Abstract
We consider the well-known nonlinear Hirota equation (NLH) with fractional time derivative and derive its periodic wave solution and approximate analytic solitary wave solution using the homotopy analysis method (HAM). We also apply HAM to two coupled time fractional NLHs and construct their periodic wave solution and approximate solitary wave solution. We observe that the obtained periodic wave solution in both cases can be written in terms of the Mittag–Leffler function when the convergence control parameter \({c}_0\) equals \(-1\). Convergence of the obtained solution is discussed. The derived approximate analytic solution and the effect of time-fractional order \(\alpha \) are shown graphically.
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Notes
Liao originally used the symbol \(\hbar \) to denote the nonzero auxiliary parameter. But it is well known as Planck’s constant. To avoid misunderstanding Liao [16] replaced \(\hbar \) by \({c}_0\).
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Acknowledgments
The authors would like to thank anonymous referees for their valuable suggestions. One of the authors (T.Bakkyaraj) would like to thank the Council of Scientific and Industrial Research (CSIR), Government of India, New Delhi, for providing Senior Research Fellowship.
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Bakkyaraj, T., Sahadevan, R. On solutions of two coupled fractional time derivative Hirota equations. Nonlinear Dyn 77, 1309–1322 (2014). https://doi.org/10.1007/s11071-014-1380-7
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DOI: https://doi.org/10.1007/s11071-014-1380-7