Bifurcations and exact solutions of generalized nonlinear Schrödinger equation

Qian Zhang; Ai Ke; Qian Zhang; Ai Ke

doi:10.3934/math.2025237

AIMS Mathematics

2025, Volume 10, Issue 3: 5158-5172. doi: 10.3934/math.2025237

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Research article

Bifurcations and exact solutions of generalized nonlinear Schrödinger equation

Qian Zhang ¹,
Ai Ke ^{2
,
,}

1.
School of Mathematics and Physics, Southwest University of Science and Technology, Mianyang 621010, Sichuan, China
2.
School of Mathematical Sciences, Zhejiang Normal University, Jinhua 321004, Zhejiang, China

Received: 19 December 2024 Revised: 26 February 2025 Accepted: 04 March 2025 Published: 07 March 2025
MSC : 34C23, 34C37, 74J30

To find the exact explicit solutions of the generalized nonlinear Schrödinger equation, we first give the corresponding differential system for the amplitude component, which constitutes a planar dynamical system featuring a singular straight line. By analyzing its corresponding traveling wave system, we can derive the dynamical behavior of the amplitude component and give the corresponding phase portraits. Under different parameter conditions, we obtain exact explicit solitary wave solutions, periodic wave solutions, as well as peakons and periodic peakons. By comparing our results with previous studies on the generalized nonlinear Schrödinger equation, we correct the error regarding the first integral and present accurate solutions to the equation.
- solitary wave,
- periodic wave,
- peakon,
- periodic peakon,
- traveling wave system
Citation: Qian Zhang, Ai Ke. Bifurcations and exact solutions of generalized nonlinear Schrödinger equation[J]. AIMS Mathematics, 2025, 10(3): 5158-5172. doi: 10.3934/math.2025237

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Abstract

To find the exact explicit solutions of the generalized nonlinear Schrödinger equation, we first give the corresponding differential system for the amplitude component, which constitutes a planar dynamical system featuring a singular straight line. By analyzing its corresponding traveling wave system, we can derive the dynamical behavior of the amplitude component and give the corresponding phase portraits. Under different parameter conditions, we obtain exact explicit solitary wave solutions, periodic wave solutions, as well as peakons and periodic peakons. By comparing our results with previous studies on the generalized nonlinear Schrödinger equation, we correct the error regarding the first integral and present accurate solutions to the equation.

References

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