow ocean-wave soliton interactions on flat beaches, Phys. Rev. E., 86 (2012), 036305.  doi: 10.1103/PhysRevE.86.036305.  Google Scholar

[2] M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, 1991.  doi: 10.1017/CBO9780511623998.  Google Scholar
[3]

M. J. Ablowitz and Z. H. Musslimani, Integrable nonlocal nonlinear Schrödinger equation, Phys. Rev. Lett., 110 (2013), 064105.  doi: 10.1103/PhysRevLett.110.064105.  Google Scholar

[4]

M. J. Ablowitz and Z. H. Musslimani, Inverse scattering transform for the integrable nonlocal nonlinear Schrödinger equation, Nonlinearity, 29 (2016), 915-946.  doi: 10.1088/0951-7715/29/3/915.  Google Scholar

[5]

M. J. Ablowitz and Z. H. Musslimani, Integrable nonlocal nonlinear equations, Stud. Appl. Math., 139 (2017), 7-59.  doi: 10.1111/sapm.12153.  Google Scholar

[6]

M. J. AblowitzB.-F. FengX-D. Luo and Z. H. Musslimani, Reverse space-time nonlocal Sine-Gordon/Sinh-Gordon equations with nonzero boundary conditions, Stud. Appl. Math., 141 (2018), 267-307.  doi: 10.1111/sapm.12222.  Google Scholar

[7]

M. J. Ablowitz and Z. H. Musslimani, Integrable space-time shifted nonlocal nonlinear equations, Phys. Lett. A, 409 (2021), 127516.  doi: 10.1016/j.physleta.2021.127516.  Google Scholar

[8] M. J. AblowitzB. Prinari and A. D. Trubatch, Discrete and Continuous Nonlinear Schrödinger Systems, Cambridge University Press, 2004.   Google Scholar
[9]

M. J. Ablowitz and H. Segur, Olitons and the Inverse Scattering Transform, SIAM, 1981.  Google Scholar

[10]

E. M. DianovP. Mamyshev and A. M. Prokhorov, Nonlinear fiber optics, Soviet J. Quantum Elect, 18 (1988), 1.   Google Scholar

[11]

A. S. Fokas, A unified transform method for solving linear and certain nonlinear PDEs, Proc. R. Soc. Lond. Ser. A., 453 (1962), 1411-1443.  doi: 10.1098/rspa.1997.0077.  Google Scholar

[12]

A. S. Fokas, Integrable multidimensional versions of the nonlocal nonlinear Schrödinger equation, Nonlinearity, 29 (2016), 319-324.  doi: 10.1088/0951-7715/29/2/319.  Google Scholar

[13]

X. Geng and H. Liu, The nonlinear steepest descent method to long-time asymptotics of the coupled nonlinear Schrödinger equation, J. Nonlinear. Sci., 28 (2018), 739-763.  doi: 10.1007/s00332-017-9426-x.  Google Scholar

[14]

X. GengH. Liu and J. Zhu, Initial-boundary value problems for the coupled nonlinear Schrödinger equation on the half-line, Stud. Appl. Math., 135 (2015), 310-346.  doi: 10.1111/sapm.12088.  Google Scholar

[15]

F.-J. HeE.-G. Fan and J. Xu, Long-time asymptotics for the nonlocal MKdV equation, Commun. Theor. Phys., 71 (2019), 475-488.  doi: 10.1088/0253-6102/71/5/475.  Google Scholar

[16] R. Hirota, The Direct Methods in Soliton Theory, Cambridge University Press, Cambridge, 2004.  doi: 10.1017/CBO9780511543043.  Google Scholar
[17]

J.-L. Ji and Z.-N. Zhu, Soliton solutions of an integrable nonlocal modified Korteweg-de Vries equation through inverse scattering transform,, J. Math. Anal. Appl., 453 (2017), 973-984.  doi: 10.1016/j.jmaa.2017.04.042.  Google Scholar

[18]

B. B. Kadomtsev and V. I. Petviashvili, On the stability of solitary waves in weakly dispersing media, Sov. Phys. Dokl., 15 (1970), 539.   Google Scholar

[19] Y. S. Kivshar and G. Agrawal, Optical Solitons: From Fibers to Photonic Crystals, Academic press, 2003.   Google Scholar
[20]

M. LakshmananK. Porsezian and M. Daniel, Effect of discreteness on the continuum limit of the Heisenberg spin chain, Phys. Lett. A, 133 (1988), 483-488.  doi: 10.1016/0375-9601(88)90520-8.  Google Scholar

[21]

W. LiuD.-Q. QiuZ.-W. Wu and J.-S. He, Dynamical behavior of solution in integrable nonlocal Lakshmanan-Porsezian-Daniel equation, Commun., Theor. Phys., 65 (2016), 671-676.  doi: 10.1088/0253-6102/65/6/671.  Google Scholar

[22]

S. Y. Lou, Alice-Bob systems, $P_s$-$T_d$-$C$ principles and multi-soliton solutions, arXiv: nlin/1603.03975. Google Scholar

[23]

W.-X. Ma, Riemann-Hilbert problems and $N$-soliton solutions for a coupled mKdV system., J. Geom. Phys., 132 (2018), 45-54.  doi: 10.1016/j.geomphys.2018.05.024.  Google Scholar

[24]

W.-X. Ma, Inverse scattering for nonlocal reverse-time nonlinear Schrödinger equations, Appl. Math. Lett., 102 (2020), 106161.  doi: 10.1016/j.aml.2019.106161.  Google Scholar

[25]

W.-X. Ma and X. Geng, Bäcklund transformations of soliton systems from symmetry constraints, CRM Proceedings and Lecture Notes, 29 (2001), 313-323.  doi: 10.1090/crmp/029/28.  Google Scholar

[26]

V. B. Matveev, Darboux transformation and explicit solutions of the Kadomtcev-Petviaschvily equation, depending on functional parameters, Lett. Math. Phys., 3 (1979), 213-216.  doi: 10.1007/BF00405295.  Google Scholar

[27]

S. Novikov, S. V. Manakov, L. P. Pitaevskii and V. E. Zakharov, Theory of Solitons: The Inverse Scattering Method, Springer Science & Business Media, 1984.  Google Scholar

[28]

W.-Q. PengS.-F. TianT.-T. Zhang and Y. Fang, Rational and semi-rational solutions of a nonlocal (2+1)-dimensional nonlinear Schrödinger equation, Math. Meth. Appl. Sci., 42 (2019), 6865-6877.  doi: 10.1002/mma.5792.  Google Scholar

[29]

K. PorsezianM. Daniel and M. Lakshmanan, On the integrability aspects of the one-dimensional classical continuum isotropic biquadratic Heisenberg spin chain, J. Math. Phys., 33 (1992), 1807-1816.  doi: 10.1063/1.529658.  Google Scholar

[30]

J. RaoY. ZhangA. S. Fokas and J. He, Rogue waves of the nonlocal Davey-Stewartson I equation, Nonlinearity, 31 (2018), 4090-4107.  doi: 10.1088/1361-6544/aac761.  Google Scholar

[31]

J. WangH. Wu and D.-J. Zhang, Solutions of the nonlocal (2+1)-D breaking solitons hierarchy and the negative order AKNS hierarchy, Commun. Theor. Phys., 72 (2020), 045002-45014.   Google Scholar

[32]

D.-S. Wang, D.-J. Zhang and J. Yang, Integrable properties of the general coupled nonlinear Schrdinger equations, J. Math. Phys., 51 (2010), 023510, 17 pp. doi: 10.1063/1.3290736.  Google Scholar

[33]

L. H. WangK. Porsezian and J. S. He, Breather and rogue wave solutions of a generalized nonlinear Schrödinger equation, Phys. Rev. E, 87 (2013), 053202.  doi: 10.1103/PhysRevE.87.053202.  Google Scholar

[34]

W. Weng and Z. Yan, Inverse scattering and N-triple-pole soliton and breather solutions of the focusing nonlinear Schrödinger hierarchy with nonzero boundary conditions, Phys. Lett. A, 407 (2021), 127472.  doi: 10.1016/j.physleta.2021.127472.  Google Scholar

[35]

Z. Yan, An initial-boundary value problem for the integrable spin-1 Gross-Pitaevskii equations with a 4$\times$ 4 Lax pair on the half-line, Chaos., 27 (2017), 053117.  doi: 10.1063/1.4984025.  Google Scholar

[36]

Z. Yan, Integrable PT-symmetric local and nonlocal vector nonlinear Schrödinger equations: A unified two-parameter model, Appl. Math. Lett., 47 (2015), 61-68.  doi: 10.1016/j.aml.2015.02.025.  Google Scholar

[37]

B. Yang and J. Yang, Transformations between nonlocal and local integrable equations, Stud. Appl. Math., 140 (2018), 178-201.  doi: 10.1111/sapm.12195.  Google Scholar

[38]

Y. Yang and E. Fan, Riemann-Hilbert approach to the modified nonlinear Schrödinger equation with non-vanishing asymptotic boundary conditions, Phys. D, 417 (2021), Paper No. 132811, 20 pp, arXiv e-prints, 2019. doi: 10.1016/j.physd.2020.132811.  Google Scholar

[39]

G. Zhang and Z. Yan, The derivative nonlinear Schrödinger equation with zero/nonzero boundary conditions: Inverse scattering transforms and N-double-pole solutions, J. Nonlinear Sci., 30 (2020), 3089-3127.  doi: 10.1007/s00332-020-09645-6.  Google Scholar

[40]

G. Zhang and Z. Yan, Inverse scattering transforms and soliton solutions of focusing and defocusing nonlocal mKdV equations with non-zero boundary conditions, Phys. D, 402 (2020), 132170.  doi: 10.1016/j.physd.2019.132170.  Google Scholar

[41]

G. Zhang and Z. Yan, Focusing and defocusing mKdV equations with nonzero boundary conditions: Inverse scattering transforms and soliton interactions, Phys. D, 410 (2020), 132521.  doi: 10.1016/j.physd.2020.132521.  Google Scholar

[42]

Z.-X. Zhou, Darboux transformations and global explicit solutions for nonlocal Davey-Stewartson I equation, Stud. Appl. Math., 141 (2018), 186-204.  doi: 10.1111/sapm.12219.  Google Scholar

Figure 1.  One-soliton solution with parameters $ \delta = 1 $, $ \theta_1 = \frac{\pi}{3} $, $ \overline{\theta}_1 = - \frac{\pi}{3} $, $ \zeta_1 = 0.8i $ and $ \overline{\zeta}_1 = -0.8i $. $ \textbf{(a)} $: the structures of the one-soliton solution, $ \textbf{(b)} $: the density plot, $ \textbf{(c)} $: the wave propagation of the one-soliton solution
Figure 2.  Two-soliton solutions with parameters $ \theta_1 = \frac{2}{3}\pi $, $ \theta_2 = \frac{3}{8}\pi $, $ \overline{\theta}_1 = \frac{2}{3}\pi $, $ \overline{\theta}_2 = \frac{3}{5}\pi $, $ \zeta_1 = 0.7+0.5i $, $ \zeta_2 = -0.7+0.5i $, $ \overline{\zeta}_1 = 0.7-0.5i $ and $ \overline{\zeta}_2 = -0.7-0.5i $. $ \textbf{(a)(b)(c)} $: the structures and the wave propagation of the two-soliton solutions with $ \delta = 5 $, $ \textbf{(d)(e)(f)} $: the structures and the wave propagation of the two-soliton solutions with $ \delta = 3 $, $ \textbf{(g)(h)(i)} $: the structures and the wave propagation of the two-soliton solutions with $ \delta = 1 $
Figure 3.  Breather-type solution with parameters $ \delta = 1 $, $ \theta_1 = \frac{2}{3}\pi $, $ \theta_2 = \frac{3}{8}\pi $, $ \overline{\theta}_1 = \frac{2}{3}\pi $, $ \overline{\theta}_2 = \frac{3}{5}\pi $, $ \zeta_1 = 0.1i $, $ \zeta_2 = 0.2i $, $ \overline{\zeta}_1 = -0.1i $ and $ \overline{\zeta}_2 = -0.2i $. $ \textbf{(a)} $: the structures of the breather-type solution, $ \textbf{(b)} $: the density plot, $ \textbf{(c)} $: the wave propagation of the breather-type solution
Figure 4.  Three-soliton solutions with parameters $ \theta_1 = \theta_2 = \theta_3 = \frac{\pi}{3} $, $ \overline{\theta}_1 = \overline{\theta}_2 = \overline{\theta}_3 = \frac{\pi}{9} $, $ \zeta_1 = 0.3i $, $ \zeta_2 = 0.5i $, $ \zeta_3 = 0.7i $, $ \overline{\zeta}_1 = -0.3i $, $ \overline{\zeta}_2 = -0.5i $ and $ \overline{\zeta}_3 = -0.7i $. $ \textbf{(a)(b)(c)} $: the structures and the wave propagation of the three-soliton solutions with $ \delta = 1 $, $ \textbf{(d)(e)(f)} $: the structures and the wave propagation of the three-soliton solutions with $ \delta = 2 $, $ \textbf{(g)(h)(i)} $: the structures and the wave propagation of the three-soliton solutions with $ \delta = 3 $
Figure 5.  Three-soliton solutions with parameters $ \theta_1 = \theta_2 = \theta_3 = \frac{\pi}{3} $, $ \overline{\theta}_1 = \overline{\theta}_2 = \overline{\theta}_3 = \frac{\pi}{9} $, $ \zeta_1 = 0.3+0.5i $, $ \zeta_2 = -0.3+0.5i $, $ \zeta_3 = 0.6i $, $ \overline{\zeta}_1 = 0.3-0.5i $, $ \overline{\zeta}_2 = -0.3-0.5i $ and $ \overline{\zeta}_3 = -0.6i $. $ \textbf{(a)(b)(c)} $: the structures and the wave propagation of the three-soliton solutions with $ \delta = 2 $, $ \textbf{(d)(e)(f)} $: the structures and the wave propagation of the three-soliton solutions with $ \delta = 1 $, $ \textbf{(g)(h)(i)} $: the structures and the wave propagation of the three-soliton solutions with $ \delta = 0.5 $
Figure 6.  Four-soliton solution with parameters $ \delta = 1 $, $ \theta_1 = \theta_2 = \theta_3 = \theta_4 = \frac{\pi}{6} $, $ \overline{\theta}_1 = \overline{\theta}_2 = \overline{\theta}_3 = \overline{\theta}_4 = \frac{\pi}{8} $, $ \zeta_1 = 0.1i $, $ \zeta_2 = 0.2i $, $ \zeta_3 = 0.3i $, $ \zeta_4 = 0.4i $, $ \overline{\zeta}_1 = -0.1i $, $ \overline{\zeta}_2 = -0.2i $, $ \overline{\zeta}_3 = -0.3i $ and $ \overline{\zeta}_4 = -0.4i $. $ \textbf{(a)} $: the structures of the four-soliton solution, $ \textbf{(b)} $: the density plot, $ \textbf{(c)} $: the wave propagation of the four-soliton solution
Figure 7.  Four-soliton solution with parameters $ \delta = 1 $, $ \theta_1 = \theta_2 = \theta_3 = \theta_4 = \frac{\pi}{6} $, $ \overline{\theta}_1 = \overline{\theta}_2 = \overline{\theta}_3 = \overline{\theta}_4 = \frac{\pi}{8} $, $ \zeta_1 = 0.1+0.2i $, $ \zeta_2 = -0.1+0.2i $, $ \zeta_3 = 0.3i $, $ \zeta_4 = 0.4i $, $ \overline{\zeta}_1 = 0.1-0.2i $, $ \overline{\zeta}_2 = -0.1-0.2i $, $ \overline{\zeta}_3 = -0.3i $ and $ \overline{\zeta}_4 = -0.4i $. $ \textbf{(a)} $: the structures of the four-soliton solution, $ \textbf{(b)} $: the density plot, $ \textbf{(c)} $: the wave propagation of the four-soliton solution
Figure 8.  Four-soliton solution with parameters $ \delta = 1 $, $ \theta_1 = \theta_2 = \theta_3 = \theta_4 = \frac{\pi}{6} $, $ \overline{\theta}_1 = \overline{\theta}_2 = \overline{\theta}_3 = \overline{\theta}_4 = \frac{\pi}{8} $, $ \zeta_1 = 0.1+0.2i $, $ \zeta_2 = -0.1+0.2i $, $ \zeta_3 = 0.4+0.3i $, $ \zeta_4 = -0.4+0.3i $, $ \overline{\zeta}_1 = 0.1-0.2i $, $ \overline{\zeta}_2 = -0.1-0.2i $, $ \overline{\zeta}_3 = 0.4-0.3i $ and $ \overline{\zeta}_4 = -0.4-0.3i $. $ \textbf{(a)} $: the structures of the four-soliton solution, $ \textbf{(b)} $: the density plot, $ \textbf{(c)} $: the wave propagation of the four-soliton solution
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