Showing papers by "Vladimir E. Zakharov published in 2001"
TL;DR: In this article, a two-parameter nonlinear dispersive wave equation proposed by Majda, McLaughlin and Tabak is studied analytically and numerically as a model for the study of wave turbulence in one-dimensional systems.
71 citations
TL;DR: In this paper, a system of one-dimensional equations describing media with two types of interacting waves is considered and it is shown that in the absence of coherent structures weak turbulence spectra can be clearly observed numerically.
10 citations
TL;DR: The dressing method as mentioned in this paper allows constructing classes of combescure-equivalent surfaces with the same rotation coefficients, each equivalence class is defined by a function of two variables (the master function of a surface) and each class of Combescure equivalence surfaces includes the sphere.
Abstract: The Gauss–Codazzi equations imposed on the elements of the first and the second quadratic forms of a surface embedded in \(\mathbb{R}^{3} \) are integrable by the dressing method. This method allows constructing classes of Combescure-equivalent surfaces with the same “rotation coefficients.” Each equivalence class is defined by a function of two variables (“master function of a surface”). Each class of Combescure-equivalent surfaces includes the sphere. Different classes of surfaces define different systems of orthogonal coordinates of the sphere. The simplest class (with the master function zero) corresponds to the standard spherical coordinates.
8 citations
TL;DR: In this paper, the results of a numerical simulation of a one-dimensional modified MMT model, which includes the processes of "one-to-three" wave interactions, were reported.
Abstract: In this article, we report the results of our numerical simulation of a one-dimensional modified MMT model, which includes the processes of “one-to-three” wave interactions. We show that this model, with properly chosen parameters, behaves according to the weak-turbulence theory. In particular, it demonstrates the validity of the Kolmogorov spectrum over a wide range of wave numbers. © 2001 MAIK “Nauka/Interperiodica”.
7 citations
01 Jan 2001
TL;DR: In this article, the weak turbulence approach is applied to find out the problem solution with the help of kinetic equation for energy spectrum, which was formulated by K.Hasselmann 1962,1963 and V Zakharov (1968).
Abstract: : Gravity waves in water surface are characterized by a small steepness value. It makes possible to apply the weak turbulence approach in order to find out the problem solution with the help of kinetic equation for energy spectrum. This equation was formulated by K.Hasselmann 1962,1963) and V Zakharov (1968).
6 citations
01 Jan 2001
TL;DR: The basic mathematical apparatus of nonlinear optics consists of an array of PDEs for the complex amplitudes of an envelope of interacting wave trains, which include linear and nonlinear dissipative terms.
Abstract: The basic mathematical apparatus of nonlinear optics consists of an array of nonlinear PDEs for the complex amplitudes of an envelope of interacting wave trains. In the general case, these equations include linear and nonlinear dissipative terms. However, in many important cases, they are small and can be neglected: therefore the equations are conservative, and the medium is transparent. According to the Kramers-Kronig relations, stemming from the principle of causality, the transparency can be realized at most in a limited spectral band, and even in this case some dissipation inevitably exists. Nevertheless, such fundamental nonlinear effects as the generation of high harmonics, induced Raman scattering, and self-focusing can be described by the conservative equations, preserving energy.
2 citations
1 citations