BIFURCATIONS AND EXACT TRAVELING WAVE SOLUTIONS FOR THE GENERALIZED NONLINEAR SCHRÖDINGER EQUATION WITH WAVE OPERATOR

Quting Chen; Yadong Shang; Huafei Di

doi:10.11948/20210216

2022 Volume 12 Issue 1

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Quting Chen, Yadong Shang, Huafei Di. BIFURCATIONS AND EXACT TRAVELING WAVE SOLUTIONS FOR THE GENERALIZED NONLINEAR SCHRÖDINGER EQUATION WITH WAVE OPERATOR[J]. Journal of Applied Analysis & Computation, 2022, 12(1): 336-346. doi: 10.11948/20210216

Citation:

Quting Chen, Yadong Shang, Huafei Di. BIFURCATIONS AND EXACT TRAVELING WAVE SOLUTIONS FOR THE GENERALIZED NONLINEAR SCHRÖDINGER EQUATION WITH WAVE OPERATOR[J]. Journal of Applied Analysis & Computation, 2022, 12(1): 336-346. doi: 10.11948/20210216

BIFURCATIONS AND EXACT TRAVELING WAVE SOLUTIONS FOR THE GENERALIZED NONLINEAR SCHRÖDINGER EQUATION WITH WAVE OPERATOR

1.
School of Mathematics and Information Science, Guangzhou University, 510006, Guangzhou, Guangdong, China

Corresponding author: Email address: gzydshang@126.com.(Y. Shang)
Fund Project: This work is supported by the NSF of China (11801108), the Scientific Program of Guangdong Province (2021A1515010314), and the College Scientific Research Project (YG2020005) of Guangzhou University

Abstract

Abstract

In this paper, we investigate the dynamical bifurcations and exact traveling wave solutions for the generalized nonlinear Schrödinger equation with wave operator under different parametric conditions by means of the theory of singular system. We analyse the high order equilibrium point and give the phase portraits. We obtain many results under different values of the parameter $ p $ reflecting the strength of the nonlinearity in the model. For $ p = 1 $, we find explicit exact solutions of Jacobian elliptic functions type which is corresponding to the curves given by $ H(\phi, y) = h $. According to the qualitative analysis of the phase portraits, we give the conclusions on the existence of solitary wave solutions and periodic wave solutions when $ p\geq\frac{1}{2} $. In addition, we obtain the only explicit exact solitary wave solution corresponding to the curves given by $ H(\phi, y) = 0 $ for any $ p $. Especially, we obtain some explicit exact double periodic solutions of elliptic functions type for $ p = \frac{1}{2} $.
- Generalized nonlinear Schrödinger equation /
- wave operator /
- bifurcations /
- solitary wave solution /
- periodic wave solution.
MSC: 35C07, 37G10

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References

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