Linear and Nonlinear Waves - WhithamCurator: Graham W Griffiths. William E. Eugene M. Andrei D. The study of waves can be traced back to antiquity where philosophers, such as Pythagoras c. However, it was not until the work of Giovani Benedetti , Isaac Beeckman and Galileo that the relationship between pitch and frequency was discovered.
Solitary waves are special solutions to nonlinear PDEs which arise due to a perfect balance between linear dispersive and nonlinear effects. They are localized disturbances that, as the name suggests, evolve without any change to their shape. In cases of completely integrable PDEs they are called solitons. Solitary waves appear in real world as, for instance, laser generated pulses, tidal bores, morning glory clouds, freak waves, tsunami, wakes of high speed ships, etc. In this seminar, after briefly covering the history of solitary wave research, we will define a plane wave, phase velocity, wavepacket, group velocity, dispersion relation and the slowly varying envelope approximation.
Historically, the study of nonlinear dispersive waves started with the pioneering work of Stokes in 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. 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.
Home Dates and deadlines Travel information Programme Contact. A family of solitary-wave solutions is found using a constrained minimisation principle and concentration-compactness methods for noncoercive functionals. The solitary waves are approximated by scalings of the corresponding solutions to partial differential equations arising as weakly nonlinear approximations; in the case of the Whitham equation the approximation is the Korteweg-deVries equation. We also demonstrate that the family of solitary-wave solutions is conditionally energetically stable. Using harmonic maps we provide an approach towards obtaining explicit solutions to the incompressible two-dimensional Euler equations.