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Resonance of Arbitrary Number of Periodic Traveling Water Waves

In: Homotopy Analysis Method in Nonlinear Differential Equations

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  • Shijun Liao

    (Shanghai Jiao Tong University)

Abstract

In this chapter, we verify the validity of the homotopy analysis method (HAM) for a rather complicated nonlinear PDE describing the nonlinear interaction of arbitrary number of traveling water waves. In the frame of the HAM, the waveresonance criterion for arbitrary number of waves is gained, for the first time, which logically contains the famous Phillips’ criterion for four small amplitude waves. Besides, it is found for the first time that, when the wave-resonance criterion is satisfied and the wave system is fully developed, there exist multiple steady-state resonant waves, whose amplitude might be much smaller than primary waves so that a resonant wave may contain much small percentage of the total wave energy. This example illustrates that the HAM can be used as a tool to deepen and enrich our understandings about some rather complicated nonlinear phenomena.

Suggested Citation

  • Shijun Liao, 2012. "Resonance of Arbitrary Number of Periodic Traveling Water Waves," Springer Books, in: Homotopy Analysis Method in Nonlinear Differential Equations, chapter 0, pages 523-562, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-25132-0_16
    DOI: 10.1007/978-3-642-25132-0_16
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