Download Nonlinear Waves 1: Dynamics and Evolution by Yu. A. Danilov (auth.), Professor Andrei V. Gaponov-Grekhov, PDF

By Yu. A. Danilov (auth.), Professor Andrei V. Gaponov-Grekhov, Professor Mikhail I. Rabinovich, Professor Jüri Engelbrecht (eds.)

Since 1972 the colleges on Nonlinear Physics in Gorky were a gathering position for Soviet scientists operating during this box. rather than generating for the 1st time English complaints it's been made up our minds to offer an outstanding pass component of nonlinear physics within the USSR. hence the members on the final university have been invited to supply English experiences and study papers for those volumes (which within the years yet to come could be by way of the court cases of approaching schools). The first volume begins with a historic evaluate of nonlinear dynamics from Poincaré to the current day and touches themes like attractors, nonlinear oscillators and waves, turbulence, development formation, and dynamics of constructions in nonequilibrium dissipative media. It then bargains with constructions, bistabilities, instabilities, chaos, dynamics of defects in 1d structures, self-organizations, solitons, spatio-temporal constructions and wave cave in in optical platforms, lasers, plasmas, reaction-diffusion structures and solids.

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Extra info for Nonlinear Waves 1: Dynamics and Evolution

Example text

Thus, in the Korteweg-de Vries equation, a closed separatrix (a soliton) exists only at V > 0; this means that in the initial variables the soliton velocity is always greater than the velocity of a linear wave. We discuss now another well-known example, dealing with the autowaves. 2). This is the known result of Kolmogorov, Petrovsky and Piskunov (see, for example, /4/). Although it is commonly said that V - Vmin is the condition for the existence of a transition solution, it is possible only at the additional condition that the sign of u is not changed.

T(x,y) are n t. 3). If the time is discrete, then t(x,y) is more than a unity. A succession of trajectory segments of DS ({ft},M) with the ends at points x i _ 1 ,x i is called an E-trajectory. In the case of discrete time any succession of points {x k } k = 0,1, ... that satisfies the inequalities dist (x i ,fx i _ 1 ) < E is an E-trajectory. Note that when we use a computer we always deal with an E-trajectory (rather than with the true trajectory of the system). If x ~ x, then x is called a chain-recurrent point.

E. a "hierarchy of lattices" is possible. On the other hand, there exist unstable lattices which readily become stochastic as in (2) (Fig. 9b) . In all these cases, the coordinates of solitons Silt) satisfy the ordinary differential equations: L j =1 f (S. 1. e. a derivative of the total field momentum of soliton p with respect to its velocity V. Thus, here we somehow return to oscillations from waves from the partial differential equations to ordinary ones. It is worthwhile to mention two of modern trends in "classical dynamics" of solitons.

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