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A numerical study on the N-periodic wave solutions of two coupled bilinear equations

Abstract

In this paper, based on the direct method proposed by Akira Nakamura, we present an efficient numerical scheme to calculate the N-periodic wave solutions to the Tzitzeica equation and the (2 + 1)-dimensional modified Bogoyavlenskii-Schiff (mBS) equation which can be transformed into a coupled bilinear system with some dependent variable transformation. By using this numerical scheme, we calculate their 2-periodic wave solutions and 3-periodic wave solutions as examples. We also show the asymptotic behaviors under a “small amplitude” limit of these quasi-periodic wave solutions numerically.

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Funding

This work was partially supported by the National Natural Science Foundation of China (Grant nos. 12071447, 11971473, 11871444, 11731014) and Fundamental Research Funds for the Central Universities (Grant no. 201964008).

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Correspondence to Jian-Qing Sun.

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Wang, XX., Sun, JQ. & Zhang, YN. A numerical study on the N-periodic wave solutions of two coupled bilinear equations. Numer Algor 88, 711–728 (2021). https://doi.org/10.1007/s11075-020-01054-w

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Keywords

  • Tzitzeica equation
  • (2 + 1)-Dimensional modified Bogoyavlenskii-Schiff equation
  • N-Periodic wave solution
  • Riemann’s 𝜃-function

Mathematics subject classification (2010)

  • Primary 37K40
  • 35B10
  • 37K10