By Vladimir V Konotop
This publication is principally dedicated to the dynamics of the one-dimensional nonlinear stochastic waves. It includes a description of the fundamental mathematical instruments in addition to the most recent ends up in the next fields: precisely integrable nonlinear stochastic equations, dynamics of the nonlinear waves in random media, evolution of the random waves in non-linear media and the elemental suggestions of the numerical simulations in non-linear random wave dynamics. a quick define of the localization phenomenon within the nonlinear medium can also be given. The strategy is interdisciplinary - describing the final equipment with program to express examples. the implications awarded will be worthwhile in the event you paintings within the parts of reliable nation physics, hydrodynamics. nonlinear optics, plasma physics, mathematical versions of micromolecules and organic buildings, ...etc. because many effects are in line with the inverse scattering method, perturbation idea for solitons and the tools of the statistical radiophysics, the terminology of the respective fields is used
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Extra info for Nonlinear random waves
The Benjamin-Feir instability. Recall that the underlying wave field u(x, f) is given by where a is a function of £ = e(x-o>oO and T= e2t and satisfies Write a = Ael
32), H = J^2q* dx. 38) is preserved. 39) as the continuous limit of 5^ 8qt A£/<;i Sq,. On the other hand, the new coordinates A and 6 are conjugate variables. 27). Gardner  was the first to note that the KdV equation could be written in a Hamiltonian framework. Later, Zakharov and Faddeev  showed how it could be interpreted as a completely integrable Hamiltonian system. N))? J = 1, • • • , N which are in involution with respect to the Poisson bracket. From this beginning, one can define N action variables (as functions of the Fj's) and N corresponding angle variables.
52) gives which when differentiated with respect to £ gives where p = A2. 52), is the equation for conservation of wave action. Next, look at the monochromatic solution which means that This is the Stokes wave. 55) is always unstable to long waves in the range 0