I read in Wikipedia (Peregrine soliton) that the NLS equation is written for deep water. Is it legitimate to use this mathematical description for shallow water? What do we mean by an anomalous dispersion and by defocusing nonlinearity?
There is one more type of solitary wave, this is called as K=dV type. This was observed by Scott Russell and reported in the year 1834. This report is the beginning nonlinear effects in wave propagation
phase velocity = constant. This constant is related to the properties of the medium. When the waves have very large amplitudes then they disturb the medium. In this case constant becomes a function of energy of the wave. This may reduce the width of wave packet which is a collection waves around a given frequency. The central frequency is taken as the reference wave. A pulse start normally shrinking. But the medium is non unform then this shrinking can stop. A constant wave packet is formed which does not change or interact with the medium. This is called as a soliton. Non Linear Schrodinger equation studies such behaviour. In side a pulse, the medium looks like a compressed spring.
There is one more type of solitary wave, this is called as K=dV type. This was observed by Scott Russell and reported in the year 1834. This report is the beginning nonlinear effects in wave propagation.
We can imagine that you send a electro-magnetic pulse in a medium. The velocity depends of each component of the pulse depend on electron density , at higher frequency region. Now if the electron density increases slowly then it may be possible that the front of the pulse may move faster and tail may move slower. This make the pulse to spread out. This is effect of in-homogeneous medium. Now if the strength of the electric field is sufficiently large then the wave may push electron and reduce the increase. This may stop the spreading the front part. When these two effect balance each other the pulse retains the shape. This condition gives rise to a soliton formation. The Nonlinear Schrodinger equation gives a stable solution. The later effect is equivalent to anomalous dispersion and the earlier one leads to focusing effect. If you remove the medium in-homogeneity then you get self focusing effect.
I hope you got some aspect clarified. The study of surface solitons is a subject in itself.
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