January  2022, 21(1): 293-313. doi: 10.3934/cpaa.2021178

Inverse scattering transform and soliton solutions of an integrable nonlocal Hirota equation

School of Mathematics, China University of Mining and Technology, Xuzhou, Jiangsu, 221116, China

*Corresponding author. *This author is contributed equally as the first author

Received  June 2021 Revised  September 2021 Published  January 2022 Early access  October 2021

Fund Project: This work was 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 Funds for the Central Universities under the Grant Nos. 2019ZDPY07 and 2019QNA35.
The first author is 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 Funds for the Central Universities under the Grant Nos. 2019ZDPY07 and 2019QNA35.

In this work, we study the inverse scattering transform of a nonlocal Hirota equation in detail, and obtain the corresponding soliton solutions formula. Starting from the Lax pair of this equation, we obtain the corresponding infinite number of conservation laws and some properties of scattering data. By analyzing the direct scattering problem, we get a critical symmetric relation which is different from the local equations. A novel left-right Riemann-Hilbert problem is proposed to develop the inverse scattering theory. The potentials are recovered and the pure soliton solutions formula is obtained when the reflection coefficients are zero. Based on the zero types of scattering data, nine types of soliton solutions are obtained and three typical types are described in detail. In addition, some dynamic behaviors are given to illustrate the soliton characteristics of the space symmetric nonlocal Hirota equation.

Citation: Yuan Li, Shou-Fu Tian. Inverse scattering transform and soliton solutions of an integrable nonlocal Hirota equation. Communications on Pure and Applied Analysis, 2022, 21 (1) : 293-313. doi: 10.3934/cpaa.2021178
References:
[1]

M. J. Ablowitz and Z. H. Musslimani, Integrable nonlocal nonlinear Schrödinger equation, Phys. Rev. Lett., 110 (2013), 064105, 5pp. doi: 10.1103/PhysRevLett.110.064105.

[2]

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.

[3]

M. J. AblowitzB. FengX. Luo and Z. H. Musslimani, Inverse scattering transform for the nonlocal reverse space-time nonlinear schrödinger equation, Theor. Math. Phys., 196 (2018), 1241-1267.  doi: 10.1134/s0040577918090015.

[4]

M. J. AblowitzX. Luo and Z. H. Musslimani, Inverse scattering transform for the nonlocal nonlinear Schrödinger equation with nonzero boundary conditions, J. Math. Phys., 59 (2018), 011501.  doi: 10.1063/1.5018294.

[5]

G. P. Agrawal, Nonlinear Fiber Optics, Springer, Berlin, 2000. doi: 10.1007/3-540-46629-0_9.

[6]

D. Anderson and M. Lisak, Nonlinear asymmetric self-phase modulation and self-steepening of pulses in long optical waveguides, Phys. Rev. A, 27 (1983), 1393-1398.  doi: 10.1103/PhysRevA.27.1393.

[7]

D. J. Benney and A. C. Newell, The propagation of nonlinear wave envelopes, J. Math. Phys., 46 (1967), 133-139.  doi: 10.1002/sapm1967461133.

[8]

J. Cen, F. Correa and A. Fring, Integrable nonlocal Hirota equations, J. Math. Phys., 60 (2019), 081508, 18pp. doi: 10.1063/1.5013154.

[9]

H. ChenY. Lee and C. Liu, Integrability of nonlinear hamiltonian systems by inverse scattering method, Phys. Scr., 20 (1979), 490-492.  doi: 10.1088/0031-8949/20/3-4/026.

[10]

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.

[11]

Martin V. Goldman, Strong turbulence of plasma waves, Rev. Mod. Phys., 56 (1984), 709-735.  doi: 10.1103/revmodphys.56.709.

[12]

Ry ogo Hirota, Exact envelope-soliton solutions of a nonlinear wave equation, J. Math. Phys., 14 (1973), 805-809.  doi: 10.1063/1.1666399.

[13]

J. Ji and Z. 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.

[14]

Y. Kodama and A. Hasegawa, Nonlinear pulse propagation in a monomode dielectric guide, IEEE J. Quantum Electron., 23 (1987), 510-524.  doi: 10.1109/JQE.1987.1073392.

[15]

Z. Q. Li and S. F. Tian, A hierarchy of nonlocal nonlinear evolution equations and $\bar{\partial}$-dressing method, Appl. Math. Lett., 120 (2021), 107254, 8pp. doi: 10.1016/j.aml.2021.107254.

[16]

M. Li and T. Xu, Dark and antidark soliton interactions in the nonlocal nonlinear Schrödinger equation with the self-induced parity-time-symmetric potential, Phys. Rev. E, 91 (2015), 033202, 8pp. doi: 10.1103/PhysRevE.91.033202.

[17]

W. Ma, Riemann-Hilbert problems and soliton solutions of nonlocal real reverse-spacetime mKdV equations, J. Math. Anal. Appl., 498 (2021), 124980, 13pp. doi: 10.1016/j.jmaa.2021.124980.

[18]

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

[19] C. Rogers and W. K. Schief, Bäcklund and Darboux transformations : geometry and modern applications in soliton theory, Cambridge University Press, Cambridge, UK, 2002. 
[20]

A. K. Sarma, M. A. Miri, Z. H. Musslimani and D. N. Christodoulides, Continuous and discrete Schrödinger systems with parity-time-symmetric nonlinearities, Phys. Rev. E, 89 (2014), 052918, 7pp. doi: 10.1103/PhysRevE.89.052918.

[21]

N. Sasa and J. Satsuma, New-type of soliton solutions for a higher-order nonlinear Schrödinger equation, J. Phys. Soc. Jpn., 60 (1991), 409-417.  doi: 10.1143/JPSJ.60.409.

[22]

C. SongD. Xiao and Z. Zhu, Solitons and dynamics for a general integrable nonlocal coupled nonlinear Schrödinger equation, Commun. Nonlinear Sci. Num. Simul., 45 (2017), 13-28.  doi: 10.1016/j.cnsns.2016.09.013.

[23]

Z. Zhou, Darboux transformations and global solutions for a nonlocal derivative nonlinear Schrödinger equation, Commun. Nonlinear Sci. Num. Simul., 62 (2018), 480-488.  doi: 10.1016/j.cnsns.2018.01.008.

show all references

References:
[1]

M. J. Ablowitz and Z. H. Musslimani, Integrable nonlocal nonlinear Schrödinger equation, Phys. Rev. Lett., 110 (2013), 064105, 5pp. doi: 10.1103/PhysRevLett.110.064105.

[2]

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.

[3]

M. J. AblowitzB. FengX. Luo and Z. H. Musslimani, Inverse scattering transform for the nonlocal reverse space-time nonlinear schrödinger equation, Theor. Math. Phys., 196 (2018), 1241-1267.  doi: 10.1134/s0040577918090015.

[4]

M. J. AblowitzX. Luo and Z. H. Musslimani, Inverse scattering transform for the nonlocal nonlinear Schrödinger equation with nonzero boundary conditions, J. Math. Phys., 59 (2018), 011501.  doi: 10.1063/1.5018294.

[5]

G. P. Agrawal, Nonlinear Fiber Optics, Springer, Berlin, 2000. doi: 10.1007/3-540-46629-0_9.

[6]

D. Anderson and M. Lisak, Nonlinear asymmetric self-phase modulation and self-steepening of pulses in long optical waveguides, Phys. Rev. A, 27 (1983), 1393-1398.  doi: 10.1103/PhysRevA.27.1393.

[7]

D. J. Benney and A. C. Newell, The propagation of nonlinear wave envelopes, J. Math. Phys., 46 (1967), 133-139.  doi: 10.1002/sapm1967461133.

[8]

J. Cen, F. Correa and A. Fring, Integrable nonlocal Hirota equations, J. Math. Phys., 60 (2019), 081508, 18pp. doi: 10.1063/1.5013154.

[9]

H. ChenY. Lee and C. Liu, Integrability of nonlinear hamiltonian systems by inverse scattering method, Phys. Scr., 20 (1979), 490-492.  doi: 10.1088/0031-8949/20/3-4/026.

[10]

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.

[11]

Martin V. Goldman, Strong turbulence of plasma waves, Rev. Mod. Phys., 56 (1984), 709-735.  doi: 10.1103/revmodphys.56.709.

[12]

Ry ogo Hirota, Exact envelope-soliton solutions of a nonlinear wave equation, J. Math. Phys., 14 (1973), 805-809.  doi: 10.1063/1.1666399.

[13]

J. Ji and Z. 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.

[14]

Y. Kodama and A. Hasegawa, Nonlinear pulse propagation in a monomode dielectric guide, IEEE J. Quantum Electron., 23 (1987), 510-524.  doi: 10.1109/JQE.1987.1073392.

[15]

Z. Q. Li and S. F. Tian, A hierarchy of nonlocal nonlinear evolution equations and $\bar{\partial}$-dressing method, Appl. Math. Lett., 120 (2021), 107254, 8pp. doi: 10.1016/j.aml.2021.107254.

[16]

M. Li and T. Xu, Dark and antidark soliton interactions in the nonlocal nonlinear Schrödinger equation with the self-induced parity-time-symmetric potential, Phys. Rev. E, 91 (2015), 033202, 8pp. doi: 10.1103/PhysRevE.91.033202.

[17]

W. Ma, Riemann-Hilbert problems and soliton solutions of nonlocal real reverse-spacetime mKdV equations, J. Math. Anal. Appl., 498 (2021), 124980, 13pp. doi: 10.1016/j.jmaa.2021.124980.

[18]

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

[19] C. Rogers and W. K. Schief, Bäcklund and Darboux transformations : geometry and modern applications in soliton theory, Cambridge University Press, Cambridge, UK, 2002. 
[20]

A. K. Sarma, M. A. Miri, Z. H. Musslimani and D. N. Christodoulides, Continuous and discrete Schrödinger systems with parity-time-symmetric nonlinearities, Phys. Rev. E, 89 (2014), 052918, 7pp. doi: 10.1103/PhysRevE.89.052918.

[21]

N. Sasa and J. Satsuma, New-type of soliton solutions for a higher-order nonlinear Schrödinger equation, J. Phys. Soc. Jpn., 60 (1991), 409-417.  doi: 10.1143/JPSJ.60.409.

[22]

C. SongD. Xiao and Z. Zhu, Solitons and dynamics for a general integrable nonlocal coupled nonlinear Schrödinger equation, Commun. Nonlinear Sci. Num. Simul., 45 (2017), 13-28.  doi: 10.1016/j.cnsns.2016.09.013.

[23]

Z. Zhou, Darboux transformations and global solutions for a nonlocal derivative nonlinear Schrödinger equation, Commun. Nonlinear Sci. Num. Simul., 62 (2018), 480-488.  doi: 10.1016/j.cnsns.2018.01.008.

Figure 1.  The single-breather solution (7.15) with $ \eta_1 = 7, \overline{\eta}_1 = -2, \theta_1 = \frac{\pi}{2}, \overline{\theta}_1 = \frac{\pi}{5}, \alpha = 5, \beta = 1 $. $ (a,b,c) $ The local structure, density and intensity profiles of the single-soliton solution $ |q (x,t)|^2 $
Figure 2.  The two-soliton solution (7.19) with $ \lambda_1 = 1.1+0.8i, \overline{\lambda}_1 = 2-i, \theta_1 = \theta_2 = \overline{\theta}_1 = \overline{\theta}_2 = 2\pi, \alpha = 1, \beta = 1 $. $ (a,b,c) $ The local structure, density and intensity profiles with different time of the two-soliton solution $ |q (x,t)|^2 $
Figure 3.  The three-soliton solution (7.24) with $ \lambda_1 = 1.2i, \lambda_2 = 1.1+2i, \overline{\lambda}_1 = -i, \overline{\lambda}_2 = 0.8-i, \theta_j = \overline{\theta}_j = \pi, (1\leq j\leq3) \alpha = \beta = 1 $. $ (a,b,c) $ The local structure, density and intensity profiles with different time of the three-soliton solution $ |q (x,t)|^2 $
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