Electron. J. Differential Equations, Vol. 2022 (2022), No. 71, pp. 1-32.

### Integrable nonlinear perturbed hierarchies of NLS-mKdV equation and soliton solutions Qiulan Zhao, Hongbiao Cheng, Xinyue Li, Chuanzhong Li

Abstract:
We propose three spectral problems for NLS-mKdV equation by combining three integrable coupling ways. Then we obtain three nonlinear perturbation terms to derive three integrable nonlinear perturbed hierarchies of the NLS-mKdV equation. We proved the Lax integrability of the integrable nonlinear perturbed hierarchies. On the basis of a special orthogonal group, we prove the Liouville integrability of a third-order integrable nonlinear perturbed hierarchy of NLS-mKdV equation by deriving its bi-Hamiltonian structures. We build three Darboux matrices for constructing the Darboux transformations of the first two equations. As applications of the Darboux transformation, we present explicit solutions of these equations, three-dimensional plots, and density profiles the evolution of solitary waves.

Submitted

Submitted April 29, 2022. Published October 13, 2022.
Math Subject Classifications: 35Q51, 37K10.
Key Words: Integrable perturbed hierarchies; nonlinear perturbation terms; Darboux transformation; soliton solutions.
DOI: https://doi.org/10.58997/ejde.2022.71

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 Qiulan Zhao College of Mathematics and Systems Science Shandong University of Science and Technology Qingdao, 266590, Shandong, China email: qlzhao@sdust.edu.cn Hongbiao Cheng College of Mathematics and Systems Science Shandong University of Science and Technology Qingdao, 266590, Shandong, China email: chenghongbiao1997@163.com Xinyue Li College of Mathematics and Systems Science Shandong University of Science and Technology Qingdao, 266590, Shandong, China email: xyli@sdust.edu.cn Chuanzhong Li College of Mathematics and Systems Science Shandong University of Science and Technology Qingdao, 266590, Shandong, China email: lichuanzhong@sdust.edu.cn