Volume 14, Issue 4
Inverse Scattering Transform and Soliton Solutions for the Hirota Equation with $N$ Distinct Arbitrary Order Poles

Adv. Appl. Math. Mech., 14 (2022), pp. 893-913.

Published online: 2022-04

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• Abstract

We employ the Riemann-Hilbert (RH) method to study the Hirota equation with arbitrary order zero poles under zero boundary conditions. Through the spectral analysis, the asymptoticity, symmetry, and analysis of the Jost functions are obtained, which play a key role in constructing the RH problem. Then we successfully established the exact solution of the equation without reflection potential by solving the RH problem. Choosing some appropriate parameters of the resulting solutions, we further derive the soliton solutions with different order poles, including four cases of a fourth-order pole, two second-order poles, a third-order pole and a first-order pole, and four first-order points. Finally, the dynamical behavior of these solutions are analyzed via graphic analysis.

• Keywords

The Hirota equation, zero boundary condition, Riemann-Hilbert problem, high-order poles, soliton solutions.

35C08, 35Q15, 35Q51

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@Article{AAMM-14-893, author = {}, title = {Inverse Scattering Transform and Soliton Solutions for the Hirota Equation with $N$ Distinct Arbitrary Order Poles}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2022}, volume = {14}, number = {4}, pages = {893--913}, abstract = {

We employ the Riemann-Hilbert (RH) method to study the Hirota equation with arbitrary order zero poles under zero boundary conditions. Through the spectral analysis, the asymptoticity, symmetry, and analysis of the Jost functions are obtained, which play a key role in constructing the RH problem. Then we successfully established the exact solution of the equation without reflection potential by solving the RH problem. Choosing some appropriate parameters of the resulting solutions, we further derive the soliton solutions with different order poles, including four cases of a fourth-order pole, two second-order poles, a third-order pole and a first-order pole, and four first-order points. Finally, the dynamical behavior of these solutions are analyzed via graphic analysis.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2020-0369}, url = {http://global-sci.org/intro/article_detail/aamm/20439.html} }
TY - JOUR T1 - Inverse Scattering Transform and Soliton Solutions for the Hirota Equation with $N$ Distinct Arbitrary Order Poles JO - Advances in Applied Mathematics and Mechanics VL - 4 SP - 893 EP - 913 PY - 2022 DA - 2022/04 SN - 14 DO - http://doi.org/10.4208/aamm.OA-2020-0369 UR - https://global-sci.org/intro/article_detail/aamm/20439.html KW - The Hirota equation, zero boundary condition, Riemann-Hilbert problem, high-order poles, soliton solutions. AB -

We employ the Riemann-Hilbert (RH) method to study the Hirota equation with arbitrary order zero poles under zero boundary conditions. Through the spectral analysis, the asymptoticity, symmetry, and analysis of the Jost functions are obtained, which play a key role in constructing the RH problem. Then we successfully established the exact solution of the equation without reflection potential by solving the RH problem. Choosing some appropriate parameters of the resulting solutions, we further derive the soliton solutions with different order poles, including four cases of a fourth-order pole, two second-order poles, a third-order pole and a first-order pole, and four first-order points. Finally, the dynamical behavior of these solutions are analyzed via graphic analysis.

Xiaofan Zhang, Shoufu Tian & Jinjie Yang. (2022). Inverse Scattering Transform and Soliton Solutions for the Hirota Equation with $N$ Distinct Arbitrary Order Poles. Advances in Applied Mathematics and Mechanics. 14 (4). 893-913. doi:10.4208/aamm.OA-2020-0369
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