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Expressions and Evolution of Traveling wave Solutions in a Generalized Two-Component Rotation b-Family System

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Abstract

In this paper we investigate traveling waves for a generalized two-component rotation b-family (R-b-family) system with \(b>1\). Based on qualitative theory and bifurcation method of dynamical systems, the traveling wave problem is converted into the dynamical analysis of the corresponding traveling wave system with 5 parameters and 2 distinct singular lines. We systematically analyze this traveling wave system with the help of the three-step method to obtain 6 bifurcation curves of its phase portraits, which enables us to draw all the phase portraits. Combining these phase portraits, we get that the R-b-family system has exactly 6 kinds of bounded traveling wave solutions and give all the explicit conditions for their existence as well as their expressions and coexistence. Finally, discussing dynamical behavior of these traveling waves, we not only provide the bifurcation wave velocities for solitary wave solutions of peak and valley type, for solitary cusp wave solutions of peak and valley type and for periodic cusp wave solutions of peak and valley type, respectively, but also find 3 other types of traveling wave evolution, namely kink wave bifurcation, solitary cusp bifurcation and periodic wave bifurcation.

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Funding

This work is supported by the Natural Science Foundation of China (No. 12271378) and the Opening Project of Sichuan Province University Key Laboratory of Bridge Non-destruction Detecting and Engineering Computing (No. 2022QZJ02).

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  1. School of Science, Civil Aviation Flight University of China, Guanghan, Sichuan, 618307, People’s Republic of China

    Feiting Fan

  2. School of Mathematics, Sichuan University, Chengdu, Sichuan, 610064, People’s Republic of China

    Xingwu Chen

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All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by Xingwu Chen and Feiting Fan. The first draft of the manuscript was written by Feiting Fan and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

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Fan, F., Chen, X. Expressions and Evolution of Traveling wave Solutions in a Generalized Two-Component Rotation b-Family System. Qual. Theory Dyn. Syst. 22, 68 (2023). https://doi.org/10.1007/s12346-023-00766-3

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