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doi: 10.3934/cpaa.2021178
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## 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 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 & Applied Analysis, 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.  Google Scholar [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.  Google Scholar [3] M. J. Ablowitz, B. Feng, X. 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.  Google Scholar [4] M. J. Ablowitz, X. 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.  Google Scholar [5] G. P. Agrawal, Nonlinear Fiber Optics, Springer, Berlin, 2000. doi: 10.1007/3-540-46629-0_9.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] J. Cen, F. Correa and A. Fring, Integrable nonlocal Hirota equations, J. Math. Phys., 60 (2019), 081508, 18pp. doi: 10.1063/1.5013154.  Google Scholar [9] H. Chen, Y. 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.  Google Scholar [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.  Google Scholar [11] Martin V. Goldman, Strong turbulence of plasma waves, Rev. Mod. Phys., 56 (1984), 709-735.  doi: 10.1103/revmodphys.56.709.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [18] W. Peng, S. Tian, T. 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.  Google Scholar [19] C. Rogers and W. K. Schief, Bäcklund and Darboux transformations : geometry and modern applications in soliton theory, Cambridge University Press, Cambridge, UK, 2002.   Google Scholar [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.  Google Scholar [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.  Google Scholar [22] C. Song, D. 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.  Google Scholar [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.  Google Scholar

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.  Google Scholar [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.  Google Scholar [3] M. J. Ablowitz, B. Feng, X. 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.  Google Scholar [4] M. J. Ablowitz, X. 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.  Google Scholar [5] G. P. Agrawal, Nonlinear Fiber Optics, Springer, Berlin, 2000. doi: 10.1007/3-540-46629-0_9.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] J. Cen, F. Correa and A. Fring, Integrable nonlocal Hirota equations, J. Math. Phys., 60 (2019), 081508, 18pp. doi: 10.1063/1.5013154.  Google Scholar [9] H. Chen, Y. 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.  Google Scholar [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.  Google Scholar [11] Martin V. Goldman, Strong turbulence of plasma waves, Rev. Mod. Phys., 56 (1984), 709-735.  doi: 10.1103/revmodphys.56.709.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [18] W. Peng, S. Tian, T. 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.  Google Scholar [19] C. Rogers and W. K. Schief, Bäcklund and Darboux transformations : geometry and modern applications in soliton theory, Cambridge University Press, Cambridge, UK, 2002.   Google Scholar [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.  Google Scholar [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.  Google Scholar [22] C. Song, D. 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.  Google Scholar [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.  Google Scholar
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 $|q (x,t)|^2$
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 $|q (x,t)|^2$
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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