Orbital stability of periodic traveling waves to some coupled BBM equations

Ye Zhao; Chunfeng Xing; Ye Zhao; Chunfeng Xing

doi:10.3934/math.20231133

AIMS Mathematics

2023, Volume 8, Issue 9: 22225-22236. doi: 10.3934/math.20231133

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Research article

Orbital stability of periodic traveling waves to some coupled BBM equations

Ye Zhao ¹,
Chunfeng Xing ^{2
,
,}

1.
Zhiyuan college, Beijing Institute of Petrochemical Technology, Beijing 102617, China
2.
Institute of Mathematics and Physics, Beijing Union University, Beijing 100101, China

Received: 18 April 2023 Revised: 02 July 2023 Accepted: 05 July 2023 Published: 13 July 2023
MSC : 74J35, 76B25, 35C08, 37K40

In this work, we show some results concerning the orbital stability of dnoidal wave solutions to some Benjamin-Bona-Mahony equations (BBM equations henceforth). First, by the standard argument, we prove the existence of a smooth curve of positive traveling wave solutions of dnoidal type. Then, we show that this type of solutions are orbitally stable by perturbations with the same period L. The major tools to obtain these results are the Grillaks, Shatah and Strauss' general theory in the periodic case. The results in the present paper extend some previous stability results for the BBM equations.
- coupled BBM equations,
- periodic traveling waves,
- orbital stability
Citation: Ye Zhao, Chunfeng Xing. Orbital stability of periodic traveling waves to some coupled BBM equations[J]. AIMS Mathematics, 2023, 8(9): 22225-22236. doi: 10.3934/math.20231133

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Abstract

In this work, we show some results concerning the orbital stability of dnoidal wave solutions to some Benjamin-Bona-Mahony equations (BBM equations henceforth). First, by the standard argument, we prove the existence of a smooth curve of positive traveling wave solutions of dnoidal type. Then, we show that this type of solutions are orbitally stable by perturbations with the same period L. The major tools to obtain these results are the Grillaks, Shatah and Strauss' general theory in the periodic case. The results in the present paper extend some previous stability results for the BBM equations.

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