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

Yuan Li; Shou-Fu Tian

doi:10.3934/cpaa.2021178

Article Contents

2022, Volume 21, Issue 1: 293-313. Doi: 10.3934/cpaa.2021178

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

Yuan Li and
Shou-Fu Tian^,

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

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

Received: June 2021

Revised: September 2021

Early access: October 2021

Published: January 2022

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.

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Abstract

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.

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Mathematics Subject Classification: Primary: 35Q15; 35Q51; 35Q55.

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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 $

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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 $

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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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