By E.A. Kuznetsov, V.E. Zakharov (auth.), P. L. Christiansen, M. P. Sørensen, A. C. Scott (eds.)

Nonlinear technological know-how is by means of now a good tested box of analysis on the interface of many conventional disciplines and attracts at the theoretical techniques built in physics and arithmetic. the current quantity gathers the contributions of prime scientists to offer the cutting-edge in lots of parts strongly motivated via nonlinear learn, comparable to superconduction, optics, lattice dynamics, biology and biomolecular dynamics. whereas this quantity is basically meant for researchers operating within the box care, has been taken that it'll even be of profit to graduate scholars or nonexpert scientist wishing to familiarize themselves with the present prestige of research.

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84) where ωce , ωpe are electron gyrofrequency and electron plasma frequency, respectively, rd = vT e / ωpe is the Debye radius, k⊥ is the component transverse to the external magnetic ﬁeld B 0 , directed along the z axis. 84), the ﬁrst term describes the potential electron plasma oscillations with a plasma frequency. Other terms are due to slower processes. 84) transforms into the dispersion law for the Langmuir waves. The nonlinear eﬀect, in a small-amplitude region (E 2 /8πnT m/M ), m and M being the electron and ion masses, respectively, corresponds to the nonlinear frequency shift due to the interaction with slow adiabatic plasma ﬂows induced by high-frequency plasma oscillations.

It is very important to note that the velocities of the 2D ground state solitons as well as their amplitudes are positive quantities. 164), move in the upstream direction and have negative amplitudes. The latter means that in the real hydrodynamic system solitons look like holes in the mean velocity proﬁle, therefore they move slower than the main ﬂow. When the soliton amplitude grows, the soliton velocity decreases and vice versa. This physical reasoning suggests a possibility for the appearance of the wave collapse in this system and the instability of 1D solitons with respect to two-dimensional perturbations as well (for details, see [78, 79]).

86), k = ∇Φ is the wave vector and n = |ψz |2 A2 Φ2z is the wave intensity. At the next order we arrive at the continuity equation for n nt + div(nV ) = 0. 97) Here V = ∂Ω/∂k is the group velocity. Eqs. 97) retain the Hamiltonian structure nt = H = δH , δΦ δH , δn n2 dr. Ω(∇Φ)n − 2 Φt = − It is possible to show that Eqs. 86) (for details, see [40, 39]). All semi-classical collapsing regimes occur to be unstable. Therefore at the initial stage of a collapse we have the formation of a weak singularity which later on serves as the origin for the appearance of a black hole.