Abstract
Historically, the study of nonlinear dispersive waves started with the pioneering work of Stokes in (Trans. Camb. Philos. Soc. 8:197–229, 1847) on water waves. Stokes first proved the existence of periodic wavetrains which are possible in nonlinear dispersive wave systems. He also determined that the dispersion relation on the amplitude produces significant qualitative changes in the behavior of nonlinear waves. It also introduces many new phenomena in the theory of dispersive waves, not merely the correction of linear results. These fundamental ideas and the results of Stokes have provided a tremendous impact on the subject of nonlinear water waves, in particular, and on nonlinear dispersive wave phenomena, in general. Stokes’ profound investigations on water waves can be considered as the starting point for the modern theory of nonlinear dispersive waves. In fact, most of the fundamental concepts and results on nonlinear dispersive waves originated in the investigation of water waves. The study of nonlinear dispersive waves has proceeded at a very rapid pace with remarkable developments over the past three decades.
The tool which serves as intermediary between theory and practice, between thought and observation, is mathematics; it is mathematics which builds the linking bridges and gives the ever more reliable forms. From this it has come about that our entire contemporary culture, in as much as it is based on the intellectual penetration and the exploitation of nature, has its foundations in mathematics. Already Galileo said: one can understand nature only when one has learned the language and the signs in which it speaks to us; but this language is mathematics and these signs are mathematical figures… Without mathematics, the astronomy and physics of today would be impossible; these sciences, in their theoretical branches, virtually dissolve into mathematics.
David Hilbert
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Debnath, L. (2012). Nonlinear Dispersive Waves and Whitham’s Equations. In: Nonlinear Partial Differential Equations for Scientists and Engineers. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-8265-1_7
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