An Integrable Nonlinear Wave Model: Darboux Transform and Exact Solutions

Jingru Geng; Minna Feng

doi:doi:10.11648/j.acm.20231201.11

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An Integrable Nonlinear Wave Model: Darboux Transform and Exact Solutions

Jingru Geng, Minna Feng

Published in Applied and Computational Mathematics (Volume 12, Issue 1)

Received: 25 January 2023 Accepted: 13 February 2023 Published: 23 February 2023

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Abstract

Soliton equations are infinite-dimensional integrable systems described by nonlinear partial differential equations. In the mathematical theory of soliton equations, the discovery of integrability of these equations has greatly promoted the understanding of their generality, and thus promoted their rapid development. A key feature of an integrable nonlinear evolution equation is the fact that it can be expressed as the compatibility condition of two linear spectral problems, i.e., a Lax pair, which plays a crucial roles in the Darboux transformation. A major difficulty, however, is the problem of associating nonlinear evolution equations with appropriate spectral problems. Therefore, it is interesting for us to search for the new spectral problem and corresponding nonlinear evolution equations. In this paper, a new integrable nonlinear wave model and its integrable nonlinear reduction are presented by using the introduced 2 × 2 matrix spectral problem. Based on the resulting gauge transforms between the 2 × 2 matrix Lax pairs, Darboux transforms are derived for the integrable nonlinear wave model and its integrable nonlinear reduction, from which an algebraic algorithm for solving this integrable nonlinear wave model and its integrable nonlinear reduction is given. As an application of the Darboux transform, explicit exact solutions of the integrable nonlinear reduction are obtained, including solitons, breathers, and rogue waves.

Published in	Applied and Computational Mathematics (Volume 12, Issue 1)
DOI	10.11648/j.acm.20231201.11
Page(s)	1-8
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

An Integrable Nonlinear Wave Model, Integrable Reduction, Darboux Transform, Exact Solutions

References

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APA Style

Jingru Geng, Minna Feng. (2023). An Integrable Nonlinear Wave Model: Darboux Transform and Exact Solutions. Applied and Computational Mathematics, 12(1), 1-8. https://doi.org/10.11648/j.acm.20231201.11

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ACS Style

Jingru Geng; Minna Feng. An Integrable Nonlinear Wave Model: Darboux Transform and Exact Solutions. Appl. Comput. Math. 2023, 12(1), 1-8. doi: 10.11648/j.acm.20231201.11

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AMA Style

Jingru Geng, Minna Feng. An Integrable Nonlinear Wave Model: Darboux Transform and Exact Solutions. Appl Comput Math. 2023;12(1):1-8. doi: 10.11648/j.acm.20231201.11

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@article{10.11648/j.acm.20231201.11,
  author = {Jingru Geng and Minna Feng},
  title = {An Integrable Nonlinear Wave Model: Darboux Transform and Exact Solutions},
  journal = {Applied and Computational Mathematics},
  volume = {12},
  number = {1},
  pages = {1-8},
  doi = {10.11648/j.acm.20231201.11},
  url = {https://doi.org/10.11648/j.acm.20231201.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20231201.11},
  abstract = {Soliton equations are infinite-dimensional integrable systems described by nonlinear partial differential equations. In the mathematical theory of soliton equations, the discovery of integrability of these equations has greatly promoted the understanding of their generality, and thus promoted their rapid development. A key feature of an integrable nonlinear evolution equation is the fact that it can be expressed as the compatibility condition of two linear spectral problems, i.e., a Lax pair, which plays a crucial roles in the Darboux transformation. A major difficulty, however, is the problem of associating nonlinear evolution equations with appropriate spectral problems. Therefore, it is interesting for us to search for the new spectral problem and corresponding nonlinear evolution equations. In this paper, a new integrable nonlinear wave model and its integrable nonlinear reduction are presented by using the introduced 2 × 2 matrix spectral problem. Based on the resulting gauge transforms between the 2 × 2 matrix Lax pairs, Darboux transforms are derived for the integrable nonlinear wave model and its integrable nonlinear reduction, from which an algebraic algorithm for solving this integrable nonlinear wave model and its integrable nonlinear reduction is given. As an application of the Darboux transform, explicit exact solutions of the integrable nonlinear reduction are obtained, including solitons, breathers, and rogue waves.},
 year = {2023}
}

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TY  - JOUR
T1  - An Integrable Nonlinear Wave Model: Darboux Transform and Exact Solutions
AU  - Jingru Geng
AU  - Minna Feng
Y1  - 2023/02/23
PY  - 2023
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DO  - 10.11648/j.acm.20231201.11
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PB  - Science Publishing Group
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AB  - Soliton equations are infinite-dimensional integrable systems described by nonlinear partial differential equations. In the mathematical theory of soliton equations, the discovery of integrability of these equations has greatly promoted the understanding of their generality, and thus promoted their rapid development. A key feature of an integrable nonlinear evolution equation is the fact that it can be expressed as the compatibility condition of two linear spectral problems, i.e., a Lax pair, which plays a crucial roles in the Darboux transformation. A major difficulty, however, is the problem of associating nonlinear evolution equations with appropriate spectral problems. Therefore, it is interesting for us to search for the new spectral problem and corresponding nonlinear evolution equations. In this paper, a new integrable nonlinear wave model and its integrable nonlinear reduction are presented by using the introduced 2 × 2 matrix spectral problem. Based on the resulting gauge transforms between the 2 × 2 matrix Lax pairs, Darboux transforms are derived for the integrable nonlinear wave model and its integrable nonlinear reduction, from which an algebraic algorithm for solving this integrable nonlinear wave model and its integrable nonlinear reduction is given. As an application of the Darboux transform, explicit exact solutions of the integrable nonlinear reduction are obtained, including solitons, breathers, and rogue waves.
VL  - 12
IS  - 1
ER  -

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Author Information

Jingru Geng

School of Computer and Artificial Intelligence, Zhengzhou University, Zhengzhou, China
Minna Feng

School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, China

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Jingru Geng, Minna Feng. (2023). An Integrable Nonlinear Wave Model: Darboux Transform and Exact Solutions. Applied and Computational Mathematics, 12(1), 1-8. https://doi.org/10.11648/j.acm.20231201.11

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ACS Style

Jingru Geng; Minna Feng. An Integrable Nonlinear Wave Model: Darboux Transform and Exact Solutions. Appl. Comput. Math. 2023, 12(1), 1-8. doi: 10.11648/j.acm.20231201.11

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AMA Style

Jingru Geng, Minna Feng. An Integrable Nonlinear Wave Model: Darboux Transform and Exact Solutions. Appl Comput Math. 2023;12(1):1-8. doi: 10.11648/j.acm.20231201.11

Copy | Download

@article{10.11648/j.acm.20231201.11,
  author = {Jingru Geng and Minna Feng},
  title = {An Integrable Nonlinear Wave Model: Darboux Transform and Exact Solutions},
  journal = {Applied and Computational Mathematics},
  volume = {12},
  number = {1},
  pages = {1-8},
  doi = {10.11648/j.acm.20231201.11},
  url = {https://doi.org/10.11648/j.acm.20231201.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20231201.11},
  abstract = {Soliton equations are infinite-dimensional integrable systems described by nonlinear partial differential equations. In the mathematical theory of soliton equations, the discovery of integrability of these equations has greatly promoted the understanding of their generality, and thus promoted their rapid development. A key feature of an integrable nonlinear evolution equation is the fact that it can be expressed as the compatibility condition of two linear spectral problems, i.e., a Lax pair, which plays a crucial roles in the Darboux transformation. A major difficulty, however, is the problem of associating nonlinear evolution equations with appropriate spectral problems. Therefore, it is interesting for us to search for the new spectral problem and corresponding nonlinear evolution equations. In this paper, a new integrable nonlinear wave model and its integrable nonlinear reduction are presented by using the introduced 2 × 2 matrix spectral problem. Based on the resulting gauge transforms between the 2 × 2 matrix Lax pairs, Darboux transforms are derived for the integrable nonlinear wave model and its integrable nonlinear reduction, from which an algebraic algorithm for solving this integrable nonlinear wave model and its integrable nonlinear reduction is given. As an application of the Darboux transform, explicit exact solutions of the integrable nonlinear reduction are obtained, including solitons, breathers, and rogue waves.},
 year = {2023}
}

Copy | Download

TY  - JOUR
T1  - An Integrable Nonlinear Wave Model: Darboux Transform and Exact Solutions
AU  - Jingru Geng
AU  - Minna Feng
Y1  - 2023/02/23
PY  - 2023
N1  - https://doi.org/10.11648/j.acm.20231201.11
DO  - 10.11648/j.acm.20231201.11
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 1
EP  - 8
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20231201.11
AB  - Soliton equations are infinite-dimensional integrable systems described by nonlinear partial differential equations. In the mathematical theory of soliton equations, the discovery of integrability of these equations has greatly promoted the understanding of their generality, and thus promoted their rapid development. A key feature of an integrable nonlinear evolution equation is the fact that it can be expressed as the compatibility condition of two linear spectral problems, i.e., a Lax pair, which plays a crucial roles in the Darboux transformation. A major difficulty, however, is the problem of associating nonlinear evolution equations with appropriate spectral problems. Therefore, it is interesting for us to search for the new spectral problem and corresponding nonlinear evolution equations. In this paper, a new integrable nonlinear wave model and its integrable nonlinear reduction are presented by using the introduced 2 × 2 matrix spectral problem. Based on the resulting gauge transforms between the 2 × 2 matrix Lax pairs, Darboux transforms are derived for the integrable nonlinear wave model and its integrable nonlinear reduction, from which an algebraic algorithm for solving this integrable nonlinear wave model and its integrable nonlinear reduction is given. As an application of the Darboux transform, explicit exact solutions of the integrable nonlinear reduction are obtained, including solitons, breathers, and rogue waves.
VL  - 12
IS  - 1
ER  -

Copy | Download