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Anatoli Babine's Abstract
Linear superposition of nonlinear waves
Nonlinear waves are described by nonlinear differential equations.
Their solutions are determined by initial data, which are functions of
the spatial variables. When the equation is linear, if the initial
function equals the sum of two functions, the solution equals the sum of
corresponding solutions. For nonlinear equations this linear
superposition principle is not valid. Nevertheless, we have proved a
theorem which shows that if the initial data of a nonlinear equation
equals a sum of several wavepackets which have different and large group
velocities, then the solution equals the sum of corresponding solutions
with a small error. Note that the nonlinearity is not small and its
influence on the time evolution of every wavepacket is not small too.
The described approximate linear superposition of nonlinear waves is
explained by a destructive wave interference between different
wavepackets in the process of their time evolution, this interference
drastically reduces nonlinear interactions between the wavepackets.
Another explanation of this phenomenon is an approximate separation of
variables in the quasimomentum space of the system. Note that in
contrast with classical theorems on complete integrability which also
produce separation of variables, in our case the variables are
separated approximately and on a finite time interval. But our results
are robust, they do not depend on the particular form of the
nonlinearity as long as equations are translation invariant in the space
or on the lattice.
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